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it is believed that the direct detection of gravitational waves ( gws ) will bring the era of gravitational wave astronomy . the interferometer detectors are now under operation and awaiting the first signal of gws @xcite . it is also known that pulsar timing arrays ( ptas ) can be used as a detector for gws @xcite . these detectors are used to search for very low frequency ( @xmath0 ) gravitational waves , where the lower limit of the observable frequencies is determined by the inverse of total observation time @xmath1 . indeed , the total observation time has a crucial role in ptas , because ptas are most sensitive near the lower edge of observable frequencies @xcite . taking into account its sensitivity , the first direct detection of the gravitational waves might be achieved by ptas . the main target of ptas is the stochastic gravitational wave background ( sgwb ) generated by a large number of unresolved sources with the astrophysical origin or the cosmological origin in the early universe . the promising sources are super massive black hole binaries @xcite , cosmic ( super)string @xcite , and inflation @xcite . previous studies have assumed that the sgwb is isotropic and unpolarized @xcite . these assumptions are reasonable for the primary detection of the sgwb , but the deviation from the isotropy and the polarizations should have rich information of sources of gravitational waves . recently , the cross - correlation formalism has been generalized to deal with anisotropy in the sgwb @xcite . result of this work enables us to consider arbitrary levels of anisotropy , and a bayesian approach was performed by using this formalism @xcite . on the other hand , for the anisotropy of the sgwb , the cross - correlation formalism has been also developed in the case of interferometer detectors @xcite . as to the polarization , there are works including the ones motivated by the modified gravity @xcite . we can envisage supermassive black hole binaries emit circularly polarized sgwb due to the chern - simons term @xcite . there may also exist cosmological sgwb with circular polarization in the presence of parity violating term in gravity sector @xcite . in this paper , we investigate the detectability of circular polarization in the sgwb by ptas . we characterize sgwb by the so called stokes @xmath2 parameter @xcite and calculate generalized overlap reduction functions ( orfs ) so that we can probe the circular polarization of the sgwb . we also discuss a method to separate the intensity ( @xmath3 mode ) and circular polarization ( @xmath2 mode ) of the sgwb . the paper is organized as follows . in section [ sec : stokes parameters for a plane gravitational wave ] , we introduce the stokes parameters for monochromatic plane gravitational waves , and clarify the physical meaning of the stokes parameters @xmath3 and @xmath2 . in section [ sec : formulation ] , we formulate the cross - correlation formalism for anisotropic circularly polarized sgwb with ptas . the basic framework is essentially a combination of the formalism of @xcite , and the polarization decomposition formula of the sgwb derived in @xcite . in section [ sec : the generalized overlap reduction function for circular polarization ] , we calculate the generalized orfs for the @xmath2 mode . the results for @xmath3 mode are consistent with the previous work @xcite . in section [ sec : separation method ] , we give a method for separation between the @xmath3 mode and @xmath2 mode of the sgwb . the final section is devoted to the conclusion . in appendixes , we present analytic results for the generalized overlap reduction functions . in this paper , we will use the gravitational units @xmath4 . let us consider the stokes parameters for plane waves traveling in the direction @xmath5 , which can be described by @xmath6 \ , \\ & & h_{xy}(t , z)=h_{yx}(t , z)={\rm re}[b_{\times}\mathrm{e}^{-iw(t - z ) } ] \ .\end{aligned}\ ] ] for an idealized monochromatic plane wave , complex amplitudes @xmath7 and @xmath8 are constants . polarization of the plane gws is characterized by the tensor , ( see @xcite and also electromagnetic case @xcite ) @xmath9 where @xmath10 take @xmath11 . any @xmath12 hermitian matrix can be expanded by the pauli and the unit matrices with real coefficients . hence , the @xmath13 hermitian matrix @xmath14 can be written as @xmath15 where @xmath16 by analogy with electromagnetic cases , @xmath17 and @xmath2 are called stokes parameters . comparing with , we can read off the stokes parameters as @xmath18= b_{+}^{\ast}b_{\times}+ b_{\times}^{\ast}b_{+},\\ v&=&-2{\rm i m } [ b_{+}^{\ast}b_{\times}]=i ( b_{+}^{\ast}b_{\times}- b_{\times}^{\ast}b_{+}).\label{stv}\end{aligned}\ ] ] apparently , the real parameter @xmath3 is the intensity of gws . in order to reveal the physical meaning of the real parameter @xmath2 , we define the circular polarization bases @xcite @xmath19 from the relation @xmath20 we see @xmath21 thus , we can rewrite the stokes parameters - as @xmath22 from the above expression , we see that the real parameter @xmath2 characterizes the asymmetry of circular polarization amplitudes . the other parameters @xmath23 and @xmath24 have additional information about linear polarizations by analogy with the electromagnetic cases . alternatively , we can also define the tensor @xmath25 in circular polarization bases @xmath26 where @xmath27 . note that the stokes parameters satisfy a relation @xmath28 next , we consider the transformation of the stokes parameters under rotations around the @xmath5 axis . the rotation around the @xmath5 axis is given by @xmath29 where @xmath30 is the angle of the rotation . the gws traveling in the direction @xmath5 @xmath31 transform as @xmath32 where we took the transverse traceless gauge @xmath33 after a short calculation , we obtain @xmath34 using and , the four stokes parameters ( [ sti])-([stv ] ) transform as @xmath35 as you can see , the parameters @xmath23 and @xmath24 depend on the rotation angle @xmath30 . this reflects the fact that @xmath23 and @xmath24 parameters characterize linear polarizations . note that this transformation is similar to the transformation of electromagnetic case except for the angle @xmath36 and can be rewritten as @xmath37 in this section , we study anisotropic distribution of sgwb and focus on the detectability of circular polarizations with pulsar timing arrays . we combine the analysis of @xcite and that of @xcite . in sec.[subsec : the spectral ] , we derive the power spectral density for anisotropic circularly polarized sgwb @xmath38 . then we also derive the dimensionless density parameter @xmath39 which is expressed by the frequency spectrum of intensity @xmath40 @xcite . in sec.[subsec : the signal ] , we extend the generalized orfs to cases with circular polarizations characterized by the parameter @xmath2 . for simplicity , we consider specific anisotropic patterns with @xmath41 expressed by the spherical harmonics @xmath42 . in the transverse traceless gauge , metric perturbations @xmath43 with a given propagation direction @xmath44 can be expanded as @xcite @xmath45 where the fourier amplitude satisfies @xmath46 as a consequence of the reality of @xmath43 , @xmath47 , @xmath48 is the frequency of the gws , @xmath49 are spatial indices , @xmath50 label polarizations . note that the fourier amplitude @xmath51 satisfies the relation @xmath52 where @xmath53 was defined by . the polarized tensors @xmath54 are defined by @xmath55 where @xmath56 and @xmath57 are unit orthogonal vectors perpendicular to @xmath58 . the polarization tensors satisfy @xmath59 with polar coordinates , the direction @xmath44 can be represented by @xmath60 and the polarization basis vectors read @xmath61 we assume the fourier amplitudes @xmath62 are random variables , which is stationary and gaussian . however , they are not isotropic and unpolarized . the ensemble average of fourier amplitudes can be written as @xcite @xmath63 where @xmath64 here , the bracket @xmath65 represents an ensemble average , and @xmath66 is the dirac delta function on the two - sphere . the gw power spectral density @xmath38 is a hermitian matrix , and satisfies @xmath67 because of the relation @xmath46 . therefore , we have the relations @xmath68 note that the stokes parameters are not exactly the same as the expression of , but they have the relation and characterize the same polarization . we further assume that the sgwbs satisfy @xmath69 we also assume the directional dependence of the sgwb is frequency independent @xcite . this implies the gw power spectral density is factorized into two parts , one of which depends on the direction while the other depends on the frequency . because of the transformations - , the parameters @xmath3 and @xmath2 have spin 0 and the parameters @xmath70 have spin @xmath71 @xcite . to analyze the sgwb on the sky , it is convenient to expand the stokes parameters by spherical harmonics @xmath72 . however , since @xmath70 parameters have spin @xmath71 , they have to be expanded by the spin - weighted harmonics @xmath73 @xcite . thus , we obtain @xmath74 in this paper , we study specific anisotropic patterns with @xmath41 for simplicity . therefore , we can neglect @xmath23 and @xmath24 from now on . thus , the gw power spectral density becomes @xmath75 where @xmath76 so , we focus on the parameters @xmath3 and @xmath2 . in what follows , we will use the following shorthand notation @xmath77 next , we consider the dimensionless density parameter @xcite @xmath78 where @xmath79 is the critical density , @xmath80 is the present value of the hubble parameter , @xmath81 is the energy density of gravitational waves , and @xmath82 is the energy density in the frequency range @xmath48 to @xmath83 . the bracket @xmath65 represents the ensemble average . however , actually , we take a spatial average over the wave lengths @xmath84 of gws or a temporal average over the periods @xmath85 of gws . here , we assumed the ergodicity , namely , the ensemble average can be replaced by the temporal average . using , , , as well as @xmath46 and @xmath86 , we get @xmath87 then we define @xmath88 hence , the dimensionless quantity @xmath39 in is given by @xmath89 where the spherical harmonics are orthogonal and normalized as @xmath90 using @xmath91 , we obtain @xmath92 without loss of generality , we normalize the monopole moment as @xmath93 so , becomes @xmath94 the time of arrival of radio pulses from the pulsar is affected by gws . consider a pulsar with frequency @xmath95 located in the direction @xmath96 . to detect the sgwb , let us consider the redshift of the pulse from a pulsar @xcite @xmath97 where @xmath98 is a frequency detected at the earth and @xmath96 is the direction to the pulsar . the unit vector @xmath44 represents the direction of propagation of gravitational plane waves . we also defined the difference between the metric perturbations at the pulsar @xmath99 and at the earth @xmath100 as @xmath101 the gravitational plane waves at each point is defined as @xmath102 for the sgwb , the redshift have to be integrated over the direction of propagation of the gravitational waves @xmath44 : @xmath103 we choose a coordinate system @xmath104 and assume that the amplitudes of the metric perturbation at the pulsar and the earth are the same . then becomes @xmath105 and therefore , reads @xmath106 where we have defined the pattern functions for pulsars @xmath107 note that our convention for the fourier transformation is @xmath108 therefore , the fourier transformation of can be written as @xmath109 in the actual signals from a pulsar , there exist noises . hence , we need to use the correlation analysis . we consider the signals from two pulsars @xmath110 where @xmath111 labels the pulsar . here , @xmath112 denotes the signal from the pulsar and @xmath113 denotes the noise intrinsic to the measurement . we assume the noises are stationary , gaussian and are not correlated between the two pulsars . to correlate the signals of two measurements , we define @xmath114 where @xmath1 is the total observation time and @xmath115 is a real filter function which should be optimal to maximize signal - to - noise ratio . in the case of interferometer , the optimal filter function falls to zero for large @xmath116 compered to the travel time of the light between the detecters . since the signals of two detectors are expected to correlate due to the same effect of the gravitational waves , the optimal filter function should behave this way . then , typically one of the detectors is very close to the other compared to the total observation time @xmath1 . therefore , the total observation time @xmath1 can be extended to @xmath117 @xcite . in contrast , in the case of pta , it is invalid that @xmath1 is very large compered to the travel time of the light between the pulsars . nevertheless , we can assume that one of the two @xmath1 can be expanded to @xmath117 , because in situations @xmath118 and @xmath119 it is known that we can ignore the effect of the distance @xmath120 of pulsars . in this case , it is clear that any locations of the pulsars are optimal and optimal filter function should behave like as the interferometer case @xcite . using these assumptions @xmath118 and @xmath119 , we can rewrite as @xmath121 where @xmath122 note that @xmath123 satisfies @xmath124 , because @xmath125 is real . moreover , to deal with the unphysical region @xmath126 we require @xmath127 . thus , @xmath123 becomes real . taking the ensemble average , using @xmath128 , @xmath118 , and assuming the noises in the two measurements are not correlated , we get @xmath129\ , \label{s2}\end{aligned}\ ] ] where we have defined @xmath130 the functions @xmath131 and @xmath132 are called the generalized orfs , which describe the angular sensitivity of the pulsars for the sgwb . note that , as we already mentioned , we consider the cases of @xmath41 for simplicity . then we have assumed @xmath118 and @xmath119 , this assumption implies that approximately becomes @xmath133 due to the rapid oscillation of the phase factor . therefore , the distance @xmath120 of the pulsars does not appear in the generalized orfs , and hence the generalized orfs do not depend on the frequency . as you can see from , the correlation of the two measurements involve both the total intensity and the circular polarization . however , the degeneracy can be disentangled by using separation method , which will be discussed in the section [ sec : separation method ] . in this section , we consider the generalized orfs for circular polarizations : @xmath134 where we defined @xmath135 in the above , we have used and the fact that the generalized orfs do not depend on frequency . for computation of the generalized orfs for circular polarizations , it is convenient to use the computational frame @xcite defined by @xmath136 where @xmath137 is the angular separation between the two pulsars . using - , , and , one can easily show that @xmath138 we therefore get @xmath139 the explicit form of the spherical harmonics reads @xmath140 where @xmath141 is the normalization factor . the associated legendre functions are given by @xmath142 and @xmath143 with the legendre functions @xmath144\ .\label{pl}\end{aligned}\ ] ] using the spherical harmonics , becomes @xmath145 where we have used the fact that the function of @xmath146 is odd parity in the case of @xmath147 and is even parity in the case of @xmath148 . note that the generalized orfs for circular polarizations are real functions . in the case of @xmath149 and/or @xmath150 , the integrand in vanishes . therefore , we can not detect circular polarizations for these cases . this fact for @xmath151 implies that we do not need to consider auto - correlation for a single pulsar . this is the reason why we neglected auto - correlation term in . integrating , we get the following form for @xmath152 : @xmath153 for @xmath154 , we have obtained @xmath155 \ , \\ \gamma^{v}_{1 - 1}&=&\gamma^{v}_{11 } \ , \end{aligned}\ ] ] recall that @xmath156 . the derivation of this formula for @xmath154 can be found in appendix [ sec : angular integral of the generalized overlap reduction function for dipole circular polarization ] . for @xmath157 , we derived the following : @xmath158\ , \\ \gamma^{v}_{2 - 1}&=&\gamma^{v}_{21}\ , \\ \gamma^{v}_{22}&=&-\frac{\sqrt{30\pi}}{6}(1-\cos\xi)\left[2-\cos\xi+6\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{v}_{2 - 2}&=&-\gamma^{v}_{22}\ , \end{aligned}\ ] ] for @xmath159 , the results are @xmath160\ , \\ \gamma^{v}_{3 - 1}&=&\gamma^{v}_{31}\ , \\ \gamma^{v}_{32}&=&\frac{\sqrt{210\pi}}{24}(1-\cos\xi)\left[8 - 5\cos\xi-\cos^2\xi+24\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{v}_{3 - 2}&=&-\gamma^{v}_{3 - 2}\ , \\ \gamma^{v}_{33}&=&-\frac{\sqrt{35\pi}}{16}\sin\xi\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\left[11 - 6\cos\xi-\cos^2\xi+32\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{v}_{3 - 3}&=&\gamma^{v}_{33}\ .\end{aligned}\ ] ] in fig . [ gv ] , we plotted these generalized orfs as a function of the angular separation between the two pulsars @xmath137 . it is apparent that considering the @xmath2 mode does not make sense when we only consider the isotropic ( @xmath152 ) orf . on the other hand , when we consider anisotropic ( @xmath161 ) orfs , it is worth taking into account polarizations . the polarizations of the sgwb would give us rich information both of super massive black hole binaries and of inflation in the early universe . as a function of the angular separation between the two pulsars @xmath137 . in fig . [ gv](a ) , we find the orf for the monopole ( l=0 ) is trivial . in fig . [ gv](b ) , the orfs for the dipole ( l=1 ) are shown . in fig . [ gv](c ) , the orfs for the quadrupole ( l=2 ) are depicted . in fig . [ gv](d ) , the orfs for the octupole ( l=3 ) are plotted . the black solid curve , the blue dashed curve , the red dotted curve , the dark - red space - dotted curve , and the green long - dashed curve represent @xmath149 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , respectively.,title="fig:",width=340 ] ( a ) @xmath152 as a function of the angular separation between the two pulsars @xmath137 . in fig . [ gv](a ) , we find the orf for the monopole ( l=0 ) is trivial . in fig . [ gv](b ) , the orfs for the dipole ( l=1 ) are shown . in fig . [ gv](c ) , the orfs for the quadrupole ( l=2 ) are depicted . in fig . [ gv](d ) , the orfs for the octupole ( l=3 ) are plotted . the black solid curve , the blue dashed curve , the red dotted curve , the dark - red space - dotted curve , and the green long - dashed curve represent @xmath149 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , respectively.,title="fig:",width=340 ] ( b ) @xmath154 as a function of the angular separation between the two pulsars @xmath137 . in fig . [ gv](a ) , we find the orf for the monopole ( l=0 ) is trivial . in fig . [ gv](b ) , the orfs for the dipole ( l=1 ) are shown . in fig . [ gv](c ) , the orfs for the quadrupole ( l=2 ) are depicted . in fig . [ gv](d ) , the orfs for the octupole ( l=3 ) are plotted . the black solid curve , the blue dashed curve , the red dotted curve , the dark - red space - dotted curve , and the green long - dashed curve represent @xmath149 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , respectively.,title="fig:",width=340 ] ( c ) @xmath157 as a function of the angular separation between the two pulsars @xmath137 . in fig . [ gv](a ) , we find the orf for the monopole ( l=0 ) is trivial . in fig . [ gv](b ) , the orfs for the dipole ( l=1 ) are shown . in fig . [ gv](c ) , the orfs for the quadrupole ( l=2 ) are depicted . in fig . [ gv](d ) , the orfs for the octupole ( l=3 ) are plotted . the black solid curve , the blue dashed curve , the red dotted curve , the dark - red space - dotted curve , and the green long - dashed curve represent @xmath149 , @xmath162 , @xmath163 , @xmath164 , @xmath165 , respectively.,title="fig:",width=340 ] ( d ) @xmath159 using the same procedure described in the above to derive the generalized orfs for circular polarizations , we can also derive the generalized orfs for the intensity @xmath166 where @xmath167 the angular integral in this case was performed in @xcite . the results are summarized in appendix [ sec : the generalized overlap reduction function for intensity ] . in this section , we separate the @xmath3 mode and @xmath2 mode of the sgwb with correlation analysis @xcite . to this aim , we use four pulsars ( actually we need at least three pulsars ) , and define correlations of @xmath168 @xmath169 where @xmath170 label the pulsars . comparing with , we obtain @xmath171 \ , \label{1c12}\\ & & c_{34}(f)=\sum_{lm}^{l=3}\left[c_{lm}^{i}i(f)\gamma_{lm,34}^{i}+c_{lm}^{v}v(f)\gamma_{lm,34}^{v}\right ] \ .\label{1c34}\end{aligned}\ ] ] if the @xmath3 mode and @xmath2 mode of the sgwb are dominated by a certain @xmath172 and @xmath173 , and become @xmath174 \ , \label{2c12 } \\ & & c_{34}(f)=\left[c _ { l m}^{i}i(f)\gamma _ { l m,34}^{i}+c _ { l ' m'}^{v}v(f)\gamma _ { l ' m',34}^{v}\right ] \ .\label{2c34}\end{aligned}\ ] ] to separate the intensity and the circular polarization , we take the following linear combinations @xmath175 where we defined coefficients @xmath176 as you can see , @xmath177 contains only @xmath40 , and @xmath178 contains only @xmath179 . for the signal @xmath180 , the formulas corresponding to and are given by @xmath181 \ , \label{sp}\end{aligned}\ ] ] where @xmath182 denotes @xmath3 and @xmath2 . we assume @xmath183 and that the noise in the four pulsars are not correlated . we also assume that the ensemble average of fourier amplitudes of the noises @xmath184 is of the form @xmath185 where @xmath186 is the noise power spectral density . the reality of @xmath187 gives rise to @xmath188 and therefore we obtain @xmath189 . without loss of generality , we can assume @xmath190 then we obtain corresponding noises @xmath191 : @xmath192\ , \label{np}\end{aligned}\ ] ] where @xmath193^{1/2 } \label{sn12 } \ , \quad s_{n,34}(f ) \equiv [ s_{n,3}(f)s_{n,4}(f)]^{1/2 } \label{sn34 } \ .\end{aligned}\ ] ] using the inner product @xmath194 \ , \end{aligned}\ ] ] we can rewrite , as @xmath195 therefore , the optimal filter function can be chosen as @xmath196 using , we get optimal signal - to - noise ratio @xmath197^{1/2}\ .\label{snr}\end{aligned}\ ] ] plugging , , and into , we obtain @xmath198^{1/2}\ , \\ { \rm snr}_{v}&=&\left[t\int_{-\infty}^{\infty}df\,\,\frac{\left(c^{v}_{{l}'{m}'}\right)^{2}v^{2}(f)\left(\gamma_{{l}'{m}',34}^{v}\gamma^{i}_{{l}{m},12}-\gamma_{{l}'{m}',12}^{v}\gamma^{i}_{{l}{m},34}\right)^2}{\left(\gamma^{i}_{{l}{m},12}\right)^2s^{2}_{n,34}(f)+\left(\gamma^{i}_{{l}{m},34}\right)^2s^{2}_{n,12}(f)}\right]^{1/2}\ .\end{aligned}\ ] ] if we assume all of the noise power spectral densities are the same , becomes @xmath199 thus , the compiled orfs can be defined as @xmath200^{1/2}}\ , \\ \gamma_{12:34}^{v}&\equiv&\frac{\gamma_{{l}'{m}',34}^{v}\gamma^{i}_{{l}{m},12}-\gamma_{{l}'{m}',12}^{v}\gamma^{i}_{{l}{m},34}}{\left[\left(\gamma^{i}_{{l}{m},12}\right)^2+\left(\gamma^{i}_{{l}{m},34}\right)^2\right]^{1/2}}\ .\end{aligned}\ ] ] this compiled orfs @xmath201 and @xmath202 describe the angular sensitivity of the four pulsars for the pure @xmath3 and @xmath2 mode of the sgwb , respectively . note that , to do this separation , we must know a priori the coefficients @xmath203 and @xmath204 . if we do not assume , the generalized orfs depend on the frequency . in this case , it seems difficult to calculate these coefficients . we next consider the case that @xmath3 mode and/or @xmath2 mode dominant in two or more @xmath205 . in this case , if we have a priori knowledge of the values of @xmath206 in each of @xmath205 for coefficients @xmath203 and @xmath204 , we can separate @xmath3 mode and @xmath2 mode . for example , assume that @xmath3 mode is dominated by @xmath207 , while @xmath2 mode is dominated by @xmath208 , then and become @xmath209\ , \label{3c12}\\ & & c_{34}(f)=\left[c^{i}_{00}i(f)\left(\gamma_{00,34}^{i}+\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,34}^{i}\right)+c_{11}^{v}v(f)\gamma_{11,34}^{v}\right]\ .\label{3c34}\end{aligned}\ ] ] thus , we can separate @xmath3 mode and @xmath2 mode by using linear combinations @xmath210\ , \\ d_{v}&\equiv&a_{v}c_{34}(f)+b_{v}c_{12}(f ) \nonumber\\ & = & c_{11}^{v}v(f)\left[\gamma_{11,34}^{v}\left(\gamma_{00,12}^{i}+\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,12}^{i}\right)-\gamma_{11,12}^{v}\left(\gamma_{00,34}^{i}+\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,34}^{i}\right)\right]\ , \end{aligned}\ ] ] where @xmath211 as in the previous calculations , we can get the compiled orfs @xmath212^{1/2}}\ , \label{gi1234}\\ \gamma_{12:34}^{v}&\equiv&\frac{\gamma_{11,34}^{v}\left(\gamma_{00,12}^{i}+\displaystyle\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,12}^{i}\right)-\gamma_{11,12}^{v}\left(\gamma_{00,34}^{i}+\displaystyle\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,34}^{i}\right)}{\left[\left(\gamma_{00,12}^{i}+\displaystyle\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,12}^{i}\right)^2+\left(\gamma_{00,34}^{i}+\displaystyle\frac{c^{i}_{11}}{c^{i}_{00}}\gamma_{11,34}^{i}\right)^2\right]^{1/2}}\ .\label{gv1234}\end{aligned}\ ] ] [ cols="^,^ " , ] in fig . [ cg ] we show some compiled orfs @xmath213 ( left panels ) and @xmath214 ( right panels ) as a function of the two angular separations @xmath137 and @xmath215 for two pulsar pairs , respectively . we used the expressions of @xmath2 mode and @xmath3 mode ( see appendix [ sec : the generalized overlap reduction function for intensity ] ) , and we assumed @xmath216 for simplicity . in fig . [ cg](a ) and [ cg](b ) , the @xmath3 mode is dominated by @xmath217 and @xmath2 mode is dominated by @xmath218 . in fig . [ cg](c ) and [ cg](d ) , the @xmath3 mode is dominated by @xmath219 and @xmath2 mode is dominated by @xmath218 . in fig . [ cg](e ) and [ cg](f ) , the @xmath3 mode is dominated by @xmath207 and @xmath2 mode is dominated by @xmath218 . in fig . [ cg](e ) and [ cg](f ) , the @xmath3 mode is dominated by @xmath220 and @xmath2 mode is dominated by @xmath218 . by definition , in the case of @xmath221 , the compiled orfs are zero . we have studied the detectability of the stochastic gravitational waves with ptas . in most of the previous works , the isotropy of sgwb has been assumed for the analysis . recently , however , a stochastic gravitational wave background with anisotropy have been considered . the information of the anisotropic pattern of the distribution should contain important information of the sources such as supermassive black hole binaries and the sources in the early universe . it is also intriguing to take into account the polarization of sgwb in the pta analysis . therefore , we extended the correlation analysis to circularly polarized sgwb and calculated generalized overlap reduction functions for them . it turned out that the circular polarization can not be detected for an isotropic background . however , when the distribution has anisotropy , we have shown that there is a chance to observe circular polarizations in the sgwb . we also discussed how to separate polarized modes from unpolarized modes of gravitational waves . if we have a priori knowledge of the abundance ratio for each mode in each of @xmath205 , we can separate @xmath3 mode and @xmath2 mode in general . this would be possible if we start from fundamental theory and calculate the spectrum of sgwb . in particular , in the case that the signal of lowest @xmath222 is dominant , we performed the separation of @xmath3 mode and @xmath2 mode explicitly . this work was supported by grants - in - aid for scientific research ( c ) no.25400251 and " mext grant - in - aid for scientific research on innovative areas no.26104708 and `` cosmic acceleration''(no.15h05895 ) . in this appendix , we perform angular integration of the generalized orf for dipole ( @xmath154 ) circular polarization ( see @xcite ) : @xmath223 where we have defined @xmath224 . it is obvious that in the case of @xmath225 , integrand of the generalized orf is zero , because of @xmath226 , then we obtain @xmath227 then , using - , we calculate @xmath228 and we find @xmath229 therefore we only have to consider the dipole generalized orf in the case of @xmath154 , @xmath230 : @xmath231 where @xmath232 first , to calculate @xmath233 , we use contour integral in the complex plane . defining @xmath234 and substituting @xmath235 into , we can rewrite @xmath233 as @xmath236 } \ , \end{aligned}\ ] ] where @xmath237 denotes a unit circle . we can factorize the denominator of the integrand and get @xmath238 where @xmath239 hereafter , the upper sign applies when @xmath240 and the lower one applies when @xmath241 . note that we only consider the region @xmath242 , so we have used the relation @xmath243 in above expression . in the region @xmath244 , @xmath245 is inside the unit circle @xmath237 except for @xmath246 and @xmath247 is outside the unit circle @xmath237 . now , we can perform the integral using the residue theorem @xmath248 where @xmath249 the residues inside the unit circle @xmath237 can be evaluated as @xmath250\right\ } = \frac{i(z_{+}+z_{-})}{2\sqrt{1-x^2}\sin\xi } \ , \end{aligned}\ ] ] @xmath251 thus , we obtain @xmath252 next , we consider @xmath253 defined in . using , we can calculate @xmath253 as @xmath254 similarly , we can evaluate @xmath255 given in . to calculate @xmath255 in the complex plane , we again substitute into and obtain @xmath256 we use the residue theorem @xmath257 where @xmath258 the residues inside the unit circle @xmath237 can be calculated as @xmath259\right\ } = \frac{i(z_{+}^2+z_{-}^2)}{4\sqrt{1-x^2}\sin\xi } \ , \end{aligned}\ ] ] @xmath260 therefore , @xmath255 becomes @xmath261 substituting to , we can calculate @xmath262 : @xmath263 finally , substituting and into , we get the generalized orf for @xmath264 @xmath265\ .\end{aligned}\ ] ] as a function of the angular separation between the two pulsars @xmath137 . [ gi](a ) shows monopole ( l=0 ) , fig . [ gi](b ) shows dipole ( l=1 ) , fig . [ gi](c ) shows quadrupole ( l=2 ) and fig . [ gi](d ) shows octupole ( l=3 ) . the black solid curve , the blue dashed curve , the dark - blue dash - dotted curve , the red dotted curve , the green long - dashed curve , the dark - green space - dashed curve represent @xmath149 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , respectively.,title="fig:",width=340 ] ( a ) @xmath152 as a function of the angular separation between the two pulsars @xmath137 . [ gi](a ) shows monopole ( l=0 ) , fig . [ gi](b ) shows dipole ( l=1 ) , fig . [ gi](c ) shows quadrupole ( l=2 ) and fig . [ gi](d ) shows octupole ( l=3 ) . the black solid curve , the blue dashed curve , the dark - blue dash - dotted curve , the red dotted curve , the green long - dashed curve , the dark - green space - dashed curve represent @xmath149 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , respectively.,title="fig:",width=340 ] ( b ) @xmath154 as a function of the angular separation between the two pulsars @xmath137 . fig . [ gi](a ) shows monopole ( l=0 ) , fig . [ gi](b ) shows dipole ( l=1 ) , fig . [ gi](c ) shows quadrupole ( l=2 ) and fig . [ gi](d ) shows octupole ( l=3 ) . the black solid curve , the blue dashed curve , the dark - blue dash - dotted curve , the red dotted curve , the green long - dashed curve , the dark - green space - dashed curve represent @xmath149 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , respectively.,title="fig:",width=340 ] ( c ) @xmath157 as a function of the angular separation between the two pulsars @xmath137 . [ gi](a ) shows monopole ( l=0 ) , fig . [ gi](b ) shows dipole ( l=1 ) , fig . [ gi](c ) shows quadrupole ( l=2 ) and fig . [ gi](d ) shows octupole ( l=3 ) . the black solid curve , the blue dashed curve , the dark - blue dash - dotted curve , the red dotted curve , the green long - dashed curve , the dark - green space - dashed curve represent @xmath149 , @xmath266 , @xmath267 , @xmath268 , @xmath269 , @xmath270 , respectively.,title="fig:",width=340 ] ( d ) @xmath159 in this appendix , we show orfs for the intensity @xcite . the following form for @xmath152 was derived in @xcite , and our expressions are identical to their expressions : @xmath271\ , \end{aligned}\ ] ] for , @xmath154 , we calculated as @xmath272\ , \\ \gamma^{i}_{11}&=&\frac{\sqrt{6\pi}}{12}\sin\xi\left[1 + 3(1-\cos\xi)\left\{1+\frac{4}{1+\cos\xi}\log\left(\sin\frac{\xi}{2}\right)\right\}\right]\ , \\ \gamma^{i}_{1 - 1}&=&-\gamma^{i}_{11}\ , \end{aligned}\ ] ] for @xmath157 , we obtain @xmath273\ , \\ \gamma^{i}_{21}&=&-\frac{\sqrt{30\pi}}{60}\sin\xi\left[21 - 15\cos\xi-5\cos^2\xi+60\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{2 - 1}&=&-\gamma^{i}_{2 - 1}\ , \\ \gamma^{i}_{22}&=&\frac{\sqrt{30\pi}}{24}(1-\cos\xi)\left[9 - 4\cos\xi-\cos^2\xi+24\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{2 - 2}&=&\gamma^{i}_{22}\ , \end{aligned}\ ] ] for @xmath159 , it is straightforward to reach the following @xmath274\ , \\ \gamma^{i}_{31}&=&\frac{\sqrt{21\pi}}{48}\sin\xi(1-\cos\xi)\left[34 + 15\cos\xi+5\cos^2\xi+\frac{96}{1+\cos\xi}\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{3 - 1}&=&-\gamma^{i}_{31}\ , \\ \gamma^{i}_{32}&=&-\frac{\sqrt{210\pi}}{48}(1-\cos\xi)\left[17 - 9\cos\xi-3\cos^2\xi-\cos^3\xi+48\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{3 - 2}&=&\gamma^{i}_{32}\ , \\ \gamma^{i}_{33}&=&\frac{\sqrt{35\pi}}{48}\frac{(1-\cos\xi)^2}{\sin\xi}\left[34 - 17\cos\xi-4\cos^2\xi-\cos^3\xi+96\left(\frac{1-\cos\xi}{1+\cos\xi}\right)\log\left(\sin\frac{\xi}{2}\right)\right]\ , \\ \gamma^{i}_{3 - 3}&=&-\gamma^{i}_{33}\ .\end{aligned}\ ] ] these are plotted in fig . the generalized orfs of total intensity are different from that of circular polarization in that the value for @xmath149 is non - trivial . then the @xmath3 mode orfs for @xmath275 have value even in the case of @xmath151 . this implies that we can consider auto - correlation for a single pulsar . 99 j. aasi _ et al . _ [ ligo scientific collaboration ] , class . grav . * 32 * , 074001 ( 2015 ) doi:10.1088/0264 - 9381/32/7/074001 [ arxiv:1411.4547 [ gr - qc ] ] . f. acernese _ et al . _ [ virgo collaboration ] , class . grav . * 32 * , no . 2 , 024001 ( 2015 ) doi:10.1088/0264 - 9381/32/2/024001 [ arxiv:1408.3978 [ gr - qc ] ] . k. somiya [ kagra collaboration ] , class . * 29 * , 124007 ( 2012 ) doi:10.1088/0264 - 9381/29/12/124007 [ arxiv:1111.7185 [ gr - qc ] ] . s. l. detweiler , astrophys . j. * 234 * , 1100 ( 1979 ) . doi:10.1086/157593 romani , r. w. 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rapid progress in the design and manufacture of optical fiber systems is a result of worldwide demand for ultra - high bit - rate optical communications . this explains the growing interest of the soliton community in soliton - based optical fiber communication systems . this area of research was considerably advanced in recent years @xcite . the most remarkable results include the application of the concept of the dispersion management to _ temporal optical solitons _ and soliton - based optical transmission systems , and the discovery of the so - called _ dispersion managed soliton_. high - speed optical communications require effective components such as high - performance broadband computer networks that can be developed by employing the concept of the bit - parallel - wavelength ( bpw ) pulse transmission that offers many of the advantages of both parallel fiber ribbon cable and conventional wavelength - division - multiplexing ( wdm ) systems @xcite . expanding development in the study of the soliton fiber systems has been observed in parallel with impressive research on their spatial counterparts , optical self - trapped beams or _ spatial optical solitons_. one of the key concepts in this field came from the theory of multi - frequency wave mixing and cascaded nonlinearities where a nonlinear phase shift is produced as a result of the parametric wave interaction @xcite . in all such systems , the nonlinear interaction between the waves of two ( or more ) frequencies is the major physical effect that can support coupled - mode multi - frequency solitary waves . the examples of temporal and spatial solitons mentioned above have one common feature : they involve the study of solitary waves in multi - component nonlinear models . the main purpose of this paper is to overview several different physical examples of multi - mode and/or multi - frequency solitary waves that occur for the pulse or beam propagation in nonlinear optical fibers and waveguides . for these purposes , we select three different cases : multi - wavelength solitary waves in bit - parallel - wavelength optical fiber links , multi - colour spatial solitons due to multistep cascading in optical waveguides with quadratic nonlinearities , and quasiperiodic solitons in the fibonacci superlattices . we believe these examples display both the diversity and richness of the multi - mode soliton systems , and they will allow further progress to be made in the study of nonlinear waves in multi - component nonintegrable physical models . because the phenomenon of the long - distance propagation of _ temporal optical solitons _ in optical fibers @xcite is known to a much broader community of researchers in optics and nonlinear physics , first we emphasize _ the difference between temporal and spatial solitary waves_. indeed , for a long time stationary beam propagation in planar waveguides has been considered somewhat similar to the pulse propagation in fibers . this approach is based on the so - called _ spatio - temporal analogy _ in wave propagation , meaning that the propagation coordinate @xmath0 is treated as the evolution variable and the spatial beam profile along the transverse direction in waveguides , is similar to the temporal pulse profile in fibers . this analogy is based on a simple notion that both beam evolution and pulse propagation can be described by the cubic nonlinear schrdinger ( nls ) equation . however , contrary to the widely accepted opinion , there is a crucial difference between temporal and spatial solitons . indeed , in the case of the nonstationary pulse propagation in fibers , the operation wavelength is usually selected near the zero point of the group - velocity dispersion . this means that the absolute value of the fiber dispersion is small enough to be compensated by a weak nonlinearity such as that produced by the ( very weak ) kerr effect in optical fibers which leads to a relative nonlinearity - induced change in the refractive index . therefore , nonlinearity in such systems is always weak and it should be well modeled by a cubic nls equation which is known to be integrable by means of the inverse - scattering technique . however , for very short ( e.g. , fs ) pulses the cubic nls equation describing the long - distance propagation of pulses should be corrected to include additional terms that would account for such effects as higher - order dispersion , raman scattering , etc . all such corrections can be taken into account with the help of the perturbation theory @xcite . thus , in fibers nonlinear effects are weak and they become important only when dispersion is small ( near the zero - dispersion point ) affecting the pulse propagation over large distances ( of order of hundreds of meters or even kilometers ) . the situation changes dramatically when we consider the propagation of multi - wavelength pulses with almost equal group velocities . the corresponding model is described by a nonintegrable and rather complicated system of coupled nls equations , which we briefly discuss below . in contrary to the pulse propagation in optical fibers , the physics underlying the stationary beam propagation in planar waveguides and bulk media is different . in this case an optical beam is generated by a continuous wave ( cw ) source and it is time independent . however , when the beam evolves with the propagation distance @xmath0 , it diffracts in the transverse spatial directions . then , a nonlinear change in the refractive index should compensate for the beam spreading caused by diffraction _ which is not a small effect_. that is why to observe spatial solitons as self - trapped optical beams , much larger nonlinearities are usually required , and very often such nonlinearities are not of the kerr type ( e.g. they saturate at higher intensities ) . this leads to the models of generalized nonlinearities with the properties of solitary waves different from those described by the integrable cubic nls equation . propagation distances involved in the phenomenon of the beam self - focusing and spatial soliton propagation are of the order of millimeters or centimeters . to achieve such large nonlinearities , one needs to use the optical materials with large nonlinearity - induced refractive index . one of the possible way to overcome this difficulty is to use the so - called _ cascaded nonlinearities _ of noncentrosymmetric optical materials where nonlinear effects are accumulated due to parametric wave interaction under the condition of the wave phase matching . such parametric wave - mixing effects generate novel classes of spatial optical solitons where resonant parametric coupling between the envelopes of two ( or more ) beams of different frequencies supports stable spatially localised waves even in a bulk medium ( see details in ref . it is this kind of multi - component solitary waves that we discuss below . a growing demand for high - speed computer communications requires an effective and inexpensive computer interconnection . one attractive alternative to the conventional wdm systems is bpw single - mode fiber optics links for very high bandwidth computer communications @xcite . they differ from the wdm schemes in that no parallel to serial conversion is necessary , and parallel pulses are launched simultaneously on different wavelengths . when the pulses of different wavelengths are transmitted simultaneously , the cross - phase modulation can produce an interesting _ pulse shepherding effect _ @xcite , when a strong ( `` shepherd '' ) pulse enables the manipulation and control of pulses co - propagating on different wavelengths in a multi - channel optical fiber link . to describe the simultaneous transmission of @xmath1 different wavelengths in a nonlinear optical fiber , we follow the standard derivation @xcite and obtain a system of @xmath1 coupled nonlinear schrdinger ( nls ) equations @xmath2 ) : @xmath3 { \displaystyle \qquad + \chi_j \left ( |a_j|^2 + 2 \sum_{m \neq j } |a_m|^2 \right ) a_j = 0 , } \end{array}\ ] ] where , for the @xmath4th wave , @xmath5 is the slowly varying envelope , @xmath6 and @xmath7 are the group velocity and group - velocity dispersion , respectively , and the nonlinear coefficients @xmath8 characterize the kerr effect . equations ( [ eq : nls_dim ] ) do not include the fiber loss , since the fiber lengths involved in bit - parallel links are only a small fraction of the attenuation length . we measure the variables in the units of the central wavelength channel ( say , @xmath9 ) , and obtain the following normalized system of the @xmath1 coupled nls equations , @xmath10 { \displaystyle \qquad + \gamma_j \left(|u_j|^2 + 2 \sum_{m\neq j } |u_m|^2\right ) u_j = 0 , } \end{array}\ ] ] where @xmath11 , @xmath12 is the incident optical power in the central channel , @xmath13 , @xmath14 , so that @xmath15 . for the operating wavelengths spaced @xmath16 nm apart within the band @xmath17 nm , the coefficients @xmath18 and @xmath19 are different but close to @xmath20 . initially , in eq . ( [ eq : nls ] ) , we omit the mode walk - off effect described by the parameters @xmath21 ( so that @xmath22 ) . this effect will be analysed later in this section . to analyze the nonlinear modes , i.e. localized states of the bpw model ( [ eq : nls ] ) , we look for stationary solutions in the form , @xmath23 and therefore obtain the system of equations for the normalized mode amplitudes , @xmath24 { \displaystyle \frac{1}{2 } \alpha_n \frac{d^2u_n}{dt^2 } + \gamma_n \left ( |u_n|^2 + 2\sum_{m \neq n } |u_m|^2 \right ) u_n = \lambda_n u_n , } \end{array}\ ] ] where @xmath25 , the amplitudes and time are measured in the units of @xmath26 and @xmath27 , respectively , and @xmath28 . system ( [ eq : nls_nn ] ) has _ exact analytical solutions _ for @xmath1 coupled components , the so - called _ bpw solitons_. indeed , looking for solutions in the form @xmath29 , @xmath30 , we obtain the constraint @xmath31 , and a system of @xmath1 coupled algebraic equations for the wave amplitudes , @xmath32 in a special symmetric case , we take @xmath33 , and the solution of those equations is simple @xcite : + @xmath34^{-1/2}$ ] . analytical solutions can also be obtained in the _ linear limit _ , when the central frequency pulse ( at @xmath35 ) is large . then , linearizing eqs . ( [ eq : nls_nn ] ) for small @xmath36 , we obtain a decoupled nonlinear equation for @xmath37 and @xmath38 decoupled linear equations for @xmath39 . each of the latter possess a localized solution provided @xmath40 , where @xmath41 ^ 2 $ ] . in this limit the central soliton pulse @xmath37 ( `` shepherd pulse '' ) can be considered as inducing an effective waveguide that supports a fundamental mode @xmath39 with the corresponding cutoff @xmath42 . since , by definition , the parameters @xmath43 and @xmath44 are close to @xmath20 , we can verify that the soliton - induced waveguide supports maximum of two modes ( fundamental and the first excited one ) . this is an important physical result that explains the effective robustness of the pulse guidance by the shepherding pulse . to demonstrate a number of unique properties of the multi - channel bpw solitons , we consider the case @xmath45 in more details . a comprehensive discussion of the case @xmath46 can be found in the preprint @xcite . we select the following set of the normalized parameters : @xmath47 , @xmath48 , and @xmath49 . solitary waves of this four - wavelength bpw system can be found numerically as localized solutions of eqs . ( [ eq : nls_nn ] ) . figure [ fig : bpw1 ] presents the lowest - order families of such localized solutions . in general , they are characterized by @xmath38 parameters , but we can capture the characteristic features by presenting power dependencies along the line @xmath50 in the parameter space @xmath51 . the power of the central - wavelength component ( @xmath35 ) does not depend on @xmath52 ( straight line @xmath53 ) . thin dashed , dotted , and dash - dotted curves correspond to the three separate single - mode solitons of the multi - channel bpw system , ( 1 ) , ( 2 ) , and ( 3 ) , respectively , shown with the corresponding branches of ( 0 + 1 ) , ( 0 + 2 ) , and ( 0 + 3 ) two - mode solitons . the latter curves start off from the bifurcation points on the @xmath37 branch at @xmath54 , @xmath55 , and @xmath56 , respectively . close separation of those curves is the result of closeness of the parameters @xmath43 and @xmath44 for @xmath57 . thick solid curves in fig . [ fig : bpw1 ] correspond to the two- ( 1 + 2 ) and three - mode ( 0 + 1 + 2 ) localized solutions . the latter solutions bifurcate and give birth to four - wavelength solitons ( 0 + 1 + 2 + 3 ) ( branch a - b ) . two examples of such four - wave composite solitons are shown in fig . [ fig : bpw1 ] ( bottom row ) . the solution b is close to an exact sech - type solution at @xmath58 ( described above ) for @xmath45 , whereas the solution a is close to that approximately described in the linear limit in the vicinity of a bifurcation point . importantly , for different values of the parameters @xmath59 , the uppermost bifurcation point for this branch ( open circle in fig . [ fig : bpw1 ] ) is not predicted by a simple linear theory and , due to the nonlinear mode coupling , it gets shifted from the branch of the central - wavelength soliton ( straight line ) to a two - mode branch ( 0 + 1 + 2 ) ( thick solid curve ) . as a result , if we start on the right end of the horizontal branch and follow the lowest branches of the total power @xmath60 in fig . [ fig : bpw1 ] , we pass the following sequence of the soliton families and bifurcation points : @xmath61 . if the modal parameters are selected closer to each other , the first two links of _ the bifurcation cascade _ disappear ( i.e. the corresponding bifurcation points merge ) , and the four - mode soliton bifurcates directly from the central - wavelength pulse , as predicted by the linear theory . note however that the sequence and location of the bifurcation points is a function of the cross - section of the parameter space @xmath51 , and the results presented above correspond to the choice @xmath62 . the qualitative picture of the cascading bifurcations preserves for other values of @xmath1 . in particular , near the bifurcation point a mixed - mode soliton corresponds to the localized modes guided by the central - wavelength soliton ( shepherd ) pulse . the existence of such soliton solutions is a key concept of bpw transmission when the data are launched in parallel carrying a desirable set of bits of information , all guided by the shepherd pulse at a selected wavelength . effects of the walk - off on the multi - channel bpw solitons seems to be most dangerous for the pulse alignment in the parallel links . for nearly equal soliton components , it was shown long time ago @xcite that nonlinearity can provide an effective trapping mechanism to keep the pulses together . for the shepherding effect , the corresponding numerical simulations are presented in figs . [ fig : bpw2](a - d ) for the four - channel bpw system . initially , we launch a composite four - mode soliton as an unperturbed solution a [ see fig . [ fig : bpw1 ] ] of eqs . ( [ eq : nls ] ) , without walk - off and centered at @xmath63 . when this solution evolves along the propagation direction @xmath0 in the presence of small to moderate relative walk - off ( @xmath64 for @xmath65 ) , its components remain strongly localized and mutually trapped [ fig . [ fig : bpw2](a , b ) ] , whereas it loses some energy into radiation for much larger values of the relative mode walk - off [ fig . [ fig : bpw2](c , d ) ] . recent progress in the study of cascading effects in optical materials with quadratic ( second - order or @xmath66 ) nonlinear response has offered new opportunities for all - optical processing , optical communications , and optical solitons @xcite . most of the studies of cascading effects employ parametric wave mixing processes with a single phase - matching and , as a result , two - step cascading @xcite . for example , the two - step cascading associated with type i second - harmonic generation ( shg ) includes the generation of the second harmonic ( @xmath67 ) followed by reconstruction of the fundamental wave through the down - conversion frequency mixing ( dfm ) process ( @xmath68 ) . these two processes are governed by one phase - matched interaction and they differ only in the direction of power conversion . the idea to explore more than one simultaneous nearly phase - matched process , or _ double - phase - matched ( dpm ) wave interaction _ , became attractive only recently @xcite , for the purposes of all - optical transistors , enhanced nonlinearity - induced phase shifts , and polarization switching . in particular , it was shown @xcite that multistep cascading can be achieved by two second - order nonlinear cascading processes , shg and sum - frequency mixing ( sfm ) , and these two processes can also support a novel class of multi - colour parametric solitons @xcite , briefly discussed below . to introduce the simplest model of multistep cascading , we consider the fundamental beam with frequency @xmath69 entering a noncentrosymmetric nonlinear medium with a quadratic response . as a first step , the second - harmonic wave with frequency @xmath70 is generated via the shg process . as a second step , we expect the generation of higher order harmonics due to sfm , for example , a third harmonic ( @xmath71 ) or even fourth harmonic ( @xmath72 ) @xcite . when both such processes are nearly phase matched , they can lead , via down - conversion , to a large nonlinear phase shift of the fundamental wave @xcite . additionally , the multistep cascading can support _ a novel type of three - wave spatial solitary waves _ in a diffractive @xmath66 nonlinear medium , _ multistep cascading solitons_. we start our analysis with the reduced amplitude equations derived in the slowly varying envelope approximation with the assumption of zero absorption of all interacting waves ( see , e.g. , ref . @xcite ) . introducing the effect of diffraction in a slab waveguide geometry , we obtain @xmath73 { \displaystyle \qquad\qquad\qquad\qquad + \chi_{2}a_{2 } a_{1}^{\ast}e^{-i\delta k_{2}z } = 0 , } \\*[9pt ] { \displaystyle 4 i k_{1 } \frac{\partial a_{2}}{\partial z } + \frac{\partial^{2 } a_{2}}{\partial x^{2 } } + \chi_{4 } a_{3 } a_{1}^{\ast } e^{-i\delta k_{3}z } } \\*[9pt ] { \displaystyle \qquad\qquad\qquad\qquad + \chi_{5 } a_{1}^{2 } e^{i\delta k_{2}z } = 0 , } \\*[9pt ] { \displaystyle 6 i k_{1}\frac{\partial a_{3}}{\partial z } + \frac{\partial^{2 } a_{3}}{\partial x^{2 } } + \chi_{3}a_{2}a_{1}e^{i\delta k_{3}z } = 0 , } \end{array}\ ] ] where @xmath74 , @xmath75 , and @xmath76 , and the nonlinear coupling coefficients @xmath77 are proportional to the elements of the second - order susceptibility tensor which we assume to satisfy the following relations ( no dispersion ) , @xmath78 , @xmath79 , and @xmath80 . in eqs . ( [ physeqns ] ) , @xmath81,@xmath82 and @xmath83 are the complex electric field envelopes of the fundamental harmonic ( fh ) , second harmonic ( sh ) , and third harmonic ( th ) , respectively , @xmath84 is the wavevector mismatch for the shg process , and @xmath85 is the wavevector mismatch for the sfm process . the subscripts ` 1 ' denote the fh wave , the subscripts ` 2 ' denote the sh wave , and the subscripts ` 3 ' , the th wave . following the technique earlier employed in refs . @xcite , we look for stationary solutions of eq . ( [ physeqns ] ) and introduce the normalised envelope @xmath86 , @xmath87 , and @xmath88 according to the relations , @xmath89 { \displaystyle a_{2 } = \frac{2 \beta k_{1}}{\chi_{2}}e^{2i\beta z + i\delta k_{2 } z } v , } \\*[9pt ] { \displaystyle a_{3 } = \frac{\sqrt{2\chi_{2}}\beta k_{1}}{\chi_{1}\sqrt{\chi_{5}}}e^{3i\beta z + i\delta k z } u , } \end{array}\ ] ] where @xmath90 . renormalising the variables as @xmath91 and @xmath92 , we finally obtain a system of coupled equations , @xmath93 { \displaystyle 2i \frac{\partial v}{\partial z } + \frac{\partial^{2 } v}{\partial x^{2 } } - \alpha v + \frac{1}{2 } w^{2 } + w^{\ast}u = 0 , } \\*[9pt ] { \displaystyle 3i\frac{\partial u}{\partial z } + \frac{\partial^{2 } u}{\partial x^{2 } } - \alpha_{1}u + \chi vw = 0 , } \\*[9pt ] \end{array}\ ] ] where @xmath94 and @xmath95 are two dimensionless parameters that characterise the nonlinear phase matching between the parametrically interacting waves . dimensionless material parameter @xmath96 depends on the type of phase matching , and it can take different values of order of one . for example , when both shg and sfm are due to quasi - phase matching ( qpm ) , we have @xmath97 $ ] , where @xmath98 . then , for the first - order @xmath99 qpm processes ( see , e.g. , ref . @xcite ) , we have @xmath100 , and therefore @xmath101 . when sfm is due to the third - order qpm process ( see , e.g. , ref . @xcite ) , we should take @xmath102 , and therefore @xmath103 . at last , when sfm is the fifth - order qpm process , we have @xmath104 and @xmath105 . dimensionless equations ( [ normal ] ) present a fundamental model for three - wave multistep cascading solitons in the absence of walk - off . additionally to the type i shg solitons ( see , e.g. , refs @xcite ) , the multistep cascading solitons involve the phase - matched sfm interaction ( @xmath106 ) that generates a third harmonic wave . two - parameter family of localised solutions consists of three mutually coupled waves . it is interesting to note that , similar to the case of nondegenerate three - wave mixing @xcite , eqs . ( [ normal ] ) possess an exact solution . to find it , we make a substitution @xmath107 , @xmath108 and @xmath109 , and obtain unknown parameters from the following algebraic equations @xmath110 valid for @xmath111 and @xmath112 . equations ( [ exactsol ] ) have two solutions corresponding to _ positive _ and _ negative _ values of the amplitude ( @xmath113 ) . this indicates a possibility of multi - valued solutions , even within the class of exact solutions . in general , three - wave solitons of eqs . ( [ normal ] ) can be found only numerically . figures [ fig : tr1](a ) and [ fig : tr1](b ) present two examples of solitary waves for different sets of the mismatch parameters @xmath114 and @xmath115 . when @xmath116 [ see fig . [ fig : tr1](a ) ] , which corresponds to an unmatched sfm process , the amplitude of the third harmonic is small , and it vanishes for @xmath117 . to summarise different types of three - wave solitary waves , in fig . [ fig : tr2 ] we plot the dependence of the total soliton power defined as @xmath118 . it is clearly seen that for some values of @xmath119 ( including the exact solution at @xmath120 shown by two filled circles ) , there exist _ two different branches _ of three - wave solitary waves , and only one of those branches approaches , for large values of @xmath119 , a family of two - wave solitons of the cascading limit ( fig . [ fig : tr2 ] , dashed ) . the slope of the branches changes from negative ( for small @xmath119 ) to positive ( for large @xmath119 ) , indicating a possible change of the soliton stability . however , the detailed analysis of the soliton stability is beyond the scope of this paper ( see , e.g. , refs . @xcite ) . another type of multistep cascading parametric processes which involve only two frequencies , i.e. _ two - colour multistep cascading _ , can occur due to the vectorial interaction of waves with different polarization . we denote two orthogonal polarization components of the fundamental harmonic ( fh ) wave ( @xmath121 ) as a and b , and two orthogonal polarizations of the second harmonic ( sh ) wave ( @xmath122 ) , as s and t. then , a simple multistep cascading process consists of the following steps . first , the fh wave a generates the sh wave s via type i shg process . then , by down - conversion sa - b , the orthogonal fh wave b is generated . at last , the initial fh wave a is reconstructed by the processes sb - a or ab - s , sa - a . two principal second - order processes aa - s and ab - s correspond to _ two different components _ of the @xmath66 susceptibility tensor , thus introducing additional degrees of freedom into the parametric interaction . different cases of such type of multistep cascading processes are summarized in table [ tab : dpm ] . to demonstrate some of the unique properties of the multistep cascading , we discuss here how it can be employed for soliton - induced waveguiding effects in quadratic media . for this purpose , we consider a model of two - frequency multistep cascading described by the principal dpm process ( c ) ( see table [ tab : dpm ] above ) in the planar slab - waveguide geometry . using the slowly varying envelope approximation with the assumption of zero absorption of all interacting waves , we obtain @xmath123 { \displaystyle 2 i k_{1}\frac{\partial b}{\partial z } + \frac{\partial^{2 } b } { \partial x^{2 } } + \chi_2 s b^{\ast}e^{-i\delta k_2 z } = 0 , } \\*[9pt ] { \displaystyle 4 i k_{1 } \frac{\partial s}{\partial z } + \frac{\partial^{2 } s}{\partial x^{2 } } + 2 \chi_1 a^2 e^{i\delta k_1 z } + 2 \chi_2 b^2 e^{i\delta k_2 z } = 0 , } \end{array}\end{aligned}\ ] ] where @xmath74 , the nonlinear coupling coefficients @xmath77 are proportional to the elements of the second - order susceptibility tensor , and @xmath124 and @xmath125 are the corresponding wave - vector mismatch parameters . to simplify the system ( [ eq_1 ] ) , we look for its stationary solutions and introduce the normalized envelopes @xmath126 , @xmath127 , and @xmath128 according to the following relations , @xmath129 , @xmath130 , and @xmath131 , where @xmath132 , @xmath133 , and @xmath134 , and the longitudinal and transverse coordinates are measured in the units of @xmath135 and @xmath136 , respectively . then , we obtain a system of normalized equations , @xmath137 { \displaystyle i \frac{\partial v}{\partial z } + \frac{\partial^{2 } v}{\partial x^{2 } } - \alpha_1 v + \chi v^{\ast}w= 0 , } \\*[9pt ] { \displaystyle 2i \frac{\partial w}{\partial z } + \frac{\partial^{2}w}{\partial x^{2 } } - \alpha w+\frac{1}{2}(u^2+v^2)= 0 , } \end{array}\ ] ] where @xmath138 , @xmath139 , and @xmath140 . first of all , we notice that for @xmath141 ( or , similarly , @xmath142 ) , the dimensionless eqs . ( [ eq_n ] ) reduce to the corresponding model for the two - step cascading due to type i shg discussed earlier @xcite , and its stationary solutions are defined by the equations for real @xmath126 and @xmath128 , @xmath143 { \displaystyle \frac{d^2 w}{d x^{2 } } - \alpha w + \frac{1}{2 } u^2 = 0 , } \end{array}\ ] ] that possess a one - parameter family of two - wave localized solutions @xmath144 found earlier numerically for any @xmath145 , and also known analytically for @xmath146 , @xmath147 ( see ref . @xcite ) . then , in the small - amplitude approximation , the equation for real orthogonally polarized fh wave @xmath127 can be treated as an eigenvalue problem for an effective waveguide created by the sh field @xmath148 , @xmath149 v = 0.\ ] ] therefore , an additional parametric process allows to propagate a probe beam of one polarization in _ an effective waveguide _ created by a two - wave spatial soliton in a quadratic medium with fh component of another polarization . however , this type of waveguide is different from what has been studied for kerr - like solitons because it is _ coupled parametrically _ to the guided modes and , as a result , the physical picture of the guided modes is valid , rigorously speaking , only in the case of stationary phase - matched beams . as a result , the stability of the corresponding waveguide and localized modes of the orthogonal polarization it guides is a key issue . in particular , the waveguide itself ( i.e. _ two - wave parametric soliton _ ) becomes unstable for @xmath150 @xcite . in order to find the guided modes of the parametric waveguide created by a two - wave quadratic soliton , we have to solve eq . ( [ eq_eigen ] ) where the exact solution @xmath148 is to be found numerically . then , to address this problem analytically , approximate solutions can be used , such as those found with the help of the variational method @xcite . however , the different types of the variational ansatz used do not provide a very good approximation for the soliton profile at all @xmath114 . for our eigenvalue problem ( [ eq_eigen ] ) , the function @xmath148 defines parameters of the guided modes and , in order to obtain accurate results , it should be calculated as close as possible to the exact solutions found numerically . to resolve this difficulty , below we suggest a novel `` almost exact '' solution that _ would allow to solve analytically many of the problems involving quadratic solitons _ , including the eigenvalue problem ( [ eq_eigen ] ) . first , we notice that from the exact solution at @xmath146 and the asymptotic result for large @xmath114 , @xmath151 , it follows that the sh component @xmath148 of eqs . ( [ eq_2 ] ) remains almost self - similar for @xmath152 . thus , we look for the sh field in the form @xmath153 , where @xmath154 and @xmath155 are to be defined . the solution for @xmath156 should be consistent with this choice of the shape for sh , and it is defined by the first ( linear for @xmath126 ) equation of the system ( [ eq_2 ] ) . therefore , we can take @xmath126 in the form of the lowest guided mode , @xmath157 , that corresponds to an effective waveguide @xmath148 . by matching the asymptotics of these trial functions with those defined directly from eqs . ( [ eq_2 ] ) at small and large @xmath158 , we obtain the following solution , @xmath159 @xmath160 here , the third relation allows us to find @xmath154 for arbitrary @xmath114 as a solution of a cubic equation , and then to find all other parameters as functions of @xmath114 . for mismatches in the interval @xmath161 , the parameter values change monotonically in the regions : @xmath162 , @xmath163 , and @xmath164 . it is really amazing that the analytical solution ( [ eq_s]),([eq_p ] ) provides _ an excellent approximation _ for the profiles of the two - wave parametric solitons found numerically , with the relative errors not exceeding 1%3% for stable solitons ( e.g. when @xmath165 ) . as a matter of fact , we can treat eqs . ( [ eq_s ] ) and ( [ eq_p ] ) as an _ approximate scaling transformation _ of the family of two - wave bright solitons . moreover , this solution allows us to capture some remarkable internal similarities and distinctions between the solitons existing in different types of nonlinear media . in particular , as follows from eqs . ( [ eq_s ] ) and ( [ eq_p ] ) , the fh component and the self - consistent effective waveguide ( created by the sh field ) have approximately the same stationary transverse profiles as for one - component solitons in a kerr - like medium with power - law nonlinear response @xcite . for @xmath166 ( @xmath167 ) and @xmath168 ( @xmath169 ) our general expressions reduce to the known analytical solutions , and the fh profile is exactly the same as that for solitons in quadratic and cubic kerr media , respectively . on the other hand , the strength of self - action for quadratic solitons depends on the normalized phase mismatch @xmath114 and , in general , the beam dynamics for parametric wave mixing can be very different from that observed in kerr - type media . now , the eigenvalue problem ( [ eq_eigen ] ) can be readily solved analytically . the eigenmode cutoff values are defined by the parameter @xmath119 that takes one of the discrete values , @xmath170 , where @xmath171^{1/2}$ ] . number @xmath172 stands for the mode order @xmath173 , and the localized solutions are possible provided @xmath174 . the profiles of the corresponding guided modes are @xmath175 where @xmath176 , @xmath177 is the hypergeometric function , and @xmath178 is the mode s amplitude which can not be determined within the framework of the linear analysis . according to these results , a two - wave parametric soliton creates , a multi - mode waveguide and larger number of the guided modes can be observed for smaller @xmath114 . figures [ fig : al1](a , b ) show the dependence of the mode cutoff values @xmath179 for a fixed @xmath180 , and @xmath181 for a fixed @xmath114 , respectively . for the case @xmath103 , the dependence has a simple form : @xmath182 ^ 2 $ ] . because a two - wave soliton creates an induced waveguide parametrically coupled to its guided modes of the orthogonal polarization , the dynamics of the guided modes _ may differ drastically _ from that of conventional waveguides based on the kerr - type nonlinearities . figures show two examples of the evolution of guided modes . in the first example [ see fig . [ fig : wave_w](a - c ) ] , a weak fundamental mode is amplified via parametric interaction with a soliton waveguide , and the mode experiences a strong power exchange with the orthogonally polarized fh component through the sh field . this process is accompanied by only a weak deformation of the induced waveguide [ see fig . [ fig : wave_w](a ) dotted curve ] . the resulting effect can be interpreted as a power exchange between two guided modes of orthogonal polarizations in a waveguide created by the sh field . in the second example , the propagation is stable [ see fig . [ fig : wave_w](d ) ] . when all the fields in eq . ( [ eq_n ] ) are not small , i.e. the small - amplitude approximation is no longer valid , the profiles of the three - component solitons should be found numerically . however , some of the lowest - order states can be calculated approximately using the approach of the `` almost exact '' solution ( [ eq_s]),([eq_p ] ) described above , which is presented in detail elsewhere @xcite . moreover , a number of the solutions and their families can be obtained in _ an explicit analytical form_. for example , for @xmath183 , there exist two _ families of three - component solitary waves _ for any @xmath152 , that describe soliton branches starting at the bifurcation points @xmath184 at : ( i ) the soliton with a zero - order guided mode for @xmath185 : @xmath186 , @xmath187 , @xmath188 , and ( ii ) the soliton with a first - order guided mode for @xmath103 : @xmath189 , @xmath190 , @xmath188 , where @xmath191 and @xmath192 . for a practical realization of the dpm processes and the soliton waveguiding effects described above , we can suggest two general methods . the first method is based on the use of _ two commensurable periods _ of the quasi - phase - matched ( qpm ) periodic grating . indeed , to achieve dpm , we can employ the first - order qpm for one parametric process , and the third - order qpm , for the other parametric process . taking , as an example , the parameters for linbo@xmath193 and aa - s @xmath194 and bb - s @xmath195 processes @xcite , we find two points for dpm at about 0.89 @xmath196 m and 1.25 @xmath196 m . this means that a single qpm grating can provide simultaneous phase - matching for two parametric processes . for such a configuration , we obtain @xmath197 or , interchanging the polarization components , @xmath198 . the second method to achieve the conditions of dpm processes is based on the idea of _ quasi - periodic qpm grating _ specifically , fibonacci optical superlattices provide an effective way to achieve phase - matching at _ several incommensurable periods _ allowing multi - frequency harmonic generation in a single structure . we describe the properties of such structures in the next section . for many years , solitary waves have been considered as _ coherent localized modes _ of nonlinear systems , with particle - like dynamics quite dissimilar to the irregular and stochastic behavior observed for chaotic systems @xcite . however , about 20 years ago akira hasegawa , while developing a statistical description of the dynamics of an ensemble of plane waves in nonlinear strongly dispersive plasmas , suggested the concept of a localized envelope of random phase waves @xcite . because of the relatively high powers required for generating self - localized random waves , this notion remained a theoretical curiosity until recently , when the possibility to generate spatial optical solitons by a partially incoherent source was discovered in a photorefractive medium @xcite . the concept of incoherent solitons can be compared with a different problem : the propagation of a soliton through a spatially disordered medium . indeed , due to random scattering on defects , the phases of the individual components forming a soliton experience random fluctuations , and the soliton itself becomes _ partially incoherent _ in space and time . for a low - amplitude wave ( linear regime ) spatial incoherence is known to lead to a fast decay . as a result , the transmission coefficient vanishes exponentially with the length of the system , the phenomenon known as anderson localization @xcite . however , for large amplitudes ( nonlinear regime ) , when the nonlinearity length is much smaller than the anderson localization length , a soliton can propagate almost unchanged through a disordered medium as predicted theoretically in 1990 @xcite and recently verified experimentally @xcite . these two important physical concepts , spatial self - trapping of light generated by an incoherent source in a homogeneous medium , and suppression of anderson localization for large - amplitude waves in spatially disordered media , both result from the effect of strong nonlinearity . when the nonlinearity is sufficiently strong it acts as _ an effective phase - locking mechanism _ by producing a large frequency shift of the different random - phase components , and thereby introducing _ an effective order _ into an incoherent wave packet , thus enabling the formation of localized structures . in other words , both phenomena correspond to the limit when the ratio of the nonlinearity length to the characteristic length of ( spatial or temporal ) fluctuations is small . in the opposite limit , when this ratio is large , the wave propagation is practically linear . below we show that , at least for aperiodic inhomogeneous structures , solitary waves can exist in the intermediate regime in the form of _ quasiperiodic nonlinear localized modes_. as an example , we consider shg and nonlinear beam propagation in _ fibonacci optical superlattices _ , and demonstrate numerically the possibility of spatial self - trapping of quasiperiodic waves whose envelope amplitude varies quasiperiodically , while still maintaining a stable , well - defined spatially localized structure , _ a quasiperiodic envelope soliton_. we consider the interaction of a fundamental wave with the frequency @xmath69 ( fh ) and its sh in a slab waveguide with quadratic ( or @xmath66 ) nonlinearity . assuming the @xmath66 susceptibility to be modulated and the nonlinearity to be of the same order as diffraction , we write the dynamical equations in the form @xmath199 { \displaystyle i\frac{\partial w}{\partial z } + \frac{1}{4 } \frac{\partial^2 w}{\partial x^2 } + d(z ) u^2 e^{i\beta z } = 0 , } \end{array}\ ] ] where @xmath200 and @xmath201 are the slowly varying envelopes of the fh and sh , respectively . the parameter @xmath202 is proportional to the phase mismatch @xmath203 , @xmath204 and @xmath205 being the wave numbers at the two frequencies . the transverse coordinate @xmath158 is measured in units of the input beam width @xmath206 , and the propagation distance @xmath0 in units of the diffraction length @xmath207 . the spatial modulation of the @xmath66 susceptibility is described by the quasi - phase - matching ( qpm ) grating function @xmath208 . in the context of shg , the qpm technique is an effective way to achieve phase matching , and it has been studied intensively @xcite . here we consider a qpm grating produced by a quasiperiodic nonlinear optical superlattice . quasiperiodic optical superlattices , one - dimensional analogs of quasicrystals @xcite , are usually designed to study the effect of anderson localization in the linear regime of light propagation . for example , gellermann _ et al . _ measured the optical transmission properties of quasiperiodic dielectric multilayer stacks of sio@xmath209 and tio@xmath209 thin films and observed a strong suppression of the transmission @xcite . for qpm gratings , a nonlinear quasiperiodic superlattice of litao@xmath193 , in which two antiparallel ferro - electric domains are arranged in a fibonacci sequence , was recently fabricated by zhu _ et al . _ @xcite , who measured multi - colour shg with energy conversion efficiencies of @xmath210 . this quasiperiodic optical superlattice in litao@xmath193 can also be used for efficient direct third harmonic generation @xcite . the quasiperiodic qpm gratings have two building blocks a and b of the length @xmath211 and @xmath212 , respectively , which are ordered in a fibonacci sequence [ fig . [ fig : d_z](a ) ] . each block has a domain of length @xmath213=l ( @xmath214=l ) with @xmath215=@xmath216 ( shaded ) and a domain of length @xmath217=@xmath218 [ @xmath219=@xmath220 with @xmath215=@xmath221 ( white ) . in the case of @xmath66 nonlinear qpm superlattices this corresponds to positive and negative ferro - electric domains , respectively . the specific details of this type of fibonacci optical superlattices can be found elsewhere @xcite . for our simulations presented below we have chosen @xmath222= @xmath223= 0.34 , where @xmath224= @xmath225 is the so - called _ golden ratio_. this means that the ratio of length scales is also the golden ratio , @xmath226= @xmath224 . furthermore , we have chosen @xmath227=0.1 . the grating function @xmath208 , which varies between @xmath216 and @xmath221 according to the fibonacci sequence , can be expanded in a fourier series @xmath228 where @xmath229=@xmath230=0.52 for the chosen parameter values . hence the spectrum is composed of sums and differences of the basic wavenumbers @xmath231=@xmath232 and @xmath233=@xmath234 . these components fill the whole fourier space densely , since @xmath231 and @xmath233 are incommensurate . figure [ fig : d_z](b ) shows the numerically calculated fourier spectrum @xmath235 . the lowest - order `` fibonacci modes '' are clearly the most intense . to analyze the beam propagation and shg in a quasiperiodic qpm grating one could simply average eqs . ( [ dynam ] ) . to lowest order this approach always yields a system of equations with constant mean - value coefficients , which does not allow to describe oscillations of the beam amplitude and phase . however , here we wish to go beyond the averaged equations and consider the rapid large - amplitude variations of the envelope functions . this can be done analytically for periodic qpm gratings @xcite . however , for the quasiperiodic gratings we have to resolve to numerical simulations . thus we have solved eqs . ( [ dynam ] ) numerically with a second - order split - step routine . at the input of the crystal we excite the fundamental beam ( corresponding to unseeded shg ) with a gaussian profile , @xmath236 we consider the quasiperiodic qpm grating with matching to the peak at @xmath237 , i.e. , @xmath238=@xmath237=82.25 . first , we study the small - amplitude limit when a weak fh is injected with a low amplitude . figures [ fig : soliton](a , b ) show an example of the evolution of fh and sh in this effectively linear regime . as is clearly seem from fig . [ fig : soliton](b ) the sh wave is excited , but both beams eventually diffract . when the amplitude of the input beam exceeds a certain threshold , self - focusing and localization should be observed for both harmonics . figures [ fig : soliton](c , d ) show an example of the evolution of a strong input fh beam , and its corresponding sh . again the sh is generated , but now the nonlinearity is so strong that it leads to self - focusing and mutual self - trapping of the two fields , resulting in a spatially localized two - component soliton , despite the continuous scattering of the quasiperiodic qpm grating . it is important to notice that the two - component localized beam created due to the self - trapping effect is quasiperiodic by itself . as a matter of fact , after an initial transient its amplitude oscillates in phase with the quasiperiodic qpm modulation @xmath208 . this is illustrated in fig . [ fig : oscillations ] , where we show in more detail the peak intensities in the asymptotic regime of the evolution . the oscillations shown in fig . [ fig : oscillations ] are in phase with the oscillations of the qpm grating @xmath208 , and we indeed found that their spectra are similar . our numerical results show that the quasiperiodic envelope solitons can be generated for a broad range of the phase - mismatch @xmath238 . the amplitude and width of the solitons depend on the effective mismatch , which is the separation between @xmath238 and the nearest strong peak @xmath235 in the fibonacci qpm grating spectrum [ see fig . [ fig : d_z](b ) ] . thus , low - amplitude broad solitons are excited for @xmath238-values in between peaks , whereas high - amplitude narrow solitons are excited when @xmath238 is close to a strong peak , as shown in fig . [ fig : soliton](c , d ) . to analyse in more detail the transition between the linear ( diffraction ) and nonlinear ( self - trapping ) regimes , we have made a series of careful numerical simulations @xcite . in fig . [ fig : transmission ] we show the transmission coefficients and the beam widths at the output of the crystal versus the intensity of the fh input beam , for a variety of @xmath238-values . these dependencies clearly illustrate the universality of the generation of localised modes for varying strength of nonlinearity , i.e. a quasiperiodic soliton is generated only for sufficiently high amplitudes . this is of course a general phenomenon also observed in many nonlinear isotropic media . however , here the self - trapping occurs for quasiperiodic waves , with the quasiperiodicity being preserved in the variation of the amplitude of both components of the soliton . we have overviewed several important physical examples of the multi - component solitary waves which appear due to multi - mode and/or multi - frequency coupling in nonlinear optical fibers and waveguides . we have described several types of such multi - component solitary waves , including : ( i ) multi - wavelength solitary waves in multi - channel bit - parallel - wavelength fiber transmission systems , ( ii ) multi - colour parametric spatial solitary waves due to multistep cascading in quadratic materials , and ( iii ) quasiperiodic envelope solitons in fibonacci optical superlattices . these examples reveal some general features and properties of multi - component solitary waves in nonintegrable nonlinear models , also serving as a stepping stone for approaching other problems of the multi - mode soliton coupling and interaction . the work was supported by the australian photonics cooperative research centre and by a collaborative australia - denmark grant of the department of industry , science , and tourism ( australia ) . for an overview of quadratic spatial solitons , see l. torner , in : _ beam shaping and control with nonlinear optics _ , f. kajzer and r. reinisch , eds . ( plenum , new york , 1998 ) , p. 229 ; yu . s. kivshar , in : _ advanced photonics with second - order optically nonlinear processes _ , a. d. boardman , l. pavlov , and s. tanev , eds . ( kluwer , dordretch , 1998 ) , p. 451

a cluster category is a certain 2-calabi - yau orbit category of the derived category of a hereditary abelian category . cluster categories were introduced in @xcite in order to give a categorical model for the combinatorics of fomin - zelevinsky cluster algebras @xcite . they are triangulated @xcite and admit ( cluster-)tilting objects , which model the clusters of a corresponding ( acyclic ) cluster algebra @xcite . each cluster in a fixed cluster algebra comes together with a finite quiver , and in the categorical model this quiver is in fact the gabriel quiver of the corresponding tilting object @xcite . a principal ingredient in the construction of a cluster algebra is quiver mutation . it controls the exchange procedure which gives a rule for producing a new cluster variable and hence a new cluster from a given cluster . exchange is modeled by cluster categories in the acyclic case @xcite in terms of a mutation rule for tilting objects , i.e. a rule for replacing an indecomposable direct summand in a tilting object with another indecomposable rigid object , to get a new tilting object . quiver mutation describes the relation between the gabriel quivers of the corresponding tilting objects . analogously to the definition of the cluster category , for a positive integer @xmath0 , it is natural to define a certain @xmath1-calabi - yau orbit category of the derived category of a hereditary abelian category . this is called the _ @xmath0-cluster category_. implicitly , @xmath0-cluster categories was first studied in @xcite , and their ( cluster-)tilting objects have been studied in @xcite . combinatorial descriptions of @xmath0-cluster categories in dynkin type @xmath2 and @xmath3 are given in @xcite . in cluster categories the mutation rule for tilting objects is described in terms of certain triangles called _ exchange triangles_. by @xcite the existence of exchange triangles generalizes to @xmath0-cluster categories . it was shown in @xcite that there are exactly @xmath1 non - isomorphic complements to an almost complete tilting object , and that they are determined by the @xmath1 exchange triangles defined in @xcite . the aim of this paper is to give a combinatorial description of mutation in @xmath0-cluster categories . _ a priori _ , one might expect to be able to do this by keeping track of the gabriel quivers of the tilting objects . however , it is easy to see that the gabriel quivers do not contain enough information . we proceed to associate to a tilting object a quiver each of whose arrows has an associated colour @xmath4 . the arrows with colour 0 form the gabriel quiver of the tilting object . we then define a mutation operation on coloured quivers and show that it is compatible with mutation of tilting objects . a consequence is that the effect of an arbitrary sequence of mutations on a tilting object in an @xmath0-cluster category can be calculated by a purely combinatorial procedure . our definition of a coloured quiver associated to a tilting object makes sense in any @xmath1-calabi - yau category , such as for example those studied in @xcite . we hope that our constructions may shed some light on mutation of tilting objects in this more general setting . in section 1 , we review some elementary facts about higher cluster categories . in section 2 , we explain how to define the coloured quiver of a tilting object , we define coloured quiver mutation , and we state our main theorem . in sections 3 and 4 , we state some further lemmas about higher cluster categories , and we prove certain properties of the coloured quivers of tilting objects . we prove our main result in sections 5 and 6 . in sections 7 and 8 we point out some applications . in section 9 we interpret our construction in terms of @xmath0-cluster complexes . in section 10 , we give an alternative algorithm for computing coloured quiver mutation . section 11 discusses the example of @xmath0-cluster categories of dynkin type @xmath2 , using the model developed by baur and marsh @xcite . we would like to thank idun reiten , in conversation with whom the initial idea of this paper took shape . let @xmath5 be an algebraically closed field , and let @xmath6 be a finite acyclic quiver with @xmath7 vertices . then the path algebra @xmath8 is a hereditary finite dimensional basic @xmath5-algebra let @xmath9 be the category of finite dimensional left @xmath10-modules . let @xmath11 be the bounded derived category of @xmath10 , and let @xmath12 $ ] be the @xmath13th shift functor on @xmath14 . we let @xmath15 denote the auslander - reiten translate , which is an autoequivalence on @xmath14 such that we have a bifunctorial isomorphism in @xmath14 @xmath16 ) \simeq d{\operatorname{hom}\nolimits}(b,\tau a).\ ] ] in other words @xmath17 \tau$ ] is a serre functor . let @xmath18 $ ] . the @xmath0-cluster category is the orbit category @xmath19 $ ] . the objects in @xmath20 are the objects in @xmath14 , and two objects @xmath21 are isomorphic in @xmath20 if and only if @xmath22 in @xmath14 . the maps are given by @xmath23 . by @xcite , the category @xmath20 is triangulated and the canonical functor @xmath24 is a triangle functor . we denote therefore by @xmath25 $ ] the suspension in @xmath20 . the @xmath0-cluster category is also krull - schmidt and has an ar - translate @xmath15 inherited from @xmath14 , such that the formula ( [ ar ] ) still holds in @xmath20 . if follows that @xmath17 \tau$ ] is a serre functor for @xmath20 and that @xmath20 is @xmath1-calabi - yau , since @xmath26 $ ] . the indecomposable objects in @xmath14 are of the form @xmath27 $ ] , where @xmath28 is an indecomposable @xmath10-module and @xmath29 . we can choose a fundamental domain for the action of @xmath18 $ ] on @xmath14 , consisting of the indecomposable objects @xmath27 $ ] with @xmath30 , together with the objects @xmath31 $ ] with @xmath28 an indecomposable projective @xmath10-module . then each indecomposable object in @xmath20 is isomorphic to exactly one of the indecomposables in this fundamental domain . we say that @xmath32 $ ] has degree @xmath33 , denoted @xmath34 ) = d$ ] . furthermore , for an arbitrary object @xmath35 in @xmath36 , we let @xmath37 $ ] be the @xmath10-module which is the ( shifted ) direct sum of all summands @xmath38 of @xmath39 with @xmath40 . in the following theorem the equivalence between ( i ) and ( ii ) is shown in @xcite and the equivalence between ( i ) and ( iii ) is shown in @xcite . let @xmath41 be an object in @xmath20 satisfying @xmath42 ) = 0 $ ] for @xmath43 . then the following are equivalent * if @xmath44 ) = 0 $ ] for @xmath43 then @xmath45 is in @xmath46 . * if @xmath47 ) = 0 $ ] for @xmath43 then @xmath45 is in @xmath46 . * @xmath41 has @xmath48 indecomposable direct summands , up to isomorphism . here @xmath46 denotes the additive closure of @xmath41 . a ( cluster-)tilting object @xmath41 in an @xmath0-cluster is an object satisfying the conditions of the above theorem . for a tilting object @xmath49 , with each @xmath50 indecomposable , and @xmath51 an indecomposable direct summand , we call @xmath52 an almost complete tilting object . we let @xmath53 denote the @xmath5-space of irreducible maps @xmath54 in a krull - schmidt @xmath5-category @xmath55 . the following crucial result is proved in @xcite and @xcite . [ p : number ] there are , up to isomorphism , @xmath1 complements of an almost complete tilting object . let @xmath51 be an indecomposable direct summand in an @xmath0-cluster tilting object @xmath56 . the complements of @xmath57 are denoted @xmath58 for @xmath59 , where @xmath60 . by @xcite , there are @xmath1 exchange triangles @xmath61 here the @xmath62 are in @xmath63 and the maps @xmath64 ( resp . @xmath65 ) are minimal left ( resp . right ) @xmath63-approximations , and hence not split mono or split epi . note that by minimality , the maps @xmath64 and @xmath65 have no proper zero summands . we first recall the definition of quiver mutation , formulated in @xcite in terms of skew - symmetric matrices . let @xmath66 be a quiver with vertices @xmath67 and with no loops or oriented two - cycles , where @xmath68 denotes the number of arrows from @xmath13 to @xmath69 . let @xmath70 be a vertex in @xmath71 . then , a new quiver @xmath72 is defined by the following data @xmath73 it is easily verified that this definition is equivalent to the one of fomin - zelevinsky . now we consider coloured quivers . let @xmath0 be a positive integer . an @xmath0-coloured ( multi-)quiver @xmath71 consists of vertices @xmath67 and coloured arrows @xmath74 , where @xmath75 . let @xmath76 denote the number of arrows from @xmath13 to @xmath70 of colour @xmath77 . we will consider coloured quivers with the following additional conditions . * no loops : @xmath78 for all @xmath79 . * monochromaticity : if @xmath80 , then @xmath81 for @xmath82 * skew - symmetry : @xmath83 . we will define an operation on a coloured quiver @xmath71 satisfying the above conditions . let @xmath70 be a vertex in @xmath71 and let @xmath84 be the coloured quiver defined by @xmath85 in an @xmath0-cluster category @xmath20 , for every tilting object @xmath86 , with the @xmath50 indecomposable , we will define a corresponding @xmath0-coloured quiver @xmath87 , as follows . let @xmath88 be two non - isomorphic indecomposable direct summands of the @xmath0-cluster tilting object @xmath41 and let @xmath89 denote the multiplicity of @xmath90 in @xmath91 . we define the @xmath0-coloured quiver @xmath87 of @xmath41 to have vertices @xmath13 corresponding to indecomposable direct summands @xmath50 , and @xmath92 . note , in particular , that the @xmath93-coloured arrows are the arrows from the gabriel quiver for the endomorphism ring of @xmath41 . by definition , @xmath87 satisfies condition ( i ) . we show in section [ s : higher ] that ( ii ) is satisfied ( this also follows from @xcite ) , and in section [ s : symmetry ] that ( iii ) is also satisfied . the aim of this paper is to prove the following theorem , which is a generalization of the main result of @xcite . [ t : main ] let @xmath86 and @xmath94 be @xmath0-tilting objects , where there is an exchange triangle @xmath95 . then @xmath96 . in the case @xmath97 the coloured quiver of a tilting object @xmath41 is given by @xmath98 and @xmath99 where @xmath100 denotes the number of arrows in the gabriel quiver of @xmath41 . then coloured mutation of the coloured quiver corresponds to fz - mutation of the gabriel quiver . let @xmath6 be @xmath101 with linear orientation , i.e. the quiver @xmath102 . the ar - quiver of the 2-cluster category of @xmath103 is @xmath104 & & { i_3 } \ar[dr ] & & * + + [ o][f-]{p_3[1 ] } \ar[dr ] & & i_1[1 ] \ar[dr ] & & p_1[2]\ar[dr ] \\ p_2[2 ] \ar[ur ] \ar[dr ] & & { p_2 } \ar[ur ] \ar[dr ] & & * + [ o][f-]{i_2 } \ar[ur ] \ar[dr ] & & p_2[1 ] \ar[ur ] \ar[dr ] & & i_2[1]\ar[ur]\ar[dr ] & & p_2[2 ] \\ & { p_3 } \ar[ur ] & & * + [ o][f-]{i_1 } \ar[ur ] & & p_1[1 ] \ar[ur ] & & i_3[1]\ar[ur ] & & p_3[2 ] \ar[ur ] & & } \ ] ] the direct sum @xmath105 $ ] of the encircled indecomposable objects gives a tilting object . its coloured quiver is @xmath106 & i_2 \ar@<0.6ex>^{(0)}[r ] \ar@<0.6ex>^{(2)}[l ] & p_3[1 ] \ar@<0.6ex>^{(2)}[l ] } \ ] ] now consider the exchange triangle @xmath107 \to i_3[1 ] \to\ ] ] and the new tilting object @xmath108 \amalg p_3[1]$ ] . the coloured quiver of @xmath109 is @xmath110 \ar@<0.6ex>^{(1)}[r ] & i_3[1 ] \ar@<0.6ex>^{(2)}[r ] \ar@<0.6ex>^{(1)}[l ] & p_3[1 ] \ar@<0.6ex>^{(0)}[l ] \ar@<0.6ex>^{(2)}@/^3.5pc/[ll ] } \ ] ] in this section we summarize some further known results about @xmath0-cluster categories . most of these are from @xcite and @xcite . we include some proofs for the convenience of the reader . tilting objects in @xmath111 give rise to partial tilting modules in @xmath9 , where a _ partial tilting module _ @xmath28 in @xmath9 , is a module with @xmath112 . [ l : partial ] * when @xmath41 is a tilting object in @xmath36 , then each @xmath113 is a partial tilting module in @xmath9 . * the endomorphism ring of a partial tilting module has no oriented cycles in its ordinary quiver . \(a ) is obvious from the definition . see ( * ? ? ? 4.2 ) for ( b ) . in the following note that degrees of objects are always considered with a fixed choice of fundamental domain , and sums and differences of degrees are always computed modulo @xmath1 . [ l : div ] assume @xmath114 . * @xmath115 for any indecomposable exceptional object @xmath39 . * we have that @xmath116 * the distribution of degrees of complements is one of the following * * there is exactly one complement of each degree , or * * there is no complement of degree @xmath0 , two complements in one degree @xmath117 , and exactly one complement in all degrees @xmath118 . * if @xmath119 , then @xmath120 . * for @xmath121 we have @xmath122 ) = \begin{cases } k & \text { if $ c'-c+t = 0 ( { \operatorname{mod}\nolimits}m+1)$ } \\ 0 & \text { else } \end{cases}\ ] ] \(a ) follows from the fact that @xmath123 for exceptional objects and the definition of maps in a @xmath0-cluster category . \(b ) follows from the fact that @xmath124 ) \neq 0 $ ] , since in the exchange triangles , the @xmath125 are not split mono and ( c ) follows from ( b ) . considering the two different possible distributions of complements , we obtain from ( c ) that if @xmath126 and @xmath127 and @xmath128 , then @xmath129 . consider the case @xmath130 . we can assume @xmath131 , since else the statement is void . hence we can clearly assume that @xmath132 . there is an exchange triangle induced from an exact sequence in @xmath9 , @xmath133.\ ] ] it is clear that @xmath134 , t_i^{(c-1 ) } ) = 0 $ ] , since @xmath131 . we claim that also @xmath135 . this holds since @xmath136 is a partial tilting object in @xmath10 , and so there are no cycles in the endomorphism ring , by lemma [ l : partial ] . hence also @xmath137 follows , and this finishes the proof for ( d ) . for ( e ) we first apply @xmath138 to the exchange triangle @xmath139 and consider the corresponding long - exact sequence , to obtain that @xmath140 ) = \begin{cases } k & \text { if $ t = 1 $ } \\ 0 & \text { if $ t=0 $ or $ t \in \{2 , \dots , m \}$ } \end{cases}.\ ] ] now consider @xmath141)$ ] . when @xmath142 , we have that @xmath143 ) \simeq { \operatorname{hom}\nolimits}(t_i^{(c+u+1 ) } , t_i^{(c)}[v+1 ] ) \simeq \\ { \operatorname{hom}\nolimits}(t_i^{(c-1 ) } , t_i^{(c)}[v+m - u ] ) \simeq { \operatorname{hom}\nolimits}(t_i^{(c ) } , t_i^{(c-1)}[1+u - v ] ) . \end{gathered}\ ] ] when @xmath144 , we have that @xmath145 ) \simeq { \operatorname{hom}\nolimits}(t_i^{(c+u-1 ) } , t_i^{(c)}[v-1 ] ) \simeq \\ { \operatorname{hom}\nolimits}(t_i^{(c ) } , t_i^{(c)}[v - u]).\ ] ] combining these facts , ( e ) follows . [ l : div2 ] the following statements are equivalent * @xmath146 ) = 0 $ ] * @xmath90 is not a direct summand in @xmath147 * @xmath50 is not a direct summand in @xmath148 furthermore , @xmath149 ) = 0 $ ] for @xmath150 . note that @xmath151 , so ( b ) and ( c ) are equivalent . consider the exact sequence @xmath152 ) \to { \operatorname{hom}\nolimits}(t_i^{(c ) } , b_j^{(0)}[1 ] ) \to \\ { \operatorname{hom}\nolimits}(t_i^{(c ) } , t_j^{(1)}[1 ] ) \to { \operatorname{hom}\nolimits}(t_i^{(c ) } , t_j^{(0)}[2 ] ) \to \end{gathered}\ ] ] coming from applying @xmath153 to the exchange triangle @xmath154 the first and fourth terms are always zero . using [ l : div](e ) we get that the second term ( and hence the third ) is non - zero if and only if @xmath155 and @xmath50 is a direct summand in @xmath148 . ( @xcite)[l : composing ] for @xmath156 , the composition @xmath157 \circ h_k^{(v-2)}[2 ] \circ \cdots \circ h_k^{(v - l+1)}[l-1 ] \colon t_k^{(v ) } \to t_k^{(v - l)}[l]\ ] ] is non - zero and a basis for @xmath158)$ ] . for @xmath97 , see @xcite . assume @xmath159 . for the first claim see @xcite , while the second claim then follows from lemma [ l : div](e ) . we include an independent proof of the following crucial property . @xcite [ l : disjoint ] @xmath160 and @xmath161 has no common non - zero direct summands whenever @xmath162 . when @xmath97 , this is proved in @xcite . assume @xmath163 . we consider two cases , @xmath164 or @xmath165 . consider first the case @xmath166 . without loss of generality we can assume @xmath167 and @xmath168 , and that @xmath169 . assume that there exists a ( non - zero ) indecomposable @xmath170 , which is a direct summand in @xmath171 and in @xmath172 . we have that @xmath173 by lemma [ l : div](b ) . assume first @xmath174 . then the exchange triangle @xmath175 is induced from the degree 0 part of the derived category , and hence from an exact sequence in @xmath9 . then the endomorphism ring of the partial tilting module @xmath176 has a cycle , which is a contradiction to lemma [ l : partial ] . assume now that @xmath177 . then @xmath178 , where 0 can only occur if @xmath179 . if @xmath180 , then clearly @xmath181 , and hence the partial tilting module @xmath182 contains a cycle , which is a contradiction . assume that @xmath183 ( and hence @xmath179 ) . then @xmath184 . if @xmath181 , we get a contradiction as in the previous case . if @xmath185 , consider the exchange triangle @xmath186 which is induced from an exact sequence in @xmath9 . hence there is a _ non - zero _ map @xmath187 obtained by composing @xmath188 with the monomorphism @xmath189 , and thus there are cycles in the endomorphism ring of the partial tilting module @xmath190 , a contradiction . this finishes the case with @xmath191 . assume now that @xmath192 . then we have @xmath193 . since @xmath194 and @xmath195 , we have by lemma [ l : div](c ) that @xmath196 . so without loss of generality we can assume @xmath197 . assume that @xmath198 . then @xmath199 using lemma [ l : div](c ) and the fact that @xmath194 . then also @xmath200 . but @xmath201 , so @xmath202 , contradicting the fact that @xmath195 . @xmath87 satisfies condition ( ii ) . let @xmath203 be a tilting object . in this section we show that the coloured quiver @xmath87 satisfies condition ( iii ) . [ p : symmetry ] with the notation of the previous section , we have @xmath204 . by lemma [ l : div2 ] we only need to consider the case @xmath205 . it is enough to show that @xmath206 . we first prove [ l : non - van ] let @xmath207 be irreducible in @xmath208 . then the composition @xmath209 \circ \gamma_i^{(0,c)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(m - c+1)}$ ] is non - zero . we have already assumed @xmath210 . assume @xmath211 \circ h_i^{(0)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(0)}[-c ] \to t_i^{(m)}[-c+1]\ ] ] is zero . this means that @xmath212 must factor through @xmath213 . since @xmath50 is by assumption a summand in @xmath214 , we have that @xmath50 is not a summand in @xmath148 by proposition [ l : disjoint ] . since @xmath215 , we have that @xmath90 is not a direct summand in @xmath147 . this means that @xmath216 is not irreducible in @xmath208 , a contradiction . so @xmath209 \circ h_i^{(0)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(m)}[-c+1]$ ] is non - zero . assume @xmath217 . if the composition @xmath209 \circ h_i^{(0)}[-c ] \circ h_i^{(m)}[-c+1]$ ] is zero , then @xmath209 \circ h_i^{(0)}[-c]$ ] factors through @xmath218 \to t_i^{(m)}[-c+1].\ ] ] we claim that @xmath219 , b_i^{(m-1)}[-c+1 ] ) \simeq { \operatorname{hom}\nolimits}(t_j^{(c ) } , b_i^{(m-1)}[1 ] ) = 0 $ ] . this clearly holds if @xmath90 is not a summand of @xmath220 . in addition we have that @xmath221 ) = 0 $ ] since @xmath217 , using lemma [ l : div](e ) . this is a contradiction , and this argument can clearly be iterated to see that @xmath209 \circ \gamma_i^{(0,c)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(m - c+1)}$ ] is non - zero , using lemma [ l : div](e ) . we now show that any irreducible map @xmath207 gives rise to an irreducible map @xmath222 . consider the composition @xmath223 \overset{g_j^{(c)}[-c]}{\longrightarrow } t_j^{(c)}[-c ] \longrightarrow t_i^{(m - c+1)}.\ ] ] since @xmath50 is a summand in @xmath214 by assumption , it is not a summand in @xmath224 . thus , @xmath224 is in @xmath225 . since @xmath226)= 0 $ ] for any @xmath39 in @xmath227 , the composition vanishes . using the exchange triangle @xmath223 \overset{g_j^{(c)}[-c]}{\longrightarrow } t_j^{(c)}[-c ] \overset{h_j^{(c)}[-c]}{\longrightarrow } t_j^{(c-1)}[-c+1],\ ] ] we see that @xmath209 \circ \gamma_i^{(0,c)}[-c ] \colon t_j^{(c)}[-c ] \to t_i^{(m - c+1)}$ ] factors through the map @xmath228 \overset{h_j^{(c)}[-c]}{\longrightarrow } t_j^{(c-1)}[-c+1]$ ] , i.e. there is a commutative diagram @xmath229 \ar^{g_j^{(c)}[-c]}[r ] & t_j^{(c)}[-c ] \ar^{h_j^{(c)}[-c]}[r ] \ar[d ] & t_j^{(c-1)}[-c+1 ] \ar[r ] \ar^{\phi_1}[dl ] & \\ & t_i^{(m - c+1 ) } & & } \ ] ] similarly , using the exchange triangle @xmath230 \overset{g_j^{(c-1)}[-c+1]}{\longrightarrow } t_j^{(c-1)}[-c+1 ] \overset{h_j^{(c-1)}[-c+1]}{\longrightarrow } t_j^{(c-2)}[-c+2]\ ] ] we obtain a map @xmath231 \to t_i^{(m - c+1)}$ ] repeating this argument @xmath79 times we obtain a map @xmath232 , such that @xmath233 \circ \phi_c = \alpha[-c ] \circ \gamma_i^{(0,c)}$ ] . @xmath234 \ar_{h_j^{(c)}[-c]}[d ] \ar[r ] & t_i^{(m - c+1 ) } \\ t_j^{(c-1)}[-c+1 ] \ar_{h_j^{(c-1)}[-c+1]}[d ] \ar^{\phi_1}[ur ] & \\ t_j^{(c-2)}[-c+2 ] \ar_{h_j^{(c-2)}[-c+2]}[d ] \ar^{\phi_2}[uur ] & \\ \vdots \ar[d ] & \\ t_j \ar^{\phi_c}[uuuur ] & \\ & } \ ] ] we claim that [ l : irred ] there is a map @xmath235 , such that @xmath233 \circ \beta = \alpha[-c ] \circ \gamma_i^{(0,c)}$ ] , and such that @xmath236 is irreducible in @xmath237 . let @xmath238 be a minimal left @xmath239-approximation , with @xmath240 in @xmath241 and @xmath242 in @xmath243 . let @xmath244 be as above , and factor it as @xmath245 since @xmath246 factors through @xmath247 $ ] , we have that @xmath233 \psi '' = 0 $ ] , so we have @xmath248(\psi ' \epsilon ' + \psi '' \epsilon'')= \gamma_j^{(c , c)}[-c ] \psi ' \epsilon'.\ ] ] hence , let we let @xmath249 and since the summands in @xmath250 are isomorphisms , it is clear that @xmath236 is irreducible . next , assume @xmath251 is a basis for the space of irreducible maps from @xmath252 to @xmath50 . then , by lemma [ l : non - van ] the set @xmath253 is also linearly independent . for each @xmath254 , consider the corresponding map @xmath255 , such that @xmath233 \circ \beta_t = \alpha_t[-c ] \circ \gamma_i^{(0,c)}$ ] , and which we by lemma [ l : irred ] can assume is irreducible . assume a non - trivial linear combination @xmath256 is zero . then also @xmath257 \circ \beta_t ) = \sum k_t \alpha_t \circ \gamma_i^{(0,c)}=0 $ ] . but this contradicts lemma [ l : non - van ] since @xmath258 is irreducible . hence it follows that @xmath259 is also linearly independent . hence , in the exchange triangle @xmath260 , we have that @xmath90 appears with multiplicity at least @xmath261 in @xmath262 . so , we have that @xmath206 , and the proof of the proposition is complete . in this section we show how mutation in the vertex @xmath70 affects the complements of the almost complete tilting object @xmath263 . as before , let @xmath264 be an @xmath0-tilting object , and let @xmath94 . we need to consider @xmath265 \ar^{(e)}[r ] & t_j \ar^{(d)}[r ] & t_k } \ ] ] for all possible values of @xmath266 . however , we have the following restriction on the colour of arrows . [ p : limits ] assume @xmath267 and @xmath268 . then @xmath269 . consider the exchange triangle @xmath270 . note that @xmath90 is a direct summand in the middle term @xmath271 by the assumption that @xmath272 . consider also the exchange triangle @xmath273 . pick an arbitrary non - zero map @xmath274 , and consider the map @xmath275 . it suffices to show that whenever @xmath276 , then @xmath277 is not irreducible in @xmath46 . so assume that @xmath276 . we claim that there is a commutative diagram @xmath278 \ar[d ] & t_j \amalg x ' \ar[r ] \ar^{\left ( \begin{smallmatrix } h & 0 \\ 0 & 0 \end{smallmatrix } \right)}[d ] & t_i^{(e+1 ) } \ar[r ] \ar[d ] & \\ t_i^{(c ) } \ar[r ] & t_k \amalg z \ar[r ] & t_i^{(c+1 ) } \ar[r ] & } \ ] ] where the rows are the exchange triangles . the composition @xmath279 is zero since * if @xmath280 @xmath281 by using @xmath276 and lemma [ l : div](e ) * if @xmath282 , there is no non - zero composition @xmath283 hence the leftmost vertical map exists , and then the rightmost map exists , using that @xmath20 is a triangulated category . then , since @xmath284 , t_i^{(c ) } ) = 0 $ ] by lemma [ l : div](e ) , there is a map @xmath285 , such that @xmath286 . hence there is map @xmath287 such that @xmath288 . by restriction we get @xmath289 under the assumption @xmath290 we have that @xmath291 can not be irreducible in @xmath292 . hence @xmath293 , where @xmath51 is not summand in @xmath294 . also , by proposition [ l : disjoint ] we have that @xmath90 is not a summand in @xmath294 . if @xmath295 was irreducible in @xmath296 , then there would be an irreducible map @xmath297 in @xmath298 , and since @xmath299 , this does not hold , by proposition [ l : disjoint ] . hence , @xmath300 , where @xmath90 is not a direct summand of @xmath301 . also by proposition [ l : disjoint ] we have that @xmath51 is not a summand of @xmath301 . by ( [ factor ] ) , this shows that @xmath274 is not irreducible in @xmath46 . let @xmath302 . for @xmath303 , let @xmath304 denote the complements of @xmath305 , where there are exchange triangles @xmath306 we first want to compare @xmath304 with @xmath307 . [ l : samecomp ] assume that @xmath308 for @xmath309 and that @xmath310 . * for @xmath311 , the minimal left @xmath312-approximation @xmath313 is also an @xmath314-approximation . * for @xmath315 , we have @xmath316 . by assumption @xmath90 is not a direct summand in any of the @xmath317 . assume there is a map @xmath318 and consider the diagram @xmath319 \ar[r ] & t_i^{(u ) } \ar[r ] \ar [ d ] & b_i^{(u ) } \ar [ r ] & \\ & t_j^{(1 ) } & & } \ ] ] since @xmath320)= 0 $ ] by lemma [ l : div2 ] , we see that the map @xmath318 factors through @xmath313 . hence the minimal left @xmath312-approximation @xmath313 is also an @xmath314-approximation , so we have proved ( a ) . then ( b ) follows directly . [ l : comp ] assume that @xmath321 and there are exchange triangles @xmath322 and @xmath323 where @xmath324 and @xmath325 , i.e. @xmath326 and @xmath327 , where @xmath51 is not isomorphic to any direct summand in @xmath328 . * the composition @xmath329 is a left @xmath314-approximation . * there is a triangle @xmath330 with @xmath331 in @xmath332 and @xmath333 . * there is a triangle @xmath334 . consider an arbitrary map @xmath335 with @xmath45 in @xmath314 . we have that @xmath336 ) = 0 $ ] , by lemma [ l : div2 ] . hence , by applying @xmath337 to the triangle ( [ i - tri ] ) we get that @xmath338 factors through @xmath339 . by applying @xmath337 to the triangle ( [ j - tri ] ) , and using that @xmath340 ) = 0 $ ] , we get that @xmath338 factors through @xmath341 . this proves ( a ) . for ( b ) and ( c ) we use the exchange triangles ( [ i - tri ] ) and ( [ j - tri ] ) and the octahedral axiom to obtain the commutative diagram of triangles @xmath342 \ar@{=}[d ] & ( t_j)^p \amalg x \ar[r ] \ar[d ] & t_i^{(e+1 ) } \ar[d ] \ar[r ] & \\ t_i^{(e ) } \ar[r ] & ( t_k)^{pq } \amalg y^p \amalg x \ar[r ] \ar[d ] & c \ar[r ] \ar[d ] & \\ & ( t_j^{(1)})^p \ar@{=}[r ] & ( t_j^{(1)})^p \ar[r ] & } \ ] ] by ( a ) the map @xmath343 is a left @xmath314-approximation , and by lemma [ l : samecomp ] we have that @xmath344 . hence @xmath345 , where @xmath331 is in @xmath346 , and with no copies isomorphic to @xmath51 in @xmath328 . note that the induced @xmath314-approximation is in general not minimal . [ l : modtri ] assume @xmath321 and @xmath347 . * then there is a triangle @xmath348 where @xmath216 is a minimal left @xmath314-approximation , and @xmath331 is as in lemma [ l : comp ] . * there is an induced exchange triangle @xmath349 where @xmath350 . * @xmath351 . consider the exchange triangle @xmath352 \to t_i^{(e+1 ) } \to b_i^{(e+1 ) } \to\ ] ] and the triangle from lemma [ l : comp ] ( b ) @xmath353 apply the octahedral axiom , to obtain the commutative diagram of triangles @xmath354 \ar[r ] \ar@{=}[d ] & t_i^{(e+1 ) } \ar[r ] \ar[d ] & b_i^{(e+1 ) } \ar[d ] \ar[r ] & \\ t_i^{(e+2)}[-1 ] \ar[r ] & ( t_i^{(e+1 ) } ) ' \amalg c ' \ar[r ] \ar[d ] & g \ar[r ] \ar[d ] & \\ & ( t_j^{(1)})^p \ar@{=}[r ] & ( t_j^{(1)})^p \ar[r ] & } \ ] ] since @xmath90 does not occur as a summand in @xmath294 by proposition [ l : disjoint ] , we have that @xmath355 ) = 0 $ ] . hence the rightmost triangle splits , so we have a triangle @xmath356 \to ( t_i^{(e+1 ) } ) ' \amalg c ' \to b_i^{(e+1 ) } \amalg ( t_j^{(1)})^p \to\ ] ] by lemma [ l : div2 ] we have that @xmath357)= 0 $ ] . by lemma [ l : div](e ) we get that @xmath358 ) = 0 $ ] , and clearly @xmath359 ) = 0 $ ] , for @xmath360 . we hence get that all maps @xmath361 , with @xmath45 in @xmath362 , factor through @xmath363 . minimality is clear from the triangle ( [ octa - tri ] ) . this proves ( a ) , and ( b ) follows from the fact that @xmath331 contains no copies of @xmath90 , and hence splits off . ( c ) is a direct consequence of ( b ) . [ p : summarize ] * if @xmath308 for @xmath364 , then @xmath365 for all @xmath366 . * if @xmath321 and @xmath347 , then @xmath365 for @xmath367 . \(a ) is a direct consequence of [ l : samecomp ] . for ( b ) note that by lemmas [ l : samecomp ] and [ l : modtri ] we have @xmath365 for @xmath368 and @xmath369 . for @xmath370 consider the exchange triangles @xmath371 since @xmath372 ) = 0 $ ] by lemma [ l : div2 ] and @xmath373 , it is clear that the map @xmath374 is a left @xmath375-approximation . hence ( b ) follows . this section contains the proof of the main result , theorem [ t : main ] . as before , let @xmath264 be an @xmath0-tilting object , and let @xmath94 . we will compare the numbers of @xmath77-coloured arrows from @xmath13 to @xmath69 , in the coloured quivers of @xmath41 and @xmath109 , i.e. we will compare @xmath376 and @xmath377 . we need to consider an arbitrary @xmath41 whose coloured quiver locally looks like @xmath265 \ar^{(e)}[r ] & t_j \ar^{(d)}[r ] & t_k } \ ] ] for any possible value of @xmath266 . our aim is to show that the formula @xmath378 holds . the case where @xmath379 is directly from the definition . the case where @xmath380 follows by condition ( ii ) for @xmath381 . for the rest of the proof we assume @xmath382 . we will divide the proof into four cases , where @xmath383 denotes the number of arrows from @xmath13 to @xmath70 , and @xmath384 . * @xmath385 * @xmath386 , @xmath321 and @xmath387 * @xmath386 , @xmath321 and @xmath388 . * @xmath386 and @xmath389 note that in the three first cases , the formula reduces to @xmath390 and in the first two cases it further reduces to @xmath391 case i. we first consider the situation where there is no coloured arrow @xmath392 , i.e. @xmath308 for all @xmath393 . that is , we assume @xmath87 locally looks like this @xmath265 & t_j \ar^{(d)}[r ] & t_k } \ ] ] with @xmath394 arbitrary . it is a direct consequence of proposition [ p : summarize ] that @xmath395 for all @xmath393 which shows that the formula holds . + + case ii . we consider the setting where we assume @xmath87 locally looks like this @xmath265 \ar^{(e)}[r ] & t_j \ar^{(d)}[r ] & t_k } \ ] ] with @xmath321 and @xmath396 . we then claim that we have the following , which shows that the formula holds . in the above setting @xmath395 for all @xmath393 . it follows directly from proposition [ p : summarize ] that @xmath395 for @xmath397 . we claim that @xmath398 . by lemma [ l : comp ] we have the ( not necessarily minimal ) left @xmath314-approximation @xmath399 first , assume that @xmath51 does not appear as a summand in @xmath326 , then the same holds for @xmath400 , and hence for @xmath401 which is a direct summand in @xmath400 . next , assume @xmath51 appears as a summand in @xmath271 , and hence in @xmath39 . then @xmath51 is by proposition [ l : disjoint ] not a summand in @xmath294 , and by lemma [ l : modtri ] we have that @xmath51 is also not a summand in @xmath331 . therefore @xmath51 appears with the same multiplicity in @xmath271 as in @xmath401 , also in this case . we now show that @xmath395 for @xmath402 . if @xmath403 , then @xmath404 for @xmath402 and we are finished . so assume @xmath405 , i.e. @xmath51 does not appear as a direct summand of @xmath39 . consider the map @xmath406 we have that @xmath407 . by assumption , @xmath51 is not a direct summand in @xmath408 , and thus not in @xmath331 . hence it follows that @xmath409 . since , by proposition [ p : summarize ] we have for @xmath410 , that @xmath316 and the left @xmath411-approximation coincide with the left @xmath412-approximations of @xmath413 , it now follows that @xmath395 for all @xmath393 . + case iii . we now consider the setting with @xmath414 non - zero , @xmath388 and @xmath321 . that is , we assume @xmath87 locally looks like this @xmath265 \ar^{(e)}[r ] & t_j \ar^{(0)}[r ] & t_k } \ ] ] where @xmath415 by proposition [ p : limits ] , and where there are @xmath416 arrows from @xmath50 to @xmath51 . [ l : formulas ] in the above setting , we have that @xmath381 is given by @xmath417 @xmath418 and @xmath419 we first deal with the case where @xmath420 and @xmath421 . by assumption @xmath39 in the triangle ( [ i - tri ] ) has @xmath422 copies of @xmath51 , so @xmath423 has @xmath424 copies of @xmath51 . hence to show ( [ form1 ] ) it is sufficient to show that @xmath331 in the triangle @xmath425 has no copies of @xmath51 . this follows directly from the lemma [ l : modtri ] and the fact that @xmath51 ( by the assumption that @xmath421 and proposition [ l : disjoint ] ) is not a summand in @xmath294 . in this case ( [ form2 ] ) and ( [ form3 ] ) follow directly from proposition [ l : disjoint ] . consider the case with @xmath426 and @xmath427 . we have that @xmath39 in the triangle ( [ i - tri ] ) does not have @xmath51 as a direct summand . assume @xmath51 appears as a direct summand of @xmath331 with multiplicity @xmath428 . we claim that @xmath429 . assume first @xmath430 , then on one hand @xmath51 appears with multiplicity @xmath431 in @xmath432 . on the other hand @xmath51 appears with multiplicity @xmath433 in @xmath401 . this contradicts proposition [ l : disjoint ] . hence @xmath429 . therefore @xmath401 has @xmath434 copies of @xmath51 and ( [ form1 ] ) and ( [ form2 ] ) hold . if @xmath435 , then ( [ form3 ] ) follows directly from the above and proposition [ l : disjoint ] . in the case @xmath436 , we also need to show that @xmath51 does not appear as a summand in @xmath437 for @xmath438 . since @xmath439 , we have @xmath440 , and the result follows from proposition [ p : summarize ] . now assume @xmath426 and @xmath441 . assume @xmath442 , where @xmath51 is not a summand in @xmath443 . now since @xmath444 with @xmath51 not a summand in @xmath328 , is a minimal left @xmath362-approximation , we have that @xmath445 and @xmath51 appears with multiplicity @xmath446 in the minimal left @xmath447-approximation of @xmath448 , hence @xmath51 can not appear as a summand in the minimal left @xmath375-approximation of @xmath449 . hence @xmath450 , and we have completed the proof of ( [ form1 ] ) and ( [ form2 ] ) in this case . the case ( [ form3 ] ) , i.e. @xmath451 follows from proposition [ l : disjoint ] . + case iv . we now consider the case with @xmath452 . assume first there are no arrows from @xmath70 to @xmath69 . then we can use the symmetry proved in proposition [ p : symmetry ] and reduce to case i. the formula is easily verified in this case . assume @xmath453 , again we can use the symmetry , this time to reduce to case iii . it is straightforward to verify that the formula holds also in this case . assume now that @xmath454 , i.e. we need to consider the following case @xmath455 \ar@<0.6ex>^{(m)}[r ] & t_j \ar@<0.6ex>^{(0)}[r ] \ar@<0.6ex>^{(0)}[l ] & t_k \ar@<0.6ex>^{(m)}[l ] \ar@<0.6ex>^{(m - c)}@/^3.5pc/[ll ] } \ ] ] now by proposition [ p : limits ] we have that @xmath79 is in @xmath456 . assume there are @xmath457 @xmath77-coloured arrows the coloured quiver of @xmath109 is of the form @xmath458 \ar@<0.6ex>^{(0)}[r ] & t_j^{(1 ) } \ar@<0.6ex>^{(m)}[r ] \ar@<0.6ex>^{(m)}[l ] & t_k \ar@<0.6ex>^{(0)}[l ] \ar@<0.6ex>^{(m - c')}@/^3.5pc/[ll ] } \ ] ] and applying the symmetry of proposition [ p : symmetry ] we have that if @xmath421 , then @xmath459 by proposition [ p : limits ] . hence for all @xmath460 we have that @xmath461 . therefore it suffices to show that @xmath462 , for @xmath463 . this is a direct consequence of the following . assume we are in the above setting . a map @xmath464 or @xmath465 is irreducible in @xmath46 if and only if it is irreducible in @xmath362 . assume @xmath464 is not irreducible in @xmath362 , and that @xmath466 for some @xmath467 , with @xmath468 the indecomposable direct summands of @xmath45 . note that by lemma [ l : div](a ) , we can assume that all @xmath469 and all @xmath470 are non - isomorphisms . if there is some index @xmath471 such that @xmath472 , the map @xmath470 factors through some @xmath473 in @xmath474 , since there are no @xmath475-coloured arrows @xmath476 or @xmath477 in the coloured quiver of @xmath41 . this shows that @xmath464 is not irreducible in @xmath46 . assume @xmath464 is not irreducible in @xmath46 , and that @xmath478 for some @xmath479 , with @xmath480 the indecomposable direct summands of @xmath481 . if there is some index @xmath471 such that @xmath482 , the map @xmath483 factors through @xmath484 , which is in @xmath485 , since there are no @xmath93-coloured arrows @xmath392 or @xmath486 in the coloured quiver of @xmath41 . this shows that @xmath464 is not irreducible in @xmath362 . by symmetry , the same property holds for maps @xmath465 . thus we have proven that the formula holds in all four cases , and this finishes the proof of theorem [ t : main ] . an @xmath0-cluster - tilted algebra is an algebra given as @xmath487 for some tilting object @xmath41 in an @xmath0-cluster category @xmath488 . obviously , the subquiver of the coloured quiver of @xmath41 given by the @xmath93-coloured maps is the gabriel quiver of @xmath487 . an application of our main theorem is that the quivers of the @xmath0-cluster - tilted algebras can be combinatorially determined via repeated ( coloured ) mutation . for this one needs transitivity in the tilting graph of @xmath0-tilting objects . more precisely , we need the following , which is also pointed out in @xcite . any @xmath0-tilting object can be reached from any other @xmath0-tilting object via iterated mutation . we sketch a proof for the convenience of the reader . let @xmath109 be a tilting object in an @xmath0-cluster category @xmath20 of the hereditary algebra @xmath489 , and let @xmath490 be the @xmath491-cluster category of @xmath10 . by @xcite , there is a tilting object @xmath41 of degree 0 , i.e. all direct summands in @xmath41 have degree 0 , such that @xmath41 can be reached from @xmath109 via mutation . it is sufficient to show that the canonical tilting object @xmath10 can be reached from @xmath41 via mutation . since @xmath41 is of degree 0 , it is induced from a @xmath10-tilting module . especially @xmath41 is a tilting object in @xmath490 . since @xmath41 and @xmath10 are tilting objects in @xmath490 , by @xcite there are @xmath490-tilting objects @xmath492 , such that @xmath50 mutates to @xmath493 ( in @xmath490 ) for @xmath494 . now each @xmath50 is induced by a tilting module for some @xmath495 where all @xmath496 are derived equivalent to @xmath497 . hence , each @xmath50 is easily seen to be an @xmath0-cluster tilting object . since @xmath493 differs from @xmath50 in only one summand the mutations in @xmath490 are also mutations in @xmath20 . this concludes the proof . a direct consequence of the transitivity is the following . for an @xmath0-cluster category @xmath111 of the acyclic quiver @xmath71 , all quivers of @xmath0-cluster - tilted algebras are given by repeated coloured mutation of @xmath71 . in this section , we discuss concrete computation with tilting objects in an @xmath0-cluster tilting category . an exceptional indecomposable object in @xmath9 is uniquely determined by its image @xmath498 $ ] in the grothendieck group @xmath499 . there is a map from @xmath500 to @xmath499 which , for @xmath501 , takes @xmath502 $ ] to @xmath503 $ ] . an exceptional indecomposable in @xmath500 can be uniquely specified by its class in @xmath499 together with its degree . the map from @xmath500 to @xmath499 does not descend to @xmath504 . however , if we fix our usual choice of fundamental domain in @xmath500 , then we can identify the indecomposable objects in it as above . let us define the combinatorial data corresponding to a tilting object @xmath41 to be @xmath87 together with @xmath505 , \deg t_i)$ ] for @xmath506 . given the combinatorial data for a tilting object @xmath41 in @xmath504 , it is possible to determine , by a purely combinatorial procedure , the combinatorial data for the tilting object which results from an arbitrary sequence of mutations applied to @xmath41 . clearly , it suffices to show that , for any @xmath13 , we can determine the class and degree for @xmath507 . if we can do that then , by the coloured mutation procedure , we can determine the coloured quiver for @xmath508 , and by applying this procedure repeatedly , we can calculate the result of an arbitrary sequence of mutations . since we are given @xmath87 , we know @xmath509 , and we can calculate @xmath510 $ ] . now we have the following lemma : [ one ] @xmath511=[b_i^{(0)}]-[t^{(0)}_i]$ ] , and @xmath512 or @xmath513 , whichever is consistent with the sign of the class of @xmath511 $ ] , unless this yields a non - projective indecomposable object in degree @xmath0 , or an indecomposable of degree @xmath1 . the proof is immediate from the exchange triangle @xmath514 . applying this lemma , and supposing that we are not in the case where its procedure fails , we can determine the class and degree @xmath515 . by the coloured mutation procedure , we can also determine the coloured quiver for @xmath516 . we therefore have all the necessary data to apply lemma [ one ] again . repeatedly applying the lemma , there is some @xmath69 such that we can calculate the class and degree of @xmath507 for @xmath517 , and the procedure described in the lemma fails to calculate @xmath518 . we also have the following lemma : [ two ] @xmath519=[b_i^{(m)}]-[t^{(0)}_i]$ ] , and @xmath520 or @xmath521 , whichever is consistent with the sign of @xmath519 $ ] , unless this yields an indecomposable in degree @xmath522 . applying this lemma , starting again with @xmath41 , we can obtain the degree and class for @xmath523 . we can then determine the coloured quiver for @xmath524 , and we are now in a position to apply lemma [ two ] again . the last complement which lemma [ two ] will successfully determine is @xmath525 . it follows that we can determine the degree and class of any complement to @xmath263 . in this section , we discuss the application of our results to the study of the @xmath0-cluster complex , a simplicial complex defined in @xcite for a finite root system @xmath526 . we shall begin by stating our results for the @xmath0-cluster complex in purely combinatorial language , and then briefly describe how they follow from the representation - theoretic perspective in the rest of the paper . for simplicity , we restrict to the case where @xmath526 is simply laced . number the vertices of the dynkin diagram for @xmath526 from 1 to @xmath48 . the @xmath0-coloured almost positive roots , @xmath527 , consist of @xmath0 copies of the positive roots , numbered @xmath491 to @xmath0 , together with a single copy of the negative simple roots . we refer to an element of the @xmath13-th copy of @xmath528 as having colour @xmath13 , and we write such an element as @xmath529 . since the dynkin diagram for @xmath526 is a tree , it is bipartite ; we fix a bipartition @xmath530 . the @xmath0-cluster complex , @xmath531 , is a simplicial complex on the ground set @xmath527 . its maximal faces are called @xmath0-clusters . the definition of @xmath531 is combinatorial ; we refer the reader to @xcite . the @xmath0-clusters each consist of @xmath48 elements of @xmath527 ( * ? ? ? * theorem 2.9 ) . every codimension 1 face of @xmath531 is contained in exactly @xmath1 maximal faces ( * ? ? ? * proposition 2.10 ) . there is a certain combinatorially - defined bijection @xmath532 , which takes faces of @xmath531 to faces of @xmath531 ( * ? ? ? * theorem 2.4 ) . it will be convenient to consider _ ordered @xmath0-clusters_. an ordered @xmath0-cluster is just a @xmath48-tuple from @xmath527 , the set of whose elements forms an @xmath0-cluster . write @xmath533 for the set of ordered @xmath0-clusters . for each ordered @xmath0-cluster @xmath534 , we will define a coloured quiver @xmath535 . we will also define an operation @xmath536 , which takes ordered @xmath0-clusters to ordered @xmath0-clusters , changing only the @xmath70-th element . we will define both operations inductively . the set @xmath537 of negative simple roots forms an @xmath0-cluster . its associated quiver is defined by drawing , for each edge @xmath538 in the dynkin diagram , a pair of arrows . suppose @xmath539 and @xmath540 . then we draw an arrow from @xmath13 to @xmath70 with colour @xmath541 , and an arrow from @xmath70 to @xmath13 with colour @xmath0 . suppose now that we have some ordered @xmath0-cluster @xmath542 , together with its quiver @xmath535 . we will now proceed to define @xmath543 . write @xmath544 for the number of arrows in @xmath535 of colour @xmath541 from @xmath70 to @xmath69 . define : @xmath545 let @xmath79 be the colour of @xmath546 . we define @xmath543 by replacing @xmath546 by some other element of @xmath527 , according to the following rules : * if @xmath546 is positive and @xmath236 is positive , replace @xmath546 by @xmath547 . * if @xmath546 is positive and @xmath236 is negative , replace @xmath546 by @xmath548 . * if @xmath546 is negative simple @xmath549 , define @xmath550 by @xmath551 , and then replace @xmath546 by @xmath552 , with colour zero . define the quiver for the @xmath0-cluster @xmath543 by the coloured quiver mutation rule from section 2 . since any @xmath0-cluster can be obtained from @xmath537 by a sequence of mutations , the above suffices to define @xmath543 and @xmath535 for any ordered @xmath0-cluster @xmath542 . the operation @xmath553 defined above takes @xmath0-clusters to @xmath0-clusters , and the @xmath0-clusters @xmath554 for @xmath555 are exactly those containing all the @xmath556 for @xmath557 . the connection between the combinatorics discussed here and the representation theory in the rest of the paper is as follows . @xmath527 corresponds to the indecomposable objects of ( a fundamental domain for ) @xmath36 . the cluster tilting objects in @xmath36 correspond to the @xmath0-clusters . the operation @xmath558 corresponds to @xmath25 $ ] . for further details on the translation , the reader is referred to @xcite . the above proposition then follows from the approach taken in section [ sec : cc ] . here we give an alternative description of coloured quiver mutation at vertex @xmath70 . 1 . for each pair of arrows @xmath559 & j\ar^{(0)}[r ] & k } \ ] ] with @xmath560 , the arrow from @xmath13 to @xmath70 of arbitrary colour @xmath79 , and the arrow from @xmath70 to @xmath69 of colour @xmath541 , add a pair of arrows : an arrow from @xmath13 to @xmath69 of colour @xmath79 , and one from @xmath69 to @xmath13 of colour @xmath561 . 2 . if the graph violates property ii , because for some pair of vertices @xmath13 and @xmath69 there are arrows from @xmath13 to @xmath69 which have two different colours , cancel the same number of arrows of each colour , until property ii is satisfied . 3 . add one to the colour of any arrow going into @xmath70 and subtract one from the colour of any arrow going out of @xmath70 . the above algorithm is well - defined and correctly calculates coloured quiver mutation as previously defined . fix a quiver @xmath71 and a vertex @xmath70 at which the mutation is being carried out . to prove that the algorithm is well - defined , we must show that at step 2 , there are only two colours of arrows running from @xmath13 to @xmath69 for any pair of vertices @xmath13 , @xmath69 . ( otherwise there would be more than one way to carry out the cancellation procedure of step 2 . ) since in the original quiver @xmath71 , there was only one colour of arrows from @xmath13 to @xmath69 , in order for this problem to arise , we must have added two different colours of arrows from @xmath13 to @xmath69 at step 1 . two colours of arrows will only be added from @xmath13 to @xmath69 if , in @xmath71 , there are both @xmath93-coloured arrows from @xmath70 to @xmath69 and from @xmath70 to @xmath13 . in this case , by property iii , there are @xmath562-coloured arrows from @xmath13 to @xmath70 and from @xmath69 to @xmath70 . it follows that in step 1 , we will add both @xmath93-coloured and @xmath562-coloured arrows . applying proposition 5.1 , we see that any arrows from @xmath13 to @xmath69 in @xmath71 are of colour 0 or @xmath0 . thus , as desired , after step 1 , there are only two colours of arrows in the quiver , so step 2 is well - defined . we now prove correctness . let @xmath563 . write @xmath76 for the number of @xmath79-coloured arrows from @xmath13 to @xmath70 in @xmath71 , and similarly @xmath564 for @xmath565 . write @xmath566 and @xmath567 for the result of applying the above algorithm . it is clear that only the final step of the algorithm is relevant for @xmath568 where one of @xmath13 or @xmath69 coincides with @xmath70 , and therefore that in this case @xmath569 as desired . suppose now that neither @xmath13 nor @xmath69 coincides with @xmath70 . suppose further that in @xmath71 there are no @xmath93-coloured arrows from either @xmath13 or @xmath69 to @xmath70 , and therefore also no @xmath0-coloured arrows from @xmath69 to @xmath13 or @xmath70 . in this case , @xmath570 . in the algorithm , no arrows will be added between @xmath13 and @xmath69 in step 1 , and therefore no further changes will be made in step 2 . thus @xmath571 , as desired . suppose now that there are @xmath93-coloured arrows from @xmath70 to both @xmath13 and @xmath69 . in this case , @xmath572 . in this case , as discussed in the proof of well - definedness , an equal number of @xmath93-coloured and @xmath562-coloured arrows will be introduced at step 1 . they will therefore be cancelled at step 2 . thus @xmath573 as desired . suppose now that there is a @xmath93-coloured arrow from @xmath70 to @xmath69 , but not from @xmath70 to @xmath13 . let the arrows from @xmath13 to @xmath70 , if any , be of colour @xmath79 . at step 1 of the algorithm , we will add @xmath574 arrows of colour @xmath79 to @xmath71 . by proposition 5.1 , the arrows in @xmath71 from @xmath13 to @xmath69 are of colour @xmath79 or @xmath575 . one verifies that the algorithm yields the same result as coloured quiver mutation , in the three cases that the arrows from @xmath13 to @xmath69 in @xmath71 are of colour @xmath79 , that they are of colour @xmath575 but there are fewer than @xmath574 , and that they are of colour @xmath575 and there are at least as many as @xmath574 . the final case , that there is a @xmath93-coloured arrow from @xmath70 to @xmath13 but not from @xmath70 to @xmath69 , is similar to the previous one . in @xcite , a certain category @xmath576 is constructed , which is shown to be equivalent to the @xmath0-cluster category of dynkin type @xmath2 . the description of @xmath576 is as follows . take an @xmath577-gon @xmath578 , with vertices labelled clockwise from 1 to @xmath577 . consider the set @xmath39 of diagonals @xmath550 of @xmath578 with the property that @xmath550 divides @xmath578 into two polygons each having a number of sides congruent to 2 modulo @xmath0 . for each @xmath579 , there is an object @xmath580 in @xmath576 . these objects @xmath580 form the indecomposables of the additive category @xmath576 . we shall not recall the exact definition of the morphisms , other than to note that they are generated by the morphisms @xmath581 which exist provided that @xmath538 and @xmath582 are both diagonals in @xmath39 , and that , starting at @xmath70 and moving clockwise around @xmath578 , one reaches @xmath69 before @xmath13 . a collection of diagonals in @xmath39 is called non - crossing if its elements intersect pairwise only on the boundary of the polygon . an inclusion - maximal such collection of diagonals divides @xmath578 into @xmath583-gons ; we therefore refer to such a collection of diagonals as an @xmath583-angulation . if we remove one diagonal @xmath550 from an @xmath583-angulation @xmath584 , then the two @xmath583-gons on either side of @xmath550 become a single @xmath585-gon . we say that @xmath550 is a _ diameter _ of this @xmath585-gon , since it connects vertices which are diametrically opposite ( with respect to the @xmath585-gon ) . if @xmath586 is another diameter of this @xmath585-gon , then @xmath587 is another maximal noncrossing collection of diagonals from @xmath39 . ( in particular , @xmath588 . ) for @xmath584 an @xmath583-angulation , let @xmath589 . then we have that @xmath590 is a basic ( @xmath0-cluster-)tilting object for @xmath576 , and all basic tilting objects of @xmath576 arise in this way . it follows from the previous discussion that if @xmath591 is a basic tilting object , and @xmath592 , then the complements to @xmath593 will consist of the objects @xmath594 where @xmath586 is a diameter of the @xmath585-gon obtained by removing @xmath550 from the @xmath583-angulation determined by @xmath584 . in fact , we can be more precise . define @xmath595 to be the diameter of the @xmath585-gon obtained by rotating the vertices of @xmath550 by @xmath13 steps counterclockwise ( within the @xmath585-gon ) . then @xmath596 . the coloured quiver @xmath598 of @xmath591 has an arrow from @xmath550 to @xmath586 if and only if @xmath550 and @xmath586 both lie on some @xmath583-gon in the @xmath583-angulation defined by @xmath584 . in this case , the colour of the arrow is the number of edges forming the segment of the boundary of the @xmath583-gon which lies between @xmath550 and @xmath586 , counterclockwise from @xmath550 and clockwise from @xmath586 . we return to the example from section 2 . the quadrangulation of a decagon corresponding to the tilting object @xmath41 is on the left . the quadrangulation corresponding to @xmath109 is on the right . passing from the figure on the left to the figure on the right , the diagonal 27 ( which corresponds to the summand @xmath599 ) has been rotated one step counterclockwise within the hexagon with vertices 1,2,3,4,7,10 .

the control and manipulation of single electrons in mesoscopic systems constitutes one of the key ingredients in nanoelectronics . the study of single - electron sources@xcite in the high - frequency regime has attracted a great interest due to their potential application in quantum electron optics experiments , in metrology , and in quantum information processing based on fermionic systems.@xcite in this work we study the time evolution of a quantum dot ( qd ) tunnel coupled to a single electronic reservoir , as depicted schematically in fig . [ fig_scheme](a ) . in the presence of some time - dependent voltage modulations , this system defines the building block of the typical single - electron source , namely the mesoscopic capacitor.@xcite in the linear - response regime , the relaxation behavior of such a mesoscopic capacitor has been extensively studied theoretically@xcite and experimentally,@xcite revealing the quantization of the charge relaxation resistance.@xcite on the other hand , the application of _ nonlinear _ periodic potentials to the mesoscopic capacitor yields the controlled emission and absorption of electrons at giga - hertz frequencies.@xcite from these experiments the average charge as well as current correlations@xcite after each cycle of the potential applied have been extracted . these results demonstrate the importance of investigating the dynamics of this kind of single - electron sources . in some of the recent realizations@xcite the coulomb interaction is weak ; however , in small - sized qds the coulomb blockade is , in general , strong and it is , therefore , desirable to include it in the theoretical analysis @xcite since it may even dominate time - dependent phenomena , see e.g. ref . . the time - evolution of interacting quantum dots after the coupling to the leads has been switched on , has , e.g. , been studied in refs . and references therein . , coupled to a normal lead with a tunneling strength @xmath0 . dot occupations can be measured via the current passing through a nearby quantum point contact ( qpc ) capacitively coupled to the dot . b ) qd attached to an additional superconducting contact . c ) qd coupled to a ferromagnetic lead . ] here we investigate the exponential relaxation of a qd towards its equilibrium state after its has been brought out of equilibrium by applying , e.g. , a voltage step pulse . we consider a voltage pulse that affects the occupation of only a single orbital energy level . the level can be spin split due to coulomb interaction . in an earlier work,@xcite some of the present authors investigated the decay of charge and spin of such a single level qd . it was found that the relaxation of charge and spin are given by rates which differ from each other due to coulomb repulsion . since the reduced density matrix of a qd with a single orbital level with spin is four dimensional , there are thus three rates which govern the relaxation of the diagonal elements of the density matrix towards equilibrium ( plus one which is always zero and corresponds to the stable stationary state ) . in addition to the rates that govern charge and spin there is a third rate that appears in the relaxation of a single level qd with spin and with interaction . this additional rate is the subject of this paper . interestingly , this additional time scale is independent of the interaction and of the dot s level position . it is shown to be related to two - particle effects and appears , e.g. , in the time - evolution of the mean squared deviations of the charge from its equilibrium value . we study in detail the perturbations leading to a relaxation of the system with the additional decay rate only , and find that it is indeed related to two - particle correlations . we also propose a procedure to separately read out the different relaxation rates occurring in the dynamics of the qd exploiting the sensitivity of a nearby quantum point contact to the occupation of the qd , see fig . [ fig_scheme ] ( a ) . in order to further clarify the properties of the additional time scale , we extend our study to two other setups : a qd proximized by an extra , superconducting electrode and tunnel coupled to a normal lead ; and a qd tunnel coupled to a ferromagnetic lead , see fig . [ fig_scheme ] ( b ) and ( c ) . we consider a quantum dot coupled to an electronic reservoir . we assume that the single - particle level spacing in the dot is larger than all other energy scales , so that only one , spin - degenerate level of the qd spectrum is accessible . at a certain time @xmath1 the system is brought out of equilibrium , e.g. by applying a gate potential , and afterwards relaxes to an equilibrium dictated by the hamiltonian @xmath2 . the hamiltonian @xmath3 of the decoupled dot @xmath4 contains the spin - degenerate level @xmath5 and the on - site coulomb energy @xmath6 for double occupation of the dot . the creation ( annihilation ) operator of an electron with spin @xmath7 on the dot is denoted by @xmath8 and @xmath9 is the corresponding number operator . the reservoir is modeled by the hamiltonian @xmath10 , in which @xmath11 creates ( annihilates ) an electron with spin @xmath12 and momentum @xmath13 in the lead . the coupling between the dot and the reservoir is described by the tunneling hamiltonian @xmath14 , where @xmath15 is a tunneling amplitude , which we assume to be independent of momentum and spin . by considering a constant density of states @xmath16 in the reservoir , the tunnel coupling strength @xmath0 is defined as @xmath17 . in the remainder of this paper , we focus on the relaxation behavior of the quantum dot to its equilibrium state and in particular on how this relaxation manifests itself in measurable quantities . we are not interested in the dynamics of the reservoir , thus the trace over its degrees of freedom is performed to obtain the reduced density matrix of the qd . the hilbert space is spanned by the four eigenstates of the decoupled dot hamiltonian , @xmath18 , where @xmath19 represents the unoccupied dot , the dot is in the state @xmath20 when being singly occupied with spin @xmath7 , and @xmath21 is the state of double occupation . the energies related to these states are @xmath22 and @xmath23 , where we set the electrochemical potential of the reservoir to zero . as we consider spin - conserving tunneling events , the off - diagonal elements of the reduced density matrix evolve independently of the diagonal ones ( which are the occupation probabilities ) . we can , therefore , consider these probabilities alone , which arranged in a vector are given by @xmath24 and fulfill the condition @xmath25 . the time evolution of the occupation probabilities is governed by the generalized master equation @xmath26 where the matrix elements @xmath27 of the kernel @xmath28 describe transitions from the state @xmath29 at time @xmath30 to a state @xmath31 at time @xmath32 . we consider now the dynamics of the system after being brought out of equilibrium at time @xmath1 . since for @xmath33 the total hamiltonian is time independent , the transition matrix elements depend only on the time difference @xmath34 , i.e. @xmath35 . furthermore , we are interested in the exponential decay towards equilibrium . to be more specific , we will therefore consider only the leading , time - independent , prefactor of the exponential functions . time - dependent corrections to the pre - exponential functions , that generally may appear,@xcite are disregarded . furthermore , when focussing on times @xmath32 distant from the switching time @xmath1 , such that the difference @xmath36 is hence much larger than the decay time of the kernel @xmath37 , we can replace the lower limit of the integral in eq . ( [ eq_master ] ) by @xmath38 . expanding the probability vector @xmath39 in eq . ( [ eq_master ] ) around the measuring time @xmath32 we find@xcite @xmath40 here we introduced the laplace transform of the kernel @xmath41 , with @xmath42 and the @xmath43-th derivative of the kernel with respect to the laplace variable @xmath44_{z=0}$ ] . the formal solution of eq . ( [ eq_masterexpand ] ) is given by @xmath45 which depends on the initial probability vector @xmath46 at @xmath47 , where the initial values for the system parameters are given by the ones just after the switching time @xmath1 . the matrix @xmath48 includes markovian and non - markovian processes.@xcite in the following , we consider the limit of weak coupling between quantum dot and reservoir and limit ourselves to a perturbation expansion up to second order in @xmath0 , which is valid for the regime where the tunnel coupling @xmath0 is much smaller than the energy scale set by the temperature @xmath49 . the perturbative expansion of @xmath48 is @xmath50 with @xmath51 and @xmath52 , where the number in the superscript represents the power of @xmath0 included in the transition matrix @xmath53 . notice that the first non - markovian correction , i.e. the term @xmath54 is present in second - order in the tunnel coupling . the evaluation of the kernel within a perturbative expansion can be performed using a real - time diagrammatic technique,@xcite which has been used in ref . in order to extract the exponential decay of spin and charge in the system studied here . considering eq . ( [ eq_exp ] ) , we see that the rates defining the decay of the state into equilibrium are found from the eigenvalues of the matrix @xmath48 , which turn out to be real and non - positive . the matrix @xmath48 is not hermitian , as expected since we deal with a dissipative system , and hence has different left and right eigenvectors , @xmath55 and @xmath56 . the time - dependent probability vector , @xmath57 , can be expressed in terms of the right eigenvectors of @xmath48 , each being related to a decay with a different rate . the left eigenvectors determine the observable that decay with a single time scale only , see also the appendix . + in the following we discuss the exponential relaxation towards equilibrium of the vector of occupation probabilities , in first order in the tunneling strength @xmath0 . we start by briefly discussing the simplest case of a single spinless particle . this limit is obtained , when a magnetic field much larger than the temperature is applied , @xmath58 . the hilbert space of the system is two dimensional and spanned by the states @xmath19 and @xmath59 for the empty and singly - occupied dot respectively , whose occupation probabilities are arranged in the vector @xmath60 . the decay to the stationary state is governed by matrix @xmath61 ( defined equivalently to @xmath62 but for the two - dimensional hilbert space for the problem at hand ) which contains a single relaxation rate , namely the tunnel coupling @xmath0 , as intuitively expected . we now include the spin degree of freedom but disregard interactions . the system is described by two independent hilbert spaces spanned by the states @xmath63 and @xmath64 with @xmath7 . the probability vector for each spin @xmath12 can be written in terms of the eigenvalues and eigenvectors of the matrix @xmath61 ( for the two - dimensional hilbert space ) as @xmath65 \label{eq_psigma}\ ] ] where the right eigenvector corresponding to the eigenvalue zero of @xmath61 defines the occupation probabilities for the equilibrium state , @xmath66 , with the fermi function @xmath67^{-1}$ ] and the inverse temperature @xmath68 . furthermore , @xmath69 is the vector representation of the number operator for dot electrons with spin @xmath12 , whose initial / equilibrium expectation value is obtained by multiplying it from the left into the initial / equilibrium probability vector , @xmath70 . the rate @xmath71 is obtained as the negative of the non - zero eigenvalue of @xmath61 , with the corresponding left eigenvector being @xmath72 . the time evolution of the occupation of each spin state is governed by a single decay rate @xmath0 , @xmath73 this equation can be obtained making use of the fact that the time evolution of the expectation value of any operator , which describes an observable of the qd , is given by projecting its vector representation from the left onto eq . ( [ eq_psigma ] ) . the time evolution of the total charge of the dot , @xmath74 , is also determined by a single relaxation rate @xmath71 . this means that both charge and spin , which are quantities related with single - particle processes , do not evolve independently from each other and the corresponding decay is given by the same rate . a similar non - interacting problem has been studied _ non - pertubatively _ in refs . and . as a next step we consider the squared deviation of the charge from its equilibrium value , @xmath75 ^ 2 $ ] . its time evolution is obtained from eq . ( [ eq_psigma ] ) as @xmath76 ^ 2\rangle(t)-[\langle\hat{n}\rangle^{\text{eq}}]^2\\ & & = \sum_{\sigma=\uparrow,\downarrow}[1+\langle\hat{n}_{\sigma}\rangle^{\text{eq}}]\langle\hat{n}_{\sigma}\rangle(t)+2\langle\hat{n}_{\uparrow}\hat{n}_{\downarrow}\rangle(t)\nonumber\end{aligned}\ ] ] the last , two - particle term of this expression exhibits a decay rate given by @xmath77 . this is in contrast to the spinless case , where such a term does not appear since double occupation is not possible . such an additional exponential decay with the rate @xmath78 appears directly in the time evolution of the probability vector , when considering the full two - particle hilbert space spanned by the basis @xmath79 . in this basis , eq . ( [ eq_exp ] ) for the non - interacting regime can be written as : @xmath80\\ \frac{1}{2}\left[1 - 2f(\epsilon)\right]\\ \frac{1}{2}\left[1 - 2f(\epsilon)\right]\\ f(\epsilon ) \end{array}\right ) e^{-\gamma t}\left(\langle\hat{n}\rangle^\mathrm{in}-\langle\hat{n}\rangle^\mathrm{eq}\right)\nonumber \\ + \left(\begin{array}{c } 0\\ \frac{1}{2}\\ -\frac{1}{2}\\ 0 \end{array}\right ) e^{-\gamma t}\langle\hat{s}\rangle^\mathrm{in } + \left ( \begin{array}{c } -1\\1\\1\\-1 \end{array } \right ) e^{-2\gamma t}\left ( \langle\hat{m}\rangle^\mathrm{in}-\langle\hat{m}\rangle^\mathrm{eq } \right ) \nonumber\\\end{aligned}\ ] ] where as before , @xmath81 defines the state at equilibrium . the decaying part of the probability vector can be divided into three contributions which appear depending on how the initial state at @xmath1 differs from the equilibrium state . deviations of charge and spin from their equilibrium value relax with the same rate @xmath0 . the corresponding expectation values are calculated by multiplying the probability vector eq . ( [ ptot ] ) from the left with the vector representation of the operators @xmath82 and @xmath83 which represent the charge and spin , respectively , in this two - particle basis . the two left eigenvectors of the matrix @xmath84 with the same eigenvalue @xmath85 , are given by @xmath86 and @xmath87 . the third contribution to the decay of the system into the equilibrium comes from the relaxation rate @xmath78 , which enters the probability vector in connection with a quantity @xmath88 , defined by the operator in vector notation @xmath89 the left eigenvector of @xmath62 with the eigenvalue @xmath90 is given by @xmath91 . in contrast to charge and spin , the quantity represented by @xmath88 does not have a straightforward intuitive interpretation , since it depends on the quantum dot parameters at @xmath33 and on the temperature and chemical potential of the reservoir via the fermi functions . from now on we assume a finite on - site coulomb repulsion @xmath6 on the dot . analogously to the noninteracting case discussed before , from eq . ( [ eq_exp ] ) we can write the time - dependent probability vector in terms of contributions exhibiting different decay times @xmath92\\ \frac{1}{2}\left[1-f(\epsilon)-f(\epsilon+u)\right]\\ \frac{1}{2}\left[1-f(\epsilon)-f(\epsilon+u)\right]\\ f(\epsilon+u ) \end{array}\right)e^{-\gamma_n t}\left(\langle\hat{n}\rangle^\mathrm{in}-\langle\hat{n}\rangle^\mathrm{eq}\right ) \nonumber \\ & & + \left(\begin{array}{c } 0\\ \frac{1}{2}\\ -\frac{1}{2}\\ 0 \end{array}\right ) e^{-\gamma_s t}\langle\hat{s}\rangle^\mathrm{in}+\left ( \begin{array}{c } -1\\1\\1\\-1 \end{array } \right ) e^{-\gamma_m t}\left ( \langle\hat{m}\rangle^\mathrm{in}-\langle\hat{m}\rangle^\mathrm{eq } \right)\ .\end{aligned}\ ] ] again , @xmath93 is the eigenvector of @xmath51 with the zero eigenvalue and represents the equilibrium state in lowest order in the tunnel coupling ( the explicit form of the four - dimensional matrix @xmath62 , together with its entire set of eigenvalues and eigenvectors , is given in the appendix ) . in the two - particle basis , again @xmath94 represents the charge operator , and @xmath95 represents the spin operator . the form of the operator @xmath88 is modified by the presence of finite coulomb interaction ; the explicit form will be discussed later in this sub - section ( see eq . ( [ eq_def_m ] ) below ) . the initial and equilibrium expectation values for these operators , entering in the above eq . ( [ eq_solution ] ) , are obtained as @xmath96 , with @xmath97 . explicit expressions for @xmath98 and @xmath99 are shown below . the negative of the other three eigenvalues of @xmath62 directly determine the decay of charge , spin,@xcite and the quantity denoted by @xmath88 . these decay rates read @xmath100\label{eq_lcharge1}\\ \gamma_s&=&\gamma\left[1-f(\epsilon)+f(\epsilon+u)\right]\label{eq_lspin1}\\ \gamma_m&=&2\gamma . \label{eq_lmal1}\end{aligned}\ ] ] notice that due to interaction , the relaxation rates for charge and spin ( @xmath101 and @xmath102 respectively ) differ from each other and depend on the level position @xmath5 , in contrast to the non - interacting case . their dependence on the level position is shown in fig . [ fig_decay ] . in the region for @xmath103 , @xmath101 is enhanced as the charge decays into the twofold degenerate state of single - occupation , whereas the spin relaxation in first order in @xmath0 is suppressed , since spin - flip processes are not possible . however , the third decay rate , @xmath104 , remains fully energy independent as in the case with @xmath105 . ( blue , dashed line ) , @xmath101 ( red , dash - dotted line ) and @xmath102 ( green , solid line ) in units of @xmath0 as a function of the dot level position @xmath5 . the temperature is @xmath106 and the interaction energy is @xmath107 . ] the right eigenvectors occurring in eq . ( [ eq_solution ] ) each represent a change to the steady state density matrix that decays exponentially with rate @xmath108 ( @xmath109 ) . therefore , a system being brought out of equilibrium by a symmetric deviation between @xmath110 and @xmath111 only , is decaying with a rate @xmath102 . a deviation from equilibrium in which the occupation of the even sector , @xmath112 is symmetrically shifted from the odd sector , @xmath113 , is governed solely by the relaxation rate @xmath104 . this right eigenvector is found to play an important role also in the low - temperature renormalization of this model . @xcite an energy - dependent change in the occupation probabilities as prescribed by the second vector in eq . ( [ eq_solution ] ) yields a decay of the total charge of the system with the rate @xmath101 . the conditions under which specific deviations from the equilibrium state should be performed in order to obtain a specific decay rate , are discussed in the following section . the attribution of these relaxation rates to the charge , spin , and @xmath88 arises from the independent decay of these quantities , due to the explicit form of the _ left _ eigenvectors of @xmath62 . the spin operator coincides with the left eigenvector associated to the eigenvalue @xmath114 and since it has a vanishing equilibrium value , the time evolution of its expectation value is given by @xmath115 equivalently , the left eigenvector corresponding to the eigenvalue @xmath116 , is @xmath117 . it contains the charge operator @xmath118 and its equilibrium value @xmath119 $ ] . hence , for the time evolution of the charge we find @xmath120 as a function of the dot level position @xmath5 . the other parameters are : @xmath106 and @xmath107 . ] the quantity decaying with the rate @xmath104 alone is related to the left eigenvector @xmath91 , where the operator @xmath88 is given by @xmath121 its expectation value follows a time evolution equivalent to the one for the charge in eq . ( [ eq_nrelax ] ) : @xmath122 . its equilibrium value @xmath123 $ ] , plotted in fig . [ fig_meh ] , is - in contrast to spin and charge - not sensitive to the regime of single occupation on the quantum dot . instead , it exhibits a feature close to the electron - hole symmetric point of the anderson model , indicating that @xmath88 represents a quantity which is affected by two - particle effects and it decays with a rate that is not modified by the coulomb interaction @xmath6 . already for the noninteracting case , we found that the rate @xmath78 appears as a consequence of introducing two particles in the system , and we considered the deviations from equilibrium charge as a quantity involving two - particle processes leading to such a decay rate . also in the case for finite coulomb interaction , the time - dependent mean squared deviations @xmath124 ^ 2\rangle(t)$ ] are suitable to reveal the relaxation rate @xmath125 . their time evolution is obtained by means of eq . ( [ eq_solution ] ) and reads @xmath126 ^ 2\rangle(t ) - [ \langle \hat{n}\rangle^\mathrm{eq}]^2 & = & c\cdot\langle \hat{n}\rangle(t)-2\cdot\langle \hat{m}\rangle(t)\nonumber\\ \label{eq_variance}\end{aligned}\ ] ] ( red , solid line ) and the coefficient @xmath127 ( blue , dashed line ) as a function of the dot level position @xmath5 . the other parameters are : @xmath106 and @xmath107 . ] where in front of the time - dependent charge @xmath128 the following coefficient appears : @xmath129 with @xmath130 the quantity @xmath131/\left[1+f(\epsilon)-f(\epsilon+u)\right]$ ] is the difference between the probability of doubly occupied and empty dot in equilibrium , which can also be related with the occupation of electrons and holes , @xmath132 . the behavior of @xmath133 is shown in fig . [ fig_coeff ] . for @xmath134 , when the dot is doubly occupied , @xmath135 ; for @xmath103 , when one electron and one hole are present in the system ( singly occupied dot ) , @xmath136 ; and for @xmath137 , when the system is completely `` filled with holes '' ( empty dot ) , @xmath138 . the quantity @xmath127 is also shown in fig . [ fig_coeff ] ( blue dashed line ) , exhibiting a sign change around @xmath139 , the point at which the anderson model is electron - hole symmetric . by replacing @xmath140 , we go from the electron - like to the hole - like behavior , finding an inversion in the sign of @xmath127 , @xmath141 . the function @xmath127 therefore indicates whether the spectrum of the quantum dot is electron - like or hole - like . the mean squared deviations of the charge from its value at equilibrium is an example for a physical quantities showing a decay with @xmath104 ; it also includes the charge relaxation rate @xmath101 , which is found independently from the time evolution of the charge . equivalently also the time - resolved charge variance , @xmath142 ^ 2\rangle(t)$ ] , or the time - resolved spin variance,@xcite @xmath143 , contain a contribution decaying with @xmath104 . we now consider in detail which external perturbations are necessary in order to induce a decay of the _ full _ occupation probability vector with one certain relaxation rate only , in a controlled way . furthermore , we address the conditions under which a single decay rate can be extracted more easily from the occupation of a single state by a measurement with a nearby quantum point contact ( qpc ) . we first address the case of an infinitesimal perturbation ( linear response ) . a small variation of the gate potential leads to a decay of the charge governed by the charge relaxation rate @xmath101 . similarly , the infinitesimal variation of the zeeman splitting in the dot yields a decay with the spin relaxation rate @xmath102 . in order to obtain a decay of the state with the rate @xmath104 only , it is not sufficient to modulate the gate voltage , also the two - particle term in the hamiltonian , @xmath144 , needs to be varied . the on - site repulsion @xmath6 could be changed , for example , by tuning the carrier density in a nearby two - dimensional electron gas , thereby controlling the screening of the electron - electron interaction in the dot . from eq . ( [ eq_solution ] ) we know that a dynamics given only by @xmath104 is obtained if the occupation of the even states are changed in the same direction , opposite to that of the single occupied states ; this condition is fulfilled if infinitesimal variations of the gate , @xmath145 , and of the interaction , @xmath146 , obey the relation : @xmath147)}{1+\exp(\beta\epsilon)}d\epsilon . \label{dvarm}\ ] ] this expression is represented in terms of field lines in fig . [ fig_field ] . an infinitesimal change tangential to the field line passing through the point corresponding to the initial values of @xmath5 and @xmath6 leads to a pure decay with @xmath104 . for parameter variations that are not infinitesimal ( beyond linear response ) , a change only of the gate voltage results in a decay of the state with both rates @xmath101 and @xmath104 . from eq . ( [ eq_solution ] ) we find that a finite variation of the energy level and the interaction from an initial condition @xmath148 to @xmath149 resulting in a relaxation containing solely @xmath101 , satisfies the equation @xmath150 . \label{varn}\ ] ] a relaxation given _ only _ by the rate @xmath104 is found when the relation : @xmath151 \label{varm}\ ] ] is fulfilled . for different values of @xmath152 and @xmath153 , eq . ( [ varm ] ) produces again the field lines shown in fig . [ fig_field ] . therefore , finite variations of the parameters between two points lying on _ the same _ field line yield a dynamics for the entire occupation probabilities vector @xmath154 governed only by @xmath104 . obviously , a generic variation in both @xmath5 and @xmath6 which does not fulfill the conditions specified by eqs . ( [ varn ] ) or ( [ varm ] ) exhibits a dynamics of the probabilities with two time scales : @xmath101 and @xmath104 . in fig . [ fig_field ] it is observed that in the region @xmath155 the field lines are approximately horizontal , i.e , only the interaction @xmath6 needs to be varied while keeping the level position constant in order to see a dynamics of the probability governed by @xmath104 only . in fact , in this regime the qd is predominantly empty and variations of the interaction strength @xmath6 do not affect the occupation of the dot . this is the reason why this variation yields a dynamics in which the rate @xmath101 does not contribute . on the other hand , in the region for @xmath156 in order to avoid that the number of particles on the dot changes , which would lead to a relaxation with rate @xmath101 , a variation in @xmath6 needs to be accompanied by an opposite variation in @xmath5 , that is @xmath157 . the crossover between the two regimes appears around the symmetry point of the anderson model , @xmath139 . importantly , it is also possible to read out either the rate @xmath101 or the rate @xmath104 by varying the gate voltage only ( and , thus , not fulfilling eqs . ( [ dvarm ] ) and ( [ varm ] ) ) , which is easier to realize in an experiment . this can be done by measuring an observable that is sensitive to only one occupation probability , for instance the probability of the quantum dot being empty . such a time - resolved read - out of the probability can be achieved by considering a qpc located nearby the system and tuned such that it conducts only if the qd is empty . @xcite in the simplest model of the qpc , which assumes a very fast response , the operator corresponding to the current in the qpc is given by @xmath158 where @xmath159 is a constant current , given by the characteristics of the qpc potential . the expectation value of the qpc current is simply @xmath160 . in this way , the qpc effectively measures the dynamics of the occupation probability @xmath161 . according to eq . ( [ eq_solution ] ) , a modulation of the gate in which the initial value @xmath162 equals the equilibrium value @xmath99 leads to a pure decay with @xmath101 . instead , for a decay given by @xmath104 either the factor @xmath163 or the factor @xmath164/\left[1-f(\epsilon)+f(\epsilon+u)\right]$ ] in eq . ( [ eq_solution ] ) has to vanish . . the on - site coulomb repulsion @xmath6 is constant and takes the value @xmath165 . dashed blue line : @xmath5 changes from @xmath166 to @xmath167 , its slope yields the relaxation rate @xmath104 . red dot - dashed line : in this case @xmath168 to @xmath169 , and the slope leads to @xmath101 . the black line is obtained if @xmath5 changes from @xmath170 to @xmath169 , in which both rates @xmath104 and @xmath101 are present . in all cases we have subtracted the corresponding value for the current in the long - time limit . ] results for the qpc current for different variations of the level position @xmath5 while @xmath6 is kept constant , are shown in the logarithmic plot in fig . [ lines ] . for clarity , we also subtracted the corresponding current in the long time limit , @xmath171 . in particular , for a fixed value of @xmath6 equal to @xmath172 , we find that if the level position is changed from @xmath166 to @xmath167 , the time evolution of @xmath161 is governed entirely by the rate @xmath104 , giving rise to the straight , blue - dashed line in fig . [ lines ] . its slope is given by @xmath104 , making it possible to extract this relaxation rate from measurements of the current in the qpc . however we can obtain a dynamics of @xmath161 given mainly by the rate @xmath101 by performing a variation in @xmath5 from @xmath166 to @xmath169 which results in the red dot - dashed straight line in fig . [ lines ] ; again , the slope yields the corresponding relaxation rate which takes the value @xmath173 . finally , we show an example in which variations from @xmath170 to @xmath169 ( solid black line ) produce a dynamics of @xmath161 which includes two exponential decays with rates @xmath104 and @xmath101 . as a result , the curve exhibits a change in the slope , showing that a single rate will not be obtained by arbitrary variations of the parameters . in the previous sections we investigated the relaxation rates in first order in the tunnel coupling strength @xmath0 . however , corrections due to higher order tunneling processes appear when the tunnel coupling gets stronger . besides quantitative corrections , this reveals an interesting new aspect . in second order in the tunnel coupling , the matrix @xmath174 included in the exponential decay takes the form @xmath175 . the second - order corrections to the relaxation rates for charge and spin are given by:@xcite @xmath176w_{\mathrm{d}0 } } { 1-f(\epsilon)+f(\epsilon+u ) } \label{eq_charge_corr}\\ \gamma_\mathrm{s}^{(2 ) } & = & \sigma(\epsilon,\gamma , u ) \frac{\partial}{\partial\epsilon}\gamma_\mathrm{s}+\sigma_\gamma(\epsilon,\gamma , u)\gamma_\mathrm{s}+2w_\mathrm{sf}\ . \label{eq_spin_corr}\end{aligned}\ ] ] these corrections contain renormalization terms as well as real cotunneling contributions . on one hand , the renormalization terms contain an effect due to the level renormalization @xmath177 , with @xmath178 $ ] , @xmath179 and @xmath180 is the digamma function . on the other hand , the renormalization of the tunnel coupling appears , @xmath181 $ ] , with @xmath182 $ ] and where @xmath127 was defined in eq . ( [ eq_coeff ] ) . real cotunneling contributions are manifest in terms of spin flips , @xmath183 , and coherent transitions changing the particle number on the dot by @xmath184 , @xmath185 and @xmath186 . these cotunneling terms read @xmath187\\ w_{\mathrm{d}0 } & = & - \frac{2 \gamma } { e^{\beta(2\epsilon+u)}-1 } \left[\gamma\phi'(\epsilon)+\gamma\phi'(\epsilon+u)-\frac{2}{u}\sigma(\epsilon , u)\right],\nonumber\\\end{aligned}\ ] ] and @xmath188 w_{\mathrm{d}0}$ ] . the way in which the cotunneling contributions enter in the respective charge and spin relaxation rates is related to the deviation of the state of the qd from equilibrium , given by eq . ( [ eq_solution ] ) in first order in @xmath0 . as an example we discuss the correction to the charge decay rate , second line of eq . ( [ eq_charge_corr ] ) . there the factor @xmath184 appears due to the change in the charge by @xmath189 in a process bringing the dot from zero to double occupation and vice versa . @xcite the fraction with which the transition from zero to double occupation , @xmath186 , enters the correction to the charge relaxation rate , @xmath190 , is given by the deviation from equilibrium of @xmath57 in the direction of @xmath161 , of the contribution which actually decays with @xmath101 only . this is the first component of the second vector in eq . ( [ eq_solution ] ) . equivalently , the transition from double to zero occupation , @xmath185 , enters with the fraction given by the fourth component of the same vector , namely by the deviation from equilibrium of @xmath57 in the direction of @xmath191 . strikingly , in contrast to the charge and spin relaxation rates , @xmath104 does not get renormalized at all by second order tunneling processes : @xmath192 the reason for this is that the contribution due to @xmath0 renormalization and those due to coherent processes between empty and doubly occupied dot , cancel each other . the lack of second order corrections , confirms that this relaxation rate is related to a quantity which is not sensitive to the coulomb interaction . the fact that corrections are missing , is also found using a renormalization - group approach . another important aspect of this missing second - order correction is that it is due to an exact cancelation of the contribution due to virtual second order processes , namely the @xmath0-renormalization , with real cotunneling contributions . this is in contrast to , e.g. the conductance , where only the real cotunneling processes contribute far from resonances , while renormalization terms are limited to the resonant regions . until now , we considered the quantum dot to be coupled to a normal conducting lead . however , the vicinity of a superconducting or a ferromagnetic reservoir induces correlations between electrons and holes or between charge and spin , respectively . in the following we study , in first order in the tunnel coupling strength @xmath0 , the influence of induced correlations on the relaxation rates of the dot . the charge response of a noninteracting mesoscopic scattering region coupled to both normal and superconducting leads has been studied in refs . . in the previous sections we have seen that the rate @xmath104 , which together with the time decay of charge and spin determines the relaxation of the qd to the equilibrium state , is independent of the level position and the coulomb interaction and that it enters in the time evolution of quantities sensitive to two - particle effects . it is therefore expected that the rate @xmath104 will directly influence the relaxation of the charge towards the equilibrium in a setup that naturally mixes the empty and doubly occupied states of the dot . this situation is obtained if the qd is not only coupled to a normal lead ( with tunnel coupling strength @xmath0 ) but also to an _ additional _ superconducting contact ( with tunnel coupling strength @xmath193 ) , as shown in fig . [ fig_scheme ] ( b ) . we consider only the case when the superconductor is kept at the same chemical potential as the normal lead and we set both chemical potentials to zero . the only purpose of the extra lead is here to induce superconducting correlations on the dot via the proximity effect . to the original hamiltonian , @xmath194 , we now add the hamiltonian for the superconducting contact and its tunnel coupling to the qd , @xmath195 where @xmath196 is the the annihilation ( creation ) operator of electrons in the lead . in the limit of a large superconducting gap @xmath197 the effect of the additional contact can be cast in an effective hamiltonian of the dot which includes a coupling between electrons and holes in the qd , @xmath198 . the eigenstates of the proximized dot are the states of single occupation @xmath20 and other two states which are superpositions of the empty and double occupied states of the dot ( due to andreev reflection ) : @xmath199 with energies given by @xmath200 , where the level detuning between @xmath19 and @xmath21 is @xmath201 and @xmath202 is the energy splitting between the @xmath203 and @xmath204 states.@xcite in the new basis @xmath205 , the vector representing the charge operator is expressed as @xmath206 and we expect that the effect of the mixing of electrons and holes will be visible in its time evolution . in first order in the tunnel - coupling strength to the normal reservoir @xmath0 and assuming @xmath207 , we find the relaxation rates @xmath208\\ \gamma_{s , s}&= & \gamma\left[1-f(\epsilon - e_-)+f(e_+-\epsilon)\right]\\ \gamma_{s,2}&=&2\gamma.\end{aligned}\ ] ] remarkably the eigenvalue @xmath209 remains unaffected , i.e. @xmath210 is not modified by the presence of the additional superconducting lead . the spin on the dot , which is determined by the occupation probabilities of singly occupied states , still decays with a single relaxation rate given by @xmath211 , i.e. @xmath212 is an eigenvector of the kernel @xmath62 ( in the proximized basis ) . in contrast , the decay of the charge to its equilibrium value is given by @xmath213 \left(e^{-\gamma_{s,2 } t}+e^{-\gamma_{s,1}t } \right ) + \langle \hat{n}\rangle^{\mathrm{eq}}\nonumber\\ & & + a_{sc } \frac{1}{2}\left[\langle x\rangle^{\mathrm{in}}-\langle x\rangle^{\mathrm{eq}}\right ] \left(e^{-\gamma_{s,2 } t}-e^{-\gamma_{s,1}t}\right ) \nonumber \\ & & + \frac{1}{2}\left[\langle y\rangle^{\mathrm{in}}-\langle y\rangle^{\mathrm{eq}}\right ] \left(e^{-\gamma_{s,2 } t}-e^{-\gamma_{s,1}t}\right ) \label{eq_nsct}\end{aligned}\ ] ] with @xmath214 and where we defined the difference in the occupation of the @xmath215 states , @xmath216 and the quantity @xmath217 , with @xmath218 . the charge evolves with two different time scales , @xmath219 and @xmath220 , instead of only one as in the normal case . this is a direct consequence of the mixing of the states @xmath19 and @xmath21 induced by the superconducting contact . this effect opens the possibility to extract this rate by measuring the time evolution of the charge in the proximized dot . even though the presence of a superconducting lead couples electrons and holes , the relaxation rate @xmath104 has not been modified . since we associate this rate with processes involving two particles each with spin @xmath12 , it is expected that if the spin symmetry is broken by introducing a ferromagnetic contact , the rate @xmath104 will now be the sum of the tunneling rates for spin up and spin down electrons . in order to verify this , we consider the hamiltonian used for the normal case and assume a spin - dependent density of states in the only reservoir attached to the quantum dot , see fig . [ fig_scheme ] ( c ) . this leads to spin - dependent tunnel couplings , @xmath221 and @xmath222 , which are included in the corresponding transition matrix @xmath62 . diagonalization of @xmath62 yields the three relaxation rates : @xmath223 ^ 2 } \\ \gamma_{f,2 } & = & \gamma-\frac{1}{2}\sqrt{\left(\delta\gamma\right)^2 + 4\gamma_\uparrow\gamma_\downarrow\left[f(\epsilon)-f(\epsilon+u)\right]^2 } \\ \gamma_{f , m}&= & 2 \gamma\end{aligned}\ ] ] with @xmath224 and @xmath225 . as in the normal case , there is an eigenvalue which does not depend on the level position nor on the interaction but on the sum of the different tunneling rates : @xmath226 . the appearance of such a combination of the spin - dependent tunneling strengths in the relaxation rate , confirms the statement that two - particle processes involving electrons with both spin polarizations are at the basis of the decay rate @xmath104 . due to the ferromagnetic lead , the dynamics of spin and charge are now mixed . the corresponding time evolution in first order in the tunnel coupling takes the form : @xmath227(e^{-\gamma_{f,1}t}-e^{-\gamma_{f,2}t } ) \nonumber \\ \langle\hat{n}\rangle_f(t ) & = & \frac{1}{2}\left[\langle \hat{n}\rangle^{\mathrm{in}}-\langle \hat{n}\rangle^{\mathrm{eq}}\right](e^{-\gamma_{f,1}t}+e^{-\gamma_{f,2}t } ) + \langle \hat{n}\rangle^{\mathrm{eq}}\nonumber \\ & & + a_c\left[\langle \hat{n}\rangle^{\mathrm{in}}-\langle \hat{n}\rangle^{\mathrm{eq}}\right](e^{-\gamma_{f,1}t}-e^{-\gamma_{f,2}t})\nonumber \\ & & + b_c\langle \hat{s}\rangle^{\mathrm{in}}(e^{-\gamma_{f,1}t}-e^{-\gamma_{f,2}t})\label{eq_decay_ferro}\end{aligned}\ ] ] where we introduced the abbreviations : @xmath228}{2\sqrt{\delta\gamma^2 + 4\gamma_\uparrow\gamma_\downarrow\left[f(\epsilon)-f(\epsilon+u)\right]^2 } } \nonumber \\ b_s & = & \frac{\delta\gamma[1+f(\epsilon)-f(\epsilon+u)]}{2\sqrt{\delta\gamma^2 + 4\gamma_\uparrow\gamma_\downarrow\left[f(\epsilon)-f(\epsilon+u)\right]^2 } } \nonumber \\ a_c & = & a_s\nonumber \\ b_c & = & \frac{\delta\gamma[1-f(\epsilon)+f(\epsilon+u)]}{2\sqrt{\delta\gamma^2 + 4\gamma_\uparrow\gamma_\downarrow\left[f(\epsilon)-f(\epsilon+u)\right]^2}}. \nonumber\end{aligned}\ ] ] the last term in eq . ( [ spin_ferro ] ) shows that at finite time @xmath32 the initial charge influences the time evolution of the spin ; similarly , the initial spin enters explicitly in the dynamics of the charge , eq . ( [ eq_decay_ferro ] ) . these terms persist in the non - interacting limit , revealing that the coupled evolution of charge and spin including two relaxation rates ( which for the non - interacting case take the form @xmath229 and @xmath230 ) is a direct consequence of the presence of the ferromagnetic contact . in contrast , the factor @xmath231 vanishes for @xmath105 implying that it stems from the combined effect of the coulomb interaction and the breaking of the spin symmetry . as expected the independent evolution of charge and spin is recovered in the limit @xmath232 . the mixing of the dynamics of both , charge and spin , induced here by a ferromagnetic lead was found in ref . for the case of lifted spin - degeneracy in the dot due to a finite zeeman splitting . note that for the hybrid as well as for the normal system , the sum of the energy - dependent relaxation rates equals @xmath78 , as long as the tunnel coupling @xmath0 is treated in first order , only . we have studied the different time scales present in the evolution of the reduced density matrix of a single - level qd with coulomb interaction and tunnel coupled to a single reservoir , after being brought out of equilibrium . besides the relaxation rates for charge and spin , we find an additional rate @xmath125 , which is independent of the energy level of the dot as well as of the interaction strength . this relaxation is related to the presence of two particles in the dot and is found to be not sensitive to the coulomb interaction . the time evolution of the square deviations of the charge from its equilibrium value is proposed as a physical quantity related with processes involving two - particles leading to the rate @xmath78 . in order to further elucidate the properties of this decay , we analyzed the response of the system to specific variations of both , the interaction strength @xmath6 and the level position @xmath5 , finding that @xmath104 can be extracted from time - resolved measurements of the current passing through a nearby quantum point contact . additionally , we analyzed two other setups : a dot proximized by a superconductor and coupled to a normal reservoir , and a dot coupled to a ferromagnetic lead . in the hybrid normal - superconducting systems , we found that the time - resolved read - out of the charge represents another possibility to get access to the rate @xmath104 . we thank michael moskalets , roman riwar and maarten wegewijs for fruitful discussion . financial support by the ministry of innovation , nrw , the dfg via spp 1285 and ko 1987/5 , the european community s seventh framework programme under grant agreement no . 238345 ( geomdiss ) , as well as the swiss national science foundation , the swiss centers of excellence manep and qsit and the european marie curie itn , nanoctm is acknowledged . the transition matrix for the normal case in the eigenbasis of the isolated qd @xmath233 , in first order in the tunneling strength @xmath0 , is calculated by means of fermi s golden rule and is given by : @xmath234 & 0 & 1-f(\epsilon+u))\\ f(\epsilon ) & 0 & -\left[1-f(\epsilon)+f(\epsilon+u)\right ] & 1-f(\epsilon+u))\\ 0 & f(\epsilon+u ) & f(\epsilon+u ) & -2\left[1-f(\epsilon+u)\right ] \end{array } \right)\end{aligned}\ ] ] as @xmath62 is non - hermitian it has different right and left eigenvectors , @xmath56 and @xmath55 . for a system with a well - defined steady state ( as the one we are considering here ) there must be at least a zero eigenvalue , @xmath238 . @xcite the other eigenvalues are found to be the negative of @xmath100 \nonumber \\ \gamma_s&=&\gamma\left[1-f(\epsilon)+f(\epsilon+u)\right ] \\ \nonumber \label{evalues}\end{aligned}\ ] ] the right eigenvector corresponding with the zero eigenvalue , @xmath239 , determines the stationary density matrix ( which we also label as @xmath81 ) , whereas each one of the rest of the right eigenvectors represents a deviation out of the equilibrium density matrix which decays exponentially with a rate given by the negative of the corresponding eigenvalue : @xmath240[1-f(\epsilon+u)]\\ f(\epsilon)[1-f(\epsilon+u)]\\ f(\epsilon)[1-f(\epsilon+u)]\\ f(\epsilon)f(\epsilon+u)\\ \end{array}\right ) , \ \mathbf{r}_s=\frac{1}{2}\left(\begin{array}{c}0\\1 \\-1 \\ 0\end{array}\right),\\ \mathbf{r}_n = \frac{1}{1-f(\epsilon)+f(\epsilon+u)}\left(\begin{array}{c } -[1-f(\epsilon)]\\ \frac{1}{2}\left[1-f(\epsilon)-f(\epsilon+u)\right]\\ \frac{1}{2}\left[1-f(\epsilon)-f(\epsilon+u)\right]\\ f(\epsilon+u ) \end{array}\right ) , \ \mathbf{r}_m = \left(\begin{array}{c } -1\\1\\1\\-1\end{array}\right)\ .\end{aligned}\ ] ] these left eigenvectors contain the operators for spin , charge and @xmath88 in vector representation , which can be understood in the following manner . while in general the expectation value of an operator @xmath247 is found from @xmath248 , with the full density matrix @xmath249 , this can be considerably simplified in the situation considered here , where only diagonal elements of the reduced density matrix of the quantum dot , collected in the vector @xmath154 , play a role . the expectation value of a quantum dot operator is then obtained by multiplying its vector representation from the left hand side onto the vector @xmath154 . to show an example the expectation value of the spin on the dot is obtained by multiplying @xmath154 from left by the vector @xmath250 , yielding @xmath251 . similarly , all other operators for quantum dot observables can be expressed in such a vector representation .

geometrical dynamics is a dynamics of elementary particles , generated by the space - time geometry . in the space - time of minkowski the geometrical dynamics coincides with the conventional classical dynamics , and the geometrical dynamics may be considered to be a generalization of classical dynamics onto more general space - time geometries . however , the geometric dynamics has more fundamental basis , and it may be defined in multivariant space - time geometries , where one can not introduce the conventional classical dynamics . the fact is that , the classical dynamics has been introduced for the space - time geometry with unlimited divisibility , whereas the real space - time has a limited divisibility . the limited divisibility of the space - time is of no importance for dynamics of macroscopic bodies . however , when the size of moving bodies is of the order of the size of heterogeneity , one may not neglect the limited divisibility of the space - time geometry . the geometric dynamics is developed in the framework of the program of the further physics geometrization , declared in @xcite . the special relativity and the general relativity are steps in the development of this program . necessity of the further development appeared in the thirtieth of the twentieth century , when diffraction of electrons has been discovered . the motion of electrons , passing through the narrow slit , is multivariant . as far as the free electron motion depends only on the properties of the space - time , one needed to change the space - time geometry , making it to be multivariant . in multivariant geometry there are many vectors @xmath1 , @xmath2, ... at the point @xmath3 , which are equal to the vector @xmath4 , given at the point @xmath5 , but they are not equal between themselves , in general . such a space - time geometry was not known in the beginning of the twentieth century . it is impossible in the framework of the riemannian geometry . as a result the multivariance was prescribed to dynamics . to take into account multivariance , dynamic variables were replaced by matrices and operators . one obtains the quantum dynamics , which differs from the classical dynamics in its principles . but the space - time conception remains to be newtonian ( nonrelativistic ) . multivariant space - time geometry appeared only in the end of the twentieth century @xcite . the further geometrization of physics became to be possible . it should note that there were numerous attempts of further geometrization of physics . they were based on the riemannian space - time geometry . unfortunately , the true space - time geometry of microcosm does not belong to the class of riemannian geometries , and approximation of real space - time geometry by a riemannian geometry can not be completely successful . in particular , the riemannian geometry can not describe such a property of real space - time geometry as multivariance . the multivariance of the space - time geometry was replaced by the multivariance of dynamics ( quantum theory ) . understanding of nature of elementary particles is the aim of the further geometr - ization of physics . this aim distinguishes from the aim of the conventional theory of elementary particles . let us explain the difference of aims in the example of history the chemical elements investigation . investigation of chemical elements reminds to some extent investigation of elementary particles . chemical elements are investigated from two sides . chemists systematized chemical elements , investigating their phenomenological properties . the results of these investigations were formulated in the form of the periodical system of chemical elements in 1870 . formulating this system , d.i.mendeleev conceived nothing about the atom construction . nevertheless the periodical system appears to be very useful from the practical viewpoint . physicists did not aim to explain the periodical system of chemical elements , they tried to understand simply the atom structure and the discrete character of atomic spectra . after construction of the atomic theory it became clear , that the periodical system of chemical elements can be obtained and explained on the basis of the atomic theory . as a result the `` physical '' approach to investigation of chemical elements appeared to be more fundamental , deep and promising , than the `` chemical '' one . on the other hand , the way of the `` physical '' approach to explanation of the periodical system is very long and difficult . explanation of the periodical system was hardly possible at the `` physical '' approach , i.e. without the intermediate aim ( construction of atomic structure ) . thus , using geometrization of physics , we try to approach only intermediate aim : explanation of multivariance of particle motion ( quantum motion ) and capacity of discrimination of particle masses . discrete character of masses of elementary particles can be understood , only if we understand the reason of the elementary particle discrimination . contemporary approach to the elementary particle theory is the `` chemical '' ( phenomenological ) approach . it is useful from the practical viewpoint . however , it admits hardly to understand nature of elementary particles , because the nature of the discrimination mechanism , leading to discrete characteristics of elementary particle , remains outside the consideration . the most general geometry is a physical geometry , which is called also the tubular geometry ( t - geometry ) @xcite , because straights in t - geometry are hallow tubes , in general . the t - geometry is determined completely by its world function @xmath6 , where @xmath7 is interval between the points @xmath8 and @xmath9 in space - time , described by the t - geometry . all concepts of t - geometry are expressed in terms of the world function @xmath0 . dynamics of particles ( geometric dynamics ) is also described in terms of the world function . the elementary particle is considered as an elementary geometrical object ( ego ) in the space - time . the elementary geometrical object @xmath10 is described by its skeleton @xmath11 and its envelope @xmath12 . the envelope @xmath12 is defined as a set of zeros of the envelope function @xmath13@xmath14the envelope function @xmath13 is a real function of arguments @xmath15 . any argument @xmath16 @xmath17 is a world function @xmath18 , @xmath19 . it is supposed that ego with skeleton @xmath20 is placed at the point @xmath5 . in t - geometry the vector @xmath21 is an ordered set of two points @xmath22 . the length @xmath23 of the vector @xmath4 is defined via the world function by means of the relation@xmath24 the scalar product @xmath25 of two vectors @xmath4 and @xmath1 is defined by the relation@xmath26equivalence @xmath4eqv@xmath1 of two vectors @xmath4 and @xmath1 is defined as follows . two vectors @xmath4 and @xmath1 are equivalent ( equal ) , if@xmath27 in the developed form we have@xmath28 skeletons @xmath20 and @xmath29 are equivalent ( @xmath30eqv@xmath31 , if corresponding vectors of both skeletons are equivalent@xmath32 the skeleton @xmath20 of ego at the point @xmath5 may exist as a skeleton of a physical body , if it may exist at any point @xmath33 of the space - time @xmath34 . it means that there is a solution for system of equations @xmath35for any point @xmath33 . further for brevity we take , that an existence of a skeleton means an existence of corresponding geometrical object . in the space - time of minkowski the problem of the skeleton existence is rather simple , because at given @xmath30 and @xmath3 the system ( [ c1.6 ] ) of @xmath36 algebraic equations has a unique solution , although the number of equations may distinguish from the number of variables to be determined . indeed , in the four - dimensional space - time the number of coordinates of @xmath37 points @xmath38 is equal to @xmath39 ( the point @xmath3 is supposed to be given ) . if @xmath40 the number @xmath36 of equations is larger than the number ( @xmath39 ) of variables . in the case of an arbitrary space - time geometry ( arbitrary world function @xmath0 ) existence of solution of the system ( [ c1.6 ] ) is problematic , and the question of existence of the skeleton as a skeleton of a physical body is an essential problem . on the contrary , if @xmath41 , the number of coordinates to be determined is less , than the number of equations , and one may have many skeletons @xmath42placed at the point @xmath3 , which are equivalent to skeleton @xmath30 , but they are not equivalent between themselves . this property is a property of multivariance of the space - time geometry . this property is actual for simple skeletons , which contain less , than four points ( @xmath41 ) . for instance , for the skeleton of two points @xmath43 , which is described by the vector @xmath4 , the problem of multivariance is actual . in the space - time of minkowski the equivalence of two vectors ( @xmath44 ) is single - variant for the timelike vectors , however it is multivariant for spacelike vectors . in the general space - time the equivalence relation @xmath44 is multivariant for both timelike and spacelike vectors . the problem of multivariance is essential for both existence and dynamics of elementary geometrical objects ( elementary particles ) . let us formulate dynamics of elementary particles in the coordinateless form . dynamics of an elementary particle , having skeleton @xmath11 , is described by the world chain @xmath45direction of evolution in the space - time is described by the leading vector @xmath4 . if the motion of the elementary particle is free , the adjacent links @xmath46 and @xmath47 are equivalent in the sense that @xmath48 relations ( [ c1.7 ] ) - ( [ c1.9 ] ) realizes coordinateless description of the free elementary particle motion . in the simplest case , when the space - time is the space - time of minkowski , and the skeleton consists of two points @xmath49 with timelike leading vector @xmath4 , the coordinateless description by means of relations ( [ c1.7 ] ) - ( [ c1.9 ] ) coincides with the conventional description . the conventional classical dynamics is well defined only in the riemannian space - time . the coordinateless dynamic description ( [ c1.7 ] ) - ( [ c1.9 ] ) of elementary particles is a generalization of the conventional classical dynamics onto the case of arbitrary space - time geometry . any geometry is constructed as a modification of the proper euclidean geometry . but not all representations of the proper euclidean geometry are convenient for modification . there are three representation of the proper euclidean geometry @xcite . they differ in the number of primary ( basic ) elements , forming the euclidean geometry . the euclidean representation ( e - representation ) contains three basic elements ( point , segment , angle ) . any geometrical object ( figure ) can be constructed of these basic elements . properties of the basic elements and the method of their application are described by the euclidean axioms . the vector representation ( v - representation ) of the proper euclidean geometry contains two basic elements ( point , vector ) . the angle is a derivative element , which is constructed of two vectors . a use of the two basic elements at the construction of geometrical objects is determined by the special structure , known as the linear vector space with the scalar product , given on it ( euclidean space ) . the scalar product of linear vector space describes interrelation of two basic elements ( vectors ) , whereas other properties of the linear vector space associate with the displacement of vectors . the third representation ( @xmath0-representation ) of the proper euclidean geometry contains only one basic element ( point ) . segment ( vector ) is a derivative element . it is constructed of points . the angle is also a derivative element . it is constructed of two segments ( vectors ) . the @xmath50-representation contains a special structure : world function @xmath0 , which describes interrelation of two basic elements ( points ) . the world function @xmath51 , where @xmath52 is the distance between points @xmath5 and @xmath53 . the concept of distance @xmath54 , as well as the world function @xmath0 , is used in all representations of the proper euclidean geometry . however , the world function forms a structure only in the @xmath0-representation , where the world function @xmath0 describes interrelation of two basic elements ( points ) . besides , the world function satisfies a series of constraints , formulated in terms of @xmath0 and only in terms of @xmath0 . these conditions ( the euclideaness conditions ) will be formulated below . the euclideaness conditions are equivalent to a use of the vector linear space with the scalar product on it , but formally they do not mention the linear vector space , because all concepts of the linear vector space , as well as all concepts of the proper euclidean geometry are expressed directly via world function @xmath0 and only via it . if we want to modify the proper euclidean geometry , then we should use the @xmath0-representation for its modification . in the @xmath0-representation the special geometric structure ( world function ) has the form of a function of two points . modifying the form of the world function , we modify automatically all concepts of the proper euclidean geometry , which are expressed via the world function . it is very important , that the expression of geometrical concepts via the world function does not refer to the means of description ( dimension , coordinate system , concept of a curve ) . the fact , that modifying the world function , one violates the euclideaness conditions , is of no importance , because one obtains non - euclidean geometry as a result of such a modification . a change of the world function means a change of the distance , which is interpreted as a deformation of the proper euclidean geometry . the generalized geometry , obtained by a deformation of the proper euclidean geometry is called the tubular geometry ( t - geometry ) , because in the generalized geometry straight lines are tubes ( surfaces ) , in general , but not one - dimensional lines . another name of t - geometry is the physical geometry . the physical geometry is the geometry , described completely by the world function . any physical geometry may be used as a space - time geometry in the sense , that the set of all t - geometries is the set of all possible space - time geometries . modification of the proper euclidean geometry in v - representation is very restricted , because in this representation there are two basic elements . they are not independent , and one can not modify them independently . formally it means , that the linear vector space is to be preserved as a geometrical structure . it means , in particular , that the generalized geometry retains to be continuous , uniform and isotropic . the dimension of the generalized geometry is to be fixed . besides , the generalized geometry , obtained by such a way , can not be multivariant . such a property of the space - time geometry as multivariance can be obtained only in @xmath0-representation . as far as the @xmath0-representation of the proper euclidean geometry was not known in the twentieth century , the multivariance of geometry was also unknown concept . transition from the v - representation to @xmath0-representation is carried out as follows . all concepts of the linear vector space are expressed in terms of the world function @xmath0 . in reality , concepts of vector , scalar product of two vectors and linear dependence of @xmath37 vectors are expressed via the world function @xmath55 of the proper euclidean geometry . such operations under vectors as equality of vectors , summation of vectors and multiplication of a vector by a real number are expressed by means of some formulae . the characteristic properties of these operations , which are given in v - representation by means of axioms , are given now by special properties of the euclidean world function @xmath55 . after expression of the linear vector space via the world function the linear vector space may be not mentioned , because all its properties are described by the world function . we obtain the @xmath0-representation of the proper euclidean geometry , where some properties of the linear vector space are expressed in the form of formulae , whereas another part of properties is hidden in the specific form of the euclidean world function @xmath55 . modifying world function , we modify automatically the properties of the linear vector space ( which is not mentioned in fact ) . at such a modification we are not to think about the way of modification of the linear vector space , which is the principal geometrical structure in the v - representation . in the @xmath0-representation the linear vector space is a derivative structure , which may be not mentioned at all . thus , at transition to @xmath0-representation the concepts of the linear vector space ( primary concepts in v - representation ) become to be secondary concepts ( derivative concepts of the @xmath0-representation ) . in @xmath0-representation we have the following expressions for concepts of the proper euclidean geometry . vector @xmath56 is an ordered set of two points @xmath8 and @xmath9 . the length @xmath57 of the vector @xmath58 is defined by the relation@xmath59the scalar product @xmath25 of two vectors @xmath4 and @xmath1 is defined by the relation@xmath60where the world function @xmath0@xmath61is the world function @xmath55 of the euclidean geometry . in the proper euclidean geometry @xmath37 vectors @xmath62 , @xmath63 are linear dependent , if and only if the gram s determinant @xmath64 where the gram s determinant @xmath65 is defined by the relation @xmath66using expression ( [ a1.1 ] ) for the scalar product , the condition of the linear dependence of @xmath37 vectors @xmath62 , @xmath63 is written in the form@xmath67 definition ( [ a1.1 ] ) of the scalar product of two vectors coincides with the conventional scalar product of vectors in the proper euclidean space . ( one can verify this easily ) . the relations ( [ a1.1 ] ) , ( [ a1.7b ] ) do not contain a reference to the dimension of the euclidean space and to a coordinate system in it . hence , the relations ( [ a1.1 ] ) , ( [ a1.7b ] ) are general geometric relations , which may be considered as a definition of the scalar product of two vectors and that of the linear dependence of vectors . equivalence ( equality ) of two vectors @xmath4 and @xmath1 is defined by the relations@xmath68where @xmath23 is the length ( [ a1.0 ] ) of the vector @xmath4@xmath69 in the developed form the condition ( [ a1.3 ] ) of equivalence of two vectors @xmath4 and @xmath1 has the form@xmath70 let the points @xmath49 , determining the vector @xmath4 , and the origin @xmath3 of the vector @xmath1 be given . let @xmath4eqv@xmath1 . we can determine the vector @xmath1 , solving two equations ( [ a1.5 ] ) , ( [ a1.6 ] ) with respect to the position of the point @xmath71 . in the case of the proper euclidean space there is one and only one solution of equations ( [ a1.5 ] ) , ( [ a1.6 ] ) independently of the space dimension @xmath37 . in the case of arbitrary t - geometry one can guarantee neither existence nor uniqueness of the solution of equations ( [ a1.5 ] ) , ( [ a1.6 ] ) for the point @xmath71 . number of solutions depends on the form of the world function @xmath0 . this fact means a multivariance of the property of two vectors equivalence in the arbitrary t - geometry . in other words , the single - variance of the vector equality in the proper euclidean space is a specific property of the proper euclidean geometry , and this property is conditioned by the form of the euclidean world function . in other t - geometries this property does not take place , in general . the multivariance is a general property of a physical geometry . it is connected with a necessity of solution of algebraic equations , containing the world function . as far as the world function is different in different physical geometries , the solution of these equations may be not unique , or it may not exist at all . if in the @xmath37-dimensional euclidean space @xmath72 , the vectors @xmath62 , @xmath63 are linear independent . we may construct rectilinear coordinate system with basic vectors @xmath62 , @xmath63 in the @xmath37-dimensional euclidean space . covariant coordinates @xmath73 of the vector @xmath74 in this coordinate system have the form@xmath75 now we can formulate the euclideaness conditions . these conditions are conditions of the fact , that the t - geometry , described by the world function @xmath0 , is @xmath37-dimensional proper euclidean geometry . i. definition of the dimension and introduction of the rectilinear coordinate system : @xmath76where @xmath77 is the gram s determinant ( a1.7 ) . vectors @xmath78 , @xmath79 are basic vectors of the rectilinear coordinate system @xmath80 with the origin at the point @xmath5 . in @xmath80 the covariant metric tensor @xmath81 , @xmath82 and the contravariant one @xmath83 , @xmath82 are defined by the relations @xmath84@xmath85 \ii . linear structure of the euclidean space : @xmath86where coordinates @xmath87 @xmath88 of the point @xmath8 are covariant coordinates of the vector @xmath74 , defined by the relation ( [ a1.8 ] ) . iii : the metric tensor matrix @xmath89 has only positive eigenvalues @xmath90 \iv . the continuity condition : the system of equations @xmath91considered to be equations for determination of the point @xmath8 as a function of coordinates @xmath92 , @xmath88 has always one and only one solution . _ _ _ _ all conditions i @xmath93 iv contain a reference to the dimension @xmath37 of the euclidean space . one can show that conditions i @xmath93 iv are the necessary and sufficient conditions of the fact that the set @xmath34 together with the world function @xmath0 , given on @xmath94 , describes the @xmath37-dimensional euclidean space @xcite . investigation of the dirac particle ( dynamic system , described by the dirac equation ) has shown , that the dirac particle is a composite particle r2004 , whose internal degrees of freedom are described nonrelativistically @xcite . the composite structure of the dirac particle may be explained as a relativistic rotator , consisting of two ( or more ) particles , rotating around their inertia centre . the relativistic rotator explains existence of the dirac particle spin , however , the problem of the rotating particles confinement appears . in this paper we try to explain the problem of spin in the framework of the program of the physics geometrization , when dynamics of physical bodies is determined by the space - time geometry . although the first stages of the physics geometrization ( the special relativity and the general relativity ) manifest themselves very well , the papers on further geometrization of physics , which ignore the quantum principles , are considered usually as dissident . dynamics in the space - time , described by a physical geometry ( t - geometry ) , is presented in @xcite . here we remind the statement of the problem of dynamics . geometrical object @xmath95 is a subset of points in the point set @xmath34 . in the t - geometry the geometric object @xmath10 is described by means of the skeleton - envelope method . it means that any geometric object @xmath10 is considered to be a set of intersections and joins of elementary geometric objects ( ego ) . the elementary geometrical object @xmath12 is described by its skeleton @xmath30 and envelope function @xmath13 . the finite set @xmath96 of parameters of the envelope function @xmath13 is the skeleton of elementary geometric object ( ego ) @xmath97 . the set @xmath97 of points forming ego is called the envelope of its skeleton @xmath30 . the envelope function @xmath13@xmath98determining ego is a function of the running point @xmath99 and of parameters @xmath100 . the envelope function @xmath13 is supposed to be an algebraic function of @xmath101 arguments @xmath15 , @xmath102 . each of arguments @xmath103 is the world function @xmath0 of two points @xmath104 , either belonging to skeleton @xmath30 , or coinciding with the running point @xmath105 . thus , any elementary geometric object @xmath12 is determined by its skeleton @xmath30 and its envelope function @xmath13 . elementary geometric object @xmath12 is the set of zeros of the envelope function @xmath106__definition . _ _ two egos @xmath107 and @xmath108 are equivalent , if their skeletons @xmath30 and @xmath109 are equivalent and their envelope functions @xmath13 and @xmath110 are equivalent . equivalence ( @xmath30eqv@xmath109 ) of two skeletons @xmath111 and @xmath112 means that @xmath113equivalence of the envelope functions @xmath13 and @xmath110 means , that they have the same set of zeros . it means that @xmath114where @xmath115 is an arbitrary function , having the property@xmath116 evolution of ego @xmath117 in the space - time is described as a world chain @xmath118 of equivalent connected egos . the point @xmath5 of the skeleton @xmath11 is considered to be the origin of the geometrical object @xmath119 the ego @xmath117 is considered to be placed at its origin @xmath5 . let us consider a set of equivalent skeletons @xmath120 @xmath121which are equivalent in pairs @xmath122the skeletons @xmath123 @xmath121are connected , and they form a chain in the direction of vector @xmath4 , if the point @xmath53 of one skeleton coincides with the origin @xmath5 of the adjacent skeleton @xmath124the chain @xmath118 describes evolution of the elementary geometrical object @xmath117 in the direction of the leading vector @xmath4 . the evolution of ego @xmath117 is a temporal evolution , if the vectors @xmath125 are timelike @xmath126 @xmath127 . the evolution of ego @xmath117 is a spatial evolution , if the vectors @xmath125 are spacelike @xmath128 @xmath127 . note , that all adjacent links ( egos ) of the chain are equivalent in pairs , although two links of the chain may be not equivalent , if they are not adjacent . however , lengths of corresponding vectors are equal in all links of the chain @xmath129we shall refer to the vector @xmath125 , which determines the form of the evolution and the shape of the world chain , as the leading vector . this vector determines the direction of 4-velocity of the physical body , associated with the link of the world chain . if the relations@xmath130@xmath131are satisfied , the relations@xmath132are not satisfied , in general , because the relations ( [ a6.11 ] ) contain the scalar products @xmath133 . these scalar products contain the world functions @xmath134 , which are not contained in relations ( [ a6.9 ] ) , ( [ a6.10 ] ) . the world chain @xmath118 , consisting of equivalent links ( [ a6.1 ] ) , ( [ a6.2 ] ) , describes a free motion of a physical body ( particle ) , associated with the skeleton @xmath30 . we assume that _ the motion of physical body is free , if all points of the body move free _ ( i.e. without acceleration ) . if the external forces are absent , the physical body as a whole moves without acceleration . however , if the body rotates , one may not consider a motion of this body as a free motion , because not all points of this body move free ( without acceleration ) . in the rotating body there are internal forces , which generate centripetal acceleration of some points of the body . as a result some points of the body do not move free . motion of the rotating body may be free only on the average , but not exactly free . conception of non - free motion of a particle is rather indefinite , and we restrict ourselves with consideration of a free motion only . conventional conception of the motion of extensive ( non - pointlike ) particle , which is free on the average , contains a free displacement , described by the velocity 4-vector , and a spatial rotation , described by the angular velocity 3-pseudovector @xmath135 . the velocity 4-vector is associated with the timelike leading vector @xmath4 . at the free on the average motion of a rotating body some of vectors @xmath136 ... of the skeleton @xmath30 are not in parallel with vectors @xmath137 , although at the free motion all vectors @xmath138 are to be in parallel with @xmath137 as follows from ( [ a6.1 ] ) . it means that the world chain @xmath118 of a freely moving body can describe only translation of a physical body , but not its rotation . if the leading vector @xmath4 is spacelike , the body , described by the skeleton @xmath30 , evolves in the spacelike direction . it seems , that the spacelike evolution is prohibited . but it is not so . if the world chain forms a helix with the timelike axis , such a world chain may be considered as timelike on the average . in reality such world chains are possible . for instance , the world chain of the classical dirac particle is a helix with timelike axis . it is not quite clear , whether or not the links of this chain are spacelike , because internal degrees of freedom of the dirac particle , responsible for helicity of the world chain , are described nonrelativistically . thus , consideration of a spatial evolution is not meaningless , especially if we take into account , that the spatial evolution may imitate rotation , which is absent at the free motion of a particle . further we consider the problem of the spatial evolution . dirac particle @xmath139 is the dynamic system , described by the dirac equation . the free dirac particle @xmath139 is described by the free dirac equation @xmath140where @xmath141 is the four - component complex wave function , and @xmath142 , @xmath143 are @xmath144 complex matrices , satisfying the relations@xmath145@xmath146 is the @xmath144 unit matrix , @xmath147 is the metric tensor . expressions of physical quantities : the 4-flux @xmath148 of particles and the energy - momentum tensor @xmath149 have the form@xmath150where @xmath151 , @xmath152 is the hermitian conjugate to @xmath141 . the classical dirac particle is a dynamic system @xmath153 , which is obtained from the dynamic system @xmath139 in the classical limit . to obtain the classical limit , one may not set the quantum constant @xmath154 in the equation ( [ f1.2 ] ) , because in this case we do not obtain any reasonable description of the particle . the dirac particle @xmath139 is a quantum particle in the sense , that it is described by a system of partial differential equations ( pde ) , which contain the quantum constant @xmath155 . the classical dirac particle @xmath153 is described by a system of ordinary differential equations ( ode ) , which contain the quantum constant @xmath155 as a parameter . may the system of ode carry out the classical description , if it contains the quantum constant @xmath155 ? the answer depends on the viewpoint of investigator . if the investigator believes that _ the quantum constant is an attribute of quantum principles and only of quantum principles _ , he supposes that , containing @xmath155 , the dynamic equations can not realize a classical description , where the quantum principles are not used . however , if the investigator consider the classical description simply as method of investigation of the quantum dynamic equations , it is of no importance , whether or not the system of ode contains the quantum constant . it is important only , that the system of pde is approximated by a system of ode . the dynamic system , described by pde , contains infinite number of the freedom degrees . the dynamic system , described by ode , contains several degrees of freedom . it is simpler and can be investigated more effectively . obtaining the classical approximation , we use the procedure of dynamic disquantization @xcite . this procedure transforms the system of pde into the system of ode . the procedure of dynamic disquantization is a dynamical procedure , which has no relation to the process of quantization or disquantization in the sense , that it does not refer to the quantum principles . the dynamic disquantization means that all derivatives @xmath156 in dynamic equations are replaced by the projection of vector @xmath157 onto the current vector @xmath148 @xmath158this dynamical operation is called the dynamic disquantization , because , applying it to the schrdinger equation , we obtain the dynamic equations for the statistical ensemble of classical nonrelativistic particles . these dynamic equations are ode , which do not depend on the quantum constant @xmath155 . applying the operation ( [ b3.1 ] ) , to the dirac equation ( [ f1.2 ] ) , we transform it to the form@xmath159the equation ( [ b3.2 ] ) is the dynamic equation for the dynamic system system @xmath160 . the equation ( [ b3.2 ] ) contains only derivative @xmath161 in the direction of the current 4-vector @xmath148 . in terms of the wave function @xmath141 the dynamic equation ( [ b3.2 ] ) for @xmath160 looks rather bulky . however , in the properly chosen variables the action for the dynamic system @xmath160 has the form @xcite @xmath162=\int \left\ { -\kappa _ { 0}m\sqrt{\dot{x}^{i}\dot{x}_{i}}+\hbar { \frac{(\dot{\mathbf{\xi } } \times \mathbf{\xi } ) \mathbf{z}}{2(1+\mathbf{\xi z})}}+\hbar \frac{(\dot{\mathbf{x}}\times \ddot{\mathbf{x}})\mathbf{\xi } } { 2\sqrt{\dot{x}^{s}\dot{x}_{s}}(\sqrt{\dot{x}^{s}\dot{x}_{s}}+\dot{x}^{0})}\right\ } d^{4}\tau \label{a5.18}\]]where the dot means the total derivative @xmath163 . the quantities @xmath164 , @xmath165 , @xmath166 , @xmath167 are considered to be functions of the lagrangian coordinates @xmath168 , @xmath169 . the variables @xmath170 describe position of the dirac particle . here and in what follows the symbol @xmath171 means the vector product of two 3-vectors . the quantity@xmath172 is the constant unit 3-vector , @xmath173 is a dichotomic quantity @xmath174 , @xmath175 is the constant ( mass ) taken from the dirac equation ( [ f1.2 ] ) . in fact , variables @xmath170 depend on @xmath176 as on parameters , because the action ( [ a5.18 ] ) does not contain derivatives with respect to @xmath177 , @xmath167 . lagrangian density of the action ( [ a5.18 ] ) does not contain independent variables @xmath178 explicitly . hence , it may be written in the form @xmath162=\int \mathcal{a}_{\mathrm{dcl}}[x,\mathbf{\xi } ] d\mathbf{\tau , \qquad } d\mathbf{\tau } = d\tau _ { 1}d\tau _ { 2}d\tau _ { 3 } \label{b3.8}\]]where @xmath179=\int \left\ { -\kappa _ { 0}m\sqrt{\dot{x}^{i}\dot{x}_{i}}+\hbar { \frac{(\dot{\mathbf{\xi } } \times \mathbf{\xi } ) \mathbf{z}}{2(1+\mathbf{\xi z})}}+\hbar \frac{(\dot{\mathbf{x}}\times \ddot{\mathbf{x}})\mathbf{\xi } } { 2\sqrt{\dot{x}^{s}\dot{x}_{s}}(\sqrt{\dot{x}^{s}\dot{x}_{s}}+\dot{x}^{0})}\right\ } d\tau _ { 0 } \label{b3.9}\ ] ] the action ( [ b3.8 ] ) is the action for the dynamic system @xmath160 , which is a set of similar independent dynamic systems @xmath153 . such a dynamic system is called a statistical ensemble . dynamic systems @xmath153 are elements ( constituents ) of the statistical ensemble @xmath160 . dynamic equations for each @xmath153 form a system of ordinary differential equations . it may be interpreted in the sense , that the dynamic system @xmath153 may be considered to be a classical one , although lagrangian of @xmath153 contains the quantum constant @xmath155 . the dynamic system @xmath180 will be referred to as the classical dirac particle . the dynamic system @xmath153 has ten degrees of freedom . it describes a composite particle @xcite . external degrees of freedom are described relativistically by variables @xmath170 . internal degrees of freedom are described nonrelativistically @xcite by variables @xmath181 . solution of dynamic equations , generated by the action ( [ b3.9 ] ) , gives the following result @xcite . in the coordinate system , where the canonical momentum four - vector @xmath182 has the form @xmath183the world line of the classical dirac particle is a helix , which is described by the relation @xmath184where the speed of the light @xmath185 , and @xmath186 is an arbitrary constant ( lorentz factor of the classical dirac particle ) . the velocity @xmath187 of the classical dirac particle is expressed as follows @xmath188 helical world line of the classical dirac particle means a rotation of the particle around some point . on the one hand , such a rotation seems to be reasonable , because it explains freely the dirac particle spin and magnetic moment . on the other hand , the description of this rotation is nonrelativistic . besides , it seems rather strange , that the world line of a free classical particle is a helix , but not a straight line . attempt of consideration of the dirac particle as a rotator , consisting of two particles @xcite , meets the problem of confinement of the two particles . although the pure dynamical methods of investigation are more general and effective , than the investigation methods , based on quantum principles , the purely dynamical methods of investigation meet incomprehension of most investigators , who believe , that the dirac particle must be investigated by quantum methods . the papers , devoted to investigation of the dirac equation by the dynamic methods , are considered as dissident . they are rejected by the peer review journals ( see discussion in @xcite ) . suddenly it is discovered that the helical world line , which is characteristic for the classical dirac particle , can be obtained as a result of a spatial evolution of geometric objects in the framework of properly chosen space - time geometry . let us consider the flat homogeneous isotropic space - time @xmath189 , described by the world function @xmath190@xmath191@xmath192where @xmath193 is the world function of the @xmath194-dimensional space - time of minkowski . @xmath195 is some elementary length . in such a space - time geometry two connected equivalent timelike vectors @xmath4 and @xmath196 are described as follows @xcite @xmath197where @xmath198 is an arbitrary unit 3-vector . the quantity @xmath199 is the length of the vector @xmath4 ( geometrical mass , associated with the particle , which is described by the vector @xmath4 ) . we see that the spatial part of the vector @xmath196 is determined to within the arbitrary 3-vector of the length @xmath200 . this multivariance generates wobbling of the links of the world chain , consisting of equivalent timelike vectors @xmath201 , @xmath196 , @xmath202 , ... statistical description of the chain with wobbling links coincides with the quantum description of the particle with the mass @xmath203 , if the elementary length @xmath204 , where @xmath205 is the speed of the light , @xmath155 is the quantum constant , and @xmath206 is some universal constant , whose exact value is not determined @xcite , because the statistical description does not contain the quantity @xmath206 . thus , the characteristic wobbling length is of the order of @xmath195 . to explain the quantum description of the particle motion as a statistical description of the multivariant classical motion , we should use the world function ( [ a4.1 ] ) however , the form of the world function ( [ a4.1 ] ) is determined by the coincidence of the two descriptions only for the value @xmath207 , where the constant @xmath208 is determined via the mass @xmath209 of the lightest massive particle ( electron ) by means of the relation @xmath210where @xmath211 is the geometrical mass of the lightest massive particle ( electron ) . the geometrical mass @xmath212 of the same particle , considered in the space - time geometry of minkowski , has the form@xmath213as far as @xmath214 , and , hence , @xmath215 , we obtain the following estimation for the universal constant @xmath206@xmath216 intensity of wobbling may be described by the multivariance vector @xmath217 , which is defined as follows . let @xmath196 , @xmath218 be two vectors which are equivalent to the vector @xmath219 . let @xmath220let us consider the vector @xmath221which is a difference of vectors @xmath196 , @xmath218 . we consider the length @xmath222 of the vector @xmath223 in the minkowski space - time . we obtain@xmath224the length of the vector ( [ a4.3a ] ) is minimal at @xmath225 . at @xmath226 the length of the vector ( [ a4.3a ] ) is maximal , and it is equal to zero . by definition the vector @xmath223 at @xmath225 is the multivariance 4-vector @xmath217 , which describes the intensity of the multivariance . we have @xmath227where @xmath198 is an arbitrary unit 3-vector . the multivariance vector @xmath217 is spacelike in the case , when @xmath228 , the corresponding wobbling length @xmath229where @xmath230 is the electron compton wave length . the relation ( [ a4.3 ] ) means that @xmath231for other values @xmath232 the form of the world function @xmath233 may distinguish from the relation ( [ a4.4 ] ) . however , @xmath234 , if @xmath235 . two equivalent connected spacelike vectors @xmath1 , @xmath236 have the form @xcite@xmath237where constants @xmath238 and @xmath239 are arbitrary . the result is obtained for the space - time geometry ( [ a4.0 ] ) . arbitrariness of constants @xmath240 generates multivariance of the vector @xmath236 even in the space - time geometry of minkowski , where @xmath241 . vectors @xmath236 , @xmath242 @xmath243are equivalent to the vector @xmath1 . the difference @xmath244 of two vectors @xmath236 , @xmath242 has the form@xmath245the vector @xmath244 may be spacelike and timelike . its length has an extremum , if @xmath246 and @xmath247 . in this case the length @xmath248 however , the length@xmath249has neither maximum , nor minimum , and one can not introduce the multivariance vector of the type ( [ a4.3c ] ) . the multivariance of the spacelike vectors equality is not introduced by the distortion @xmath250 , defined by ( [ a4.0a ] ) . it takes place already in the space - time of minkowski . in the conventional approach to the geometry of minkowski one does not accept the multivariance of spacelike vectors equivalence . furthermore , the concept of multivariance of two vectors parallelism ( and equality ) is absent at all in the conventional approach to the geometry . for instance , when in the riemannian geometry the multivariance of parallelism of remote vectors appears , the mathematicians prefer to deny at all the fernparallelism ( parallelism of two remote vectors ) , but not to introduce the concept of multivariance . this circumstance is connected with the fact , that the multivariance may not appear , if the geometry is constructed on the basis of a system of axioms . the world chain , consisting of timelike equivalent vectors , imitates a world line of a free particle . this fact seems to be rather reasonable . considering the vectors @xmath1 and @xmath236 in ( [ a6.50 ] ) from the viewpoint of the geometry of minkowski , we see that the vector @xmath236 is obtained from the vector @xmath1 as a result of spatial rotation and of an addition of some temporal component . one should expect , that the world chain , consisting of spacelike equivalent vectors , imitates a world line of a free particle , moving with a superluminal velocity . the motion with the superluminal velocity seems to be unobservable . such a motion is considered to be impossible . however , if the spacelike world line has a shape of a helix with timelike axis , such a situation may be considered as a free rotating particle . the fact , that the particle rotates with the superluminal velocity is not so important , if the helix axis is timelike . the world line of a classical dirac particle is a helix . it is not very important , whether the rotation velocity is tardyon or not . especially , if we take into account that the dirac equation describes the internal degrees of freedom ( rotation ) nonrelativistically , ( i.e. the description of internal degrees of the classical dirac particle is incorrect from the viewpoint of the relativity theory ) . we investigate now , whether a world chain of equivalent spacelike vectors may form a helix with timelike axis . if it is possible , then we try to investigate , under which world function such a situation is possible . we consider the world function @xmath233 of the form @xmath251where the function @xmath252 should be determined from the condition , that the world chain , consisting of spacelike links , forms a helix with timelike axis . to estimate the form of @xmath233 as a function of @xmath193 at @xmath232 , we consider the chain , consisting of equivalent spacelike vectors ... @xmath4 , @xmath196 , @xmath253 ... we suppose that the chain is a helix with timelike axis in the space - time . let the points @xmath254 . have the coordinates@xmath255all points ( [ a4.6 ] ) lie on a helix with timelike axis . in the space - time of minkowski the step of helix is equal to @xmath256 , and @xmath105 is the radius of the helix . the constants @xmath257 and @xmath258 are parameters of the helix . all vectors @xmath259 have the same length . introducing designations [ 02beg]@xmath260we obtain coordinates of vectors @xmath259 in the form @xmath261where @xmath262 are parameters of the helix . let us investigate , under which conditions the relation @xmath263eqv@xmath259 takes place . we suppose that all vectors of the helix are spacelike @xmath264 . it is evident , that it is sufficient to investigate the situation for the case @xmath265 , when @xmath4eqv@xmath196 . let coordinates of vectors @xmath4 , @xmath196 be @xmath266 in this case the coordinates of the points @xmath267 may be chosen in the form@xmath268and the vector @xmath219 has coordinates @xmath269we choose the world function ( [ a4.5 ] ) in the form@xmath270and introduce the quantity @xmath271 thus , we have @xmath272the space - time geometry ( [ a4.15 ] ) is a special case of the space - time geometry ( [ a4.4 ] ) . we do not pretend to the claim , that ( [ a4.15 ] ) is the world function of true space - time geometry . we shall show only that in the space - time geometry ( [ a4.15 ] ) the spacelike vectors ( [ a4.9 ] ) may be equivalent at some proper choice of parameters @xmath273 , @xmath274 and @xmath257 . in our calculations we shall use two geometries : the geometry @xmath275 of minkowski and the space - time geometry @xmath276 , described by the world function @xmath233 , determined by ( [ a4.15 ] ) . then expressions of the geometry @xmath276 may be reduced to expressions of the geometry @xmath275 by means of relations@xmath277@xmath278@xmath279the geometrical relations in @xmath276 are expressed via the same relations , written in @xmath275 with additional terms , containing the distortion @xmath250 . this additional terms in dynamic relations are interpreted as additional metric interactions , acting on a particle , when the real space - time geometry @xmath276 is considered to be the geometry @xmath275 . appearance of additional interactions reminds appearance of inertial forces at a use of accelerated coordinate systems instead of inertial ones . condition @xmath4eqv@xmath196 of equivalence of vectors @xmath4 , @xmath196 is written in the form of two equations@xmath280@xmath281where index m means , that the corresponding quantities are calculated in @xmath275 . the function @xmath250 is determined by the relation ( [ a4.15 ] ) , and the quantity @xmath282 is determined by the relation @xmath283which follows from the definition of the scalar product ( [ a4.15c ] ) . using the conventional relations for the scalar product in @xmath275 , we can rewrite the relations ( [ a4.16 ] ) , ( [ a4.17 ] ) in the form @xmath284@xmath285where@xmath286 to obtain the relation ( [ a4.21 ] ) from ( [ a4.18 ] ) , we use the relations@xmath287@xmath288the equation ( [ a4.20 ] ) is the identity . let us introduce pure quantities @xmath289 , @xmath290 , defining them by relations@xmath291@xmath292then the equation ( [ a4.19 ] ) takes the form@xmath293where the function @xmath294 is defined by the relation ( [ a4.15])@xmath295and the constant @xmath296 is defined by the relation ( [ a4.14a ] ) . let us note , that in the case , when @xmath297 is a linear function @xmath298 , for @xmath299 $ ] , the equation ( [ a4.25a ] ) has the unique solution @xmath300 . the solution with @xmath301 describes a straight but not a helix . considering solutions of equation ( [ a4.25a ] ) with respect to @xmath302 , we are interested in positive values of @xmath290 , because the quantity @xmath290 is nonnegative by definition ( [ a4.25 ] ) . at @xmath303 numerical solutions of equation ( [ a4.25a ] ) with respect to @xmath290 are presented in the form@xmath304 according to ( [ a4.7 ] ) , ( [ a4.24 ] ) and ( [ a4.25 ] ) we have the following relations for the helix radius @xmath105 @xmath305we obtain the helix step @xmath306 in the form@xmath307 negative values of @xmath289 correspond to helix with timelike vectors @xmath263 . positive solutions of equation ( [ a4.25a ] ) take place only for @xmath308 ( spacelike vectors ) and @xmath309 ( timelike vectors ) . the values of parameter @xmath310 belong to intervals @xmath311 , \qquad a\in \left ( 0,3\right ) \label{a4.38}\ ] ] for spacelike and timelike vectors correspondingly . thus , we see that in the space - time geometry with the world function ( a4.15 ) the spatial evolution , determined by the spacelike vectors @xmath259 , may lead to a helical world chain with timelike axis . however , equivalence of spacelike vectors @xmath259 is multivariant even in the space - time of minkowski . it is valid for the space - time geometry ( [ a4.15 ] ) also . as a result the wobbling of the spacelike vectors appears . it may lead to destruction of the helix . in reality the conditions @xmath312 determines vector @xmath313to within the vector @xmath314 , and we have instead of equations ( [ a4.9 ] ) @xmath315where @xmath316 , @xmath317 , @xmath318 are 4-vectors with coordinates@xmath319 instead of equations ( [ a4.19 ] ) ( [ a4.21 ] ) we have the following equations@xmath320@xmath321where @xmath322 and @xmath323 mean scalar products of vectors @xmath324 in the space - time of minkowski . the relation ( [ a4.25a ] ) is the necessary condition of the fact , that @xmath325 is a solution of equations ( [ a4.41 ] ) , ( [ a4.42 ] ) . we obtain from ( [ a4.41 ] ) @xmath326where @xmath327 means the scalar product of 3-vectors @xmath328 and @xmath329 . taking into account the relation ( [ a4.25a ] ) , we obtain from relation ( [ a4.42 ] ) @xmath330supposing , that @xmath322 is a small quantity we obtain from ( [ a4.44 ] ) by means of ( [ a4.43])@xmath331 the relation ( [ a4.45 ] ) may be transformed to the equation@xmath332where@xmath333as far as @xmath334 , we obtain , that @xmath335 , and the surface ( [ a4.46 ] ) is a hyperboloid in the 3-space of quantities @xmath336 . it means that solutions of the equations ( [ a4.43 ] ) , ( [ a4.44 ] ) may deflect arbitrarily far from the helix solution ( [ a4.9 ] ) . this deflection is a manifestation of the multivariance of the space - time geometry . suppression of multivariance and stabilization of the world chain , consisting of spacelike vectors , can be achieved , if we consider the world chain with composed links , whose skeleton consists of three points @xmath337 , @xmath338 ( see figure 1 ) . let @xmath259 be a spacelike vector , whereas the vector @xmath339 be a timelike vector in @xmath275 . to investigate the effect of stabilization , it is sufficient to consider the points @xmath340 , having coordinates@xmath341the following vectors are associated with these points of the skeletons@xmath342where the quantities @xmath343 are the given 4-vectors , whereas the quantities @xmath344 are 4-vectors , which are to be determined from the condition @xmath345 expressions for 4-vectors @xmath317 and @xmath54 are chosen in such a way , that vectors @xmath196 and @xmath346 ( at @xmath347 ) were a result of rotation of vectors @xmath4 and @xmath348 in the plane @xmath349 by the angle @xmath350 . the quantities @xmath351are supposed to be given . the 4-vectors @xmath352are to be determined from the relations ( [ a5.3a ] ) . the 4-vectors @xmath316 and @xmath317 coincide with vectors ( [ a4.9 ] ) . we are interested in the following question , whether the stabilizing vector @xmath353 can be chosen in such a way , that equations ( [ a5.3a ] ) have the unique solution @xmath347 . if such a stabilizing vector @xmath354exists , the world chain will have a shape of a helix without wobbling . it may be , that the complete stabilization is impossible . then , maybe , a partial stabilization is possible , when the quantities @xmath318 , @xmath355 are small , although they do not vanish . in any case the problem of the stabilizing vector existence is a pure mathematical problem . solving this problem , we shall use relations ( [ a4.15a ] ) , ( [ a4.15b ] ) to reduce all geometrical relations to the geometrical relations in the space - time of minkowski . working in the space - time of minkowski , we shall use the conventional covariant formalism , where the expressions of the type @xmath356 and @xmath357 mean@xmath358index " will be omitted for brevity . it follows from the condition @xmath4eqv@xmath196@xmath359where@xmath360after transformations we obtain@xmath361@xmath362these equations coincide with ( [ a4.41 ] ) , ( [ a4.42 ] ) . if @xmath363 the equations ( [ b5.11 ] ) , ( [ b5.12 ] ) coincide with ( [ a4.20 ] ) , ( a4.25a ) respectively . we obtain from the condition @xmath348eqv@xmath346@xmath364@xmath365where@xmath366 the equations ( [ b5.14 ] ) and ( [ b5.15 ] ) are transformed to the form@xmath367@xmath368@xmath369 let us suppose that the stabilizing vector @xmath101 is long in the sense that @xmath370then in ( [ b5.17 ] ) the functions @xmath250 , which contains @xmath101 in its argument will be equal to @xmath371 and all terms , containing @xmath101 compensate each other . the necessary condition of the fact , that @xmath372 is a solution of equations ( [ b5.16 ] ) , ( [ b5.17 ] ) , has the form@xmath373@xmath374the equation ( [ b5.18 ] ) is satisfied identically by the choice ( b5.4a ) , ( [ b5.4b ] ) of vectors @xmath101 and @xmath54 . we obtain from the condition @xmath375eqv@xmath376@xmath377@xmath378@xmath379@xmath380 the necessary conditions of the fact , that @xmath381 is a solution of equations ( [ b5.23 ] ) , ( [ b5.24 ] ) , have the form@xmath382@xmath383 the equation ( [ b5.27 ] ) is satisfied identically by the relations ( b5.4a ) , ( [ b5.4b ] ) . the difference of equations ( [ b5.19 ] ) and ( b5.28 ) leads to the equation @xmath384 let us substitute expressions for @xmath385 , determined by the relations ( [ b5.4a ] ) , ( [ b5.4b ] ) , in ( [ b5.29 ] ) . after transformations we obtain the connection between the quantities @xmath386 and @xmath387 in the form@xmath388 the equation ( [ b5.19 ] ) by means of ( [ b5.30 ] ) is reduced to the form@xmath389where according to ( [ a4.15 ] ) the function @xmath390 is substituted by @xmath391 . setting @xmath392and using designations ( [ a4.24 ] ) , ( [ a4.25 ] ) , we transform the equation ( [ b5.31 ] ) to the form @xmath393 the equations ( [ a4.25a ] ) and ( [ b5.33 ] ) form a system of two necessary conditions , imposed on parmeters of the helical world chain . each link of the chain consists of two vectors : leading vector @xmath394 and stabilizing vector @xmath395 . parameter @xmath396 is determined by the space - time geometry . the quantity @xmath397 describes the length of the spacelike leading vector @xmath394 . parameter @xmath398 describes the length of the projection of the leading vector @xmath394 on the plane of rotation . finally , @xmath399 describes the angle @xmath350 of rotation of the leading vector in the plane of rotation . numerical solutions of equations ( [ a4.25a ] ) and ( [ b5.33 ] ) are presented for the parameter @xmath303 solutions of equations , which describe the necessary conditions of the fact , that the world chain may be a helix , are not unique . there may be solutions of ( [ a5.3a ] ) , described by nonvanishing @xmath318 and @xmath355 , which generate wobbling and violate the helical character of world chain . we write six equation ( [ a5.3a ] ) as equation for @xmath401 with parameters @xmath343 , satisfying the necessary conditions ( [ a4.25a ] ) and ( b5.33 ) . we obtain instead of equations ( [ b5.11 ] ) , ( [ b5.12 ] ) the following two equations@xmath402@xmath403where the quantities @xmath404 satisfy the necessary conditions ( [ b5.33 ] ) ( [ a4.25a ] ) , and @xmath405 is a derivative of the function ( [ a4.26 ] ) , which is always nonnegative . then it follows from ( [ b6.2 ] ) @xmath406 equations ( [ b6.1 ] ) , ( [ b6.3 ] ) contain only the variable @xmath318 ( but not @xmath355 ) and coincide with the equations ( [ a4.41 ] ) , ( a4.42 ) . however , there are additional constraints , containing @xmath318 . as a result the constraints on @xmath318 distinguish from the relation ( a4.46 ) , describing values of @xmath318 without the stabilizing vector @xmath395 . in the developed form the relations ( [ b5.16 ] ) , ( [ b5.17 ] ) have the form @xmath407@xmath408they contain only the variable @xmath355 ( but not @xmath318 ) finally the relations ( [ b5.23 ] ) , ( [ b5.24 ] ) in the developed form can be written as follows@xmath409@xmath410the relation ( [ b6.7 ] ) is a linear combination of equations ( [ b5.17 ] ) and ( [ b5.24 ] ) , which does not contain the function @xmath294 . relations ( b6.6 ) and ( [ b6.7 ] ) contain both quantities @xmath401 and @xmath411 . the constraints ( [ b6.6 ] ) and ( [ b6.7 ] ) modify the constraints ( [ a4.46 ] ) , transforming the hyperboloid into ellipsoid . we suppose for simplicity , that the vector @xmath395 is very long ( @xmath412 ) . we suppose , that @xmath413 . in this case we obtain from the relation ( [ b6.5 ] ) , that @xmath414 . it follows from ( [ b6.7 ] ) , that @xmath415 . besides , it follows from ( [ b6.3 ] ) , that @xmath416 . thus , solutions of the equations ( [ b6.5 ] ) , ( [ b6.7 ] ) and ( [ b6.3 ] ) have the form @xmath417 at the constraints ( [ b6.8 ] ) three other equations ( [ b6.1 ] ) , ( b6.4 ) and ( [ b6.6 ] ) take the form @xmath418@xmath419@xmath420solution of equation ( [ b6.9 ] ) has the form @xmath421where @xmath422 is an arbitrary angle . solution of equation ( [ b6.10 ] ) has the form@xmath423where the quantities @xmath424 @xmath425 are arbitrary . and the quantity @xmath426 is determined by the relation ( [ b5.30 ] ) . substituting ( [ b6.12 ] ) - ( [ b6.14c ] ) in ( [ b6.11 ] ) , one obtains a constraint on the quantities @xmath427 @xmath424 @xmath425 . independently of this constraint the 3-vector @xmath428has the same 3-length @xmath274 , as the length of 3-vector @xmath429 . the angle between the 3-vectors @xmath430 and @xmath431is equal to @xmath350 . if @xmath432 , then @xmath325 , and vectors @xmath394 and @xmath433 are elements of the same helix . we see that the stabilizing vector @xmath395 reduces wobbling of vector @xmath394 . in the case of equation ( [ a4.46 ] ) the spatial component @xmath329 of the 4-vector @xmath318 may be infinite . in the case of the equation ( [ b6.9 ] ) the length @xmath434 of the spatial component @xmath329 of the 4-vector @xmath318 is less , than @xmath435 thus , the stabilizing vector @xmath395 reduces the wobbling of the world chain . one can not be sure , that this wobbling does not destroy the helical character of the world chain . however , the main question is , whether or not the evolution of the world chain in the spacelike direction lead to the world chain , which is timelike on the average . any next point @xmath436 of the world chain jumps along the timelike direction at the distance @xmath273 and in the 3-space , which is orthogonal to this direction , the point jumps at the distance @xmath437 . direction of the jump in the 3-space is described by the vector @xmath430 , which is given by the relation ( [ c5.17 ] ) . the length of @xmath430 is @xmath274 . if the direction of jump is completely random , the displacement @xmath438 for @xmath37 steps ( @xmath439 is proportional to @xmath440 , whereas displacement in the temporal direction is @xmath441 . it means that the mean velocity @xmath442tends to zero for @xmath443 , although @xmath444 . in the case , if @xmath363 and the 3-vector @xmath430 * * * * determined by ( [ c5.17 ] ) is not random , the world chain form a helix with timelike axis . in this case the mean velocity tends to zero also . it should expect that in the case , when the vector ( [ c5.17 ] ) is stochastic , but its stochasticity is restricted by the relation ( [ c5.17 ] ) ( the angle @xmath422 is completely random ) , the mean world chain will be also timelike on the average . we can not prove this fact strictly now , but this result seems to be very probable . the obtained classical helical world chain ( [ a4.6 ] ) associates with the classical dirac particle , which has alike world line ( [ e6.33 ] ) , ( e6.34 ) . the direction of the mean momentum distinguishes from the direction of the 4-velocity . this fact is characteristic for both particles ( the dirac particle , and the particle , described by the world chain ( a4.6 ) ) . both particles have angular moment . for the dirac particle the mass @xmath175 , which enters in the dirac equation , distinguishes from the mass @xmath445 of the particle moving along the world line ( [ e6.33 ] ) , ( [ e6.34 ] ) @xcite . as to the mass of the particle , described by the world chain ( [ a4.6 ] ) , it is not yet determined . for determination of the mass , one needs to consider the world chain ( [ a4.6 ] ) of charged particle with the skeleton @xmath446 in the distorted space - time of klein - kaluza , containing electromagnetic field . existence of helical world chain with timelike axis seems to be rather unexpected , because leading vectors @xmath259 of the chain are spacelike , and it corresponds to superluminal motion of a particle . superluminal motion seems to be incompatible with the relativity principle , which admits only motion with the speed less , than the speed of the light . however , this constraint is valid only for continuous space - time geometry , which admits unlimited divisibility . in a discrete geometry there are no distances less , than some elementary length , and it is difficult to formulate the relativity principle statement on impossibility of superluminal motion . one needs another more adequate formulation of the relativity principle . is the space - time geometry ( [ a4.14 ] ) discrete ? at @xmath447 the space - time geometry ( [ a4.14 ] ) turns to the space - time geometry@xmath448which is certainly discrete , because in the space - time there no timelike intervals @xmath57 , which are less , than @xmath195 , and there are no spacelike intervals @xmath449 , which are less , than @xmath195 . in such a space - time geometry there are no particles , whose geometrical mass @xmath199 is less than @xmath195 . however , if @xmath214 , is the space - time geometry discrete ? to answer this question , we introduce the parameter of discreteness : the relative density of points in the space - time with respect to the point density in the space - time of minkowski . let us define the quantity @xmath450 by means of the relation @xmath451 in the case ( [ a4.14 ] ) we have for @xmath452 $ ] @xmath453resolving ( [ a7.3 ] ) with respect to @xmath193 , we obtain @xmath454where@xmath455 taking into account ( [ a7.4 ] ) we obtain the world function @xmath193 as a function of @xmath233@xmath456the relative density of points in the space - time geometry @xmath276 with respect to the standard geometry @xmath275 of minkowski is given by the relation ( [ a7.2 ] ) . the expression for @xmath457 is given by the relation@xmath458where @xmath459 is given by the relation @xmath460if @xmath461 and @xmath462 , we have approximately@xmath463 in the limit @xmath461 , when the world function ( [ a4.14 ] ) turns into the world function ( [ a7.1 ] ) of the completely discrete geometry , we obtain for the relative density@xmath464 thus , @xmath465 for @xmath466 , and this fact correspond to the space - time geometry ( [ a7.1 ] ) , where close points , for which @xmath467 , are absent . the relative density @xmath450 of points may serve as quantity , describing the discreteness of the space - time geometry and the character of this discreteness . the discreteness may be complete , when the density @xmath468vanishes in some region as in the case ( [ a7.10 ] ) . but the discreteness may be incomplete , as in the case ( [ a7.7 ] ) . in this case for @xmath469 we have @xmath470{g_{1}\left ( \sigma _ { \mathrm{d}}\right ) } + \frac{1}{3}\frac{1}{\sqrt[3]{g_{1}\left ( \sigma _ { \mathrm{d}}\right ) } } \right ) , \qquad \sigma _ { \mathrm{d}}\in \left ( -2\lambda _ { 0}^{2},2\lambda _ { 0}^{2}\right ) \label{a7.11}\]]where@xmath471the expression ( [ a7.11 ] ) is a symmetric function of @xmath472 , as one can see from ( [ a7.3 ] ) . it is symmetric , indeed , although it does not look formally as a symmetric function of @xmath233 . numerical values of @xmath473 @xmath474 are presented in the table @xmath475the relative density @xmath450 is less , than unity . it may be interpreted in the sense , that the space - time geometry is discrete only partly . nevertheless the incompletely discrete space - time geometry discriminates most of world chains with spatial leading vector , remaining only some of them . multivariance of particle motion and discrimination of some states of motion play the crucial role in structure of elementary particles , as well as in the structure of atoms . let us explain this circumstance in the example of the hydrogen atom . according to laws of the classical mechanics the electron of the hydrogen atom is to fall onto the nucleus due to the coulomb attraction . two reasons prevent from this falling : ( 1 ) multivariant ( stochastic ) motion of the electron , and ( 1 ) rotation of electron around the nucleus . the multivariant motion of the electron leads to escape of the electron from the nuclear surface . this process has the same nature , as an escape of dust from the earth surface . moving multivariantly ( as brownian particles ) , the flecks of dust form a stationary distribution in the gravitational field of the earth . if multivariance of their motion is cut out , the flecks of the dust fall onto the surface of the earth . statistical description of the electron distribution and the dust distribution are different , because the multivariant electron motion is conceptually relativistic , whereas the brownian particles motion is nonrelativistic . one may describe brownian particles by means of probabilistic statistical description , whereas one may use only dynamical conception of statistical description for statistical description of multivariant motion of relativistic particles . rotation of the electron around the nucleus creates the field of centrifugal force , which is added to the coulomb force . as a result additional distributions of the electrons appear . if the obtained distribution of electrons is nonstationary , the electrons emanate the electromagnetic radiation until the electron distribution becomes to be stationary . thus , the electromagnetic radiation carries out discrimination of nonstationary states ( electron distributions ) . the multivariance of the electron motion and mechanism of discrimination of non - stationary states generates the structure of the hydrogen atom and discrete character of the radiation spectra . from the mathematical viewpoint the discrete character of the electron states is conditioned by procedure of the eigenstates determination . only eigenstates of the hamilton operator appear to be stationary and stable . the multivariance of the particle motion and some mechanism of discrimination play also the crucial role in the understanding of the structure of elementary particles . however , in the case of the elementary particle structure the discrimination mechanism is conditioned by some metric ( geometric ) forces , which appear , when we use space - time geometry of minkowski instead of the real multivariant space - time geometry . formally these forces have the form of additional terms of the type ( [ a4.18 ] ) in dynamic equations . these additional terms are expressed via the space - time distortion @xmath250 . they describe both multivariance of motion and the discrimination mechanism . the multivariance of motion is associated with the multivariance of the vector equivalence definition ( [ a1.3 ] ) , whereas the discrimination mechanism is associated with the zero - variance of the same definition ( [ a1.3 ] ) for some vectors . besides , as we have seen , the zero - variance ( discrimination ) is associated with the discreteness ( or partial discreteness ) of the space - time geometry . it is very important , that consideration of multivariant space - time geometry _ is not a hypothesis _ , which needs an experimental test . consideration of the multivariant space - time geometry is a corollary of correction of our imperfect conception of geometry . conception of geometry , based on supposition that any space - time geometry may be axiomatized ( i.e. may be concluded from some system of axioms ) , is imperfect , because it does not admit one to construct multivariant geometry conceptually . however , the motion of electrons and other elementary particles is multivariant . multivariance of this motion is an experimental fact , which can not be ignored . as far as the imperfect conception of geometry did not admit one to construct multivariant space - time geometry , investigators were forced to ascribe multivariance to dynamics , introducing quantum principles with all their attributes . the quantum principles look enigmatic and artificial , because multivariance is ascribed to dynamics , whereas it should be ascribed to the space - time geometry . multivariance and zero - variance as properties of the space - time geometry look as quite natural properties of the definition ( [ a1.3 ] ) . indeed , it does not follow from anywhere , that equations ( [ a1.3 ] ) are to have unique solution for arbitrary world function , which determines the form of these equations . absence of any hypotheses is a very important property of the geometrical approach to the structure of elementary particles . besides , the geometrical dynamics is very general and simple . dynamic equations of the geometric dynamics do not use even differential equations . formulation of dynamic equations does not contain a reference to the coordinate system . on the other hand , when the geometric dynamics in the real space - time is described in terms of the space - time of minkowski , one obtains additional metric forces , which look rather exotic . they can be obtained hardly in the framework of the conventional approach . the conventional approach to the theory of elementary particles contains a lot of secondary concepts and properties . one may not see any discrimination mechanism in wave functions , field equations , branes , symmetries and other remote corollaries of the unknown structure of elementary particles . but it is impossible to obtain and to understand the discrete properties of elementary particles without a reliable mechanism of discrimination . even if investigating and systematizing these remote corollaries , one succeeds to obtain a perfect systematization of elementary particles , one can obtain structure of elementary particles from the perfect systematization with the same success , as one can obtain the atomic structure from the periodical system of chemical elements . consideration of t - geometry as a space - time geometry admits one to obtain dynamics of a particle as corollary of its geometrical structure . evolution of the geometrical object in the space - time is determined by the skeleton @xmath476 of the geometrical object and by fixing of the leading vector @xmath4 . the skeleton and the leading vector determine the world chain , which describes the evolution completely . the world chain may wobble , it is manifestation of the space - time geometry multivariance . quantum effects are only one of manifestation of the multivariance . it is remarkable , that for determination of the world chain one does not need differential equations , which may be used only on the space - time manifold . one does not need space - time continuity ( continual geometry ) . of course , one may introduce the continual coordinate system and write dynamic differential equation there . one may , but it is not necessary . in general , the geometrical dynamics ( i.e. dynamics generated by the space - time geometry ) is a discrete dynamics , where step of evolution is determined by the length of the leading vector . it is possible , that one will need a development of special mathematical technique for the geometrical dynamics . the real space - time geometry contains the quantum constant @xmath155 as a parameter . as a result the geometric dynamics explains freely quantum effects , but not only them . the particle mass is geometrized ( the particle mass is simply a length of some vector ) . as a result the problem of mass of elementary particles is simply a geometrical problem . it is a problem of the structure of elementary geometrical object and its evolution . one needs simply to investigate different forms of skeletons of simplest geometrical objects . in general , not all skeletons are possible , because at the spatial evolution the world chain is observable ( helical ) only for several skeletons . additional points of skeleton lead to additional ( sometimes unexpected ) properties of corresponding elementary geometrical objects ( elementary particles ) . note that the geometric dynamics does not contain a rotational motion . it contains only a shift . all vectors of the skeleton @xmath477 of the link @xmath478 are equivalent to vectors of the skeleton @xmath479 of the adjacent link @xmath480 . such a situation is quite reasonable , because the geometrical dynamics describes evolution of free particles . the rotating particle can not be completely free , because in the rotating particle there is centripetal acceleration . however , acceleration of all parts of the body has to be absent for completely free motion . on the other hand , the geometric dynamics contains the spatial evolution , which absent in the conventional dynamics . from the geometrical viewpoint we may not discriminate spatial evolution on the basis , that the leading vector @xmath4 is spacelike and its length is imaginary . in fact the spatial evolution discriminates itself , by the fact , that the corresponding world chain is unobservable , in general . it appears to be observable only for some complex skeletons , consisting of more , than two points . the world chain , describing the spatial evolution is observable only in the case , when it may be localized near the world chain of the observer . it takes place , when the world chain has a shape of a helix with timelike axis , or some other shape , which may be localized near the world chain of the observer . as a result not all skeletons appear to be observable . although the geometric dynamics does not contain a rotation , but the corollaries of the rotation ( angular momentum , magnetic momentum ) may be obtained as a result of the spatial evolution , when the world chain is a helix . apparently , the fact , that such a `` particle rotation '' is a corollary of the spatial evolution , leads to the spin discreteness of the dirac particle . of course , such statements are to be tested by exact mathematical investigations of different types of skeletons and of different space - time geometries . however , such a statement of the problem is quite concrete and realizable . note , that the geometric dynamics in the real ( non - minkowskian ) space - time contains additional terms with respect to dynamics in the space - time of minkowski . from viewpoint of the space - time of minkowski these additional terms may be interpreted as some ( metric ) interactions , which take place inside the elementary particles . from the conventional viewpoint these interactions look very exotic and strange . it is impossible ( or very difficult ) to guess at them , starting from conventional conception of the space - time and dynamics . in the geometric dynamics there are no additional interactions , if we use the true space - time geometry . however , additional interactions appear , if we use inadequate geometry ( for instance , geometry of minkowski , or riemannian geometry ) . in other words , it is possible to compensate false space - time geometry by introduction of additional interactions . it is well known from the general relativity , that the motion of free body in the curved space - time looks as a motion in the gravitational field , if one interprets this motion as a motion in the space - time of minkowski . description of conceptually new unknown phenomena by means of a change of the space - time geometry is simpler , than an introduction of additional interactions , because the space - time geometry is described by the world function , which is a function of two points . the form of the world function for large distances is determined by the necessity of obtaining the nonrelativistic quantum mechanics . restrictions , imposed on the world function at small distances , are determined by the condition , that the spatial evolution may describe the dirac particle . ( very many elementary particles are the dirac particles ) . besides , the condition of localization of the world chain ( helical world chain ) imposes restrictions on parameters of the particle . not all parameters of particles appear to be possible . this condition is a condition of `` peculiar quantization '' of the particle parameters , which include the particle mass . let us note that the contemporary theory of elementary particles returns to geometrical considerations ( strings , branes ) . however , these considerations are restricted by the framework of the riemannian geometries and geometries close to the riemannian geometry . for instance , the quantum geometry , which uses operators instead of the point coordinates . this is some way of introduction of multivariance in the geometry . however , this geometry is developed on the basis of the linear vector space , which is a restriction on the space - time geometry . in any case the conventional approach to the space - time geometry considers only a part of all possible space - time geometries . one can not be sure , that the class of considered geometries contains true space - time geometry . of course , if one uses a false space - time geometry , there is a possibility to correct the false space - time geometry by means of additional interaction , generated by difference with the true space - time geometry . but such a correction is difficult , especially if the true geometry is discrete or close to the discrete geometry . note , that the geometry ( [ a4.0 ] ) is discrete , although it is given on the continuous manifold of minkowski . it is discrete , because the module of distance between any two points is more , than @xmath195 . it is very unexpected , because it is a common practice to consider any geometry on the manifold as a continuous geometry , although in reality the geometry is determined by the world function and only by the world function . a discrete geometry is associated with a grid . of course , a geometry , given on a grid , can not be continuous . however , a geometry , given on the continuous set of points ( manifold ) , may be discrete . why the microcosm physics of the twentieth century did leave the successful program of the physics geometrization and choose the alternative program of quantum theory ? discovery of the electron diffraction need of multivariance of the microcosm physics . multivariance may be taken into account either on the level of the space - time geometry , or on the level of dynamics . the multivariant space - time geometry was not known in the thirtieth , when the electron diffraction was discovered . the nonrelativistic quantum mechanics had been constructed already , and it was applied successfully for explanation of the electron diffraction . the space - time geometry is a basis of dynamics . introducing multivariance in dynamics , one can describe not only nonrelativistic phenomena of microcosm . one can describe also relativistic phenomena and that part of the microcosm physics , which is known as the theory of elementary particles . the principles of quantum mechanics , which introduce multivariance in the microcosm physics , were invented for the newtonian conception of the space - time , and their extrapolation to the relativistic phenomena appeared to be problematic . of course , some properties of the true space - time geometry may be taken into account by introduction of additional interactions . however , it is very difficult to invent and introduce additional interactions without understanding of these innovations . capacities of the geometrical approach are very large , especially if one takes into account all possible space - time geometries . the theory of elementary particles returns to the geometrical description , but this description is burthened by such concepts as wave function , string , brane , which have very abstracted relation to the structure of elementary particles and microcosm physics . a. rylov , tubular geometry construction as a reason for new revision of the space - time conception . in _ what is geometry ? _ polimetrica publisher , italy , pp.201 - 235 http://www.polimetrica.com/polimetrica/406/ yu . a. rylov , author s comments to referee s reports on the paper by y. a. rylov `` dynamical methods of investigation in application to the dirac particle '' , submitted to a scientific journal . ( available at _ http://rsfq1.physics.sunysb . edu / rylov /comm1e . pdf _ )

the rationale for , capabilities of , and scientific context of the mid - infrared instrument ( miri ) on jwst are described in rieke et al . ( 2014 ; hereafter paper i ) and the overall optical , thermal , mechanical / structural electronic and control aspects of the design are summarized in wright et al . ( 2014 ; hereafter paper ii ) . this paper describes in more detail the miri medium resolution spectrometer ( mrs ) , which is an integral field unit ( ifu ) spectrometer that covers the wavelength range from 5 to 28.5 @xmath0 m at spectral resolving powers of a few thousand . the mrs consists of 4 channels that have co - aligned fields of view , simultaneously observing the wavelength ranges listed in table 1 with individually optimised ifus , collimators and gratings . section [ sec : optical ] of this paper provides a description of the optical design , including the rationale for choosing the ifu concept and its impact on how observations are carried out . section [ sec : measured ] then describes the expected on - orbit optical performance of the mrs as measured during cryogenic testing , and with a description of the procedure used to construct calibrated spectral data cubes from the raw measured images . the impact of particular characteristics of the mrs , including spectral fringing and straylight are also discussed here . an ifu based design was preferred to a long - slit design for the miri spectrometer for the following reasons . firstly , for point source observations , the need to centre the source in a narrow slit ( via a peak - up procedure ) is relaxed , simplifying and accelerating the target acquisition procedure . there is an additional benefit that there is no loss of light at the slices ( ` slit losses ' in a conventional long slit spectrometer ) . the mrs slice width is set to be less than or equal to the fwhm of the diffraction limited point spread function at the slicing mirror . an equivalent long - slit spectrometer would vignette the light outside this region , losing about 50% of the total . the amount of lost light could be reduced by making the slit wider , but doing so would in turn reduce the spatial and spectral resolution and could also decrease the signal to noise because of increased background radiation . second , from a scientific perspective , the wavelength range covered by the miri spectrometer is sufficiently broad ( a factor of nearly six ) that different emission mechanisms may dominate in different regions of the spectrum . in cases where these mechanisms do not share a common centre ( e.g. , stellar output compared with infrared re - emission in a starburst galaxy ) , a simple slit spectrograph poses a dilemma in placing the `` source '' on the slit . an ifu implements 3d spectroscopy , which solves this problem by giving accurately registered spatially resolved spectroscopy over the entire field . these considerations , combined with the wavelength coverage and resolving power requirements defined by the jwst mission science goals , and the mass , volume and electrical power limitations set by the jwst spacecraft environment , have resulted in a system with the characteristics summarised below and illustrated in block diagram form in figure 1 . as shown in the figure , the full 5 to 28.5 @xmath0 m wavelength range is divided within the spectrometer pre - optics ( spo ) into four simultaneous spectral channels @xcite , each with its own ifu @xcite and using a simple scheme of pass - band separation by dichroic filters whose design is described in hawkins et al . ( 2007 ) and wells et al . we denote the short and long wavelength limits of each channel as @xmath1 and @xmath2 . each channel serves an optimised spectrometer . this separation provides several benefits : ( 1 ) it enables the use of diffraction gratings in first order , allowing each to be used near peak efficiency around the blaze wavelength , and ( 2 ) it also allows the ifu slice widths to be tailored to the wavelength - scaled fwhm in each channel . electrical and thermal constraints limit the mrs to two 1024 x 1024 si : as detectors ( ressler et al . , 2014 , hereafter paper viii ) , resulting in the spectrometer being split into two sets of optics , each with its own detector array , one for the two short - wave channels and one for the two long - wave channels . in each case , the spectra of two wavelength channels are imaged simultaneously onto the left and right halves of a detector array . the product of spatial and spectral coverage that can be achieved in a single instantaneous exposure is ultimately set by the number of detector pixels . for the mrs , this results in a single exposure providing a spectral sub - band that covers only one third of each channel . full wavelength coverage then requires three exposures , with a pair of mechanisms being used to select the gratings and dichroics in each case . we refer to the short and long wavelength limits of each sub - band as @xmath3 and @xmath4 . this design choice allows the grating performance to be optimised over a narrow wavelength range ( @xmath4/@xmath5 @xmath61.2 ) . the gratings only have to work over 20% of the 1@xmath7 order near the blaze wavelength and the dichroics do not have to be used near the cross over between reflection and transmission . we will now step through the optical train as shown in figure 1 : the input optics and calibration ( ioc ) modules pick off the mrs fov from the jwst focal surface and pass it on to the spo . the spo spectrally splits the light into the 4 spectrometer channels and spatially reformats the rectangular fields of view into slits at the entrance of the spectrometer main optics ( smo ) . the smo comprises fixed optics that collimate the light for presentation to diffraction gratings mounted in the spo and then re - images the dispersed spectra onto the two focal plane arrays . key features of this optical system are described individually in the following subsections . the spectral and spatial coverage of the mrs is summarised in table 1 . the field of view of the mrs is adjacent to the miri imager field and picked off from the jwst focal surface using the miri pickoff mirror ( pom ) , which is common to both ( see paper ii ) . the sky and the jwst pupil are reimaged by the ioc so that there is a pupil image at the spo input and a sky image further on . a cold stop that is 5% oversized with respect to the jwst pupil is placed at the spo input pupil for straylight control . the light is directed towards the first dichroic filter via a fold mirror placed 10 mm beyond the pupil . the focal plane is formed 535 mm beyond the pupil , providing the long path length and narrow beam waist needed between the pupil and the inputs of the four integral field units ( ifus ) , for mounting a chain of dichroics to divide the light among the four spectral channels . a hole in the fold mirror , sized to be smaller than the footprint of the telescope central obscuration , acts as an aperture for the injection of light from the on - board spectrometer calibration unit ( scu ) , shown in figure 2 . the scu provides spatially and spectrally uniform blackbody illumination for flux calibration and pixel flat fielding functions , using as its light source a tungsten filament heated to a temperature of @xmath61000 k by the application of an 8 ma drive current . the filament is mounted inside a non - imaging flux concentrator that generates spatially uniform focal plane illumination at the exit port of a 25 mm diameter reflective hemisphere , ( described in glasse et al . , 2006 ) . two cadmium telluride lenses within the scu then re - image the exit port onto the ifu input focal planes . the lenses provide a pupil image that coincides with the hole in the input fold mirror . this hole is positioned to lie within the footprint of the central obscuration of the jwst primary mirror and so has no impact on the science beam . in this way , the scu can provide flood illumination of the full mrs field of view without any need for mechanisms or additional optical elements . the overall layout of the dichroic assembly is shown in figure 3 , with the input fold mirror and scu situated at the left - hand end ( but not shown ) . the three dichroics needed to divide the spectral band among the four spectrometer channels for one of the three sub - bands are indicated as d1 , d2 , and d3 in figures 1 and 3 . taking sub - band a as an example , the required reflective band for dichroic d1 is the wavelength range for sub - band a in channel 1 , while its transmission band needs to extend from the short end of sub - band a in channel 2 to the long end of sub - band a in channel 4 . the bands are listed in table 2 for all nine dichroics . in all cases , the mean reflectivity is above 0.95 and the mean transmission is above 0.74 . all of the gratings work in first diffractive order so additional blocking is needed to reject second and higher orders . because the dichroics work in series it is possible to use the combined blocking of dichroics 1 and 2 to remove the need for blocking filters in channels 3 and 4 . for channels 1 and 2 , dedicated blocking is provided by the fixed filters shown as bf1 and bf2 in figure 3 , with light traps lt1 and lt2 absorbing unwanted reflections . the path length required to reach the input of channel 4 is greater than the 535 mm discussed above , so the light transmitted by the final dichroic is re - imaged via an intermediate focal plane to the entrance of the channel 4 ifu . the 21 mm diameter dichroic filters are mounted on two wheels . first is the nine - sided wheel a containing the three channel 1 dichroic filters and three flat mirrors to direct the light towards channel 1 . the second , six - sided wheel , contains the six dichroics needed to divert the light into channels 2 and 3 . each dichroic filter is mounted onto a diamond machined facet on the wheel that provides the required alignment accuracy and reduces the magnitude of any print - through of surface form errors from the wheel to the filter substrate . the filters are held in place with a spring loaded bezel with a clear aperture of 17 mm compared with the coated area of 17.4 mm . the bezel prevents light from reaching the uncoated area of the filter , which might result in un - filtered light entering the optical path . the blocking filters are mounted directly to the spo chassis with a stand and bezel , similar to the mounting arrangement for the wheels . the mechanisms that carry the dichroic wheels also carry the corresponding wheels with diffraction gratings for each sub - band ( three per channel ) , as discussed in section 2.4 . by arranging for the gratings to be mounted on the same mechanisms as their band - selecting dichroics , the number of moving mechanisms within the mrs is kept to two . before the light in the four channels reaches the gratings it is sliced and reformatted by the integral field units and so we describe these next . the parameters pertaining to the spatial and spectral coverage and sampling of the spectrometer , which are largely determined by the design of the four ifus , are given in table 1 . spatially , the image is sampled in the dispersion direction by the ifu slicing mirrors and in the slice direction by the detector pixels . spectrally , the width of the slices defines the spectrometer entrance slit and the width of the image ( in pixels ) of the slice at the detector defines the width of the spectral sample . inspection of table 1 then shows that channels 1 to 3 are all slightly undersampled spectrally , with slice widths less than 2 pixels at the detector . the bandwidths of the spectral channels listed in table 1 were chosen such that the ratio @xmath8/@xmath1 in all four channels was the same : @xmath2/@xmath9 ( 28.3 / 5)@xmath10 1.54 . this means that , if each channel is assigned the same number of spectral pixels , the resolving power in each will be approximately the same . each sub - band then has a width that is slightly larger than one third of the band - width of its parent channel , where the excess provides sufficient overlap of adjacent spectra to allow their concatenation . in practice , the overlap is typically 10 to 15% of the spectral range , depending on the specific sub - bands and position in the field of view . for full nyquist sampling of the telescope psf the four ifu slice widths should ideally be matched to the half width at half maximum intensity of the psf at the shortest wavelength in each channel . for the jwst telescope pupil this would equate to a@xmath11 0.088 ( @xmath12 / 5 @xmath0 m ) arcsec . however , the large increase in the beam size produced by diffraction at the ifu slicer mirror would then necessitate large apertures for the spectrometer optics to avoid vignetting . to control the size , mass and optical aberrations of the spectrometer , the starting point for determining the slice widths was therefore to set them equal to 2a@xmath13 , with fully sampled psfs to be achieved using two or more pointings of the telescope . optimum spatial sampling , which minimises the number of telescope pointings needed to fully sample all four channels simultaneously in the across - slice direction , is achieved by having a single pointing offset equal to n@xmath14 + 1/2 slice widths for all channels where n@xmath14 is a different integer for each channel . this led us to adopt a set of slice widths that follow a scheme where the slices are factors of 1 , 11/7 , 11/5 and 11/3 wider than the narrowest 0.176 arcsec slice . these values closely match the increasing fwhm of the psf in the 4 channels and allow full nyquist sampling to be achieved using a single ` diagonal ' offset whose magnitude in the across slice direction is equal to 11/2 times the width of the narrowest slice ( 0.97 arcseconds ) and ( n + 1/2 ) detector pixels in the along slice direction ( where n is an integer ) . with these slice widths fixed the design was interated to , as far as possible , meet the following criteria : 1 . ) each channel would occupy half a 1024 x 1024 pixel detector with borders to allow for tolerances in alignment and gaps between slices to avoid crosstalk ; 2 . ) the fov of each channel should be as large as possible and approximately square ; and 3 . ) the long and short wavelength arms of the spectrometer should be identical ( in practice one is the mirror image of the other ) . this iteration resulted in the parameters in table 1 . the ifu design was developed from the ifu deployed in uist , a 1 - 5 @xmath0 m spectrometer for ukirt @xcite . other examples of all - reflecting image slicers are the spectrometer for infrared faintfield imaging ( spiffi ) @xcite , and the slicer in the gemini near - infrared spectrograph ( gnirs ) @xcite . the miri ( and uist ) ifu design allows excellent control over stray light by providing through - apertures for baffling . it consists of an entrance pupil , an input fold mirror , an image slicer mirror , a mask carrying exit pupils for the individual sliced images , a mask carrying slitlets for the individual images , and an array of re - imaging mirrors behind the slitlets . these components ( except for the last ) are shown in figure 4 . the design is all - reflecting and constructed entirely of aluminium , and hence is well - suited for operation in the infrared and at cryogenic temperatures , as discussed in paper ii . @xcite describe the manufacture of the slicer . the optical path through the ifus begins with the four toroidal mirrors , which comprise the anamorphic pre - optics ( apo ) module , ( not shown in figure 4 ) . the apo re - images an area of up to 8 by 8 arcseconds of the input focal plane onto the image slicer mirror , with anamorphic magnification . the image slicer ( at the relayed image plane ) consists of a stack of thin mirrors angled to divide the image along the dispersion direction of the spectrometer into separate beams . in the across slice ( dispersion ) direction one slice width is matched to the fwhm of the airy pattern at the shortest operating wavelength for the ifu . in the along slice direction the magnification is chosen to provide the required plate scale at the detector . these two magnifications are not the same and so the apo exit pupil is elliptical , as illustrated in figure 5 for channel 3 , where the footprint of the jwst pupil is shown in blue . the light exits the ifu through individual pupil masks for each beam , then through individual slitlets ( see figure 4 ) . re - imaging mirrors behind the slitlets relay the beam to the input of the appropriate spectrometer . that is , each ifu takes the rectangular fov and transforms it into a pattern of slitlets ( as shown in figure 4 ) that are re - imaged onto the entrance aperture for its corresponding grating spectrometer . the slicing mirror comprises a number of spherical optical surfaces , diamond turned onto a common substrate . this manufacturing approach was used previously for the ifu in gnirs @xcite . figure 6 is a photograph of the image slicer for channel 1 , which has 21 slicer mirrors arranged about the centre of the component with 11 slices on one side and 10 on the other . each mirror is offset to direct its output beam towards the corresponding re - imaging mirror . the design for channels 2 to 4 is similar but with correspondingly fewer and larger slices as the wavelength increases , as listed in table 1 . the mirrors are arranged in a staircase - like manner to aid manufacturing . each of the slices re - images the ifu entrance pupil at a scale of @xmath151:1 and because the centres of curvature of the mirrors are offset laterally with respect to each other , the pupils that the slices produce are separated , as can be seen in figure 4 . the slicing mirrors truncate the anamorphically magnified image of the sky along the dispersion direction , as shown in figure 7 ( a ) to ( c ) . this truncation results in the pupil image being diffraction broadened in the dispersion direction as shown in figure 7 ( d ) . to reduce vignetting , the baffle at this pupil image and subsequent optical components ( including the diffraction gratings ) are oversized . this is illustrated in figure 7(d ) where the geometric footprints for 3 field positions are overlaid along a slice to appear as the blurred green patch in the image . the notional , geometric pupil is shown in red and the oversized mask aperture is shown as a green rectangle . the re - imaging mirror array forms images of the slicing mirrors as input slits of the spectrometer and also images the pupils at infinity , making the output telecentric . finally , a roof mirror redirects light from the re - imaging mirrors so that the output consists of the two lines of staggered slitlets relayed from those seen in figure 4 . a slotted mask at the position of the ifu output slits defines the field of view along the individual slices , while in the spectral direction the slots are oversized to ensure they do not vignette the image , but still act as baffles to reduce scattered light between the slices . the numbers of detector columns covered by the four ifus are approximately 476 , 464 , 485 , 434 for channels 1 to 4 respectively . the size of the gap between individual slits for channel 1 is set to be approximately equal to the diameter of the first dark diffraction ring in the telescope psf , ( about 4 pixels ) . for channels 2 4 the size of the gap is slightly greater than the diameter of the first dark ring . the smo ( whose layout is shown in figure 8) , comprises four grating spectrometers in two arms . the development and test of these spectrometers are described by @xcite . each of them performs three functions : collimation of the telecentric output beams of one of the four ifus , dispersion of the collimated beam , and imaging of the resulting spectrum onto one half of one of the two focal plane arrays . one of the two spectrometer arms includes the two short wavelength channels ( 1 and 2 ) , and the other the long wavelength channels ( 3 and 4 ) . we will now step through the optical path from the ifus to the detectors . the ifu output beam for each channel is collimated and its light directed towards the corresponding diffraction gratings as shown in figure 9 . each spectrometer arm uses 6 gratings ( two wavelength channels , three sub - band exposures ) . figure 8 then shows the optical paths from the gratings to the detectors . the dispersed beams are imaged by three - mirror - anastigmat ( tma ) camera systems ( m1-m2-m3 ) . in each spectrometer arm the channels have separate , but identical , m1 camera mirrors that provide intermediate images of the spectra between m1 and m2 . folding flats at these intermediate focus positions reflect the channel 1 and channel 4 beams such that the combined ( ch . 1 + 2 ) and ( ch . 3 + 4 ) beam pairs are imaged onto opposite halves of the detectors by common m2 and m3 mirrors . the symmetry through the centre plane of the camera optics and the positioning of the optics allow for an opto - mechanical design of two mirror - imaged boxes , with identical optics and almost identical structures . the combination of symmetry and an all aluminium design ( optics as well as structures ) allowed the very strict alignment and stability requirements to be met fully with an adjustment - free mounting ( i.e. positional accuracy by manufacturing accuracy only except for the focus shim at the detector interface ) , a major advantage regarding the total design , manufacture and test effort . the mirror substrates are all light - weighted aluminium with diamond turned optical surfaces that are gold coated for maximum reflectivity at the miri operating wavelengths . they are mounted through holes in the housing walls using stress - relieving lugs under light - tight covers to prevent any high temperature radiation from the instrument enclosure ( at @xmath640 k ) reaching the detectors . the two sets of gratings for the a , b and c sub - bands of the two channels are mounted on a single wheel , as shown in figure 10 , that is mounted on a mechanism that also carries the dichroics wheel on a lower level . the mechanisms are located in the spo and each mechanism is rotated by a single actuator to give the correct combination of dichroics and gratings across all channels for each exposure . the gratings are master rulings on aluminium substrates and are gold coated . table 3 lists the design parameters of the gratings . all gratings operate in first order . the angular values quoted in figure 1 are calculated for the ( virtual ) nominal input beam , originating from the centre of the ifu output area . the spo is designed to be a well baffled optical system . to intercept stray light from the telescope the first optical element in the mrs is a cold stop placed at the entrance pupil . a number of features within the mrs help to control stray light : 1 . ) stops at each pupil and sky image in the optical train ; 2 . ) light traps where significant stray light occurs ( 0@xmath16 order from the gratings ) ; 3 . ) labyrinths at the edges of apertures between modules and tight control of all external apertures , e.g. electrical feed throughs ; 4 . ) black coating of all surfaces not in the main optical path , which is an inorganic black anodising applied to a roughened surface of aluminium . the process was carried out by protection des metaux , paris ; and 5 . ) avoidance of surfaces at grazing incidence near the main optical path . in addition , for each slice of each ifu there are output pupil and slit masks . all the diamond turned aluminium mirrors in miri were specified to have a surface roughness of < 10 nm rms . this results in a total integrated scatter per surface of < 0.06% ; non - sequential optical modelling indicated that this level of scatter should not significantly degrade the psf of the mrs . all mirrors have an overcoated gold surface with reflectivity > 98.5% in the miri wavelength range . straylight analysis shows that the extensive baffling combined with the low scattering optical surfaces and blackened structure should reduce unwanted light ( cross talk between channels , degraded psf and out - of - beam light at the output image ) to levels that are estimated to be less than 0.1% of the background radiation included in the science beam . due to the nature of an ifu spectrograph with its slicing and dispersing optics , the resulting detector images are not straightforward to analyze : spatial , spectral , and with it photometric information , are spread over the entire detector array . further , the gratings are used in an optically fast , non - littrow conguration . the resulting anamorphic magnification varies from the short to long wavelength limit for each sub - band which , when combined with the curved spectral images of the dispersed slices on the detector as shown in figure 11 , makes the optical distortion significant and complex . the anamorphic and slicing optics add other components of distortion that vary the plate scale in the along - slice direction for each slice individually . this leads to a very complicated variation of the spatial plate scales and finally to a dependency of the spatial and spectral axes on each other and on the location on the detector array . to enable scientific studies based on mrs data , the flux measured in each detector pixel needs to be associated with a wavelength and location on the sky . since the ifu provides two spatial dimensions , each detector pixel corresponds to a position within a three - dimensional cube with two spatial and one spectral dimension . due to the optical distortion , the edges of these cubes are neither orthogonal nor constant in length . consequently , we have developed a process ( called image or cube reconstruction ) , which allows a transformation of the detector pixels onto orthogonal cubes . cube reconstruction requires a thorough understanding of the optical distortion . it is not possible to just characterize the curved images of the dispersed slices ( as shown in the left plot of figure 11 ) by approximating the curvature with polynomial functions . due to the distortion caused by the slicing optics , the spatial plate scale varies within a slice non - uniformly . the characterization of the plate scale and the association of each detector pixel with its projected location on the sky at any given wavelength is a task for the astrometric and wavelength calibration , described in the next sections . once the sky - coordinates and wavelength are known for each detector pixel , the cube can be reconstructed as discussed by glauser et . 2010 ) . to describe the optical distortion on the image plane of the detectors we began from the optical model , using ray - tracing techniques to project each detector pixel through the slicing optics backwards onto the sky . when the as - built geometry was taken into account , we achieved a high spatial accuracy , as demonstrated for the miri verification model ( glauser , et al . , 2010 ) . to verify this approach , a dedicated astrometric calibration campaign was conducted during the instrument test campaign of the miri flight model . as outlined in glauser et al . ( 2010 ) , the intra - slice spatial distortion can be approximated with a 2@xmath17 order polynomial to accuracies of a few milli - arcseconds - much better than required for the astrometric accuracy given the minimum plate scale of 0.196 arcseconds in the along - slice direction for channel 1 . we conducted the astrometric calibration by placing a broad - band point source at three field positions for each slice and each channel and recording the dispersed signature on the detector . the central pixel of the spatial profile was determined and correlated with the known position of the point source ( our reference on the sky ) . with this method we were able to determine the spatial plate scale at any location on the detector . due to limitations of the test setup , i.e. , strong field distortion of the steerable point - source , the achievable relative astrometric accuracy was very limited . however , within an estimated upper limit of @xmath60.5 pixels ( 0.098 arcsecond for channels 1 and 2 , 0.123 arcsecond for channel 3 , and 0.137 arcsecond for channel 4 ) , the approach of using the optical model for the reconstruction was validated . a further result from this calibration campaign was the measurement of the fov for each sub - band . figure 12 shows the mrs fov in the jwst coordinate frame and its position relative to the imager field . the fov common to all mrs channels is 3.64 arcsec in the along - slice ( @xmath18 and 3.44 arcsec in the across - slice direction ( @xmath19 . to determine the absolute wavelengths for the mrs channels we conducted a series of calibration measurements during the instrument test campaign . fabry - prot etalon filters were used to create a dense pattern of unresolved spectral lines on the detector . many tens of lines were typically visible at a signal to noise ratio of more than 50 in a single sub - band exposure . an example etalon measurement is shown in figure 13 . to provide a reference wavelength to distinguish between adjacent etalon lines , the telescope simulator used in the test was also equipped with a pair of ` edge ' filters , one of which cut - on at around @xmath20 6.6 @xmath0 m ( in sub - band 1c ) and the other which cut off at @xmath21 21.5 @xmath0 m ( in sub - band 4b ) . the absolute wavelengths of the etalon lines and edge filters had previously been measured at ambient and 77 k temperatures using a laboratory standard fourier transform spectrometer with a spectral resolving power of r @xmath22 100,000 . the extrapolation of the wavelength scale to the 34 k operating temperature for the test campaign was based on published measurements ( browder and ballard , 1969 , browder and ballard , 1972 and smith and white , 1975 ) . this extrapolation resulted in a typical wavelength correction of less than 1% of the width of the mrs spectral resolution element . the repeatability of the scale after multiple mechanism reconfigurations has been measured to be 0.02 resolution elements . the wavelength calibration process then involved the assignment of an absolute wavelength to an etalon line by fitting the calibrated spectra of the edge filters to their mrs measured spectra in sub - band 1c and 4b . the known separation of the etalon lines was used to extend the wavelength scale across the full spectral image in each of these sub - bands . to extend the scale to other sub - bands , pairs of measured etalon spectra were co - added and the positions of unique identifying features ( due to spectral beating between the two patterns ) were used . a set ( 2 per sub - band ) of second order polynomial fits to the positions of the etalon lines was used to generate a wavelength value for the corners of all illuminated pixels . the wavelength calibration derived in this way was encoded for use in the mrs calibration pipeline by forming six images ( one for each mrs detector and all three grating wheel settings ) where the image signal values were set equal to the wavelengths at the corner of each detector pixel . we note that as a result , these wavelength reference images have one more row and column than the detector images . the relative accuracy of this wavelength scale ( within and between sub - bands ) is estimated to be better than 0.02 spectral resolution elements which , for example , corresponds to 0.03 nm at @xmath21 5 @xmath0 m . the absolute accuracy of the wavelength scale is estimated to be comparable to this figure but this will need to be confirmed during on - orbit commissioning using spectral standards . the wavelength calibrated etalon spectra were also used to measure the spectral resolving power of the mrs . the results are shown in figure 14 , where the sub - band averaged values are shown in red and the spread of values seen across the field of view is indicated by the black band . we note that figure 14 does not take account of the intrinsic spectral width of the etalon features , which was determined using the r @xmath22 100,000 calibrated spectra , described above . initial efforts to deconvolve the intrinsic line profiles from the mrs measured spectra suggest that the resolving powers quoted in figure 14 may be underestimated by around 10 % . we have therefore used the band averaged measurements scaled by a factor 1.1 to generate the summary values quoted in table 1 . as outlined in section [ sec : optical ] , the psf of the mrs is under - sampled by design , with full sampling in both spatial and spectral dimensions requiring that the object be observed in at least two dither positions that include an offset in the across - slice direction of 11/2 times the channel 1 slice width ( which corresponds to 7/2 slices in channel 2 , 5/2 slices in channel 3 , and 3/2 slices in channel 4 ) . due to the curved shape of the distorted spectrum on the detector and the variable plate scale along the individual slices , the exact dither offset in the along - slice direction is less well determined ( but also less critical ) . figure 15 shows the nominal mrs dither pattern to be used in a single observation to sample point sources fully , as derived during test campaigns , and as proposed for in - flight operations . to achieve a fully sampled psf , these dithered observations must be combined . we anticipate that this could most readily be achieved using the reconstructed cubes . to avoid loss of spatial resolving power caused by any shift- and co - adding algorithm due to the re - binning of the data ( for example , fruchter & hook , 2002 ) , we minimize the necessary re - binning steps by incorporating dither offsets parallel to the slice into the reconstruction algorithm itself . this has the advantage that only one re - binning step is required from detector data to reconstructed cubes , while dither offsets in the across slice direction can be corrected and combined afterwards using an interlacing method . we expect more sophisticated techniques than we have developed so far to be incorporated into the data reduction pipeline before launch . we attempted to measure the mrs psf during flight model testing at ral using the miri telescope simulator ( mts ) , described in paper ii . however , optical aberrations and vignetting in the mts led to the generated point source being elongated and extended at short wavelengths , such that it did not provide a sufficiently point - like image . even at longer wavelengths , where aberrations became less apparent as diffraction started to dominate , the measured psf was broader than nominal due to vignetting in the telescope simulator . we therefore repeated the psf measurement on the flight model during the first cryo - vacuum test campaign at nasa - goddard ( cv1rr ) , where a compact and well defined point source was available . deep exposures at 17 different locations in the mrs field were combined and reconstructed to form the psf image for channel 1a , shown in figure 16 . for these data , the spectral coverage was limited to a narrow ( 0.125 @xmath0 m wide ) wavelength range around 5.6 @xmath0 m . a comparison with the model psf ( pure diffraction limited fourier - transformed jwst pupil ) shows a very good match across the slices . in the along - slice direction , a broadening of approximately 50% is observed . currently , possible causes considered for the broadening include scattering in the detector substrate ( the detector halo effect , which is also observed in the imager and is discussed in rieke et al . , ( 2014 , paper vii ) or a side - effect of the straylight discussed in section [ subsec : mylabel6 ] . in the case of the halo effect , the larger degree of broadening seen in the mrs may be due to the larger spatial sample per pixel . more detailed modelling is required to confirm the root cause . spectral fringes are a common characteristic of infrared spectrometers . they originate from interference at plane - parallel surfaces in the light path of the instrument . these surfaces act as fabry - prot etalons , each of which can absorb light from the source signal with a unique fringe pattern . in the infrared wavelength range , surfaces separated by a fraction of a mm up to a few cm may form very efficient etalons . the most obvious source of fringes in the mrs is the detector itself with a physical thickness of 500 @xmath0 m ( paper vii ) . similar fringes have also been observed in spectra measured with the miri verification model during testing and with the spitzer - irs instrument , which employs comparable ( though smaller ) si : as bib detectors , lahuis & boogert ( 2003 ) . for the initial analysis we follow the formalism as defined in kester et al . , ( 2003 ) and lahuis & boogert , ( 2003 ) . the sine approximation , @xmath23 _ _ cos(2__@xmath24 _ _ sin(2__@xmath25 , is used with @xmath26 being the wavenumber and @xmath27 the optical thickness of the instrument component . of primary interest to help in the identification and first characterization of the fringes is the optical thickness @xmath27 . in the mrs test data three distinct fringe components are seen , with key parameters listed in table 4 . of the three fringe components two are directly matched to optical components in the instrument . the main fringe component ( # 1 in table 4 ) has a derived optical thickness of approximately 3.5 mm for all sub - bands . this corresponds to the optical thickness of the detector substrate which has d @xmath6 500 @xmath28 m and n @xmath6 3.42 , giving d @xmath6 0.34 cm . figure 17 gives an illustration of this fringe pattern , showing the main detector and dichroic fringes in more detail for channel 4 . the optical thickness derived for the second , high frequency , component ( d @xmath6 2.7 cm ) is matched to that of the cdte dichroic filters ( section 2.2 ) . for the third set of low frequency components ( d = 0.01 to 0.1 cm ) , no unique surfaces in the instrument are identified ; instead these are likely to originate from beating between primary fringe components . the fringe variations come from the layered structure of the detector substrate and optical thickness differences between individual dichroic filters . figure 18 shows the peak normalized fringes over the entire wavelength range of the mrs . the two curves in figure 18 show predictions for the expected fringe amplitudes based on representative anti - reflection ( ar ) coating profiles as applied to the miri flight detectors . the solid curve assumes a pure two - sided etalon while the dashed - dotted curve simulates a back - illuminated surface with a fully reflective front surface and photon absorption in the active layer ( adopted from woods et al . , 2011 ) . though not a perfect match , this approximate detector model does reproduce the general trend and magnitude of the fringes . fringe removal will be achieved using the techniques developed for and applied to iso and spitzer data ( see lahuis & van dishoeck , 2000 ; kester et al . , 2003 ; lahuis & boogert , 2003 ) . this involves dividing the observed spectra by a fringe flat - field followed by the removal of fringe residuals using the sine fitting method . this approach has proven to be reliable and robust for most spectra and the experience with iso and spitzer has shown that it allows the removal of fringe residuals down to the noise level . the main limitations with this technique are the definition of the spectral continuum in the presence of spectral features and isolating the fringe spectrum from broad molecular ( vibration-)rotation bands ( e.g. those of c@xmath29h@xmath29 , hcn , co@xmath29 and h@xmath29o ) . for the miri ifu other effects play a role and may limit the fringe removal for point and compact source measurements . the two major effects are ; i ) the illumination ( and its effective angle ) on the detector depends on the source morphology ( position and extent ) and ii ) the wavelength depends on the spatial location of the point source in the ifu field . this results in i ) changes in the effective optical thickness from source to source and ii ) a wavelength shift with spatial offset . both modify the detailed fringe pattern for individual cases . figure 19 illustrates this with point source measurements from the cv1rr test campaign and using extended source blackbody spectra from flight model testing at ral . small sub - pixel pointing offsets are seen to have a discernable impact on the fringe pattern . these effects can be mitigated by traditional fringe removal techniques using optimized and iterative reduction algorithms ( e.g. by modifying , shifting and stretching the reference fringe spectrum before applying it ) . this has been used in individual iso and spitzer cases and the miri team will work on developing optimized methods for the mrs . for miri an alternative model - based approach is also under study which uses the observed source morphology to define the fringe spectrum . this method will be applicable to both the mrs and the lrs , and will complement the traditional fringe removal techniques . this requires both a well - defined and well - calibrated fringe model ( an ongoing miri team activity ) and a flexible and iterative reduction pipeline ( in development at stsci based on input from the miri team ) . we determined the pixel - to - pixel variation of the response for both short and long wavelength detectors , using measurements taken during testing at ral . the illumination was provided by an external , extended source in the mts . a number of exposures measured over the period of the whole test run and covering the full mrs wavelength range were included in this calculation . we first de - fringed the data using the prescription described in section [ subsec : spectral ] , with the results illustrated in figure 17 . we then calculated the mean value of each pixel , @xmath30 , by averaging over 5 x 5 pixel boxes centered on pixel ( @xmath31 and accounting for pixels close to the edges of the slices or near bad pixel clusters . the original data were then divided by these averaged values to create a map of the normalized pixel gain . we checked the distribution of the pixel - to - pixel variation for each exposure in all the available data for both detectors and all the sub - bands and found that the distribution is gaussian with a full - width - half - maximum of @xmath62% ( figure 20 ) . the variation among different datasets has a standard deviation of @xmath32 0.2% for 16 different observations . the pixel - to - pixel variation does not appear to be wavelength dependent as both short and long wavelength detectors show the same overall flatness . this uniformity suggests that the correction of pixel response variations on orbit can be achieved using infrequent calibration measurements . the absolute responsivity of the mrs is expressed in terms of the quantity referred to as the photon conversion efficiency ( pce ) , which is equal to the number of electrons detected by the focal plane array for each photon incident at the miri entrance focal plane . the wavelength - dependent pces for the mrs were derived during testing at ral . this was done by configuring the mts to provide extended illumination of the miri entrance pupil with the spectral energy distributions of blackbodies of 400 , 600 and 800 kelvin . for each mts blackbody configuration , mrs spectra were obtained in all 12 spectral bands , together with background measurements using the blank position in the mts filter wheel . the data were processed using the standard dhas tool ( paper ii ) , which converts the raw readouts of the integration ramps to slopes in physical units ( electrons / sec ) . the dhas miricube module , which performs the reconstruction technique described in section [ subsec : mylabel4 ] , was then used to construct spectral cubes from the slope images , re - gridding the focal plane array pixel signals onto an equidistant spectral cube . the spectral cubes of the background measurements ( mts filter blank position ) were subtracted from the flood illumination measurements to correct for the test facility background . the spectral cubes produced with the dhas miricube routine have a fully calibrated wcs with the plate scale and wavelength coverage of every cube pixel . when combined with an estimate for the absolute flux delivered to the entrance focal plane by the mts ( paper ii ) , this allows us to calculate the photon conversion efficiency in every pixel of the spectral cube for all mrs spectral bands . table 5 lists the mean pces for all mrs bands . we estimate the fractional error to be 20 % in these figures , due to systematic effects , primarily in estimating the absolute flux from the mts . one obvious feature of table 5 is the sharp drop in pce from channel 3 to channel 4 . the pce in channel 4 is roughly a factor of 2.5 lower than was expected from sub - system measurements of the nominal mrs optical train . the extra loss was identified ( m. te plate , esa , private communication ) as being caused by a fault in the groove profiles of the channel 4 gratings , which can not be corrected before launch . following the procedure to determine the photon conversion efficiency , we can establish a first spectrophotometric calibration . by comparing the flux conversion factors derived from blackbody measurements of the mts at 400 , 600 and 800k , we can assess the achievable spectrophotometric calibration accuracy . figure 21 shows the ratios of the different obtained flux conversion factors as a function of wavelength . we are encouraged by the good agreement ( less than 2 % variation ) over a large swathe of the mrs wave - band . as described in sections 2.3 and 2.5 , great care was taken to minimise sources of straylight and optical cross - talk within the mrs ifus . however , a source of straylight was detected during ral testing , which was identified as being caused by scattering in optical components within the smo . the stray light is manifested as a signal that extends in the detector row direction . its magnitude is proportional to that of bright illuminated regions of the spectral image , at a ratio that falls with increasing wavelength , from about 2 % in channel 1a to undetectably low levels longward of channel 2b . [ fig23 ] emphasises the impact of the straylight in channel 1b , using the wavelength - averaged , reconstructed image of a bright source ( seen at the top - left ) . the straylight signature is seen as the two horizontal bands , where the variation in brightness with the ` alpha ' coordinate is well explained by the mapping between alpha and detector row coordinate . the development of algorithms for correction of this straylight is underway . they take advantage of the stray - light being the dominant signal in the inter - slice regions of the detector , thereby making it amenable to accurate characterisation . initial indications ( figure 22 ) suggest that an effective correction algorithm will be available before launch . testing at ral revealed a gap in the performance of the train of dichroics described in section 2.2 . the set of filters that define the pass - band for channel 3 , sub - band a , have an unwanted ( but small ) transmission peak at a wavelength of 6.1 @xmath0 m , which allows light in the second diffraction order to reach the detector at the position where 12.2 @xmath0 m light is detected in the first diffraction order . this leak was confirmed using fabry - prot etalon data and characterised to produce the leak profile plotted in figure 23 . this curve can be interpreted as the transmission profile by which the 6.1 @xmath0 m spectrum of a target object should be multiplied to determine the leakage signal at 12.2 @xmath0 m . there are two options for mitigating the effects of the spectral leak . first , the channel 3a spectrum can be corrected by multiplying an observation of the 6.1 @xmath0 m ( channel 1b ) spectrum of the target object by the leak profile , resampling the wavelength scale of the resulting spectrum to the wavelength grid of channel 3a and then subtracting it from the contaminated channel 3a spectrum . this requires that a separate channel 1b observation has been taken . the second method would be to make the channel 3a observation with dichroic wheel 1 set to use the sub - band c dichroics ( see section 2.2 to see what this means in terms of the optical train ) . this combination of dichroics reduces the spectral leak by a factor of more than 1000 at @xmath33 = 6.1 @xmath28 m . the unwanted side - effect of this solution is up to a factor of three loss of pce at the short wavelength ends of channel 2a and channel 4a . we have presented the key parameters that describe the performance of the miri mrs spectrometer as designed or measured , in a form that both provides our best estimate of the behaviour of the instrument on - orbit and also that is accessible to the prospective user . the optical design behind the parameters is presented at a level of detail that is intended to provide the astronomer with an understanding of what to expect in terms of operating restrictions and data format when planning observations . the impact of straylight and spectral leaks in contaminating the spectral images has been discussed , along with proposals of operational and analytical techniques that should mitigate their effects . when combined with the latest sensitivity estimates ( glasse et al . , 2014 , paper ix ) , we are confident that the miri mrs will meet all of its scientific objectives as part of the jwst observatory . the work presented is the effort of the entire miri team and the enthusiasm within the miri partnership is a significant factor in its success . miri draws on the scientific and technical expertise many organizations , as summarized in papers i and ii . a portion of this work was carried out at the jet propulsion laboratory , california institute of technology , under a contract with the national aeronautics and space administration . we would like to thank the following national and international funding agencies for their support of the miri development : nasa ; esa ; belgian science policy office ; centre nationale detudes spatiales ; danish national space centre ; deutsches zentrum fur luft - und raumfahrt ( dlr ) ; enterprise ireland ; ministerio de economi y competividad ; netherlands research school for astronomy ( nova ) ; netherlands organisation for scientific research ( nwo ) ; science and technology facilities council ; swiss space office ; swedish national space board ; uk space agency . lccccc slice width & arcsec & 0.176 & 0.277 & 0.387 & 0.645 + number of slices & | & 21 & 17 & 16 & 12 + @xmath35 & pixels & 1.405 & 1.452 & 1.629 & 2.253 + slice width at detector & & & & & + @xmath36 & pixels & 1.791 & 1.821 & 2.043 & 2.824 + pixel size along slice & arcsec / pixel & 0.196 & 0.196 & 0.245 & 0.273 + fov ( across @xmath37 along slices ) & arcsec & 3.70 @xmath37 3.70 & 4.71 @xmath37 4.52 & 6.19 @xmath37 6.14 & 7.74 @xmath37 7.95 + & & sub - band a & & & + wavelength range @xmath38 - @xmath39 & @xmath28 m & 4.87 - 5.82 & 7.45 - 8.90 & 11.47 - 13.67 & 17.54 - 21.10 + resolution & @xmath40 & 3320 - 3710 & 2990 - 3110 & 2530 - 2880 & 1460 - 1930 + & & sub - band b & & & + wavelength range @xmath41 - @xmath42 & @xmath28 m & 5.62 - 6.73 & 8.61 - 10.28 & 13.25 - 15.80 & 20.44 - 24.72 + resolution & @xmath40 & 3190 - 3750 & 2750 - 3170 & 1790 - 2640 & 1680 - 1770 + & & sub - band c & & & + wavelength range @xmath41 - @xmath42 & @xmath28 m & 6.49 - 7.76 & 9.91 - 11.87 & 15.30 - 18.24 & 23.84 - 28.82 + resolution & @xmath40 & 3100 - 3610 & 2860 - 3300 & 1980 - 2790 & 1630 - 1330 + lcc dichroic 1a & 4.84 - 5.83 & 7.40 - 21.22 + dichroic 1b & 5.59 - 6.73 & 8.55 - 24.73 + dichroic 1c & 6.45 - 7.77 & 9.87 - 28.5 + dichroic 2a & 7.40 - 8.91 & 11.39 - 21.22 + dichroic 2b & 8.55 - 10.29 & 13.16 - 24.73 + dichroic 2c & 9.87 - 11.88 & 15.20 - 28.5 + dichroic 3a & 11.39 - 13.68 & 17.45 - 21.22 + dichroic 3b & 13.16 - 15.80 & 20.34 - 24.73 + dichroic 3c & 15.20 - 18.25 & 23,72 - 28.5 + cccccc & & channel 1 & & & + a & 266.67 & & & & + b & 230.77 & 28 & 44 & 55.46 & 29.2 + c & 200.00 & & & & + & & channel 2 & & & + a & 171.43 & & & & + b & 148.45 & 30 & 63 & 54.46 & 28.5 + c & 128.57 & & & & + & & channel 3 & & & + a & 112.06 & & & & + b & 96.97 & 30 & 63 & 54.46 & 28.5 + c & 83.96 & & & & + & & channel 4 & & & + a & 71.43 & & & & + b & 61.28 & 34 & 64 & 53.46 & 27.2 + c & 52.55 & & & & + m ( top left ) compared with the fft of the jwst pupil ( top right ) and normalized peak profiles of the measured ( black ) and modelled ( red ) for the along - slice ( bottom left ) and across - slice direction ( bottom right ) . the x - axis is multiplied by two for the lower figures.,width=480 ]

the direct and precise measurement of the self - coupling between the electroweak gauge bosons in @xmath2-pair production will be a crucial step in testing the standard model of electroweak interactions and searching for physics beyond it . it will form an important part of the physics programme at lep2 and at a planned linear @xmath3-collider ( lc ) . as is well known there are three diagrams at tree level that contribute to the amplitude of @xmath4 in the standard model , one with @xmath5-channel neutrino exchange and the other two with a @xmath6 or @xmath7 in the @xmath8-channel , involving the vertices @xmath1 and @xmath0 . one can parametrise the corresponding vertex functions in order to quantify the couplings and to compare them with their form in the standard model . in the most general form respecting lorentz covariance each vertex involves seven complex form factors @xcite , three of which give couplings that violate @xmath9 symmetry . without further physical assumptions one is thus left with 28 real parameters whose simultaneous extraction in one experiment looks quite hopeless . given the limited event statistics expected at both lep2 and the lc one will only obtain meaningful errors on a reduced number of coupling parameters at one time . this may be achieved by imposing certain constraints on the full set of coupling constants ; various suggestions for such constraints based on symmetry considerations have been made in the literature @xcite . one must however keep in mind that experimental values or bounds on couplings that have been obtained with particular constraints can not be converted into results without constraints or with different ones ; the information lost by assuming relations between couplings can not be retrieved . although imposing such constraints is certainly legitimate and can be useful we stress that a data analysis with independent couplings will be valuable , both from the point of view of model independence and the capability to compare results of different experiments . we remark that of course one can also give ( reasonably small ) errors on _ single or few _ couplings in a multi - parameter analysis . in this paper we propose a parametrisation of the couplings which is well adapted to this end , the statistical errors on the different measured parameters being approximately uncorrelated . we will work in the framework of optimal observables , a way to extract unknown coupling parameters introduced for the case of one parameter in @xcite that has since been used for various reactions @xcite . general aspects of this method , in particular its extension to an arbitrary number of parameters , as well as its application to @xmath10 production were discussed in @xcite . in this paper we investigate again the reaction @xmath11 . we concentrate here on the decay channels , where one @xmath2 decays hadronically and the other into an electron or muon and its neutrino . calculated with the born level cross section of the standard model the statistics of these channels is about 3000 events for a collision energy of @xmath12 and @xmath13 integrated luminosity , which are typical planned lep2 parameters , and about 22000 events with @xmath14 at @xmath15 , which might be achieved at the lc . a complementary source of information is the integrated cross section , which is a quadratic function of the triple gauge couplings . the combination of information from the total event rate and from observables that make use of the detailed distribution in the final state has for example been used in @xcite , where @xmath9 violation in the decay @xmath16 was investigated . in sec . [ sec : method ] of this paper we will further develop some aspects of the method of optimal observables , in particular we will show how to apply it without the linear approximation in the coupling parameters that was used in @xcite . in sec . [ sec : diagon ] we then propose a parametrisation of the couplings that simultaneously diagonalises certain matrices connected with our observables and with the integrated cross section . these parameters achieve two goals : their quadratic contribution to the total cross section is a simple sum of squares and the covariance matrix of the corresponding optimal observables is diagonal . the methods which we use for this purpose are borrowed from the theory of small oscillations of a system with @xmath17 degrees of freedom ( cf . e.g. @xcite ) . our parameters correspond to `` normal coordinates '' and their use in an experimental analysis should in our view present several advantages . we give some numerical examples for @xmath2-pair production at lep2 and the lc in sec . [ sec : numeric ] and make some further remarks on how our proposal might be implemented in practice in sec . [ sec : practice ] . the last section of this paper gives a summary of our main points . the method of optimal observables has previously been presented in the approximation that the couplings to be extracted are sufficiently small to allow for a leading order taylor expansion of various expressions . here we show how to use it beyond this approximation . let us denote by @xmath18 the real and imaginary parts of the @xmath1 and @xmath0 form factors minus their values in the standard model at tree level . as the amplitude of our process is linear in these couplings we can write the differential cross section as @xmath19 where @xmath20 is a positive semidefinite symmetric matrix . @xmath21 collectively denotes the set of measured phase space variables . the integrated cross section is @xmath22 with the standard model cross section @xmath23 and coefficients @xmath24 the idea of using integrated observables is to define suitable functions @xmath25 of the phase space variables and to extract the unknown couplings from their measured mean values @xmath26 . let us give the details . from ( [ diffxsection ] ) and ( [ intxsection ] ) we obtain the expectation value @xmath27 $ ] of @xmath28 as @xmath29 - e_0[{{\cal o}}_i ] = \frac{\displaystyle \sum_{j } c_{ij } \ , g_j + \sum_{jk } q_{ijk } \ , g_j g_k}{\displaystyle 1 + \sum_{j } { \hat{\sigma}_{1,j } } \ , g_j + \sum_{jk } { \hat{\sigma}_{2,jk } } \ , g_j g_k}\ ] ] with the standard model expectation value @xmath30 = ( \int d\phi \ , { { \cal o}}_i s_0 ) / \sigma_0 $ ] and coefficients @xmath31 \ , { \hat{\sigma}_{1,j } } { \hspace{6pt},}\nonumber \\ q_{ijk } & = & \frac{1}{\sigma_0 } \int d\phi \ , { { \cal o}}_i s_{2,jk } - e_0[{{\cal o}}_i ] \ , { \hat{\sigma}_{2,jk } } { \hspace{6pt}.}\end{aligned}\ ] ] we remark in passing that the coefficients in ( [ expect ] ) can be written in a compact form as @xmath32 { \hspace{6pt},}\hspace{3em } q_{ijk } { \hspace{0.4em } = \hspace{0.4em}}v_0[{{\cal o}}_i \ , , \ ; s_{2,jk } /s_0 ] { \hspace{6pt},}\nonumber \\ { \hat{\sigma}_{1,j } } & = & e_0[s_{1,j } /s_0 ] { \hspace{6pt},}\hspace{4.4em } { \hat{\sigma}_{2,jk } } { \hspace{0.4em } = \hspace{0.4em}}e_0[s_{2,jk } /s_0 ] { \hspace{6pt},}\end{aligned}\ ] ] where @xmath33 = e_0[f g ] - e_0[f ] \ , e_0[g]$ ] is the covariance of @xmath34 and @xmath35 in the standard model . note that @xmath36 is symmetric and positive definite , whereas @xmath37 as a matrix in @xmath38 and @xmath39 is symmetric but in general indefinite . an estimation of the couplings can now be obtained by solving the system ( [ expect ] ) with @xmath27 $ ] replaced by the mean values @xmath26 , @xmath40 = \frac{\displaystyle \sum_{j } c_{ij } \ , g_j + \sum_{jk } q_{ijk } \ , g_j g_k}{\displaystyle 1 + \sum_{j } { \hat{\sigma}_{1,j } } \ , g_j + \sum_{jk } { \hat{\sigma}_{2,jk } } \ , g_j g_k } { \hspace{6pt},}\ ] ] provided of course one has @xmath41 observables for @xmath41 unknown couplings . when the system ( [ mean ] ) is linearised in the @xmath18 it is easily solved by inversion of the matrix @xmath42 . one is however not constrained to do so and can instead solve the exact set of equations ( [ mean ] ) . by multiplication with the denominator it can be rearranged to a coupled set of quadratic equations in the @xmath18 and will in general have several solutions . some of these may be complex and thus ruled out , but from the information of the @xmath43 alone one can not tell which of the remaining real ones is the physical solution . we will come back to this point . the measured mean values @xmath43 are of course only equal to the @xmath27 $ ] up to systematic and statistical errors . we only consider the latter here , which are given by the covariance matrix @xmath44 of the observables @xmath28 divided by the number @xmath45 of events in the analysis . to convert the errors on the observables into errors on the extracted couplings we use the quantity @xmath46 \right ) n v({{\cal o}})^{-1 } { } _ { ij } \left ( { \bar{{{\cal o}}}}_j - e[{{\cal o}}_j ] \right ) { \hspace{6pt},}\ ] ] which depends on the @xmath18 through the @xmath27 $ ] given in ( [ expect ] ) . solving ( [ mean ] ) is tantamount to minimising @xmath47 with @xmath48 , and a confidence region on the couplings is as usual given by @xmath49 with the constant determined by the desired confidence level . there are several possible choices for the covariance matrix @xmath50 in ( [ chi ] ) . it can be 1 . determined from the measured distribution of the observables @xmath28 , 2 . calculated from the differential cross section ( [ diffxsection ] ) , taking for the @xmath18 the values extracted in the measurement , 3 . calculated for vanishing couplings @xmath18 , 4 . calculated as a function of the couplings . choices 1 . and 2 . should lead to the same results in the limit of large @xmath45 where the statistical errors on the measured @xmath44 and @xmath18 become small . comparison of the covariance matrices obtained by these two methods might indeed be helpful to rule out unphysical solutions of ( [ mean ] ) . . in turn will be a good approximation of 2 . if the couplings are small enough . we consider possibility 4 . as the least practical one , except maybe for the case of one coupling . for several couplings the expression of @xmath44 as a function of the @xmath18 involves tensors of rank up to four and is even more complicated than the one for the expectation values ( [ expect ] ) , and the inverse matrix is yet more clumsy . for this reason we will discard choice 4 . in the following . in @xcite we considered an analysis at leading order in the @xmath18 , where one uses the linearised form of ( [ mean ] ) to estimate the couplings : @xmath51 \right ) { \hspace{6pt}.}\ ] ] correspondingly the linear approximation of ( [ expect ] ) is used in the expression ( [ chi ] ) of @xmath47 which then reads @xmath52 where @xmath53 is the inverse covariance matrix of the estimated couplings @xcite . as one works to leading order in the @xmath18 one can approximate @xmath50 by its value for zero couplings , i.e. choose possibility 3 . above . the confidence regions @xmath54 for the measured couplings are then ellipsoids in the space of the @xmath18 with centre at @xmath55 . the optimal observables @xmath56 discussed in @xcite have the property that to leading order the statistical errors on the estimated couplings are the smallest possible ones that can be obtained with _ any _ method , including e.g. a maximum likelihood fit to the full distribution of @xmath21 given by the differential cross section ( [ diffxsection ] ) . note that one can still use the linearised expressions ( [ linearestim ] ) and ( [ linearchi ] ) in an analysis beyond leading order . the error @xmath57 on the couplings will be given by an ellipsoid with defining matrix ( [ couplingscov ] ) , where @xmath50 is the covariance matrix at the actual values of the couplings . these errors will in general no longer be optimal , so that when the leading order approximation is not good one might obtain better errors with a different choice of observables . more importantly , however , the extracted values of the couplings are biased : averaged over a large number of experiments the measured couplings differ from the actual ones by terms quadratic in the @xmath18 . if instead one uses the full expressions ( [ expect ] ) , ( [ mean ] ) and ( [ chi ] ) one has no bias on the extracted coupling parameters , provided the number @xmath45 of events in the analysis is large enough . let us see if we can find optimal observables for this case . to this end we expand the differential cross section around some values @xmath58 of the couplings : @xmath59 the corresponding zeroth order cross sections and mean values are @xmath60 and @xmath61 = ( \int d\phi \ , { { \cal o}}_i \widetilde{s}_0 ) / \widetilde{\sigma}_0 $ ] , respectively . we then can re - express @xmath27 $ ] in ( [ expect ] ) , replacing @xmath18 with @xmath62 , @xmath63 with @xmath64 , and using new coefficients @xmath65 etc . constructed as in ( [ xsectioncoeffs ] ) , ( [ obscoeffs ] ) . making the same replacements in ( [ mean ] ) we have an alternative set of equations to extract the coupling parameters . it can be shown that for sufficiently large @xmath45 the confidence regions obtained from ( [ chi ] ) , ( [ confidence ] ) in a nonlinear analysis are again ellipsoids given by @xmath66 one can then write @xmath67 as in ( [ linearchi ] ) , but with @xmath68 of ( [ couplingscov ] ) replaced by @xmath69 where @xmath70 corresponds to an expansion ( [ newdiffxsection ] ) of @xmath71 about the _ actual _ values of the couplings . the main point of the argument is that for large @xmath45 the statistical errors on the @xmath26 become small , so that the extracted couplings will be sufficiently close to the actual ones to allow for a linearisation of ( [ expect ] ) and ( [ mean ] ) , cf . @xcite , p.695 , and @xcite . finally one can construct new observables @xmath72 from ( [ newdiffxsection ] ) . they will be optimal , i.e. have minimum statistical error if the @xmath58 are equal to the actual values of the @xmath18 . in the appendix we show that , up to linear reparametrisations given in ( [ reparam ] ) , this is the only set of @xmath41 integrated observables that measures the @xmath41 couplings with minimum error . there is hence no choice of observables that would be optimal for _ all _ values of the actual coupling parameters . as these are unknown one can in practice not write down the truly `` optimal '' observables , but our argument tells us how one can improve on the choice in ( [ optimal ] ) if one has some previous estimates @xmath58 of the couplings ( cf . also @xcite ) . one may then choose to perform a leading order analysis as described above , linearising about @xmath73 instead of @xmath74 . a practical way to proceed could be to estimate the parameters @xmath18 at first using the linearised method around @xmath75 . suppose this gives as best estimate some values @xmath76 . then in a second step one could set @xmath77 and use the linearised method around @xmath78 to improve the estimate etc . at this point we wish to comment on the `` optimal technique '' for determining unknown parameters in the differential cross section that has been proposed in @xcite . the `` weighting functions '' @xmath79 there depend on the actual values of the parameters one wants to extract and are thus not `` observables '' . only if one sets the unknown parameters in the @xmath79 equal to some previous estimates of them can one use these functions to weight individual events ; the better these estimates are the more sensitive the functions will be . if one does this then the set @xmath79 is equivalent to our observables ( [ newoptimal ] ) defined for some estimates @xmath58 of the coupling parameters . we finally remark that if @xmath45 is not large enough the statistical errors on the mean values @xmath26 and thus on the measured couplings might be so large that they lead into a region where a linearisation of ( [ expect ] ) is not a good approximation . the covariance matrix @xmath80 is then no longer given by ( [ newcouplingscov ] ) . moreover the errors on the couplings might be asymmetric and the shape of the confidence region defined by ( [ chi ] ) , ( [ confidence ] ) very different from an ellipsoid , so that knowledge of @xmath80 is not sufficient to estimate the errors on the @xmath18 . in such a case we can not say on general grounds how sensitive our observables are . incidentally this also holds for other extraction methods such as maximum likelihood fits , whose optimal properties are realised in the limit @xmath81 . if one is rather far from this limit the sensitivity of a method will have to be determined by other means , e.g. by detailed monte carlo simulations . the method we have outlined can of course also be applied if one chooses to reduce the number of unknown parameters by imposing certain linear constraints on the couplings . one may still use the observables ( [ optimal ] ) corresponding to the _ full _ set of couplings but minimise @xmath47 in ( [ chi ] ) for the _ reduced _ set ; in this case one can of course not take choice 2.for @xmath50 . in general @xmath82 is then different from zero and its value indicates to which extent the particular constraints on the couplings are compatible with the data . if @xmath45 is large enough @xmath82 follows in fact a @xmath47-distribution with @xmath83 degrees of freedom for @xmath41 observables and @xmath84 independent couplings so that its value can be converted into a confidence level @xcite . we conclude with a remark on the use of optimal observables in practice . a realistic data analysis will not be good enough if the born approximation of the differential cross section ( [ diffxsection ] ) is used . both higher - order theoretical corrections , such as initial state radiation and the finite @xmath2 width , and experimental effects like detection efficiency and resolution will modify the observed distribution of the phase space parameters @xmath21 . if they are taken into account in the determination of the coefficients in ( [ expect ] ) , ( [ mean ] ) and of the covariance matrix @xmath50 they will _ not _ lead to any bias in the extraction of the couplings and their errors . while this will presumably be done with sets of generated events and might be computationally intensive one still has to determine only a rather limited number of `` sensitivity '' constants . on the other hand one needs to know the observables @xmath25 of ( [ optimal ] ) as functions over the entire experimental phase space , so that the expressions of @xmath85 and @xmath86 used to construct them will in practice be taken from a less sophisticated approximation to the actual distribution of @xmath21 in order to keep them manageable . the observables are then no longer optimal , and it will depend on the individual case which approximations of @xmath85 , @xmath86 are good enough to obtain observables with a sensitivity close to the optimal one . in @xcite it was shown how with a suitable combination of all semileptonic @xmath87 decay channels one can define observables that are either even or odd under the discrete transformations @xmath9 and @xmath88 , where @xmath89 denotes charge conjugation , @xmath90 the parity transformation , and @xmath91 the `` naive '' time reversal operation which flips particle momenta and spins but does not interchange initial and final state . under the conditions on the experimental setup and event selection spelt out in @xcite we have two important symmetry properties : 1 . a @xmath9 odd observable can only have a nonzero expectation value if @xmath9 symmetry is violated in the reaction . 2 . if the expectation value of a @xmath88 odd observable is nonzero the transition amplitude must have an absorptive part whose phase must satisfy certain requirements in order to give an interference with the nonabsorptive part of the amplitude . we assume in this analysis that any nonstandard physics in the reaction is due to the triple gauge vertices . in the standard model one needs at least two loops to violate @xmath9 ; to a good accuracy the triple gauge couplings are therefore the only possible source of @xmath9 violation . for our process , i.e. @xmath3 annihilation into four fermions , an absorptive part that satisfies the requirements mentioned in point 2 . will appear in the standard model already at next - to - leading order in the electroweak fine structure constant , either through nonresonant diagrams or through loop corrections . to leading order , however , they are only due to the imaginary parts of triple gauge couplings . in this approximation the optimal observables ( [ optimal ] ) are @xmath9 even ( odd ) if they correspond to @xmath9 conserving ( violating ) couplings , and @xmath88 even ( odd ) if they correspond to the real ( imaginary ) parts of form factors . the coefficient matrix @xmath92 is then block diagonal in four symmetry classes of observables and three - boson - couplings : 1 . @xmath9 and @xmath88 even 2 . @xmath9 even and @xmath88 odd 3 . @xmath9 odd and @xmath88 even 4 . @xmath9 and @xmath88 odd . in the leading order analysis one thus can treat these four classes of couplings separately and benefit from a great reduction of unknown parameters . beyond leading order , however , form factors of any symmetry can contribute to @xmath27 $ ] : * in the integrated cross section and thus in the denominator of @xmath27 - e_0[{{\cal o}}_i]$ ] in ( [ expect ] ) couplings of all four classes enter quadratically , couplings of class @xmath93 also appear linearly ; * if @xmath28 belongs to class @xmath93 the numerator of @xmath27 - e_0[{{\cal o}}_i]$ ] has terms linear in the couplings of this class but couplings of all four classes enter quadratically through @xmath94 ; * if @xmath28 belongs to a @xmath9 ( @xmath88 ) odd coupling then the numerator in ( [ expect ] ) is only linear in @xmath9 ( @xmath88 ) odd couplings , but it contains also quadratic terms where a @xmath9 ( @xmath88 ) odd coupling is multiplied with a @xmath9 ( @xmath88 ) even one . we remark that this leads to different behaviours of @xmath27 $ ] as one or more couplings @xmath18 become large : whereas for observables in classes @xmath95 , @xmath96 and @xmath97 the expectation value goes to zero when a coupling of the same class goes to plus or minus infinity the corresponding limit of an observable in class @xmath93 can be a positive or negative constant or zero . in a nonlinear analysis one will therefore in principle have to consider couplings with all symmetries at the same time . in practice one might choose simpler procedures if the linear approximation is expected to be not too bad and if one wants to calculate corrections to it . one might for instance first analyse the four symmetry classes separately , neglecting in each case the contributions of the three other classes at the r.h.s . of ( [ mean ] ) and then refine the analysis of a class by taking the values obtained in the first step for the couplings in the other classes as fixed in ( [ mean ] ) . we emphasise that even beyond the leading order approximation it is still true that a nonzero mean value of a @xmath9 or a @xmath88 odd observable is an unambiguous sign of @xmath9 violation or the presence of absorptive parts in the process , respectively . the extraction of the values of the couplings , however , becomes more involved than in leading order . we shall now propose a method to analyse the data which presents several advantages in view of the basic problem posed by the large number of unknown three - boson couplings : with limited event statistics significant error bounds can only be obtained for subsets of the coupling parameters , but imposing constraints on the couplings to reduce their number entails a loss of information that can not be retrieved . in view of this it should be advantageous to use a parametrisation of the couplings which in a given process and at a given c.m . energy has the following properties : 1 . it allows to find observables which are only sensitive to one particular coupling parameter . 2 . the induced errors on the couplings determined from these observables are statistically independent . with this we can on the one hand give single errors for each parameter , on the other hand we can recover from the single errors the multidimensional error of the full set of couplings , having avoided the loss of information incurred by imposing constraints . from the single errors we can also directly see which combinations of couplings in more conventional parametrisations can be measured with good accuracy and to which one is rather insensitive . let us remark that in the leading order analysis there is a set of observables satisfying point 1 . in _ any _ parametrisation of the couplings . the linear combinations @xmath98 of our optimal observables ( [ optimal ] ) are only sensitive to @xmath18 for each @xmath99 ( cf . also @xcite ) . the errors on the couplings determined from these observables are , however , in general not uncorrelated ; in fact their correlations are the same as those obtained with the original set @xmath28 . this can be seen as follows : going from the @xmath28 to the @xmath100 we must replace @xmath101 so that we have from ( [ couplingscov ] ) @xmath102 in such a case the single errors give an incomplete picture of the situation if correlations are large . this is illustrated in fig . [ fig : correlations ] @xmath93 , where the 1@xmath103 ellipsis for two parameters is shown . their single errors are given by its projection on the coordinate axes and in our example are both rather large . some linear combinations of them are however measurable with much better precision , which one can only recognise if both errors and their correlations are given . in fig . [ fig : correlations ] @xmath95 where a set of couplings leading to uncorrelated errors is used the situation is much simpler . note also that the number of correlations , i.e. off - diagonals in @xmath80 , is yet modest for two couplings but increases rapidly with their number . ( 0,0 ) ( 5502,3031)(889,-4805 ) ( 1981,-2401)(0,0)[lb ] ( 5131,-2401)(0,0)[lb ] ( 6391,-3706)(0,0)[lb ] ( 3241,-3706)(0,0)[lb ] ( 1116,-2006)(0,0)[lb ] ( 4176,-2006)(0,0)[lb ] we will now first see that a parametrisation of the couplings satisfying both points 1 . and 2 . above can be found in idealised circumstances , and then mention the restrictions one will encounter under more realistic assumptions . if the leading order analysis is a sufficiently good approximation the solution to our problem is easily found . starting from a set of couplings @xmath18 and the corresponding optimal observables @xmath28 in ( [ optimal ] ) we can go to another set @xmath104 by @xmath105 where we use vector and matrix notation . the coefficients in the expansion of the differential cross section and the optimal observables transform as follows : @xmath106 let now @xmath28 be an arbitrary set of observables related to @xmath18 and define the corresponding @xmath107 related to @xmath104 as in ( [ obstransf ] ) . then we have for the matrices relevant for our analysis the following transformation properties : @xmath108 as shown in @xcite our optimal observables satisfy @xmath109 and @xmath110 so that for them one can choose a transformation @xmath111 which diagonalises all three matrices . this new set @xmath112 of parameters obviously has the properties 1 . and 2 . we were looking for . beyond the linear approximation of ( [ expect ] ) the expectation value of @xmath107 will still receive contributions from several couplings . in fact there is no set of observables for which the full nonlinear expression in ( [ expect ] ) satisfies point 1 . exactly , because the denominator involves quadratic terms in _ all _ couplings , and this can not be changed by any linear transformation of the couplings . if on the other hand the statistical errors are too large the covariance matrix @xmath80 will not give a good picture of the errors as we discussed in sec . [ sec : method ] , and its diagonalisation will not ensure point 2 . in the case however where nonlinear effects in the determination of the couplings and their errors are not too large , i.e. where the leading order expressions are a good first approximation both points 1 . and 2 . above will still be _ approximately _ satisfied in a full nonlinear analysis . we remark that if one has some previous estimates @xmath58 of the couplings that considerably deviate from zero one may reduce nonlinear effects in the determination of the @xmath18 by working with an expansion of @xmath113 around the @xmath58 as shown in sec . [ sec : method ] ; in our diagonalisation programme one will then use couplings @xmath114 instead of @xmath18 , the matrix @xmath70 instead of @xmath115 etc . to the extent that the observables ( [ optimal ] ) are constructed from expressions of @xmath85 and @xmath116 which are only approximations of those that determine the experimentally observed kinematical distribution the matrices @xmath115 , @xmath50 and @xmath117 will not quite be the same and can not be diagonalised at the same time . one can then diagonalise either @xmath50 or @xmath68 because they are by definition symmetric and positive definite , whereas @xmath115 is not necessarily so . again , unless such effects are large one will end up with a matrix @xmath118 that is not diagonal but has relatively small off - diagonals . it should also be borne in mind that the covariance matrix @xmath80 only gives the statistical errors on the couplings , so that even if it is exactly diagonal the final errors may be correlated due to systematics . the choice of transformation in ( [ couplingstransf ] ) to ( [ matrixtransf ] ) is not unique if one does not require @xmath111 to be orthogonal . we see in fact no strong argument in favour of an orthogonal transformation and remark that the various parametrisations of the @xmath1 and @xmath0 couplings in the literature are related by non - orthogonal linear transformations . the freedom to choose @xmath111 can be used to impose additional conditions on the transformation , and the one we propose here is that the transformed quadratic coefficient @xmath119 in the integrated cross section be the unit matrix . in terms of the new couplings one then has @xmath120 where we choose the numbering such that @xmath121 to @xmath122 belong to symmetry class @xmath93 introduced in sec . [ sec : symmetries ] , i.e. they are the @xmath9 and @xmath88 even couplings . only these appear linearly in the cross section , whereas all couplings give a quadratic contribution with coefficient one . having @xmath123 leads to a convenient simplification of ( [ expect ] ) , ( [ mean ] ) . moreover , the measurement of the total cross section gives complementary information on the unknown couplings . rewriting ( [ xsectransf ] ) as @xmath124 we see that measuring a cross section @xmath125 within an error @xmath126 constrains the couplings to be in a shell between two hyperspheres with centre at @xmath127 in the space of all couplings as shown in fig . [ fig : shell ] . their radii are given by @xmath128 here @xmath129\ ] ] is the smallest value the cross section can attain ; that such a minimum exists has been pointed out in @xcite . if in ( [ radii ] ) @xmath130 is positive but @xmath131 negative the couplings are inside the hypersphere with radius @xmath132 , and if both @xmath130 and @xmath131 are negative the ansatz ( [ diffxsection ] ) for the cross section is inconsistent with the data within the error @xmath126 . ( 0,0 ) ( 4647,3843)(1339,-5473 ) ( 4231,-3391)(0,0)[lb ] ( 2611,-1726)(0,0)[lb ] ( 5986,-3661)(0,0)[lb ] ( 3401,-3796)(0,0)[lb ] ( 3601,-3166)(0,0)[lb ] such constraints can be useful to find the physical set of couplings when the solution of ( [ mean ] ) from the measurement of the optimal observables is not unique . if they are strong enough they might even restrict the couplings to the region where ( [ mean ] ) can be linearised and thus simplify their extraction . one can of course use the information from the integrated cross section working with any set of couplings , but again the situation is particularly simple with the form ( [ xsectransf ] ) . we note that the information from the total rate is complementary to what is extracted from the mean values of our observables , which involve normalised kinematical distributions . from an experimental point of view their respective measurements will presumably have quite different systematic errors . let us also recall that the the measurement of the mean values @xmath26 times the number @xmath45 of events obtained with a fixed integrated luminosity combines the information of both @xcite . a nonlinear data analysis as presented in sec . [ sec : method ] can also be done in this case . we draw attention to the fact that unphysical solutions of equation ( [ mean ] ) for @xmath26 and of its analogue for @xmath133 will in general be different . we shall however not elaborate on this point here . another aspect of the couplings with the property ( [ xsectransf ] ) is the following . it is well known that constant coupling parameters deviating from the standard model tree level values lead to amplitudes that violate unitarity @xcite . the coefficients @xmath134 and @xmath135 in the total cross section @xmath103 increase strongly with the @xmath3 c.m . energy @xmath136 and the couplings @xmath18 must vanish as @xmath8 becomes large to ensure a decent high - energy behaviour of @xmath103 . in our new parametrisation the quadratic coefficients in @xmath137 are energy independent , and in this sense the new couplings are at a `` natural scale '' at every energy . to complete this section we show that a transformation with the properties we require always exists , i.e. that we can find a matrix @xmath111 that diagonalises @xmath68 in ( [ matrixtransf ] ) and transforms @xmath138 in ( [ quadrattransf ] ) to the unit matrix . the argument is analogous if one replaces @xmath68 with @xmath50 . by construction both @xmath68 and @xmath138 are symmetric and positive definite , so our problem is the same as finding normal coordinates for a multidimensional harmonic oscillator in classical mechanics ( cf . e.g. @xcite ) . to make this analogy transparent let us write @xmath139 and @xmath140 ; we then have to find @xmath111 so that @xmath141 with @xmath142 being diagonal . the elements @xmath143 of @xmath142 are generalised eigenvalues of @xmath144 satisfying @xmath145 where @xmath146 is the @xmath99-th column vector of @xmath111 . the solution is well known to be @xmath147 where @xmath148 is the orthogonal matrix that transforms @xmath149 to @xmath142 . of course one need not use ( [ solution ] ) in practice as there are convenient algorithms available to find @xmath111 and @xmath142 . in our numerical calculations we have used the routine ` eigenvals ` of the algebraic package maple . we will now give some numerical examples of our method described in the previous section . in this section we will stay within the framework of a leading order analysis of the observables . we start from the results in @xcite , where the sensitivity of optimal observables for semileptonic @xmath150-decays was calculated . we assume a full kinematical reconstruction of the final state , except for the ambiguity one is left with if the jet charge is not known . for the standard model cross section we use the born approximation and neglect effects of the finite @xmath2 width . to describe the triple boson couplings we take the form factors @xmath151 , @xmath152 ( @xmath153 ) of @xcite ; deviations from their standard model tree level values will be referred to as `` anomalous couplings '' . let us first look at a c.m . energy of @xmath12 , which will be attained at lep2 . the coefficient matrix @xmath115 can be found in table 4 of @xcite and in table [ tab : coeff190 ] here we give the diagonal elements of the transformed matrix @xmath118 , ordered according to the symmetry of the corresponding observables . the one standard deviation ellipsoid is diagonal in the couplings @xmath104 , thus its intersections with the coordinate axes equal its projections on these axes . the errors @xmath154 setting all other @xmath155 to zero are then equal to the errors @xmath156 where all other @xmath155 are arbitrary . they are given by @xmath157 and are listed in table [ tab : sens190 ] for an integrated luminosity of @xmath13 . we immediately remark that a negative diagonal element occurs in the transformed coefficient matrix , which is not allowed because @xmath158 is a covariance matrix and thus positive definite . we encounter here a problem of numerical instability : small errors in the calculation of the original matrices @xmath115 and @xmath138 can have a large effect on the smallest generalised eigenvalues @xmath159 and their eigenvectors , even to the point that eigenvalues come out with the wrong sign . this is not only a problem of our particular way of diagonalisation , but also occurs if one diagonalises @xmath115 with an orthogonal matrix ; we find that one of the usual eigenvalues of @xmath115 in the subspace of couplings with symmetry @xmath95 is negative . such instabilities can cause large errors in the matrix inversion of @xmath115 and @xmath50 . one needs @xmath160 to calculate the error on the extracted couplings as can be seen from ( [ chi ] ) and ( [ couplingscov ] ) , and large errors on @xmath161 can lead to large uncertainties in the extracted couplings , irrespective of whether @xmath161 is explicitly used to solve the system ( [ mean ] ) . one will of course aim to calculate @xmath115 and @xmath50 with best possible precision , but such an effort has limits , in particular if they are determined from simulated events and include for instance radiative corrections or detector effects . on a more fundamental level any calculation of these matrices will only be an approximation of the `` exact '' ones that correspond to the kinematical distributions seen in experiment . in this sense it seems quite inevitable that small eigenvalues ( the usual or our generalised ones ) of @xmath115 and @xmath50 and their eigenvectors are sensitive to imprecisions in the calculation and can lead to large errors or uncertainties in the data analysis . this holds of course even if one does not obtain eigenvalues with the wrong sign . we think that also in view of this a diagonalisation is useful , not because it solves the problem but because it makes it explicit ! it allows to easily identify those combinations of couplings which have small corresponding eigenvalues in @xmath115 and @xmath50 and will be the most unsafe ones in the analysis . from ( [ onesigma ] ) we see that they are those combinations for which the statistical errors will be largest . here the most unsafe coupling parameter is @xmath162 . one might thus choose to exclude it , and possibly other couplings , from the analysis and work in the remaining subspace of the @xmath104 where the numerics is more stable and where in any case the experiment is most sensitive . we will come back to this in sec . [ sec : practice ] . .[tab : lr190]diagonal elements @xmath159 of the coefficient matrix restricted to the left or right handed subspace of the couplings as explained in the text . the values in the left handed subspace differ from the corresponding ones in table 1 by at most 3% . [ cols="^ , > , > , > , > , > , > , > , > " , ] crrrrrrrrr & & & + @xmath93 & 1.4 & 1.1 & 0.72 & 0.63 & & 0.50 & 0.17 & 0.062 & 0.044 + @xmath95 & 1.3 & 1.0 & 0.79 & 0.26 & & 0.11 & 0.083 & 0.023 & @xmath163 + @xmath96 & 1.2 & 0.58 & 0.32 & & & 0.076 & 0.031 & 0.013 & + @xmath97 & 1.4 & 1.0 & 0.83 & & & 0.24 & 0.040 & 0.026 & + finally we remark that like in the case for @xmath12 those couplings @xmath104 which give the largest statistical errors in the optimal observable analysis are predominantly related to right handed combinations of form factors as can be seen from the comparison of tables [ tab : coeff500 ] and [ tab : lr500 ] . let us sketch how our method of simultaneous diagonalisation might be used in practice . 1 . one first has to choose which matrix to diagonalise simultaneously with @xmath138 . these matrices need not be the same ones to be used in the data analysis itself but may be calculated under further approximations . covariance matrices for the observables and extracted couplings can be evaluated for zero @xmath18 as our entire procedure will only have its desired properties if nonlinear effects are not too large . if one uses the same approximation of the differential cross section ( [ diffxsection ] ) for the construction of the optimal observables ( [ optimal ] ) and the calculation of @xmath115 , @xmath50 and @xmath117 then the latter are all equal and can be diagonalised at the same time . otherwise one has to choose a positive definite symmetric matrix for the diagonalisation , i.e. one of the covariance matrices . the calculation of @xmath80 or of its inverse from ( [ couplingscov ] ) involves however a matrix inversion and might suffer from numerical instabilities , so presumably the best choice will be @xmath50 . 2 . in the next step one carries out the simultaneous diagonalisation of the chosen matrix and @xmath138 as described in sec . [ sec : simultan ] and determines the transformation matrix @xmath111 in ( [ couplingstransf ] ) to ( [ matrixtransf ] ) . at this point it will be useful to test the numerical stability of the transformed matrices , for instance by re - calculating them in the new basis of couplings or by repeating the diagonalisation procedure with slightly modified initial matrices . one might choose to discard some of the new couplings @xmath104 and the corresponding observables from the analysis if the corresponding matrix elements are found to be instable . this does not mean that one has to set these couplings to zero . from the measurement of the total cross section one will obtain limits on them , which will also allow to control the contribution they can give to the mean values of those observables that are kept in the analysis because the matrix @xmath118 is not exactly diagonal and because of nonlinear terms in ( [ mean ] ) . 3 . in the new parametrisation of the couplings one then carries out the analysis of the data . here @xmath164 , @xmath118 , @xmath165 and the other coefficients in ( [ expect ] ) will be determined under the most realistic assumptions and with the best precision one can afford . they will not be exactly diagonal in practice , but should have small off - diagonal elements if the approximations made in step 1 . and in the construction of the optimal observables are sufficiently good . 4 . one can then give both single and multidimensional errors on the measured coupling parameters @xmath104 . at this stage one can also present the results in other , more conventional parametrisations of the couplings and in particular compare with the measurements of other experiments , restricting oneself to whatever subspace of couplings might have been chosen there . in the first part of this paper we have shown how to extract coupling parameters from the measured mean values @xmath26 of appropriate observables without the approximation that the couplings are small . errors on the couplings can be obtained from a @xmath47-fit of the @xmath26 . if one puts constraints on the couplings in order to reduce their number the method also gives an indication of how compatible these constraints are with the data . the `` optimal observables '' discussed in @xcite have statistical errors equal to the smallest possible ones to leading order in the coupling parameters @xmath18 . beyond the leading order approximation one can obtain more sensitive observables if one has some previous estimate @xmath58 for the couplings , expanding the differential cross section around @xmath58 instead of zero and constructing observables from the corresponding expansion coefficients . in the appendix we show that up to linear reparametrisations the choice of optimal observables is unique : any other set of observables must give bigger ( statistical ) errors . in a second part we have proposed to perform the data analysis using a particular parametrisation @xmath104 of the couplings , which is specific to the process and its c.m . energy . it is obtained from the initial set @xmath18 by a linear transformation which diagonalises the covariance matrix @xmath50 of the observables and transforms the matrix @xmath138 of quadratic coefficients in the integrated cross section ( [ intxsection ] ) to unity . in an idealised framework each optimal observable @xmath107 for this parametrisation is only sensitive to one coupling , and the statistical errors on the extracted couplings are uncorrelated . under realistic circumstances both properties can be approximately satisfied provided that the analysis stays in a region of parameter space where the dependence of the mean values @xmath166 on the couplings is not far from linear . various matrices are then approximately diagonal which should generally facilitate the data analysis . in particular one can directly give errors on single or a small number of couplings , which will be necessary to obtain statistically significant results with a limited number of events . at the same time one can readily present multidimensional errors in parameter space , which is essential to compare with the results of measurements that impose various different constraints on the couplings . having approximately diagonal matrices also allows to easily identify those directions in parameter space which can be measured best and those for which the statistical errors will be large and which are likely to be associated with numerical instabilities , for example in matrix inversions . one can thus recognise and seek to remedy such problems in an early stage of the analysis . the measurement of the total cross section @xmath103 gives valuable complementary information on the coupling parameters . its dependence on the couplings is particularly simple in the parametrisation we propose since the quadratic contributions are @xmath167 times the standard model cross section @xmath168 , i.e. they have the same form for all couplings . a measurement of @xmath103 will then restrict the @xmath104 to a shell between two hyperspheres in parameter space . we have given some numerical examples of our method applied to the semileptonic decay channels in @xmath4 . in particular we find that the couplings @xmath104 which can be measured best with unpolarised beams predominantly appear in the amplitude for left handed electrons ( or right handed positrons ) , and that the @xmath104 with the largest statistical errors mainly correspond to the opposite lepton helicity . comparing our results at lep2 and lc energies we see that the coefficients in the linear contributions of the couplings @xmath104 to our observables and to the integrated cross section change much less with energy than in usual parametrisations . this is because in the new parametrisation the quadratic coefficients in the normalised cross section @xmath169 are by construction energy independent . we would like to thank ch . hartmann and m. kocian for their continued interest in optimal observables for triple gauge couplings . we gratefully acknowledge discussions with and remarks by j. blmlein , p. overmann , n. wermes , , and p. m. zerwas . this work has in part been financially supported by the eu programme `` human capital and mobility '' , network `` physics at high energy colliders '' , contracts chrx - ct93 - 0357 ( dg 12 coma ) and erbchbi - ct94 - 1342 , and by bmbf , grant . it was started while one of us ( md ) was at the university of cambridge , and we acknowledge support by the arc programme of the british council and the german academic exchange service ( daad ) , grant 313-arc - viii - vo / scu , which made mutual visits of the cambridge and heidelberg groups possible . in this appendix we show that the set of observables ( [ newoptimal ] ) , obtained from expanding the differential cross section about the actual values of the couplings , is unique in the sense that up to the linear reparametrisations ( [ reparam ] ) it is the only set of @xmath41 integrated observables which in the limit of large @xmath45 leads to the minimum error on the @xmath41 extracted parameters . to keep our notation simple we give the proof for the case that the actual values of the @xmath18 are zero . the expectation value and covariance of functions @xmath34 and @xmath35 are then given by @xmath170 = \frac{\int d\phi \ , f(\phi ) s_0(\phi)}{\int d\phi \ , s_0(\phi ) } { \hspace{6pt},}\hspace{3em } v_0[f , g ] = e_0[f g ] - e_0[f ] \ , e_0[g ] { \hspace{6pt}.}\ ] ] in the general case one has instead of @xmath85 the zeroth order coefficient @xmath171 ( [ newdiffxsection ] ) from the expansion about the appropriate values @xmath58 . for large @xmath45 the covariance matrix for the extracted couplings is given by ( [ couplingscov ] ) . under a linear reparametrisation of observables , @xmath172 where @xmath173 and @xmath174 are constants and the matrix @xmath173 is nonsingular , the matrices @xmath115 from ( [ obscoeffcompact ] ) and @xmath50 transform according to @xmath175 from ( [ couplingscov ] ) we see that the covariance matrix @xmath80 is unchanged under such a transformation . for our proof we can hence restrict ourselves to observables with mean value @xmath176 = 0\ ] ] and with a coefficient matrix @xmath177 . from ( [ obscoeffcompact ] ) we then have the condition @xmath178 = \delta_{ij}\ ] ] and the error on the extracted couplings is given by @xmath179 { \hspace{6pt}.}\ ] ] from @xcite we know that the optimal observables ( [ optimal ] ) lead to the smallest possible error on the @xmath18 , given by the cramr - rao bound . to satisfy our conditions ( [ centred ] ) and ( [ normalised ] ) we take the linear combinations @xmath180 \right)\ ] ] with @xmath181 { \hspace{6pt}.}\ ] ] we assume that the functions @xmath86 are linearly independent . otherwise some of the parameters @xmath18 are superfluous and can be eliminated ; our assumption is thus that the @xmath18 are an independent set of parameters for the anomalous couplings . linear independence of the @xmath182 guarantees that @xmath183 is nonsingular , which has tacitly been used at several instances in our paper . the set @xmath184 is related to the optimal observables @xmath185 by a linear transformation ( [ reparam ] ) and thus gives the same optimal error matrix @xmath80 . the covariance @xmath186 $ ] defines a scalar product on the hilbert space @xmath187 of sufficiently smooth functions of @xmath21 with the property @xmath188 = 0 $ ] . the functions @xmath184 span a subspace @xmath189 of @xmath187 , and we define @xmath190 as the orthogonal complement of @xmath189 with respect to the scalar product @xmath33 $ ] . any set of @xmath41 observables satisfying ( [ centred ] ) can then be written as @xmath191 with @xmath192 , @xmath193 . further decomposing @xmath194 and using the constraint ( [ normalised ] ) we obtain @xmath195 , i.e.@xmath196 finally , we have from ( [ simplecov ] ) , ( [ decompose ] ) , ( [ first ] ) @xmath197 + n^{-1 } v_0[{{\cal o}}^{\it ii}_i , { { \cal o}}^{\it ii}_j ] { \hspace{6pt}.}\ ] ] the first term gives the error on the couplings for the optimal observables @xmath184 , which is minimal . if the observables @xmath28 have minimal error , too , the second term must be zero , so that for each @xmath99 we have @xmath198 = 0 $ ] and thus @xmath199 which completes our proof . in sec . [ sec : simultan ] we mentioned that instead of @xmath26 one may use the product @xmath200 measured with fixed luminosity to extract the couplings @xcite . by an argument analogous to the one of this appendix one finds that up to linear reparametrisations our observables ( [ newoptimal ] ) are again the only optimal ones . in this case linear reparametrisations have to be homogeneous , i.e. one must have @xmath201 in ( [ reparam ] ) , since adding constants to the observables can change the induced errors on the coupling parameters . g. gounaris et al . , `` triple gauge boson couplings '' , report of the `` triple gauge couplings '' working group during the lep2 workshop 19941995 , hep - ph/9601233 , in _ physics at lep2 _ , vol . 1 , g. altarelli and f. zwirner eds . , cern report 1996 , and references therein p. overmann , dortmund preprint do - th 93/24 ; + w. bernreuther , g. w. botz , d. bru , p. haberl and o. nachtmann , z. phys . c68 ( 1995 ) 73 ; + d. bru , o. nachtmann and p. overmann , `` cp violation in radiative @xmath7 decays '' , univ . heidelberg report hd thep9702 ( 1997 ) r. akers et al . ( opal coll . ) , z. phys . c66 ( 1995 ) 31 ; + a. stahl , nucl . b ( proc . suppl . ) 40 ( 1995 ) 505 ; + aleph coll . , `` search for cp violation in the decay @xmath202 '' , paper contributed to the ichep96 conference , warsaw , 2531 july 1996 , report pa 08 - 030 ; + delphi coll . , `` improved test of cp - violation in @xmath203 using optimized observables '' , paper contributed to the ichep96 conference , warsaw , 2531 july 1996 , report pa 07 - 014 ( 1996 ) ; + n. wermes , `` cp tests and dipole moments in @xmath204-pair production experiments '' , talk given at tau96 workshop , estes park , colorado , usa , 1619 september 1996 , report bonn - he-96 - 10

the standard model is greatly successful but it still has many free parameters which must be small to describe nature . while its supersymmetric extensions , e.g. , the minimal supersymmetric standard model , are attractive scenarios , small couplings are also required to explain observed facts such as the fermion mass hierarchy and mixing angles . in recent years , extra dimensions have cast a new perspective on physics beyond the standard model . one of the important aspects of extra dimensional models is that bulk fields can be localized with finite - width wave - function profiles . this fact provides us with a geometrical explanation for small numbers . that is , with a configuration where some fields are separated from each other in the extra dimensional space , the couplings among them are generally suppressed . then how and where fields are localized is an issue to be considered . from this viewpoint , extra dimensional models with a curved background are interesting because fields could be localized depending on the shape of the background geometry . one of the most famous examples of curved geometries is the randall - sundrum ( rs ) model with the ads@xmath0 warped metric @xcite . field theories of vectors , spinors , and scalars have been studied on this background @xcite-@xcite . the localization behavior of zero - mode wave functions has interesting applications to phenomenology such as the suppression of unwanted operators . for example , hierarchical forms of yukawa couplings and proton decay were studied in @xcite . the localization of kaluza - klein ( kk ) excited modes also leads to interesting phenomena . for instance , the localization of higher kk gauge bosons could realize a composite scalar ( higgs ) condensation , which induces dynamical ( electroweak ) symmetry breaking on the brane where the kk gauge bosons localize @xcite . in addition , models on more complicated backgrounds where a warp factor oscillates generate bulk fields which localize at some points in extra dimensions @xcite . this type of localization might be useful in explaining the observed phenomena . however extra dimensional theories are generally nonrenormalizable and the calculations depend on the regularization scheme that one adopts . furthermore , extra dimensional theories are constrained by symmetries of higher dimensions . for example , in the supersymmetric case , bulk theories are constrained by @xmath5 supersymmetry in five dimensions . motivated by these facts , recently a four - dimensional ( 4d ) description of extra dimensional models was proposed @xcite . with this method , the phenomena of higher dimensional models are reproduced in terms of 4d theories , and several interesting models have been proposed along this line @xcite . in this paper , we present 4d gauge theories that describe physics on 5d curved geometries . as will be discussed below , taking generic values of gauge couplings and gauge - symmetry - breaking vacuum expectation values ( vevs ) , the models provide vector , spinor , and scalar fields on curved extra dimensions . as a good and simple illustration , we compare our 4d model with the rs one . we particularly focus on the `` localization '' behaviors of mass eigenstates in `` index spaces '' of gauge groups . it will be shown that the localization profiles and the exponentially suppressed massive spectrum are certainly reproduced . in addition , our formulation gives a localization mechanism even for massless vector fields . as a phenomenological application , hierarchical yukawa matrices are derived in our approach ; that is a hierarchy without symmetries in four dimensions . the localization behavior depends on the required conditions for gauge - symmetry - breaking vevs and gauge and other couplings . if these values are determined in the underlying theories , it may be said that the physics on warped backgrounds is dynamically generated within a four - dimensional framework . we consider several possibilities to realize the conditions by utilizing , for example , strongly coupled gauge theories . thus this could provide a purely 4d dynamical approach for small numbers . we will proceed with the argument as follows . in sec . [ sec : dwd ] , we describe our 4d gauge theories , which have generic ( nonuniversal ) values of gauge - symmetry - breaking vevs and couplings . the models provide vector , spinor , and scalar fields in warped extra dimensions . it is also shown that supersymmetry multiplets in flat 4d models generate supersymmetry multiplets on warped backgrounds . in sec . [ sec : ne ] , we then numerically determine with a finite number of gauge groups that the formulation given in sec . 2 certainly reproduces various properties of bulk fields on the rs background . in addition , a phenomenological application to quark mass matrices is also given . finally , we discuss possibilities of dynamically realizing the conditions required for curved geometries in sec . [ sec : ddwd ] . we conclude the discussion in sec . [ sec : conclusion ] . the appendix is devoted to a brief review of 5d bulk fields on a rs background . following refs . @xcite , we introduce @xmath6 gauge theories with gauge couplings @xmath7 ( @xmath8 ) , and scalar fields @xmath9 [ @xmath10 which are in bifundamental representations of @xmath11 . the system is schematized by the segment diagram in fig . [ fig : links ] . the gauge invariant kinetic term of the scalars @xmath9 is written by @xmath12 where the covariant derivative is given by @xmath13 . we assume that the scalar fields @xmath9 develop vevs proportional to the unit matrix , @xmath14 , which break the gauge symmetry to a diagonal @xmath15 . from the kinetic term ( [ eq : qkin ] ) , the mass terms for the vector fields @xmath16 are obtained : @xmath17 where the @xmath18 matrix @xmath19 is defined as @xmath20 the consequence of these mass terms is that we have a massless gauge boson corresponding to the unbroken gauge symmetry , which is given by the following linear combination : @xmath21 where @xmath22 and @xmath23 is the gauge coupling of the low - energy gauge theory @xmath15 . the profile of @xmath24 is independent of the values of @xmath25 . it is found from this that the massless vector field is `` localized '' at the points with smaller gauge couplings . if the gauge couplings take a universal value @xmath26 , the massless mode @xmath27 has a constant `` wave function '' along the `` index space '' of gauge groups . as seen below , this direction labeled by @xmath28 becomes the fifth spatial dimension in the continuum limit ( @xmath29 ) . the localization behavior can easily be understood from the fact that , for smaller gauge coupling @xmath7 , the symmetry - breaking scale @xmath30 of @xmath6 becomes lower , and hence the corresponding vector field @xmath16 becomes the more dominant component in the low - energy degree of freedom @xmath27 . it is interesting to note that this vector localization mechanism ensures charge universality . suppose that there is a field in a nontrivial representation of @xmath6 only . that is , it couples only to @xmath31 with strength @xmath7 . this corresponds to a four - dimensional field confined on a brane . if there are several such fields , they generally have different values of gauge couplings . however , note that these fields couple to the massless modes @xmath24 with a _ universal _ gauge coupling @xmath23 defined above . this is because , in the presented mechanism , the vector fields are localized depending on the values of the gauge couplings . as for massless eigenstates , the mass eigenvalues and wave functions are obtained by diagonalizing the mass matrix ( [ eq : diffop ] ) . the simplest case is the universal couplings @xmath32 in this case , one obtains the mass eigenvalues of @xmath19 as @xmath33 in the limit @xmath29 with @xmath34 fixed ( the limit to continuum 5d theory ) , the eigenvalues become @xmath35 these are the same spectrum as that of the bulk gauge boson in the @xmath36 extra dimension with radius @xmath37 . with generic values of vevs @xmath25 and gauge couplings @xmath7 , the situation is rather complicated . in this case , the mass term ( [ eq : gau - mass ] ) of the vector fields becomes @xmath38(a_\mu^{i+1})^2 \nonumber \\[1 mm ] & & + \frac{1}{2}v_n^2g_n(g_{n+1}-g_n)(a_\mu^n)^2 -\frac{1}{2 } v_1 ^ 2g_1(g_2-g_1)(a_\mu^1)^2 . \label{eq : gbbm}\end{aligned}\ ] ] the first term becomes the kinetic energy transverse to the four dimensions in the continuum limit . on the other hand , the second and third terms are bulk and brane mass terms , respectively . it should be noted that these mass terms vanish in the case of universal gauge coupling , which corresponds to a flat massless vector field in 5d theory as discussed above . in other words , nonuniversal gauge couplings generate bulk / brane mass terms and cause a localization of the wave function . first we consider the series of vevs @xmath25 and couplings @xmath7 that generates a vector field on the rs warped background , namely , the ads@xmath0 background . this model can be obtained by choosing a universal @xmath7 and by varying @xmath25 as @xmath39 substituting this and taking the continuum limit , eq . ( [ gkt ] ) becomes @xmath40 ^ 2 , \label{eq : contlimrs}\end{aligned}\ ] ] where @xmath41 represents the coordinate of the extra dimension : @xmath42 ( @xmath8 ) and @xmath43 , etc . it is found that eq . ( [ eq : contlimrs ] ) successfully induces the kinetic energy term along the extra dimension and mass terms for the vector field on the warped background metric @xmath44 where @xmath45 with @xmath46 . we here conclude that we can obtain the vector field on a rs warped background by varying only the vevs @xmath25 . in the following we will see that nonuniversal gauge couplings @xmath7 induce other interesting results beyond the effects from the background metric . now let us compare the 4d model with generic couplings ( [ eq : gbbm ] ) to extra dimensional ones . we define the dimensionless parameters @xmath47 and @xmath48 as @xmath49 first we restrict ourselves to the case that the gauge group is @xmath50 , namely , abelian theory with _ no vector self - couplings_. similarly substituting eq . ( [ eq : deffh ] ) and taking the continuum limit , eq . ( [ gkt ] ) becomes @xmath51\bigr]^2 . \label{eq : contlim}\end{aligned}\ ] ] equation ( [ eq : contlim ] ) induces the kinetic energy term along the extra dimension and mass terms for the vector field on the warped background metric : @xmath52 ^ 2 \eta_{\mu\nu}dx^\mu dx^\nu - dy^2 . \label{eq : generalbkg}\ ] ] the bulk and boundary mass terms are @xmath41 dependent and proportional to the derivatives of @xmath53 . this is also seen from the 4d model [ the second and third terms in eq . ( [ eq : gbbm ] ) ] . the above is a generic correspondence between our 4d case and continuum 5d theory . as an example , consider the following forms of the parameters : @xmath54 where @xmath55 is a positive constant with mass dimension @xmath56 . equation ( [ eq : contlim ] ) leads to @xmath57 -\frac{1}{2}\biggl[\zeta ke^{-2(\zeta+\eta)ky } ( a_\mu)^2\biggr]_0^{l/2}. \label{eq : wgmbbm}\end{aligned}\ ] ] the first term on the right hand side is the kinetic term of the gauge boson along the extra dimension with the warped background @xmath58 the second and third terms correspond to the bulk and boundary masses announced before . as easily seen , the above equation includes the expression for vector fields on the rs background . in the 5d rs model , the lagrangian for vector fields is written as ( see the appendix ) @xmath59 where the @xmath60 gauge fixing condition is chosen . by comparing eq . ( [ eq : wgmbbm ] ) with eq . ( [ rsgauge ] ) , we find that the case with @xmath61 realizes vector fields on the rs background . also , a special limit , @xmath62 , produces the flat zero - mode solution . that corresponds to the form of the parameters ( [ eq : rsparaf ] ) in the previous special argument . the other solutions which satisfy @xmath63 correspond to nonflat wave functions of the zero - mode vector field on the rs background . it is clearly understood in our formulation that such nonflat wave functions are caused by introducing bulk and/or boundary mass terms in the rs model . for example , in the case of @xmath64 , the vector field has bulk and boundary mass terms , and is localized with a peak at the @xmath65 point . it should be noticed that with these bare mass terms the zero mode is still massless . this is understood from our formulation where the gauge symmetry @xmath66 is left unbroken in the low - energy effective theory . in the above abelian case we discussed interpretation of the nonuniversal @xmath47 as @xmath41-dependent bulk or boundary masses in the warped extra dimension . next we treat the non - abelian theory with vector self - couplings . since in this case we also have @xmath41-dependent vector self - couplings in addition to the @xmath41-dependent bulk or boundary masses , it may be convenient and instructive to see @xmath47 as a @xmath41-dependent coefficient of the vector kinetic term . to this end , we define the four - dimensional field @xmath67 , @xmath68 where @xmath69 is the lattice spacing , which goes to zero in the continuum limit . rescaling the gauge fields @xmath70 , the kinetic term @xmath71 and eq . ( [ eq : qkin ] ) become @xmath72 \nonumber \\ & & \qquad \qquad \qquad \qquad \qquad \qquad + i \hat{g } a_5^{i+1/2 } ( a^{i+1}_\mu - a^i_\mu ) + { \cal o}(a^{1/2 } ) \bigg|^2 , \label{eq : gqkin2}\end{aligned}\ ] ] where @xmath73 and @xmath74 . in the continuum limit @xmath75 with @xmath76 and @xmath77 fixed , eq . ( [ eq : gqkin2 ] ) results in @xmath78 ^ 2 \eta^{\mu \nu } f_{\mu 5 } f_{\nu 5 } \big\},\end{aligned}\ ] ] where @xmath79 $ ] . this completely reproduces the 5d yang - mills kinetic term with a @xmath41-dependent coefficient @xmath80 on the warped - background metric ( [ eq : generalbkg ] ) , provided that @xmath81 . this is the generic correspondence between the present 4d model and continuum 5d theory . from eq . ( [ eq:5dym ] ) , we thus find the @xmath41-dependent factor @xmath82 in front of the canonical yang - mills term , which corresponds to a 5d dilaton vev . the factor does carry the origin of the massless vector localization shown in eq . ( [ zero - a ] ) . with the constant gauge coupling @xmath83 ( @xmath47 = 1 ) , one obtains a bulk vector field with a constant zero mode on the warped metric ( [ eq : generalbkg ] ) . a field redefinition @xmath84 in eq . ( [ eq:5dym ] ) gives the previous bulk and boundary mass terms but one then has @xmath41-dependent vector self - couplings in non - abelian cases . we next consider spinor fields by arranging fermions of fundamental or antifundamental representation in each gauge theory @xmath6 . we introduce two weyl ( one dirac ) spinors to construct a 5d bulk fermion . the orbifold compactification in continuum theory requires that one spinor obeys the neumann boundary condition and the other the dirichlet one . in the present 4d model , this can be achieved by having asymmetrical numbers of fundamental and antifundamental spinors , resulting in chiral fermions in the low - energy gauge theory . here we consider the fundamental weyl spinors @xmath85 ( @xmath8 ) in the @xmath6 theory and the antifundamental @xmath86 ( @xmath87 ) . as seen below , @xmath88 corresponds to the bulk fermion with the neumann boundary condition and @xmath89 to that with the dirichlet one . the generic gauge - invariant mass and the mixing terms of @xmath85 and @xmath86 are written as @xmath90 where @xmath91 and @xmath92 are dimensionless coupling constants . we assume that @xmath9 develop vevs @xmath93 . the mass matrix for spinors is then given by @xmath94 the spinor mass eigenvalues and eigenvectors ( wave functions ) are read from this matrix . one easily sees that the massless mode is contained in @xmath88 and given by the following linear combination : @xmath95 therefore the localization profile of zero mode depends on the ratio of dimensionless couplings @xmath91 and @xmath92 . a simple case is @xmath96 for all @xmath28 . in this case , @xmath97 corresponds to a chiral zero mode obtained from a 5d bulk fermion on the flat background . if @xmath98 , the system describes a fermion with a curved wave - function profile . for example , if @xmath99 ( @xmath100 ) , @xmath97 has a monotonically increasing ( decreasing ) wave - function profile . as another interesting example , taking @xmath101 ( @xmath102 , @xmath103 are constants and @xmath104 ) , @xmath97 has a gaussian profile with a peak at @xmath105 . other profiles of massless chiral fermions could also be realized in our approach . let us discuss the 5d continuum limit . the relevant choice of couplings @xmath91 and @xmath92 is @xmath106 the parameters @xmath107 give rise to a bulk bare mass in the continuum limit as will be seen below . the only difference between the vector and spinor cases is the existence of possible bulk mass parameters [ see eqs . ( [ eq : diffop ] ) and ( [ eq : mass - spi ] ) ] . the mass and mixing terms ( [ eq : fermionmoose ] ) then become @xmath108 + \textrm{h.c.},\end{aligned}\ ] ] where @xmath109 and @xmath110 are the same as defined in the case of vector fields ( [ eq : deffh ] ) . similar to the vector case , this form is compared with the bulk spinor lagrangian in the rs space - time ( see the appendix ) @xmath111 . \label{eq : rsspinor}\end{aligned}\ ] ] here the kinetic terms have been canonically normalized in order to compare them to the 4d model . in eq . ( [ eq : rsspinor ] ) , @xmath102 is a possible 5d dirac mass , and the `` 1/2 '' contribution in the mass terms comes from the spin connection with the rs metric . it is interesting that the 5d spinor lagrangian ( [ eq : rsspinor ] ) is reproduced by taking the exact same limit of parameters as that in the vector case , defined by eq . ( [ eq : rsparaf ] ) . furthermore , the relation between the mass parameters @xmath102 should be taken as @xmath112 that is , the @xmath107 s take a universal value . now the localization behavior of the spinors is easily understood . in the present 4d model , the spinor mass matrix ( [ eq : mass - spi ] ) becomes with eq . ( [ eq : spinorcond ] ) @xmath113 a vanishing bulk mass parameter @xmath114 corresponds to @xmath115 , that is , @xmath116 in our model . then the mass matrix @xmath117 is exactly the same as @xmath19 , and their mass eigenvalues and eigenstates are the same . in particular , the massless mode @xmath97 has a flat wave function with universal gauge couplings as considered here . this is consistent with the expression ( [ zero - eta ] ) , where the ratio @xmath118 determines the wave - function profile . on the other hand , in the case of @xmath119 ( @xmath120 ) , the rs zero - mode spinor is localized at @xmath121 ( @xmath65 ) @xcite , which in turn corresponds to @xmath122 ( @xmath123 ) in our model . one can see from the spinor mass matrix ( [ eq : mass - rsspi ] ) that the zero - mode wave function is monotonically increasing ( decreasing ) with respect to the index @xmath28 . in this way , we have a 4d localization mechanism for the spinor fields . nonuniversal gauge couplings or nonuniversal masses give rise to a nonflat wave function for a chiral massless fermion . the latter option is not realized for vector fields . notice that the charge universality still holds in the low - energy effective theory . that is , with any complicated wave - function profiles , zero modes interact with a universal value of the gauge coupling . this is because curved profiles of vector fields depend only on the gauge couplings . finally we consider scalar fields . we may introduce two types of scalar field @xmath124 and @xmath125 in the fundamental and antifundamental representations of @xmath6 , respectively . in addition , for each type of scalar , there are two choices of the @xmath126 parity assignment in the continuum limit . this orbifolding procedure is incorporated by removing @xmath127 or @xmath128 . the gauge invariant mass and mixing terms for @xmath129 and @xmath130 are written as @xmath131 where @xmath132 , @xmath133 , and @xmath134 are the dimensionless coupling constants . it is implicitly assumed that nonintroduced fields are appropriately removed in the sums . we have included the mixing mass terms up to the nearest - neighbor interactions . other invariant terms such as @xmath135 or terms containing @xmath9 correspond to nonlocal interactions in 5d theories , and we do not consider these in this paper . notice , however , that for a supersymmetric case , these terms may be suppressed due to renormalizability and holomorphy of the superpotential . the zero - mode eigenstates are given in the same form as that of the spinor shown in the previous section , replacing @xmath132 and @xmath133 by @xmath136 and @xmath137 ( @xmath138 and @xmath139 ) . therefore the ratio @xmath140 ( @xmath141 ) determines the zero - mode wave function . let us consider the continuum 5d limit . in what follows , we remove @xmath128 , which corresponds to the @xmath126 assignment @xmath142 and @xmath143 . the 5d limit can be achieved by taking the following choices of couplings : @xmath144 & & \bar\alpha'_i \,=\ , g_{i+1 } , \qquad \bar\beta'_i \,=\ , g_{i+1}(1-\bar c'_{i+1}),\end{aligned}\ ] ] where @xmath145 and @xmath146 correspond to the bulk mass parameters , as in the spinor case . then the mass terms ( [ eq : nsp ] ) and ( [ eq : dsp ] ) for the scalars take the following forms with the parametrization ( [ eq : deffh ] ) : @xmath147\{f(y)\phi(x , y)\}\bigr|^2 + [ gv\gamma(y)]^2|f(y)\phi(x , y)|^2 \biggr],\ ] ] @xmath148\{h(y)\varphi(x , y)\ } \bigr|^2 + [ gv\bar\gamma(y)]^2|h(y)\varphi(x , y)|^2 \biggr].\ ] ] as a special case , we compare @xmath129 and @xmath130 with the scalar fields in the rs space - time . the scalar lagrangian on the rs background is ( see also the appendix ) @xmath149\end{aligned}\ ] ] where the 4d kinetic term is canonically normalized , and @xmath150 and @xmath151 are the bulk and boundary mass parameters , respectively , defined in the appendix [ eq . ( [ eq : smass ] ) ] . by substituting the rs limit in our model given by eq . ( [ eq : rsparaf ] ) , we find the relations between the mass parameters in 4d and 5d : @xmath152 in this subsection , we discuss 5d supersymmetry on warped backgrounds . generally a supersymmetric theory may be obtained by relevant choices of couplings from a nonsupersymmetric theory . we here examine whether it is possible to construct supersymmetric 4d models which describe 5d supersymmetric ones on warped backgrounds . this is a nontrivial check for the ability of our formalism to properly describe 5d nature . in ref . @xcite , the 5d theory on the ads@xmath0 rs background was studied . there , supersymmetry on ads@xmath0 geometry was identified and then the conditions on the mass parameters imposed by this type of supersymmetry were derived ( also given in the appendix here ) . as seen below , these relations among mass parameters for ads@xmath0 supersymmetry are indeed satisfied in our _ 4d formalism_. this fact seems remarkable in the sense that the present analyses do not include gravity . first consider vector supermultiplets in 5d . the scalar fields @xmath9 and the gauge bosons @xmath31 are extended to chiral and vector superfields in 4d , respectively . notice that the vevs that were discussed above , @xmath153 are in the ( baryonic ) @xmath154-flat direction . we start with the following 4d supersymmetric lagrangian @xmath155 the bilinear terms of the component fields are written in the unitary gauge ( we follow the conventions of @xcite ) : @xmath156 where we have rescaled @xmath157 for canonical normalization of the kinetic terms . the mass matrix @xmath158 is defined in eq . ( [ gkt ] ) . the first term is nothing but eq . ( [ gkt ] ) , that is , the mass terms for vector fields . by also canonically normalizing @xmath159 and @xmath160 and integrating out the auxiliary fields , we find the mass terms for the adjoint spinors and scalar @xmath161 these masses have the same forms as that of the vector field because we started from a supersymmetric theory . we thus have a model with @xmath162 for the spinors and @xmath163 for the scalar . ( note that @xmath164 , which originate from @xmath9 , have @xmath126 odd parity . ) it is a nontrivial check to see whether the above mass terms satisfy the conditions for 5d ads@xmath0 supersymmetry . we find from the relations ( [ eq : ci - spi ] ) and ( [ eq : massd ] ) that the mass terms for @xmath165 and @xmath166 imply @xmath167 indeed , these relations are just those required by ads@xmath0 supersymmetry @xcite . in this way , _ 5d vector supermultiplets on a rs warped background are automatically derived from a 4d supersymmetric model on a flat background_. we also construct a 5d hypermultiplet in the warped extra dimension starting from a 4d supersymmetric theory . in order to have a hypermultiplet we introduce the chiral superfields @xmath124 and @xmath125 in the fundamental and antifundamental representations of the @xmath6 gauge theory . in the following , @xmath128 is removed to implement @xmath126 orbifolding which leaves a chiral zero mode of the fundamental representation . the fermionic components of @xmath124 and @xmath125 , then correspond to @xmath85 and @xmath168 , respectively , in sec . [ sec : spinor ] . the generic renormalizable superpotential is written as @xmath169 this superpotential just leads to a spinor mass term of the form ( [ eq : fermionmoose ] ) . in addition , the mass and mixing terms of the scalars @xmath129 and @xmath130 also have the same form as those of the spinors : @xmath170 supersymmetry induces equivalence between the boson and fermion mass matrices . in turn , this implies in our formulation given in the previous sections that the mass parameters are equal , @xmath171 and also @xmath172 . thus , there is only one parameter @xmath102 left . it is found from eqs . ( [ eq : ci - spi ] ) , ( [ eq : massn ] ) , and ( [ eq : massd ] ) that if one take the continuum limit the relations @xmath173 are generated . these mass relations are exactly those imposed by supersymmetry on the ads@xmath0 geometry @xcite . thus hypermultiplets on the rs background are properly incorporated in our formalism with a flat background . it may be interesting that the mass relation for vector multiplets is the one for chiral multiplets with dirichlet boundary conditions ( [ eq : abchypd ] ) with @xmath114 . this value of @xmath102 is the limit of vanishing bulk mass parameters . it should be noticed that our analyses have been performed for generic warped backgrounds , including the rs case as a special limit . we thus found that even in generic warped backgrounds the conditions on the bulk mass parameters required for 5d warped supersymmetry should be the same as for the rs case . here we perform a numerical study to confirm our formulation of the curved extra dimension discussed in the previous sections . we will also give a phenomenological application to the hierarchy among yukawa couplings . in the following , we consider the case that corresponds to the rs model in the continuum limit , as a good and simple application . the gauge couplings and vevs are specified as given in sec . [ sec : dwd ] ; @xmath174 the universal gauge coupling @xmath26 implies that vector zero modes have flat wave functions as shown in eq . ( [ zero - a ] ) . the following is a summary of the mass terms for various spin fields , which were derived in the previous sections : @xmath175 { \cal l}_{fm } & = & -\psi d_c \eta + \textrm{h.c . } , \\[2 mm ] { \cal l}_{sm}^\phi & = & -\bigl|d_{3/2-b } \phi\bigr|^2 -|m \phi|^2 , \\[1 mm ] { \cal l}_{sm}^\varphi & = & -\bigl|d_{-(3/2-b)}^\dagger \varphi\bigr|^2 -|m^\dagger \varphi|^2.\end{aligned}\ ] ] the parameters @xmath151 and @xmath102 represent the bulk mass parameters for scalars and spinors , respectively . the @xmath18 mass matrices @xmath176 and @xmath177 are defined as follows : @xmath178 m & = & \sqrt{a+4b - b^2}\,\frac{k}{v } \pmatrix { v_1 & & \cr & \ddots & \cr & & v_{n-1 } } \pmatrix { 1 & 0 & & \cr & \ddots & \ddots & \cr & & 1 & 0 } , \end{aligned}\ ] ] where @xmath179 for supersymmetric cases , the mass matrices @xmath176 for bosons and fermions take the same form and , moreover , @xmath180 , as discussed previously . we define the matrices @xmath181 that diagonalize the mass matrices for gauge , fermion , and scalar fields , respectively . for example , @xmath182 satisfies @xmath183 where @xmath184 are the mass eigenvalues which should correspond to the kk spectrum of vector fields . in the following , we use the notation @xmath185 that is , the coefficients of @xmath31 in the @xmath186th massive eigenstates @xmath187 . in the continuum limit , this corresponds to the value of the wave function at @xmath188 for the @xmath186th kk excited vector field . similar definitions are made for spinors and scalars . for vector fields , we illustrate the resultant eigenvalues @xmath189 and eigenvectors @xmath190 in figs . [ fig : bulkgaugesys](a)[fig : bulkgaugesys](f ) . for comparison , we also show in the figures the wave functions and kk mass eigenvalues of vector fields on the rs background . it is found from the figures that our 4d model completely reproduces the mode function profiles [ figs . [ fig : bulkgaugesys](b ) , [ fig : bulkgaugesys](d ) , and [ fig : bulkgaugesys](f ) ] . localization becomes sharp as @xmath191 increases ; this situation is similar to the continuum case . the warp - suppressed spectra of kk excited modes are also realized [ figs . [ fig : bulkgaugesys](a ) , [ fig : bulkgaugesys](c ) , and [ fig : bulkgaugesys](e ) ] . for a larger @xmath192 ( the number of gauge groups ) , the model leads to a spectrum more in agreement with the continuous rs case . note , however , that the localization profiles of wave functions can be seen even with a rather small @xmath192 . it is interesting that even with a finite number of gauge groups the massive modes have warp - suppressed spectra and localization profiles in the index space of gauge theory . for fermion fields , there is another interesting issue to be examined . it is the localization behavior via dependence on the mass parameters @xmath102 , which was discussed in sec.[sec : spinor ] . we show the @xmath102 dependence of the zero - mode wave function @xmath193 in fig . [ fig : cdepwf ] . the figure indicates that the zero - mode wave functions surely give the expected localization nature of the continuum rs limit [ eq . ( [ eq : hzero ] ) in the appendix ] . we find that the values of the wave functions are exponentially suppressed at the tail of localization profile even with a finite number of gauge groups . the profiles of massive modes can also be reproduced . now we apply our formulation to phenomenological problems in four dimensions . let us use the localization behavior , which has been shown above , to obtain the yukawa hierarchy . this issue has been studied in the 5d rs framework @xcite . we consider a model corresponding to the ( supersymmetric ) standard model in the bulk . the yukawa couplings for quarks are given by @xmath194 where @xmath195 , @xmath196 , and @xmath197 denote the left - handed quarks and the right - handed up and down quarks , respectively , and @xmath198 are the family indices . for simplicity , we study a supersymmetric case and introduce two types of higgs scalars @xmath199 and @xmath200 . then the mass parameters of the higgs scalars satisfy eq . ( [ eq : abchypn ] ) and they are denoted by @xmath201 in the following . similarly , the quark behaviors are described by their mass parameters @xmath202 . we assume @xmath203 . generally , in supersymmetric 5d models , yukawa couplings such as eq . ( [ eq : yukawaorg ] ) are prohibited by 5d supersymmetry . however , since the present model is 4d , one may apply 5d - like results to yukawa couplings without respecting 5d consistency . this is one of the benefits of our scheme . we are now interested in the zero - mode part of eq . ( [ eq : yukawaorg ] ) , which generates the following mass terms @xmath204 where the fields with tildes @xmath205 stand for the @xmath186th mass eigenstate given by @xmath206 ( similarly for @xmath196 , @xmath197 , and @xmath207 ) . the effective yukawa couplings are @xmath208 and similarly for @xmath209 . a typical behavior of @xmath210 is shown in fig . [ fig : cdepwf ] for several values of the bulk mass parameter @xmath102 . in fig . [ fig : cdepy ] , we show the behaviors of the zero - mode yukawa couplings @xmath211 against the quark mass parameters . two limiting cases with @xmath212 and 1 are shown . the former corresponds to a bulk higgs scalar localized at @xmath65 and the latter to one at @xmath121 in the continuum rs limit . from the figures , we see that if there is a @xmath213 difference of mass ratio among the generations , it generates a large hierarchy between yukawa couplings . combined with the mechanisms that control mass parameters discussed in the next section , one obtains a hierarchy without symmetries within the four - dimensional framework . \(a ) @xmath212 \(b ) @xmath214 \(c ) @xmath212 \(d ) @xmath214 in the case of @xmath214 , the yukawa coupling depends exponentially on the quark bulk mass parameters @xmath202 when @xmath215 .. for @xmath216 , the higgs scalars have a peak at @xmath217 ( @xmath121 ) . this situation is different from the one discussed in ref . @xcite where the higgs field is localized at @xmath218 ( @xmath65 ) . ] this implies that if @xmath202 exist in this region one obtains the following form of the yukawa matrices : @xmath219 where their exponents satisfy @xmath220 this form is similar to that obtained by the froggatt - nielsen mechanism @xcite with a @xmath50 symmetry . as an illustration , let us take the following mass parameters @xmath221 and @xmath222 and @xmath223 , which generates the low - energy yukawa matrices @xmath224 where @xmath225 . this pattern of quark mass textures leads to realistic quark masses and mixing angles @xcite with a large value of the ratio @xmath226 . if the above analysis were extended to su(5 ) grand unified theory , realistic lepton masses and mixing may be derived . other forms of yukawa matrices that may be realized by the froggatt - nielsen mechanism are easily incorporated in our formulation . for more complicated patterns of mass parameters , we could realize yukawa matrices that are different from those derived from the froggatt - nielsen mechanism . in general , off - diagonal entries tend to be rather suppressed , that is , we have @xmath227 for the yukawa matrix ( [ yij ] ) . such a form may lead to realistic fermion masses and mixing angles . for example , one could derive the yukawa matrix @xmath228 if initial values of @xmath229 are sufficiently suppressed . in this case , the @xmath230 submatrix for the second and third generations does not satisfy eq . ( [ fntype ] ) . the yukawa matrix ( [ eq : compli ] ) may be relevant to the down - quark sector , indeed studied in ref . we do not pursue further systematic studies on these types of yukawa matrices in this paper . we have shown that 4d models with nonuniversal vevs and gauge and other couplings can describe 5d physics on curved backgrounds , including the rs model with an exponential warp factor . in the continuum 5d theory , this factor is derived as a solution of the equation of motion for gravity . on the other hand , in the 4d viewpoint , warped geometries are generated by taking the couplings and vevs as appropriate forms . in the previous sections , we have just assumed their typical forms and examined its consequences . if one could identify how to control these couplings by the underlying _ dynamics _ , the resultant 4d theories turn out to provide attractive schemes to discuss low - energy physics such as tiny coupling constants . first we consider the scalar vevs @xmath231 . a simple way to dynamically control them is to introduce additional strongly coupled gauge theories @xcite . consider the following set of asymptotically free gauge theories : @xmath232 where @xmath233 and @xmath234 denote the dynamical scales . we have , for simplicity , assumed common values of @xmath235 and @xmath233 for all @xmath6 . in addition , two types of fermions are introduced : @xmath236 where their representations under @xmath237 gauge groups are shown in parentheses . if @xmath238 , the @xmath239 theories are confined at a higher scale than @xmath6 , and the fermion bilinear composite scalars @xmath240 appear . their vevs @xmath25 are given by the dynamical scales @xmath234 of the @xmath239 gauge theories through a dimensional transmutation as @xmath241 where @xmath242 is a universal one - loop gauge beta function for @xmath239 ( @xmath243 ) . the gauge couplings @xmath244 generally take different values and thus lead to different values of @xmath25 . for example , a linear dependence of @xmath245 on the index @xmath28 is amplified to an exponential behavior of @xmath25 . that is , @xmath246 which reproduces the bulk fields on the rs background as shown before . the index dependences of the gauge couplings are actually generic situations , and may also be controlled , for example , by some mechanism fixing dilatons or the radiatively induced kinetic terms discussed below . a supersymmetric extension of the above scenario is achieved with quantum - deformed moduli spaces @xcite . another mechanism that dynamically induces nonuniversal vevs is obtained in supersymmetric cases . consider the gauge group @xmath247 and the chiral superfields @xmath9 with charges @xmath248 under @xmath249 . it is assumed that the scalar components @xmath250 of @xmath9 develop their vevs @xmath251 . the @xmath154 term of each @xmath252 is given by @xmath253 where @xmath254 is the coefficient of the fayet - iliopoulos ( fi ) term , and the ellipsis denotes contributions from other fields charged under @xmath252 , which are assumed not to have vevs . given nonvanishing fi terms , @xmath255 , the @xmath154-flatness conditions mean @xmath256 and nonuniversal vevs @xmath25 are indeed realized . in this case , the dynamical origin of nonuniversal vevs is the nonvanishing fi terms . these may be generated at the loop level . furthermore , if the matter content is different for each gauge theory , the @xmath254 themselves have complicated forms . above , we supposed that the charges of @xmath9 are @xmath248 under @xmath257 . alternatively , if @xmath9 have charges @xmath258 under @xmath257 and other matter fields have integer charges , the gauge symmetry @xmath247 is broken to the product of a diagonal @xmath50 gauge symmetry and the discrete gauge symmetry @xmath259 . such discrete gauge symmetry would be useful for phenomenology @xcite . models with nonuniversal gauge couplings @xmath7 are also interesting in the sense that they can describe the localization of massless vector fields . a nonuniversality of gauge couplings is generated , e.g. , in the case that the @xmath6 gauge theories have different matter content from each other . then radiative corrections to gauge couplings and their renormalization - group running become nonuniversal , even if their initial values are equal . this fact is also applicable to the above - mentioned mechanism for nonuniversal @xmath25 . suppose that the @xmath239 theory contains ( @xmath260 ) vectorlike quarks which decouple at @xmath261 . the gauge couplings @xmath262 are then determined by @xmath263 where we have assumed that the @xmath239 theories are strongly coupled at a high - energy scale @xmath264 ( @xmath265 ) . tuning of the relevant matter content thus generates the desired linear dependence of @xmath245 . with these radiatively induced couplings ( [ eq : nonuni - g ] ) at hand , the vevs are determined from eq . ( [ eq : dtvev ] ) : @xmath266 we have formulated 4d models that provide 5d field theories on generic warped backgrounds . the warped geometries are achieved with generic values of symmetry - breaking vevs , gauge couplings , and other couplings in the models . we focused on field localization behaviors along the index space of gauge theory ( the fifth dimension in the continuum limit ) , which is realized by taking relevant choices of the mass parameters . as a good and simple application , we constructed 4d models corresponding to bulk field theories on the ads@xmath0 randall - sundrum background . the localized wave functions of massless modes are completely reproduced with a finite number of gauge groups . in addition , the exponentially suppressed spectrum of the kk modes is also generated . these results imply that most properties of brane world models can be obtained within 4d gauge theories . supersymmetric extensions were also investigated . in 5d warped models , the bulk and boundary mass terms of spinors and scalars satisfy complicated forms imposed by supersymmetry on the rs background . however , we show in our formalism that these forms of the mass terms are derived from a 4d _ global _ supersymmetric model on a _ flat _ background . as an application of our 4d formulation , we derived hierarchical forms of yukawa couplings . the zero modes of scalars and spinors with different masses have different wave - function profiles as in the 5d rs cases . therefore by varying the @xmath213 mass parameters for each generation , one can obtain realistic yukawa matrices with a large hierarchy from the overlaps of the wave functions in a purely 4d framework . other phenomenological issues such as proton stability , grand unified theory ( gut ) symmetry breaking , and supersymmetry breaking can also be discussed . the conditions on the model parameters should be explained by some dynamical mechanisms if one considers the models from a fully 4d viewpoint . one interesting way is to include additional strongly coupled gauge theories . in this case , a small @xmath213 difference between gauge couplings is converted to exponential profiles of symmetry - breaking vevs via dimensional transmutation , and indeed generates a warp factor of the rs model . a difference of gauge couplings is achieved by , for example , the dynamics controlling dilatons , or radiative corrections to gauge couplings . supersymmetrizing models provide a mechanism for dynamically realizing nonuniversal vevs with @xmath154-flatness conditions . our formulation makes sense not only from the 4d points of view but also as a lattice - regularized 5d theory . in this sense , effects such as the ads / conformal field theory ( cft ) correspondence might be clearly seen with our formalism . as another application , it can be applied to construct various types of curved backgrounds and bulk or boundary masses . for example , we discussed massless vector localization by varying the gauge couplings @xmath7 . furthermore , one might consider models in which some fields are charged under only some of the gauge groups . these seem not like bulk or brane fields , but `` quasi - bulk '' fields . applications including these phenomena will be studied elsewhere . this work is supported in part by the japan society for the promotion of science under the postdoctoral research program ( grants no . @xmath267 and no . @xmath268 ) and a grant - in - aid for scientific research from the ministry of education , science , sports and culture of japan ( no . @xmath269 ) . here we briefly review the field theory on a rs background , following ref . one of the original motivations for introducing a warped extra dimension by randall and sundrum is to provide the weak planck mass hierarchy via the exponential factor in the space - time metric . this factor is called `` warp factor , '' and the bulk space a `` warped extra dimension '' . such a nonfactorizable geometry with a warp factor distinguishes the rs brane world from others . consider the fifth dimension @xmath41 compactified on an orbifold @xmath270 with radius @xmath271 and two three - branes at the orbifold fixed points @xmath121 and @xmath272 . the einstein equation for this five - dimensional setup leads to the solution @xcite @xmath273 where @xmath55 is a constant with mass dimension @xmath56 . let us study a vector field @xmath274 , a dirac fermion @xmath275 , and a complex scalar @xmath129 in the bulk specified by the background metric ( [ metric ] ) . the 5d action is given by @xmath276 , \label{kin}\ ] ] where @xmath277 and the covariant derivative is @xmath278 where @xmath279 is the spin connection given by @xmath280 and @xmath281 . from the transformation properties under @xmath282 parity , the mass parameters of scalar and fermion fields are parametrized as is taken as @xmath283 . here we adopt @xmath284 , and then the boundary mass parameter @xmath151 in eq . ( [ eq : smass ] ) is different from that in ref . @xcite by the factor @xmath285 . ] @xmath286 where @xmath150 , @xmath151 , and @xmath102 are dimensionless parameters . referring to @xcite , the vector , scalar and spinor fields are cited together using the single notation @xmath287 . the kk mode expansion is performed as @xmath288 by solving the equations of motion , the eigenfunction @xmath289 is given by @xmath290 , \label{eq : modefunction}\end{aligned}\ ] ] where @xmath291 , @xmath292 , and @xmath293 for each component in @xmath294 . @xmath295 is the normalization factor and @xmath296 and @xmath297 are the bessel functions . the corresponding kk spectrum @xmath298 is obtained by solving @xmath299 a supersymmetric extension of this scenario was discussed in @xcite . the on - shell field content of a vector supermultiplet is @xmath300 where @xmath274 , @xmath301 , and @xmath302 are the vector , two majorana spinors , and a real scalar in the adjoint representation , respectively . also a hypermultiplet consists of @xmath303 , where @xmath304 @xmath305 are two complex scalars and @xmath275 is a dirac fermion . by requiring the action ( [ kin ] ) to be invariant under supersymmetric transformation on the warped background , one finds that the five - dimensional masses of the scalar and spinor fields have to satisfy @xmath306 where @xmath102 remains as an arbitrary dimensionless parameter . that is , @xmath307 , @xmath308 , and @xmath114 for vector multiplets and @xmath309 and @xmath310 for hypermultiplets . there is no freedom to choose the bulk masses for vector supermultiplets and only one freedom parametrized by @xmath102 for the bulk hypermultiplets . it should be noted that in warped 5d models fields contained in the same supermultiplet have different bulk and boundary masses . that is in contrast with the flat case . the @xmath126 even components in supermultiplets have massless modes with the following wave functions : @xmath311 \frac{e^{(1/2- c)\sigma}}{n_0\sqrt{2\pi r } } & & \textrm { for } \;h^{1\ , ( 0 ) } \ , \textrm { and } \,\psi^{(0)}_l . \label{eq : hzero}\end{aligned}\ ] ] the subscript @xmath76 means the left - handed ( @xmath126 even ) component . the massless vector multiplet has a flat wave function in the extra dimension . on the other hand , the wave function for massless chiral multiplets involves a @xmath41-dependent contribution from the space - time metric , which induces a localization of the zero modes . the zero modes with masses @xmath119 and @xmath120 localize at @xmath121 and @xmath312 , respectively . the case with @xmath114 corresponds to the conformal limit where the kinetic terms of the zero modes are independent of @xmath41 . l. randall and r. sundrum , phys . lett . * 83 * , 3370 ( 1999 ) [ hep - ph/9905221 ] ; 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the dynamical density matrix renormalization group ( dmrg ) @xcite and the closely related correction vector dmrg @xcite have been widely used in the last decade to compute the dynamical correlation functions and spectral functions of low - dimensional strongly correlated quantum systems . @xcite although more powerful dmrg approaches have been developed recently , @xcite dynamical dmrg ( ddmrg ) often remains the method of choice because it offers two practical advantages over the other approaches : it is simpler and it can be easily parallelized . for instance , it has been recently shown that ddmrg allows us to investigate features with small spectral weights such as power - law pseudo - gaps in luttinger liquids . @xcite the main drawback of ddmrg is that it always yields the convolution of the desired spectrum with a lorentzian distribution of finite width . therefore , the true spectrum can only be obtained through a deconvolution of the ddmrg spectrum . ( in principle , there are some methods to get around this problem @xcite but they are rarely used in practice . ) deconvolution is a typical ill - conditioned inverse problem , however . @xcite a direct solution of the deconvolution equation usually yields a very noisy and thus useless spectrum . nevertheless , various regularization methods have been successfully used to deconvolve ddmrg spectra for one - dimensional systems and quantum impurity problems . @xcite astonishingly , some of these deconvolution methods even allow us to bypass the finite - size scaling analysis and to obtain the piecewise smooth spectrum of an infinite systems directly from a broadened finite - system ddmrg spectrum . unfortunately , regularization also smooths out the sharp features of the true spectrum . this is a serious issue as the spectra of one - dimensional systems and quantum impurities often exhibit very interesting ( power - law ) singularities . in this paper we present a method , which allows us to determine sharp spectral features in the thermodynamic limit starting from a broadened finite - system ddmrg spectrum . for this purpose we consider the extrapolation to the thermodynamic limit and the deconvolution for the lorentzian kernel to be a single blind deconvolution , @xcite i.e. an inverse problem with an unknown kernel including both the lorentzian broadening and the finite - size effects . the key idea to preserve sharp spectral features in a piecewise smooth spectrum is to impose a minimal distance @xmath0 between extrema of the deconvolved spectrum . to illustrate our method we investigate the single - particle density of states ( dos ) of one - dimensional paramagnetic mott insulators represented by the half - filled hubbard model . @xcite we confirm that this dos has the step - like onset predicted by field - theoretical studies @xcite at least at weak to intermediate coupling up to @xmath1 . the hubbard model @xcite with on - site interaction @xmath2 and nearest - neighbor hopping @xmath3 is a basic lattice model for the physics of strongly interacting electrons , in particular the mott metal - insulator transition . @xcite at half filling ( i.e. , the number of electrons equals the number of sites @xmath4 ) the ground state is a mott insulator for strong interaction @xmath5 , while it is a fermi gas in the non - interacting limit @xmath6 . the hamiltonian of the hubbard model is defined by @xmath7 where the operator @xmath8 ( @xmath9 ) creates ( annihilates ) an electron with spin @xmath10 on the site @xmath11 , @xmath12 , and @xmath13 . the first sum runs over all pairs @xmath14 of nearest - neighbor sites while the other two sums run over all sites @xmath15 . here we will only consider half - filled systems and thus set the chemical potential @xmath16 to have electron - hole - symmetric spectra and a fermi energy @xmath17 . the bulk single - particle dos @xmath18 can be measured experimentally using photoemission spectroscopy or scanning tunneling spectroscopy . theoretically , it can be defined as the average of the local dos @xmath19 where the sum runs over both spins and all sites @xmath15 in the lattice , while @xmath20 is the local single - particle dos at site @xmath15 for spin @xmath10 and can be calculated using @xmath21 for @xmath22 and @xmath23 for @xmath24 . here @xmath25 denotes the eigenstates of the hamiltonian @xmath26 and @xmath27 their eigenenergies in the fock space . the ground state for the chosen number of particles corresponds to @xmath28 . the total spectral weight is @xmath29 we will consider only lattice geometry for which the hamiltonian ( [ eq : hamiltonian ] ) is invariant under the electron - hole transformation @xmath30 . therefore , for half filling the density of states is symmetric , @xmath31 . if the system is translation invariant , the bulk dos and the local dos are identical . for dmrg simulations , however , open boundary conditions are preferred to periodic boundary conditions . in this case , the bulk dos can be identified with the local dos on one of the two equivalent middle sites of the system , i.e. as far as possible from the system boundaries . inverse problems such as ( blind ) deconvolutions @xcite occur in many scientific fields and are among the most challenging numerical computations . experimental measurements and computer simulations often yield approximations of the true quantities which are measured or computed , respectively . it is often assumed that the deviations from exact results can be modelled by a convolution with a smoothing function and an additive noise due to the finite accuracy and resolution of the measurement or simulation process . a typical example of a blind deconvolution is the reconstruction of an original signal from a degraded copy using incomplete information about the degradation process . @xcite here we want to compute sharp spectral features in the piecewise smooth spectrum of an infinite system from a broadened finite - system spectrum calculated with ddmrg . in this section we first show that this task can be formulated as a blind deconvolution problem , then present an algorithm for solving it . let @xmath32 be a spectrum of a finite lattice model with @xmath4 sites . this spectrum is a dirac - comb ( a finite sum of dirac - peaks ) @xmath33 where the sum runs over all hamiltonian eigenstates @xmath25 which contributes to the spectrum , i.e. with a nonzero spectral weight @xmath34 . here @xmath35 denotes the corresponding excitation energies . this spectrum can be broadened with a lorentzian distribution of width @xmath36 @xmath37 to obtain a smooth spectral function @xmath38 with the ddmrg method we can calculate this spectrum for a discrete set of excitation energies @xmath39 . as numerical calculations are always affected by errors , ddmrg actually yields values @xmath40 which are related to the true spectral function by @xmath41 for @xmath42 , where @xmath43 represents the unknown errors . ( it should be noted that ddmrg errors @xmath43 include significant systematic contributions , for instance due to the variational nature of the procedure . @xcite ) in principle , one could determine the true spectrum , i.e. , the excitation energies @xmath35 and the corresponding weights @xmath44 , through this system of equations . in practice , however , this is an ill - conditioned problem except for simple discrete spectra . moreover , we are not interested in resolving the discrete peaks of small systems but in calculating the piecewise smooth spectra of macroscopic systems . the spectrum in the thermodynamic limit is given by @xmath45 note that , generally , the order of the two limits can not be exchanged . typically , the spectral function @xmath46 is piecewise smooth , i.e. , it exhibits one or more continua as well as isolated sharp features such as steps , power - law singularities or cusps . in principle , one should carry out several ddmrg simulations with varying system size @xmath4 and broadening @xmath36 and then extrapolate the numerical data to obtain @xmath46 . in most cases , a simultaneous extrapolation for @xmath47 and @xmath48 is possible @xcite using a constant value of @xmath49 . nevertheless , the computational cost of ddmrg simulations increases very rapidly with smaller @xmath36 and the overall cost of this approach is prohibitive for a full spectrum . indeed , this approach has been mostly used to study isolated spectral features in the thermodynamic limit such as power - law singularities and steps . @xcite as all operations used to define @xmath46 from @xmath50 are linear , the broadened spectrum of the finite system can also be written explicitly as a function of the infinite system spectrum @xmath51 the kernel @xmath52 includes both the finite - size effects and the lorentzian smoothing . its form is not known but it is clear that we must recover a pure lorentzian smoothing in the thermodynamic limit @xmath53 combining eqs . ( [ eq : deconv ] ) and ( [ eq : spec - inf ] ) we obtain a system of equations @xmath54 for @xmath55 , relating the ddmrg data set @xmath56 to the infinite system spectrum @xmath46 . determining @xmath46 from these equations is a so - called inverse problem . @xcite this kind of problem is also called blind deconvolution since our knowledge of the kernel is incomplete . [ strictly speaking , it is not a deconvolution because eq . ( [ eq : inverse - problem ] ) is not a convolution . however , as the kernel approaches the form @xmath57 in the thermodynamic limit , we will use the terminology of deconvolution problems . ] it should be obvious that this is an ill - posed problem . first , the errors @xmath43 and the kernel @xmath52 are not known . second , the problem is sorely underdetermined as we try to reconstruct the function of a continuous variable from a finite number @xmath58 of data points . finally , a convolution with a lorentzian is a smoothing operation and thus the corresponding deconvolution is an extremely ill - conditioned inverse problem : the solution will be extremely sensitive to small changes or errors in the input . various deconvolution methods have been used successfully to deduce piecewise smooth spectra from the broadened finite - system spectra calculated with ddmrg . they include , direct inversion at low resolution , @xcite linear regularization methods , @xcite fourier transform with low - pass filtering , @xcite nonlinear regularization methods such as the maximum entropy method , @xcite parametrization with piecewise polynomial functions , @xcite and a deconvolution ansatz for the self energy . @xcite however , this task has not been viewed as a blind deconvolution so far . instead , it has been considered as the deconvolution of a perfectly known kernel . the need for regularization or filtering techniques has been viewed as the consequence ill - conditioning and under - determination of the problem ( [ eq : inverse - problem ] ) with a lorentzian kernel . all of these methods offer some advantages for particular spectral forms . however , their common drawback is that they are ill - suited for sharp spectral features , such as steps or power - law singularities , within or at the edge of a continuum . either the regularization procedure smooths out true sharp features excessively or it allows the occurrence of deconvolution artifacts ( artificial sharp structures , rapid oscillations or negative spectral weight ) , especially in the vicinity of the true spectrum singularities . naturally , better results can be obtained if we can use _ a priori _ knowledge about the properties of the spectrum @xcite but , in practice , this is a rare occurrence . therefore , we need a better method for solving the inverse problem ( [ eq : inverse - problem ] ) which allows us to determine isolated sharp spectral features accurately while preserving the positivity and the piecewise smoothness of @xmath46 . let the ddmrg data ( [ eq : dmrg - data ] ) be evenly distributed in the energy interval @xmath59 $ ] . the difference between two consecutive energies is @xmath60 . additionally , consider a set of equidistant energies @xmath61 in the interval @xmath62 \subset [ \epsilon_a,\epsilon_b]$ ] . the distance between these energies is @xmath63 . as we will always use @xmath64 , we have @xmath65 . ( typical values are @xmath66 and @xmath67 . ) as in a least - square approach we define a cost function @xmath68 as the sum of the squares of the differences between the ddmrg data and an approximate representation parametrized by a discrete set of variables @xmath69 @xmath70 the absolute minimum of @xmath68 is zero and the corresponding parameters @xmath71 are determined by a linear system of @xmath58 equations @xmath72 using this equation system to determine the parameters @xmath71 would be an unconstrained least - square fit . in the limit @xmath73 ( followed by @xmath74 and @xmath75 ) this equation system becomes equivalent to the inverse problem ( [ eq : inverse - problem ] ) with vanishing errors @xmath76 . thus the absolute minimum of @xmath68 yields the spectrum @xmath46 through @xmath77 if we substitute a lorentz kernel @xmath78 for the unknown kernel , @xmath79 in ( [ eq : linear - system ] ) , we recover the finite - system deconvolution problem defined by equations ( [ eq : spec - finite ] ) and ( [ eq : deconv ] ) for vanishing errors @xmath43 . thus the absolute minimum of the cost function corresponds to the discrete finite - system spectrum @xmath32 through @xmath80 if @xmath81 and @xmath82 otherwise . physically , the solution of the deconvolution problem is unique for vanishing errors @xmath43 and thus the cost function should have a unique absolute minimum . from a mathematical point of view , however , the equation system ( [ eq : linear - system ] ) could have no solution or infinitely many solutions . then any small error @xmath43 can generate wildly different ( and mostly unphysical ) solutions . therefore , as ( [ eq : kernel - limit ] ) holds in the thermodynamic limit , it is possible and preferable to obtain a reasonable approximation of the infinite - system spectrum @xmath46 from the minimization of the cost function ( [ eq : cost - function ] ) with a lorentz kernel under the constraint that the spectral function @xmath46 is physically allowed . for instance , @xmath46 should be positive semidefinite and piecewise smooth . generally , this solution does not correspond to the absolute minimum or even a local minimum of @xmath68 . indeed , the solution of the inverse problem ( [ eq : inverse - problem ] ) corresponds to the value @xmath83 if we assume that the relation ( [ eq : solution ] ) holds . of course , it could be possible to lower the cost function with other configurations @xmath84 but in that case the relation ( [ eq : solution ] ) would no longer hold . note that , this idea has been implicitly assumed in all previous deconvolution schemes of ddmrg data aiming at piecewise smooth spectra so far . however , in these approaches the agreement between solution @xmath84 and numerical data @xmath85 in eq . ( [ eq : linear - system ] ) with a lorentzian kernel is considered essential while the regularization of the solution and the errors @xmath43 are seen as perturbations which should deteriorate the agreement as little as possible . yet a blind deconvolution requires equal balancing of the agreement between solution and numerical data and of the smoothness and stability of the solution . @xcite hence we must take a different point of view : the lorentzian kernel ( [ eq : kernel - limit ] ) is only an approximation of the true kernel @xmath52 and the physical constraints on the deconvolved spectrum @xmath46 are essential in the minimization of the cost function . thus we accept significant deviations of @xmath84 from the conditions ( [ eq : linear - system ] ) yielding the absolute minimum of the cost function , or , equivalently , we assume that the errors @xmath43 can be substantial . therefore , the blind deconvolution problem can be formulated as a least - square optimization under non - linear constraints . we want to minimize the cost function ( [ eq : cost - function ] ) with the lorentz kernel under the constraints that the spectrum @xmath46 has the following properties : 1 . finite band width , i.e @xmath86 for all @xmath87 and all @xmath88 for some finite @xmath89 , 2 . positive semi - definite , @xmath90 , and 3 . piecewise smooth . the first two conditions are easily expressed for the parameters @xmath91 . the somewhat fuzzy concept of a piecewise smooth spectrum must now be formulated more precisely . in principle , we wish that @xmath46 is piecewise continuous and that the distance between discontinuities is larger than a minimal energy difference @xmath0 . as we must work with a finite number @xmath92 of points @xmath93 , we only have a discrete representation of @xmath46 and we have to formulate a `` continuity '' condition for the discrete set of variables as well . therefore , we require that the distance between two significant extrema is larger than a parameter @xmath94 . two neighboring extrema at energy @xmath95 and @xmath96 are significant if there relative height difference is larger than a parameter @xmath97 , @xmath98 this condition can easily be formulated for the parameters @xmath91 . the minimal extremum distance @xmath0 must be chosen carefully . it should be smaller than the distance between actual singularities in the spectrum @xmath46 but a too small value allows many artificial peaks in a deconvolved spectrum . in practice , we have found that we can obtain reasonable solutions to the blind deconvolution problem which look piecewise smooth using @xmath99 . in all examples discussed in this paper every local extremum is considered to be significant ( i.e. , we have used the precision of floating - point arithmetic @xmath100 ) . the cost function is minimized iteratively . iterations are repeated until the procedure converges . each iteration consists in two steps . in the first step the cost function @xmath101 is minimized with respect to each variable @xmath102 successively . this minimization under constraint does not present any difficulty as @xmath101 is a second - order polynomial in each variable @xmath103 . in the second step , we first find the positions @xmath104 of all significant extrema pairs in @xmath105 which are separated by less than a distance @xmath0 . then we interpolate the data @xmath105 linearly from @xmath106 to @xmath107 to smooth out the spectrum around the extrema . in doing so we take care to preserve the total spectral weight @xmath108 the search for extrema and their smoothing is repeated until there is no more close significant extrema in @xmath109 . then we start the next iteration . by design the first step results in a decrease of @xmath101 . the second step nearly always results in an increase of @xmath101 . without the second step , however , we would perform an under - determined ( @xmath110 ) deconvolution devoid of any regularization mechanism and thus obtain a completely useless result . typically , we observe a rapid and monotonic decrease of @xmath101 in the initial iterations followed by a saturation or oscillations in further iterations . therefore , we monitor the changes in the parameters @xmath103 and the normalized cost function @xmath111 to determine converged configurations @xmath71 . convergence requires typically @xmath112 to @xmath113 iterations depending on the quality of the ddmrg data and the complexity of the spectrum . finally , the solution @xmath91 can be smoothened using a narrow lorentzian distribution to obtain a continuous function @xmath114 with @xmath115 . alternatively , we can use a gaussian distribution @xcite of width @xmath116 . the second approach yields sharper ( real or artificial ) features because the tail of a gaussian distribution decreases faster than that of the lorentz distribution . as our minimization problem possesses many local minima , the final results @xmath71 depend somewhat on the criteria for convergence . however , if the final smoothening function is broad enough , the differences are canceled out . if the ddmrg data ( [ eq : dmrg - data ] ) are not evenly distributed in the interval @xmath59 $ ] or if they already exhibit numerous close extrema , it is useful to regularize them before starting the deconvolution iterations using an interpolation and the smoothening procedure described above . the computational effort required by this procedure is negligible compared to the computational cost of the ddmrg simulations yielding the original data . ( our code in the programming language c contains less than 400 lines of instructions and the deconvolution of one spectrum takes less than 30 minutes on a single cpu . ) therefore , we have not bothered to optimize the algorithm . nevertheless , it should be implemented in such a way that it only requires @xmath117 operations rather than the @xmath118 operations of a straightforward implementation . the method described here can be generalized in several ways . for instance , it is possible to use a variable spacing @xmath119 of the ddmrg data points , such as a finer mesh close to sharp spectrum features . however , this does not seem to improve the results in practice because the broadening parameter @xmath36 , not @xmath119 , is the limiting scale . a generalization to variable @xmath36 and @xmath119 , as proposed in ref . for quantum impurity problems , should also be possible but we have not tested it yet . to introduce information about the variation of the spectrum with @xmath36 one could combine ddmrg data obtained for different values of @xmath36 by defining an overall cost function as the sum of the cost functions for each @xmath36 . these generalizations will be tested in future works . as an illustration of our deconvolution procedure we discuss its application to the dos of one - dimensional paramagnetic mott insulators . the nature of mott insulators is a long - standing open problem in the theory of strongly correlated quantum systems . @xcite in a paramagnetic mott insulator quantum fluctuations or frustration of the antiferromagnetic spin exchange coupling prevents the formation of a long - range magnetic order . experimentally , non - magnetic mott insulators have been found in layered organic insulators @xcite as well as in quasi - one - dimensional cuprate chains @xcite and ladders @xcite . despite decades of extensive research the properties of mott insulators , in particular their single - particle dos , are still poorly understood and thus actively investigated . the half - filled hubbard model @xcite with repulsive on - site interaction @xmath2 is a basic lattice model for describing mott insulators and the mott metal - insulator transition . here we consider the case of the one - dimensional hubbard model , which is exactly solvable by bethe ansatz . @xcite at half- filling it describes a paramagnetic mott insulator with a charge gap ( mott - hubbard gap ) @xmath120 for @xmath121 . however , the dos can not be calculated directly from the bethe ansatz . all ddmrg spectra used here have been calculated with a variable number of density - matrix eigenstates kept ( up to 512 ) to reach a discarded weight lower than @xmath122 and to check dmrg truncation errors . typically , convergence was reached after three sweeps for each frequency interval of size @xmath36 . the ddmrg method is presented in detail in ref . . for @xmath6 the exact dos of the tight - binding chain in the thermodynamic limit is @xmath123 for @xmath124 while @xmath125 vanishes for larger @xmath126 . for finite coupling @xmath127 the spectrum consists in two symmetric hubbard bands separated by the gap @xmath128 . low - order strong - coupling perturbation theory @xcite predicts a square - root divergence at the dos threshold for @xmath129 , namely @xmath130 for @xmath131 and @xmath132 otherwise with @xmath133 . because of this strong - coupling result and the result of the hartree - fock ( hf ) approximation it has often been assumed that the dos of one - dimensional mott insulators exhibits a square - root divergence at the spectrum onset @xmath134 like in a one - dimensional band insulator . indeed , in the unrestricted hf approximation the one - dimensional half - filled hubbard model is an antiferromagnetic mott insulator for @xmath121 . its dos is given by @xmath135 for @xmath136 and vanishes otherwise . here @xmath137 is the hf gap . for @xmath6 , @xmath138 and this dos reduces to the dos of the tight - biding chain ( [ eq : dos - tb ] ) . for @xmath139 , @xmath140 and the hf dos shows a square - root divergence at the onset of the spectrum . however , in the weak - coupling limit @xmath141 a field - theoretical analysis @xcite predicts that the dos of one - dimensional mott insulators is constant above the threshold energy @xmath142 , @xmath143 at least for @xmath144 . thus there is a discrepancy between the field - theoretical and strong - coupling predictions for the behavior of @xmath18 just above the threshold energy @xmath142 . ( color online ) dos of a tight - binding chain : ( a ) ddmrg spectrum for a @xmath145-site chain with a lorentzian broadening @xmath146 ( red dashed line ) and result of our deconvolution method with a gaussian broadening @xmath147 ( black line ) . ( b ) enlarged view close to the singularity at @xmath148 on a double logarithmic scale : exact results ( black long - dashed line ) , ddmrg data ( red short - dashed line ) , and the deconvolved spectra @xmath109 for @xmath149 ( black solid line ) and @xmath150 ( blue dash - dot line).,title="fig:",scaledwidth=48.0% ] ( color online ) dos of a tight - binding chain : ( a ) ddmrg spectrum for a @xmath145-site chain with a lorentzian broadening @xmath146 ( red dashed line ) and result of our deconvolution method with a gaussian broadening @xmath147 ( black line ) . ( b ) enlarged view close to the singularity at @xmath148 on a double logarithmic scale : exact results ( black long - dashed line ) , ddmrg data ( red short - dashed line ) , and the deconvolved spectra @xmath109 for @xmath149 ( black solid line ) and @xmath150 ( blue dash - dot line).,title="fig:",scaledwidth=48.0% ] figure [ fig : tb](a ) shows the dos of a tight - binding chain calculated with ddmrg and the result of our deconvolution procedure . the ddmrg spectrum has been calculated in the middle of an open chain with @xmath151 sites using a broadening @xmath146 , which is just broad enough to hide its discreteness . we see that the square - root divergences at @xmath152 have been smoothed into two broad peaks and that there is substantial spectral weight at energies @xmath153 . the deconvolved dos has been determined from these same ddmrg data using a minimal extremum distance @xmath154 and a final gaussian broadening with @xmath155 . we see now that the singularities at @xmath152 are clearly visible as sharp peaks and that there is not any spectral weight at @xmath153 . overall the deconvolved dos is in excellent agreement with the exact spectrum in the thermodynamic limit ( [ eq : dos - tb ] ) . in particular , we do not observe any unphysical artefact such as negative spectral weight . however , in fig . [ fig : tb](a ) we observe two shoulders in the deconvolved dos at energies @xmath156 , which are not present in the exact solution ( [ eq : dos - tb ] ) . an enlarged view close to the singularity at @xmath157 is shown in fig . [ fig : tb](b ) on a double logarithmic scale . we see that the ddmrg data agree with the exact result only at some distance from the singularity . in this figure we also show deconvolved spectra @xmath109 for two different values of the normalized cost function @xmath158 . clearly , they reproduce the square - root divergence at @xmath157 much better than the original ddmrg data . the overall divergent behavior is visible on a broader energy scale for the smaller value of @xmath158 but we see that the reduction of the cost function is also accompanied by stronger oscillations around the exact result . these oscillations correspond to the shoulder seen in fig . [ fig : tb](a ) . the occurrence of artificial shoulder - like structures is the main drawback of our deconvolution procedure . any deconvolution method magnifies the noise ( numerical errors ) which is present in the original data . this the main issue that existing methods try to solve in different ways . @xcite we have systematically tested our deconvolution procedure using exact results for non - interacting systems and purposely adding random numerical errors . we have found that by preventing the formation of local maxima in the deconvolved spectrum our procedure allows us to control the noise magnification only partially . unfortunately , it can not handle extrema ( @xmath159 oscillations ) in the spectrum derivative . thus the magnified noise shows up as shoulder - like structures ( but not as local maxima , discontinuities , or sharp angles ) on an energy scale @xmath0 and gives a rough appearance to some deconvoled spectra presented here . in principle , we should be able to correct this deficiency with a higher - order interpolation procedure in the smoothening step or with a smoothening of the derivative of @xmath46 ( i.e. , the finite differences between the parameters @xmath103 ) . however , we have not yet succeeded in developing a practical algorithm based on these ideas . ( color online ) ( a ) dos of the one - dimensional half - filled hubbard model at @xmath160 calculated with ddmrg in a 128-site chain using a broadening @xmath146 ( red dashed line ) and the result of our deconvolution procedure ( black line ) with a minimal extremum distance @xmath161 and a gaussian broadening @xmath162 . the vertical dashed lines show the exact position of the dos threshold calculated from the bethe ansatz solution . ( b ) enlarged view of the same data around the dos threshold for @xmath163 . additionally , the result of our procedure with @xmath164 ( blue dot - dash line ) and of a deconvolution with linear regularization @xcite ( green dots ) are also shown.,title="fig:",scaledwidth=48.0% ] ( color online ) ( a ) dos of the one - dimensional half - filled hubbard model at @xmath160 calculated with ddmrg in a 128-site chain using a broadening @xmath146 ( red dashed line ) and the result of our deconvolution procedure ( black line ) with a minimal extremum distance @xmath161 and a gaussian broadening @xmath162 . the vertical dashed lines show the exact position of the dos threshold calculated from the bethe ansatz solution . ( b ) enlarged view of the same data around the dos threshold for @xmath163 . additionally , the result of our procedure with @xmath164 ( blue dot - dash line ) and of a deconvolution with linear regularization @xcite ( green dots ) are also shown.,title="fig:",scaledwidth=48.0% ] figure [ fig : onedim4](a ) shows the single - particle dos of the half - filled one - dimensional hubbard model at @xmath160 calculated with ddmrg and the result of our deconvolution procedure . the ddmrg spectrum has been computed @xcite in the middle of an open chain with @xmath151 sites and a broadening @xmath146 . the deconvolved spectrum has been obtained from these ddmrg data using a minimal extremum distance @xmath165 and a final gaussian broadening @xmath166 . the effects of the broadening are clearly visible in the ddmrg data . for instance , although one can recognize the opening of the mott - hubbard gap @xmath128 , spectral weight is clearly visible inside the gap . a point - wise analysis @xcite of the scaling for @xmath47 with @xmath167 is required to confirm that the spectral weight jumps from @xmath168 to a finite value at the onset @xmath169 and that the gap width agrees with the exact results @xmath170 calculated with the bethe ansatz . @xcite however , the behavior for @xmath171 remains uncertain because of the relatively large broadening used in the ddmrg calculation . on the contrary the deconvolved dos clearly shows a gap with the step - like onset ( [ eq : dos - ft ] ) predicted by field theory @xcite at the position @xmath172 given by the bethe ansatz solution . we obtain similarly unambiguous results for @xmath127 up to @xmath173 ( see below ) . therefore , our numerical investigation confirms the field - theoretical prediction @xcite for the onset of the dos in one - dimensional mott insulators . the superiority of the deconvolved spectrum over the original ddmrg data is even more obvious in fig . [ fig : onedim4](b ) which shows an enlarged view of the dos around @xmath172 . in this figure we also show the result of a deconvolution of the ddmrg data with a standard linear regularization method @xcite . ( note that this method yields negative spectral weights for some energies @xmath174 but we show the positive parts only . ) we see that the result of the deconvolution procedure proposed in this work is much superior to that of the standard one , which is too blurred to allow us to determine the true form of the dos at the onset @xmath172 . in addition , fig . [ fig : onedim4](b ) shows the result of our deconvolution procedure for a minimal extremum distance @xmath175 which is deliberately too small . in that case , artificial oscillations on energy scales @xmath176 are clearly visible in the deconvolved spectrum . ( color online ) upper hubbard band in the dos of the one - dimensional half - filled hubbard model at @xmath177 calculated with ddmrg in a 64-site chain using a broadening @xmath178 ( red dashed line ) and the result of our deconvolution procedure ( black line ) with a gaussian broadening @xmath179 . , scaledwidth=48.0% ] in the strong - coupling limit @xmath180 our results are less conclusive . for instance , we show the ddmrg and deconvolved upper hubbard and for a very strong coupling @xmath177 in fig . [ fig : onedim40 ] . the ddmrg spectrum has been calculated in the middle of an open chain with @xmath181 sites using a broadening @xmath178 . the deconvolved spectrum has been obtained from these ddmrg data using a minimal extremum distance @xmath182 and a final a gaussian broadening @xmath179 . two broad peaks are clearly visible at energies @xmath183 as predicted for the strong - coupling limit . however , the widths and heights of these peaks after deconvolution are not compatible with the square - root divergences ( [ eq : dos - scpt ] ) predicted by the low - order strong - coupling expansion . actually , higher - order corrections indicate @xcite that some spectral weight is present below the peak at @xmath184 on a scale set by the effective spin exchange coupling @xmath185 . this width is compatible with our deconvolved spectra for @xmath186 . thus our numerical results agree at least qualitatively with strong - coupling perturbation theory and suggest that the square - root divergences in the dos ( [ eq : dos - scpt ] ) is an artifact of a truncated strong - coupling expansion . nevertheless , for strong coupling such as @xmath177 we are not able to determine the shape of the dos at the onset @xmath172 . in particular , it is not clear if the field - theoretical prediction ( [ eq : dos - ft ] ) is still valid . indeed , if the distance @xmath187 between onset at @xmath169 and peak at @xmath188 becomes smaller than the minimal extremum distance @xmath0 , we can no longer distinguish both structures in the deconvolved spectrum . in practice , the distance @xmath0 must be comparable to the broadening @xmath36 of the ddmrg data to obtain piecewise smooth spectra . therefore , although we can reduce the broadening of isolated spectral structures by several orders of magnitude , we can not resolve distinct spectral features on a scale lower than the original broadening of the ddmrg data . this is a limitation of our deconvolution method that one has to keep in mind . ( color online ) deconvolved dos of the half - filled hubbard model for energies @xmath189 for @xmath6 ( dashed line ) , and from left to right @xmath190 ( red line ) , @xmath191 ( blue line ) and @xmath192 ( green line ) . the result for @xmath177 is shifted to the left by @xmath193 . , scaledwidth=48.0% ] finally , fig . [ fig : onedim ] recapitulates the evolution of the dos as a function of the interaction strength @xmath5 . as the spectrum is symmetric @xmath194 , we show only the deconvolved spectra for positive energies ( i.e. , the upper hubbard band ) . at @xmath6 the spectrum consists in a single band with two clearly visible square - root singularities at the band edges @xmath195 . for weak coupling @xmath127 the band splits into two symmetric hubbard bands separated by a gap @xmath128 , which agrees perfectly with the charge gap calculated from the bethe ansatz solution . at the dos onsets @xmath196 the spectrum exhibits the step - like behavior ( [ eq : dos - ft ] ) predicted by field theory . @xcite there is an apparent plateau between the onset and a strong first peak , which evolves from the square - root singularities at @xmath195 for @xmath6 . additionally , we observe substantial spectral weight and a small second peak at higher excitation energy . however , for weak enough @xmath127 most of the spectral weight lies between the spectrum onset and the first peak . as @xmath127 increases ( compare the spectra for @xmath160 and @xmath1 in fig . [ fig : onedim ] ) , the spectrum and all its features shift to higher excitation energy and the spectral weight becomes more concentrated between the visible peaks . in addition , we note that the separation between onset energy @xmath142 and the strong first peak becomes systematically smaller until it is no longer resolvable with our method , the peak separation increases monotonically from about @xmath197 for @xmath198 to approximately @xmath199 for @xmath200 , and the strength of both peaks become more equal . comparing the dos with the momentum - resolved spectral function and the bethe ansatz dispersion ( see figs . 4 and 5 in ref . ) we note that the strong first peak corresponds to the edge of the spinon branch at momentum @xmath201 , the weak second peak corresponds to the edge of the holon branch at @xmath202 , and the upper edge of the dos spectrum coincide with the edge of the single spinon - holon continuum . finally , we do not observe any spectral weight outside the first lower and upper hubbard bands and these two bands account for the full spectral weight ( [ eq : sum ] ) . thus we conclude that higher - energy hubbard bands do not carry any spectral weight in the bulk single - particle dos . we have presented a blind deconvolution procedure which allows us to obtain piecewise smooth spectral functions for infinite - size systems from the ddmrg spectra of finite systems . it involves a trade - off between the agreement of the deconvolved spectrum to the original ddmrg data and the piecewise smoothness and positivity of spectral functions . in practice , the method reduces to a least - square optimization under non - linear constraints which enforce the positivity and piecewise smoothness . we have tested this deconvolution method on many spectra which are known exactly in the thermodynamic limit , such as the single - particle density of states and the optical conductivity of correlated one - dimensional insulators . @xcite we have found that our method works well for several kinds of singularities ( e.g. power - law band edges , steps , excitonic peaks ) in piecewise smooth spectra . in particular , it allows us to reduce the broadening by orders of magnitude and even to substitute the lorentzian broadening by a gaussian one . its main drawback is the frequent appearance of artificial shoulder - 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we are poised on the threshold of unprecedented technical growth in wide - field time domain astronomy , where ground - based observations yield very precise measurements of stellar brightness from high - volume data streams . so far , wide - field time - series surveys has been spearheaded by relatively small telescopes since they are supported by large field of view ( fov ) instruments operating with high duty cycle ( see @xcite for a summary of optical variability surveys ) . within the last decade , the advent of large mosaic ccds has facilitated the coverage of large sky area even for large - aperture telescopes ( e.g. , mmt megacam : @xcite ; eso very large telescope omegacam : @xcite ; subaru suprime - cam : @xcite ; chft megacam : @xcite ; iptf : @xcite ) . although these facilities are generally devoted to imaging surveys , researchers are attempting to utilize them for short- and long - term variability surveys with short - cadence exposures ( e.g. , @xcite ) . such wide - field imaging systems have enabled us to observe hundred of thousands of target stars simultaneously and also to detect various variability phenomena . a remarkable thing about these surveys is that the fraction of variable sources increases as the photometric precision of the survey improves . for this reason , it is important to improve the accuracy in photometry . another key issue in wide - field time - series photometry is the removal of temporal systematics from a single image frame or several consecutive image frames . it has recently become known that systematic trends in time - series data can be different and localized within the image frame when the fov is large . such spatially localized patterns may be related to subtle point spread function ( psf ) differences and sky condition within the detector fov ( e.g. , @xcite ) . as these patterns change in time , we can see how the temporal variations of systematic trends affect the brightness and shape of light curves directly . the time - scale of systematic variation is sometimes comparable to short - term variability , such as transits or eclipses , and in some cases even long - term variability . thus , it is often difficult to identify and characterize true variabilities . in this paper , we introduce a new photometry procedure , called multi - aperture indexing , which is suited to analyzing well - sampled wide - field images of non - crowded fields with a highly varying psf , such as those produced by wide - field mosaic imagers on large telescopes . we apply this procedure to archival imaging data from the mmt / megacam transit survey of the open cluster m37 ( hartman et al . 2008a ) , demonstrating a substantial improvement over the existing photometry . section 2 describes the mmt imaging database and identifies problems in the existing photometry which motivated the development of our new methods . section 3 describes the multi - aperture photometry that utilize newly defined contamination index and carefully tuned calibration procedures , including the results of the basic tests to validate our approach . section 4 gives an in - depth discussion about systematic trends in time - series data and suggests an efficient way for identifying , measuring , and removing spatio - temporal trends . section 5 describes the effects of new calibration on period search , and we summarize our main results in the last section . @xcite have conducted a study to find neptune - sized planets transiting solar - like stars in the rich open cluster m37 . the observing strategy was carefully designed for a transiting planet search by several considerations ( e.g. , the reliability of exposure time per frame , the effects of pixel - to - pixel sensitivity variations , and sensitivity of filter ) . their work did not reveal any transiting planets , but it did provide a rare opportunity to explore photometric variability at relatively high temporal resolution with 3090 s. @xcite discovered 1430 new variable stars , including very short - period eclipsing binaries ( e.g. , v37 , v706 , v1160 ) and @xmath1 sct - type pulsating stars ( e.g. , v397 , v744 , v1412 ) . we used the same data set on the open cluster m37 . a detailed discussion of the observations , original data reduction , and light curve production is described in @xcite . the data archive consists of approximately 5000 @xmath2-filter images taken over 24 nights with the wide - field mosaic imager ( megacam ) mounted at the @xmath3 cassegrain focus of the 6.5 m mmt telescope . note that megacam is made up of 36 2048 @xmath4 4608 pixel ccd chips in a @xmath5 pattern , covering a 24@xmath6@xmath424@xmath6 fov @xcite . this instrument has an unbinned pixel scale of @xmath7 , but it was used in @xmath8 binning mode for readout . the observation logs are summarized in table 1 of @xcite . in brief , the @xmath9-band time - series observations were undertaken between 2005 december 21 and 2006 january 21 , with a median fwhm of @xmath10 arcsec . exposure times are chosen to keep an @xmath11 mag star as close to the saturation limit , which is expressed as a function of seeing conditions . with an average seeing @xmath00@xmath12.89 on images , the quality of the images is good to achieve high - precision light curves ( less than 1% rms value ) down to 20 . in addition to the imaging data set , this database includes light curve data sets for a total of 23,790 sources detected in a co - added reference image . theses light curves are obtained by the image subtraction technique using a modified version of isis software ( @xcite ; @xcite ) . as shown in figure [ fig : fig1 ] , however , the raw light curves from the original image subtraction procedures exhibit many unusual outliers , and more than @xmath015% of data get rejected by a simple filtering algorithm after cleaning procedures . in practice , brutal filtering that is often applied to remove outlying data points can result in the loss of vital data , with seriously negative impact to short - term variations such as flares and deep eclipses . we also find that the image subtraction technique often resulted in measurement failures from several frames due to poor seeing or tracking problem . after removing these bad frames , it leads to loss of additional @xmath05% data points from most light curves . in order to overcome this problem , we have re - processed the entire image database with new photometric reduction procedures . 3-@xmath13 control limits ( dotted lines ) . these light curves contain outlier points that significantly increase the rms scatter of the raw light curves . ] we followed the standard ccd reduction procedures of the bias correction , overscan trimming , dark correction , and flat - fielding as described in @xcite . the individual ccd frames were calibrated in iraf , using the mosaic data reduction package megared . the first step is to correct the pixel - to - pixel zero - point differences that are usually described by the sum of a mean bias level and a bias structure . as the bias frames were not separately taken during the time of the observations , the mean bias level was subtracted from each image extension using an overscan correction and so we can not remove any remaining bias structure from all overscan - subtracted data frames . according to description in matt ashby s megacam reduction guide , the bias structure can be very significant in some small regions such as the portions of the arrays close to the readout leads . the dark currents are normally insignificant for megacam so that corrections are not needed even for long exposures . the next step is to correct pixel - to - pixel variations in the sensitivity of the ccd ; we used the program ` domegacamflat2 ` . this program determines the scaling factors to correct the gain difference between the two amplifiers of each chip by finding the mode in the quotient of pixels to the left and right of the amplifier boundary , and then flattens each of the frames with a master flat field frame . it is worth mentioning that the sky conditions were rarely photometric during the observing run , with persistent light cirrus for most of the nights . therefore it was only possible to obtain twilight sky flats on a handful of nights ( dome flats were not possible ) . we removed bad pixels using the megacam bad pixel masks distributed with the megared package . the values of bad pixels are replaced with interpolated value of the surrounding pixels using the iraf task ` fixpix ` . the numerous single pixel events ( cosmic - rays ) were identified and removed using the lacosmic package @xcite . the megacam data already have a rough world coordinates system ( wcs ) solution that is based on a single value of the telescope pointing . to update these with a more precise solution , we applied astrometric correction to each ccd in the mosaic using the wcstools imwcs program @xcite . the new solution is derived by minimizing the differences between the r.a . and decl . positions of sources in a single ccd chip and their positions listed in the 2mass point source catalog @xcite . the resulting astrometric accuracy is typically better than 0@xmath14.1 rms in both r.a . and decl . typically , a point or extended source detection algorithm is applied to each frame independently and it always requires criteria for what should be regarded as a true detection . in obtaining the pixel coordinates for all objects in the m37 fields , this procedure often misses some objects when the detection threshold approaches the noise level . also it needs a substantial effort to match the objects that are detected in only some of the frames . our approach is as follows : a complete list of all objects is obtained from a co - added reference frame , and then the photometry is performed for each frame using the fixed positions of the sources detected on the reference . since the relative centroid positions of all objects are the same for all frames in the time series , we can easily place an aperture on each target and measure the flux even for the stars at the faint magnitude end . we constructed the reference frame for each chip from the best seeing frames using the swarp software . benefiting from a highly accurate astrometric calibration of input frames , we were able to improve the quality of co - added images . in the swarp implementation , the pixels of each frame were resampled using the lanczos3 convolution kernel , then combined into the reference frame by taking a median or average . after this was done , sources were detected and extracted on the reference frame using the sextractor software @xcite . when configured with a lower detection threshold , sextractor extracts the number of spurious detections ( e.g. , diffraction spikes around bright stars , or outer features of bright galaxies ) . these false detections were removed by careful visual inspection for each chip . the final catalog contains a total of 30,294 objects including both point and extended sources . prior to the photometry , the initial centroid coordinates of the target objects for each frame were computed by using the wcstools sky2xy routine @xcite . the stored world coordinate system for each frame is used to convert the ( r.a . , decl . ) coordinates from the master source catalog to the @xmath15 pixel locations . however , actual positions of objects for each frame can be slightly moved from its original locations depending on the focus condition of instrument , seeing condition , and the signal - to - noise ratio ( s / n ) in the individual observations . these types of positioning errors ( i.e. , centroid noise ) will lead to the internal error for a photometric measurement that results from placement of the measuring aperture on the object being measured . the situation gets worse for faint stars because the centroid position of them is itself subject to some uncertainty . the fractional error in the measured flux as a result of mis - centering is given by:@xmath16 where @xmath17 is the positioning error , @xmath18 is the radius of aperture , @xmath13 is the profile width for a source with a gaussian psf , and total flux @xmath19 ( see appendix a in irwin et al . , 2007 for details ) . this expression shows that if we set the aperture radius equal to the fwhm ( @xmath18=2.35-@xmath13 ) , even small differences in placement of the aperture ( e.g. , @xmath17=0.1-@xmath13 ) may increase the uncertainty in the flux measurements ( @xmath20 1 mmag ) . thus , accurate centroid determination is important to achieve the high - precision photometry . following the windowed centroid procedure in the sextractor , a refined centroid of each object is calculated iteratively . on average , the rms uncertainty in the coordinate transformation using the wcs information was @xmath21 pixels for the bright reference stars . after the coordinate transformation from sky to @xmath22 , however , the centroid coordinates ( @xmath23 , @xmath24 ) are slightly misaligned from their actual ones ( @xmath25 , @xmath26 ) . the refined centroid values ( @xmath27 , @xmath28 ) are used only if the maximum displacement is at least less than 1.5 pixels . this condition prevents arbitrary shifting of a source centroid , especially for faint stars . we estimated a local sky background by measuring the mode of the histogram of pixel values within a local annulus around each object , which is suitable choice for our uncrowded field ( less than @xmath01000 stars per chip ) . this process is a combination of @xmath29-@xmath13 clipping and mode estimation . the background histogram is clipped iteratively at @xmath303-@xmath13 around its median , and then the mode value is taken as : @xmath31 it represents the most probable sky value of a randomly chosen pixel in the sample of sky pixels @xcite . for relatively crowded regions , we utilized a background map created by sextractor package using a mesh of @xmath32 pixels and a median filter box of @xmath33 pixels . this map is used to confirm the properness of individual sky values from annulus estimates . modern data reduction techniques aim to reach photon noise limit and minimize systematic effects . for example , differential photometry technique can be achieved better than 1% precision for brighter stars ( e.g. , @xcite ) , and the deconvolution - based photometry algorithm leads to the minimization of systematic effects in very crowded fields ( e.g. , @xcite ) . however , conventional data reduction methods often fail to handle various artifacts in wide - field survey data . we present below a new photometric reduction method for precise time - series photometry of non - crowded fields , without the need to involve complicated and cpu intensive process ( e.g. , psf fitting or difference image analysis ) . our photometry is similar to standard aperture photometry , except in that we compute the flux in a sequence of several apertures and then determine the optimum aperture individually to each object at each epoch . this multi - aperture photometry is an efficient way to determine the optimum aperture size that gives the maximum s / n for a flux measurement . the maximum s / n is not necessarily at the same aperture for all objects , and it can be obtained from a relatively small aperture @xcite . this _ photometric _ aperture is to achieve the optimal balance between flux loss and noises based on a relationship derived from the ccd equation ( see @xcite ) . figure [ fig : fig2 ] shows how the optimum apertures vary with the stellar magnitude . there is an obvious trend of decreasing aperture sizes with increasing magnitudes down to the faint magnitude limit in the example frame . once we measure the flux of each object with the optimum aperture , we need to apply the aperture correction for small apertures . the aperture correction terms are estimated from the growth curve analysis of selected isolated bright stars ( i.e. , reference stars ) . the average curve - of - growth for each frame is calculated by measuring the difference in magnitude between different pairs of apertures ( up to 10 pixels aperture radius ) and then an automatic correction is applied to all objects for each photometric aperture . the use of a common aperture correction for each ccd assumes that there is no variation in the correction across the ccd . this flux correction method gives nearly the same brightness within the measurement uncertainties for all apertures . any psf variation across the ccd causes systematic errors , however , and we deal with this in section 3.4 and section 4 . 100 thumbnail images of the target star . _ middle panels _ : normalized s / n as a function of aperture size . _ bottom panels _ : aperture corrected magnitude as a function of aperture size . the arrows represent the peak locations in the aperture - s / n diagram ( see text for details ) . ] we performed the multi - aperture photometry based on the concentric aperture photometry algorithm in daophot package @xcite , using several circular apertures ( up to 10 pixels aperture radius ) with a fixed sky annulus from 35 to 45 pixels . the initial results of multi - aperture photometry are stored in ascii - format photometry tables , including the date of the observations ( mjd ) , the pixel @xmath15 coordinates , the aperture - corrected magnitudes with errors for each aperture , the sky values and its errors . figure [ fig : fig3 ] shows the details of the multi - aperture photometry for one star at different epochs . in the former two epochs , the photometric apertures can be properly selected by the s / n cuts , while in the latter two epochs , s / n increases for lager apertures . this unusual behavior is due to contamination by a moving object . we automatically identifies similar unusual cases by the method of aperture indexing . -filter light curve of same star ( id=10213 ) as shown in figure [ fig : fig3 ] . bottom panels show the multi - aperture indexing scheme . the @xmath34-axis is the aperture size and @xmath35 @xmath36-axis is the differential magnitude between pairs of apertures @xmath37(=@xmath38 ) . we can see whether and at what aperture the differential magnitude ( solid lines ) begins to deviate from the model curve ( dashed lines ) for each epoch . ] our multi - aperture indexing method is similar to the basic concept of the discrete curve - of - growth method @xcite . each object is indexed based on the difference in aperture - corrected magnitude between pairs of apertures @xmath37(=@xmath38 ) with mean trend for stars of similar brightness ( see solid and dotted lines in bottom panels of figure [ fig : fig4 ] , respectively ) . the aperture with a 10 pixel radius is used as the fixed reference aperture . the mean trend is determined by computing the rms curve of the aperture correction values for all measured apertures , and used to evaluate whether magnitude at a given aperture significantly differs from the mean trend . since the rms value depends on the chosen magnitude interval , all stars are divided into groups according to their brightness in the individual frames . we determine the rms curve for each magnitude group using an iterative @xmath13-clipping until convergence is reached . objects lying within @xmath303-@xmath13 of the model curve are indexed as contamination - free , and those above @xmath303-@xmath13 as a contaminated source . figure [ fig : fig4 ] shows that multi - aperture indexing guides us to throw out some photometric measurements if they are discrepant from the mean trend . this approach also gives us a chance to recover a measurement that would be otherwise thrown out . the problematic aperture can be simply replaced by one of the smaller apertures if it is indexed as contamination - free . this help us make a full use of the information offered by the data . we present a new photometric calibration to convert the instrumental magnitudes onto the standard system , including a relative flux correction of the left and right half - region of each ccd chip . as mentioned in the section 3.1 , mmt / megacam shows the temporal variations in the gain between two amplifiers on each ccd , as well as between ccds that are part of the same mosaic . it may have been caused by unstable bias voltage of the ccd output drain which has a profound impact on the gain of the output amplifier . the level of readout noise is also unstable between two amplifiers . to correct for this effect , the photometric calibration needs to be performed individually for each amplifier region . we use a sufficient number of ( pre - selected ) bright isolated stars as standard stars and compute the relative flux correction terms . these terms were derived for each frame using the mean magnitude offset ( @xmath39 ) of standard stars ( @xmath40 ) with respect to corresponding magnitudes ( @xmath41 ) in the master frame chosen as an internal photometric reference @xmath42 where @xmath43 is the aperture - corrected instrumental magnitudes in other frames and @xmath44 is the photometric zero - points for the left- and right - side of each chip , respectively . to calculate the zero - points , we solve a linear calibration relation of the form : @xmath45 where @xmath46 and @xmath47 are standard magnitudes from the photometric catalog of m37 @xcite and @xmath48 is an airmass term . the fit is performed iteratively using a sigma - clipping method . figure [ fig : fig5 ] shows the difference in the magnitude offsets between the left- and right - side of each chip ( @xmath49 ) for the whole data set , which is within @xmath50 magnitude level for all 36 ccd chips . the histograms are normalized by the total number of data frames @xmath51 and are described by a gaussian function with slightly different mean values and shapes ( dashed line ) . we clearly see a significant variation in difference between a pair of magnitude offsets for all ccds . the photometric calibration for wide - field imaging systems is also affected by position - dependent systematic errors due to a psf variation across the fov ( e.g. , @xcite ) . we derive psf variations across the fov with the sextractor package . the change of the psf shapes in the image plane is represented by spatial distribution of psf fwhm values for several bright stars , with parameters of ` class\_star ` @xmath52 0.9 , ` magerr\_auto ` @xmath53 0.01 mag , and ` flags ` = 0 ( i.e. , isolated point sources with no contamination ) . note that the psf fwhm values are defined as the diameter of the disk that contains half of the object flux based on a circular gaussian kernel . figure [ fig : fig6 ] presents the variation of the psf fwhm as a function of distance from the image center for various seeing conditions . for each quadrant , the dashed lines represent the weighted spline approximation of the median value of each distance bin ( 1 arcmin ) . the result shows that the psf fwhm varies significantly as a function of position on the single - epoch image frames and variations are at the level of @xmath010% to 20% ( @xmath54 ) across the fov . as the field distortion is not negligible from the center of field to its edges , such variations limit the accuracy of stellar photometry . to address this issue , we perform a 2d polynomial fitting technique . for each frame , the correction terms are described by a linear or quadratic polynomial depending on the position @xmath55 only . @xmath56 where @xmath57 are the pixel coordinates of @xmath58 bright isolated stars , @xmath59 is the statistical weight in the fitting procedure , @xmath60 are the sets of polynomial coefficients for each aperture size , and @xmath61 are the difference in magnitude between the reference aperture and @xmath62 aperture , @xmath63 , at the position @xmath55 for each chip @xmath64 . we derived the optimal parameter values from a nonlinear least - squares fit using the levenberg marquardt algorithm and automatic differentiation , and choose between two models that best fit the data . figure [ fig : fig7 ] shows the field - dependent magnitude offsets and the distortion correction by 2d polynomial fitting method for one example mosaic ccd . for the outer and the central region of the mosaic , we compare the magnitude offsets between the reference aperture and the relatively smaller apertures as a function of @xmath55 coordinates . here @xmath34-axis is in the declination direction and @xmath36-axis is opposite to the right ascension direction . we find that the magnitude difference depends on position @xmath15 and is most discrepant in the outer part of the fov . this effect is usually more significant in the @xmath36-direction than in the @xmath34-direction , especially for the case of aperture photometry performed with small apertures . the correction for field - dependent psf variation reduces the initial @xmath010% variation ( gray lines ) to less than @xmath01% ( black lines ) . . ] we compare the photometric performance of the re - calibrated light curves with the non - de - trended archival light curves by means of the two representative measures : ( i ) the rms photometric precision @xmath65 , and ( ii ) the data recovery rate @xmath66 . the former is defined as the standard deviation of light curves around the mean value as a function of @xmath46 magnitude : @xmath67 where @xmath68 is the number of data points in each light curve , @xmath69 is the observed magnitude , and @xmath70 is the mean magnitude of the object @xmath71 , and the latter refers to the number of _ analyzed _ data frames normalized by the total number of observed data frames @xmath58 for each object . in typical cases , the data recovery rate should be near unity in the bright magnitude regime and decreases with magnitude for fainter objects . for comparison , we decided to select light curve samples which show either no significant variability or seeing - correlated variations induced by image blending . we remove all known variable stars from the sample list based on a new catalog of variable stars in m37 field ( s .- w . chang et al . 2015 , hereafter paper ii ) . to remove the light curves of blended objects , we use an empirical statistical technique to quantify the level of blending by looking for seeing - correlated shifts of the object from its median magnitude @xcite . @xmath72 where @xmath73 for light curve points @xmath69 with uncertainties @xmath74 , and @xmath75 is the same statistic measured with respect to a fourth order polynomial in fwhm fitted to the data . we adopt the value @xmath76 for the selecting light curves with no blending . the last selection criterion is that the light curves must exit both in the archive and our database . in the bottom panels of figure [ fig : fig8 ] and figure [ fig : fig9 ] , we plot the rms photometric precision of light curves for the two megacam ccd chips in the outer ( ccd 1 ) and central ( ccd 21 ) part of the fov , respectively . the black points show the rms values of the re - calibrated light curves , while the gray points are for the raw and filtered light curves in archive . the first impression from this comparison is that the typical rms scatter is overestimated from the raw light curves because of many outliers in the photometric data ( bottom left panels ) . for the better results , these light curves were filtered out in two steps : ( i ) clipping 5-@xmath13 outliers from each light curve and ( ii ) removing every data points that are outliers in a large number of light curves @xcite . in the second step , the outlier candidates are estimated by choosing a cutoff value for each ccd chip . the cutoff value is defined as the fraction of light curves for which a given image is a 3-@xmath13 outlier . this filtering was applied to remove bad measurements due to image artifacts or poor conditions , which were previously thought to be unrecoverable , but resulting data loss is up to 20% of the total number of data points ( bottom right panels ) . as shown in the top panels of the two figures , the data recovery rate for the re - calibrated light curves is close to 100@xmath77 over a wide range of magnitude and it appears to be more complete compared with the raw ( top left panels ) and filtered ( top right panels ) light curves . at bright magnitudes ( @xmath78 ) the data recovery rate does not reach 100@xmath77 because the exposure time was chosen to be saturated at a magnitude of @xmath11 . this comparison proves that our approach is a powerful strategy for improving overall photometric accuracy without the need to throw out many outlier data points . finally , we compare the light curves themselves for selected variable stars between the archive and our own . this comparison serves to illustrate how the photometric precision and data recovery rate of the time - series data affect the ability to address a variety of variability characteristics . figure [ fig : fig10 ] shows a direct comparison with the filtered light curves ( top panels ) and our re - calibrated light curves ( bottom panels ) for four variable stars . it is shown that our method recovers more data points ( black ) from the same data set of images . and @xmath36 coordinates , respectively , for selected ccd chips . the filled squares are relatively bright stars with @xmath79 mag , while the open squares are stars with @xmath80 mag . ] in order to further investigate possible systematics in our approach , we conducted psf - fitting photometry with daophot ii and allstar @xcite . for each mosaic frame , we select bright , isolated , and unsaturated stars to make the psf model varying quadratically with @xmath55 coordinates . after psf modeling , we run allstar to perform iterative psf photometry of all detected sources in the frame with initial centroids set to the same values used for our own photometry . we then calculated aperture corrections using the package daogrow @xcite after subtraction of all but psf stars , which creates aperture growth curves for each frame and then integrates them out to infinity to obtain a total magnitude for each psf star . the final step is to convert the instrumental magnitudes into the standard photometric system . for each frame , the initial zero - point correction is applied by correcting the magnitude offset with respect to the master frame . this places photometry for all frames on a common instrumental system . following the same procedure in section 3.3 , the photometric calibration is performed individually for each amplifier region . figure [ fig : fig11 ] shows the residual magnitudes and sky values between our multi - aperture photometry and the psf - fitting photometry as a function of position in the selected ccd chips . there are no position - dependent trends in the magnitude residuals . for the brighter stars with @xmath79 mag , the rms magnitude difference between the two methods is very small ( @xmath81 = @xmath82 and @xmath83 = @xmath84 , respectively ) , while for the relatively faint stars the rms difference is somewhat larger ( @xmath81 = @xmath85 and @xmath83 = @xmath86 , respectively ) . the results of this example indicate that we can reliably correct for the psf variations by our calibration procedures . meanwhile , our sky values are slightly higher than the allstar sky values , but not to the degree that can seriously affect photometric measurements . magnitude in the central region of the open cluster m37 ( left panel ) . the arrows indicate the three different magnitude levels from bright to faint in our sample shown in the right panels . note that the variable object ids are taken from the new variable catalog of the m37 ( see paper ii ) . ] figure [ fig : fig12 ] shows a comparison of the rms dispersion of the light curves obtained with our photometry with respect to the that of the psf - fitting photometry in the central region of the open cluster m37 . we only compare the light curves of non - blended objects as described in section 3.5 . our multi - aperture photometry does not reach the same level of precision as psf - fitting photometry for the faintest stars , while the psf - fitting approach results in poorer photometry for bright stars . as shown in the right panels of figure [ fig : fig12 ] , it is clear that our photometry tend to have smaller measurement errors with respect to the psf photometry for the bright stars . -axis indicate the corresponding timestamps in each frame . the deviations from the mean value , @xmath87 , are less than @xmath300.01 mag level ( with a rms values of 0.0021 , 0.0025 , and 0.0033 mag from the top panel down ) . these stars show a similar pattern of light variations over the observation span . ] from a visual inspection of the re - calibrated light curves in the same ccd chip , we found that some light curves tend to have the same pattern of variations over the observation span ( figure [ fig : fig13 ] ) . this kind of systematic variation ( i.e. , _ trend _ ) is often noticed in other studies . for example , the importance of minimizing known ( or unknown ) systematics have been recognized by several exo - planet surveys because planet detection performance can be easily damaged by them ( e.g.,@xcite ) . also space - based time - series data ( e.g. , corot and kepler ) are no exception to this behavior although it is completely free from systematics caused by the turbulent atmosphere . most of the raw light curves are affected by a secular ( or a sudden ) variation of flux without any obvious physical reason @xcite . in order to check the properties of temporal systematics , we examined the correlation coefficients as measure of similarity between two light curves @xmath47 and @xmath71 obtained from a single ccd chip ( ccd 1 ) . @xmath88 where @xmath89 is the flux of each star at time @xmath90 , @xmath58 is the total number of measurements , @xmath91 is the mean flux of each star , and @xmath13 is the standard deviation of @xmath89 . this comparison is a point - by - point comparison and is done for every pair of light curves in the data set . the resultant similarity matrix can be used to identify correlated pairs of light curves and to determine which light curve is least like all other light curves ( e.g. , @xcite ) . after that , we selected stars showing most systematics based on a hierarchical clustering method with the correlation coefficients ( see @xcite , for more details ) . figure [ fig : fig14 ] represents spatial distribution of the most prominent trend groups on the ccd plane ( top panel ) and its strongly correlated features determined by the weighted sum of normalized light curves ( bottom panel ) . there are two interesting features in this figure : the first one is that each trend covers only a certain part of the sky area and the second one is that some portions of neighboring trends show different variation patterns even at the same moment in time ( shaded gray region in the figure ) . in particular , we found an anti - correlated variation for the trends between the group 1 ( @xmath92 ) and the group 2 ( @xmath93 ) , so we might expect to find possible noise sources that are responsible for these discrepancies . why the trends are different and localized within a single ccd frame is a subject of further study , but it is probably related to subtle changes in point spread function and sky condition within the detector fov . and @xmath94 ) of stars . the numbers on the @xmath34-axis indicate the corresponding timestamps between the frame 1 and the frame 400 . from the top to bottom , the panels show the average difference of trend , flux concentrations in which we applied a @xmath95 mag shift to the @xmath96 value , sky level , and psf fwhm value between the two groups . ] , for the original multi - aperture magnitude measurements as a function of aperture size , which is marked with a gray dots in all panels . a positive @xmath97 indicates that the photometric measurements with the corresponding aperture size , @xmath98 , are brighter than those of reference aperture @xmath99 , while a negative @xmath97 indicates vise versa . the dashed lines are the rms model profiles introduced in section 3.2.2 . there is a noticeable distinction between the group 1 ( _ top panels _ ) and group 2 ( _ bottom panels _ ) when looked at different concentration levels . but the trends are not fully explained by the difference patterns in @xmath97 because those correlated variations become small after the correction for the psf variation ( black dots ) . ] for these two groups , we consider a possible causal relationship between the systematic trends and average object / image properties . figure [ fig : fig15 ] shows the differences in trend , differential magnitudes , sky level , and psf fwhm between the two groups , respectively . in the top panel , we plot the magnitude difference in trends , which shows variations in the range of @xmath50 mag . we suspect that this may be due to the different concentration of star light between these two groups . it can be checked by using the magnitude difference @xmath100 , where @xmath99 and @xmath98 are the reference aperture and the relatively small aperture , respectively . in fact , we already know that there is a magnitude variation in @xmath37 depend on the aperture size due to the field - dependent psf variation ( see section 3.4 ) . for example , figure [ fig : fig16 ] shows the response of multi - aperture photometry for the two groups of stars at one epoch ( mjd = 53726.14817 ) before and after applying the distortion corrections . although the magnitude variation between the group 1 and the group 2 seems to have different behavior as a function of aperture size ( gray points ) , it is negligible after the removal of the psf variation ( black points ) . we also check that the possible contribution of sky level ( @xmath101 ) and psf fwhm differences ( @xmath102 ) to the systematic trends on the re - calibrated light curves . as mentioned by @xcite , sky over - subtraction may lead to the systematic trends as a function of the psf fwhm , the amplitude of which increase for fainter stars . the third and forth panels of figure [ fig : fig15 ] show the variation of the mean @xmath101 and @xmath102 , respectively . in our case , however , the form and amplitude of trends seem independent of sky level and psf fwhm . some other possible sources that may contribute to the observed systematic trends include : higher order variations in the psf shape beyond just the fwhm ; cross - talk from other amplifiers , or ghosts from bright stars undergoing multiple reflections within the optics ; non - uniform variations in the gain ; and unmodeled temporal atmospheric variations that are dominated by rayleigh scattering , molecular absorption by ozone and water vapor , and aerosol scattering @xcite . while we find a clear presence of trends that should be removed , we are not able to identify their exact cause . in order to reduce systematic effects in photometric time - series data , several methods were introduced ( e.g. , tfa : @xcite ; sys - rem : @xcite ; pdt : @xcite ; cda : @xcite ; pdc : @xcite ) . all of these algorithms share a common advantage that they work without any prior knowledge of the systematic effects . we use the pdt algorithm , which has been designed to detect and remove spatially localized patterns . by default , this algorithm works with a set of light curves that contain the same number of data points distributed in the same series of epoch . in many cases , however , missing data occur when no photometric measurements are available for some stars in a given observed frame . these missing data can be simply replaced by means , medians , or the values from the interpolation of adjacent data points in each light curve ( e.g. , @xcite ) . although using the replaced value is the easiest way to reconstruct the light curve to be analyzed , it is not appropriate if the time separation between two subsequent observations is too large . instead we use more straightforward approach by applying the pdt algorithm in two separate steps : ( i ) we construct the master trends from the subset of bright stars , and ( ii ) de - trend light curves of all stars with most similar master trend and matching time line . -axis indicate the corresponding timestamps between the frame 1 and the frame 2000 . the @xmath36-axis is @xmath46-filter magnitude ( normalized by its mean value ) . while the morphology of the two light curves in the left panels appear to be variable stars of some kind , these turn out to be non - variable after applying the photometric de - trending method . in the case of the right panels , all true variabilities are preserved from the raw light curves . from upper left to lower right : id=10032 , id=10039 , id=170088 , and id=170108 . ] we briefly describe the main procedure of our de - trending process following the algorithm derived by @xcite . we first select the template light curves from bright stars that show the highest correlation in the light - curve features . the total length of template light curves should be long enough to cover the whole time span of observations . in this step , we take a sequence of data points , @xmath103 as the reference time line . using the correlation matrix calculated from equation ( 7 ) , we extract all subset of light curves that show spatio - temporally correlated features ( i.e. , clusters ) . each cluster is determined by hierarchical tree clustering algorithm based on the degree of similarity . next , we obtain master trends @xmath104 for each cluster by weighted average of the normalized differential light curves , @xmath105 : @xmath106 @xmath107 @xmath108 where @xmath109 is the total number of light curves in each cluster @xmath64 , @xmath110 is mean value of _ i light curve , and @xmath111 is the standard deviation of @xmath105 . after determining the master trends , we de - trend the light curves of all stars with matching master trend and time line . we adjust the temporal sequence of measurements for the master trends @xmath112 by that of individual light curves to be de - trended @xmath113 . because each light curve is assumed as a linear combination of master trends and noise , we can determine the optimal solution by minimizing the residual between the master trends and the light curve : @xmath114 @xmath115 where @xmath116 is the total number of master trends and @xmath117 are free parameters to be determined by means of minimization of noise term @xmath118 . _ figure [ fig : fig17 ] shows examples of our light curves before and after removing the systematic trends . the algorithm we used for de - trending removes only the systematic variations that are shared by light curves of stars in the adjacent sky regions ( left panels ) , while all kinds of true variabilities are preserved ( right panels ) . ( indicated by arrow ) . the phased diagram of this candidate frequency is shown in the low - right panel . we see that the model fits well the overall pulsating variability ( red line ) . ] , but the subfigure of low - right panel is 200@xmath4200 thumbnail image of the target star . there is no potential sources of contamination to hamper the interpretation of the power spectrum . ] the usefulness of our photometry is tested for a set of variable stars . we immediately find abundant cases of improvements in the following three aspects : ( i ) refinement of the derived period , ( ii ) detection of a new significant peak in the periodogram , and ( iii ) separation of non - variable candidates where systematics in the light curves were mistaken for true variability . for each case , we compare light curves and power spectra for archival data and our data . for the first example , we show that a new photometric measurement and calibration allowed us to derive a much improved refinement of the light curves and of the derived periods ( figure [ fig : fig18 ] ) . we performed a lomb - scargle ( l - s : @xcite ) search of both archival and new light curves for periodic variable star ( v427 ) . the light curves are folded by the best - fit period of 5.4615 and 4.4158 days , respectively . we also calculated the false - alarm probability ( @xmath119 fap ) for each peak and its signal to noise ratio ( s / n ) : @xmath119 fap@xmath120 = @xmath121 , s / n@xmath120 = 43.2 for the archival data and @xmath119 fap@xmath122 = @xmath123 , s / n@xmath122 = 84.5 for the new data . since we can get better estimation with much lowered minimum fap value , our new period is the most likely result . in the bottom panels , the resulting amplitude spectrum was calculated with ` period04 ` package . since the archival data is more noisy than the new one , it is rather complicated to interpret the peaks of its power spectrum . for the second example , we show the newly discovered low - amplitude pulsating variable star ( figure [ fig : fig19 ] ) . we used the ` period04 ` package to find multiple pulsation periods . the whole process of identifying , fitting , and pre - whitening successive frequencies was repeated until no significant frequencies were found . we adopt a conservative approach in selecting the statistical significant peaks from the amplitude spectrum . a s / n amplitude ratio of 4.0 is a good criterion for independent frequencies , equivalent to 99.9% certainty of variability @xcite . while no clear periodicity was found in the archive data , our amplitude spectrum shows a clear excess of power centered at 22.3979 days@xmath124 with peak amplitudes of about 1 mmag ( s / n@xmath122 = 9.02 ) . the last example is the opposite case of the second . figure [ fig : fig20 ] shows that this object is unlikely to be a variable source because there is no evidence for any significant peaks , which indicates that the variations are mostly noise . extensive study on variabilities will be presented in paper ii . in this paper , we introduce a new time - series photometry with multi - aperture indexing and spatio - temporal de - trending techniques , together with complex corrections to minimize instrumental biases . we used the archival , high - temporal time - series data from one - month long mmt / megacam transit survey program . the re - calibration of the archival data has made several improvements as follows : ( i ) the photometric information derived from the multi - aperture indexing measurements is useful to obtain the best s / n measurement , but also to diagnose whether or not the targets are contaminated ; ( ii ) the resulting light curves utilize nearly 100% of available data and reach precisions down to sub mmag level at the bright magnitude end without the need to throw out many outlier data points , which makes it possible to preserve data points that show intrinsic sudden variations such as flare events ; ( iii ) corrections for position - dependent psf variations and de - trending of spatio - temporal systematic trends improve the quality of light curves ; and ( iv ) new photometry enables us to determine the variability nature and period estimate more accurately . while this study deals with a particular set of data from mmt , we find our approach has a potential for other wide - field time - series observations . multi - aperture indexing measurement is a powerful tool in isolating and even correcting various contaminations . spatio - temporal de - trending is also very useful in removing systematics caused by psf variation and even non - uniform extinction of thin clouds across the fov . @xcite proved this for different sets of archival survey data . this research was supported by basic science research program of the national research foundation of korea ( 2011 - 0030875 ) . i.b . is grateful for support from kasi - yonsei drc program of korea research council of fundamental science and technology ( drc-12 - 2-kasi ) . we thank the mmt / m37 survey team for the kind provision of raw image data . kim , d .- w . helped us to test the modified photometric de - trending algorithm . additionally , we would like to thank the anonymous referee for many helpful comments , including the suggestion to use psf photometry to further justify the successful removal of systematics ( section 3.5.2 ) . alard , c. 2000 , , 144 , 363 alard , c. & lupton , r. h. 1998 , , 503 , 325 becker , a. c. , wittman , d. m. , boeshaar , p. c. , et al . 2004 , , 611 , 418 bertin , e. & arnouts , s. 1996 , , 117 , 393 bianco , f. b. , protopapas , o. , mcleod , b. a. , et al . 2009 , , 138 , 568 boulade , o. , charlot , x. , abbon , p. , et al . 2003 , , 4841 , 72 bramich , d. m. & freudling , w. 2012 , , 424 , 1584 breger , m. , stich , j. , garrido , r. , et al . 1993 , , 271 , 482 chang , s .- w . & byun , y .- i . 2013 , in astronomical society of the pacific conference series , vol . 475 , astronomical data analysis software and systems xxii , ed . d. n. freidel , 41 everett , m. e. & howell , s. b. 2001 , , 113 , 1428 gillon , m. , pont , f. , moutou , c. , et al . 2007 , , 466 , 743 hartman , j. d. , stanek , k. z. , gaudi , b. s. , holman , m. j. , & mcleod , b. a. 2005 , , 130 , 2241 hartman , j. d. , gaudi , b. s. , holman , m. j. , et al . 2008a , , 675 , 1233 hartman , j. d. , gaudi , b. s. , holman , m. j. , et al . 2008b , , 675 , 1254 hartman , j. d. , gaudi , b. s. , holman , m. j. , et al . 2009 , , 695 , 336 hodgkin , s. t. , irwin , m. j. , hewett , p. c. , & warren , s. j. 2009 , , 394 , 675 howell , s. b. 1989 , , 101 , 616 irwin , j. , irwin , m. , aigrain , s. , et al . 2007 , , 375 , 1449 ivezi , z. , smith , j. a. , miknaitis , g. , et al . 2007 , , 134 , 973 jenkins , j. m. , caldwell , douglas a. , chandrasekaran , h. , et al . 2010 , , 713 , l87 kim , d .- w . , protopapas , p. , alcock , c. , byun , y .- i . , & bianco , f. 2009 , , 397 , 558 kim , d .- w . , protopapas , p. , alcock , c. , byun , y .- i . , et al . 2010 , , 139 , 757 kovcs , g. , bakos , g. , & noyes , r. w. 2005 , , 356 , 557 kuijken , k. , bender , r. , cappellaro , e. , et al . 2002 , the eso messenger , 110 , 15 kulkarni , s. r. 2013 , the astron . telegram , 4807 , 1 magain , p. , courbin , f. , gillon , m. , et al . 2007 , , 461 , 373 mazeh , t. , guterman , p. , aigrain , s. , et al . 2009 , , 506 , 431 mcleod , b. a. , conroy , m. , gauron , t. m. , geary , j. c. , & ordway , m. p. 2000 , in further developments in scientific optical imaging , ed . m. b. denton ( cambridge : royal society of chemistry ) , 11 merline , w. & howell , s. b. 1995 , exp . 6 , 163 . mink , d.j . 2002 in astronomical society of the pacific conference series , vol . 281 , astronomical data analysis software and systems xi , ed . d. a. bohlender , d. durand , & t. h. handley , 169 mislis , d. , schmitt , j. h. m. m. , carone , l. , guenther , e. w. , & ptzold , m. 2010 , , 522 , 86 miyazaki , s. , komiyama , y. , sekiguchi , m. , et al . 2002 , pasj , 54 , 833 padmanabhan , n. , schlegel , d. j. , finkbeiner , d. p. , et al . 2008 , , 674 , 1217 pepper , j. , stanek , k. z. , pogge , r. w. , et al . 2008 , , 135 , 907 pietrukowicz , p. , minniti , d. , fernndez , j. m. , et al . 2009 , , 503 , 651 pont , f. , zucker , s. , & queloz , d. 2006 , , 373 , 231 protopapas , p. , giammarco , j. m. , faccioli , l. , et al . 2006 , , 369 , 677 randall , s. k. , calamida , a. , fontaine , g. , et al . 2011 , , 737 , 27 scargle , j. d. 1982 , , 263 , 835 skrutskie , m. f. , cutri , r. m. , stiening , r. , et al . 2006 , , 131 , 1163 stetson , p. b. 1987 , , 99 , 191 stetson , p. b. 1990 , , 102 , 932 tamuz , o. , mazeh , t. , & zucker , s. 2005 , , 356 , 1466 twicken , j. d. , chandrasekaran , h. , jenkins , j. m. , et al . 2010 , , 7740 , 62 van dokkum , p. g. 2001 , , 113 , 1420

in nature , there exists a variety of systems that involve chemical reactions . some are on a geographical - scale , while others on a nano - scale . chemical reactions are an integral part of life , including all living forms of life . to study the dynamics of reaction systems , we often adopt rate equations in order to observe the change in chemical concentrations . in rate equations , we regard the concentrations as continuous variables ; the rate of the reaction as a function of the concentrations . in macroscopic systems , there are a vast number of molecules ; thus , continuous representations are usually applicable . when the concentration of a certain chemical is small , fluctuations in the reactions or flow can be significant . we often handle such systems with the help of stochastic differential equations , in which we regard noise as a continuum description of the fluctuations @xcite . such an approximation is useful when the number of molecules is intermediate . the employment of stochastic differential equations led to some important discoveries such as noise - induced order @xcite , noise - induced phase transitions @xcite , and stochastic resonance @xcite . in stochastic differential equations , still quantities of chemicals are regarded as continuous variables . essentially , on a microscopic level , chemicals are composed of molecules . the number of molecules should be an integer ( @xmath0 , @xmath1 , @xmath2 , @xmath3 ) , which changes discretely . fluctuations are derivatives of discrete stochastic changes ; thus , continuum descriptions of fluctuations are not always appropriate and can be doubted . for chemicals with a small number of molecules of the order of @xmath1 , a single molecule is extremely significant ; therefore , the discreteness in the number is significant . biological cells appear to be a good example . the size of the cells is of the order of microns , in which nano - scale `` quantum '' effects can be ignored . however , in cells , some chemicals act at extremely low concentrations of the order of pm or nm . assuming that the typical volume of a cell ranges from @xmath1 to @xmath4 @xmath5 , the concentration of one molecule in the cell volume corresponds to @xmath6 pm@xmath6 nm . it is probable that the molecular numbers of some chemicals in a cell are of the order of @xmath1 , or sometimes reach @xmath0 . if such chemicals play only a minor role , we can safely ignore these chemicals . however , this is not always the case . in biological systems , chemical species with a small number of molecules may critically affect the behavior of the entire system . for example , there exist only one or a few copies of genetic molecules such as dna , which are important to characterize the behavior , in each cell . further , some experiments show that doses of particular chemicals at concentrations of the order of pm or fm may alter the behavior of the cells ( e.g. , @xcite ) . biological systems also include positive - feedback mechanisms such as autocatalytic reactions , which may amplify single molecular changes to a macroscopic level . the effects due to small molecular numbers in cells have been noticed only recently , both theoretically @xcite and experimentally @xcite . at present , we focus on the possible effects of molecular discreteness . to study such effects , we should adopt an appropriate method to handle molecular discreteness . some numerical methods to investigate reaction systems that take into account discreteness and stochasticity already exist ( we briefly review these methods ; see appendix ) . among the methods , we adopted gillespie s direct method , which is popular and frequently used . furthermore , some works related to molecular discreteness also exist . for example , blumenfeld et al . showed that the mass action law may breakdown in a small system @xcite . stange et al . studied the synchronization of the turnover cycle of enzymes @xcite . we regard it important to identify the phenomena for which molecular discreteness is essential . through stochastic simulations , we show that discreteness can induce transitions to novel states in autocatalytic systems @xcite , which may affect macroscopic chemical concentrations @xcite . we consider a simple autocatalytic network ( loop ) with @xmath7 chemicals . we consider @xmath8 chemicals and assume @xmath9 reactions between these chemicals ( @xmath10 , @xmath2 , @xmath3 , @xmath7 ; @xmath11 ) . all the reactions are irreversible . for the reactor , we assume a well - stirred container with volume @xmath12 . the set of @xmath13 , the number of @xmath8 molecules , determines the state of the system . the container is in contact with a chemical reservoir , in which the concentration of @xmath8 is fixed at @xmath14 . the flow rate of @xmath8 between the container and the reservoir is @xmath15 , which corresponds to the probability of the flowing out of a molecule per time unit is the diffusion rate across the surface of the container . here , we choose the flow proportional to @xmath12 , to have a well - defined continuum limit . one might assume the flow proportional to @xmath16 , considering the area of the surface . by rescaling @xmath17 , the model can be rewritten into the case with @xmath16 for finite @xmath12 . ] . we can consider the continuum limit as @xmath18 . in the continuum limit , the change of @xmath19 , the chemical concentration @xmath8 in the container , follows the rate equation @xmath20 where @xmath21 is the rate constant of the reaction @xmath22 , and @xmath23 . for simplicity , we consider the case with equivalent chemical species , given as @xmath24 , @xmath25 , and @xmath26 for all @xmath27 ( @xmath28 , @xmath17 , @xmath29 ) . by this assumption , the rate equation has only one attractor : a stable fixed point @xmath30 for all @xmath27 . for any initial condition , each @xmath19 converges to @xmath31 , the fixed point value . around the fixed point , @xmath19 vibrates with the frequency @xmath32 . if the number of molecules is finite but fairly large , we can estimate the dynamical behavior of the system using a langevin equation , obtained by adding a noise term to the rate equation . each concentration @xmath19 fluctuates and vibrates around the fixed point . an increase in the noise ( corresponding to a decrease in the number of molecules ) merely boosts the fluctuation . however , when the number of molecules is small , the behavior of the system is completely different . first , we investigate the case when @xmath33 , which is the smallest number of species to show the novel states described below . subsequently , we investigate the dynamical behavior of the system with a small number of molecules . in order to detect the phenomena for which the discreteness of the number of molecules is crucial , we employ stochastic simulations . here , we adopt gillespie s direct method . the frequency ( expected number per unit time ) * of the reaction @xmath22 is @xmath34 ; * of the outflow of @xmath8 is @xmath35 ; * of the inflow of @xmath8 is @xmath36 . in the continuum limit ( @xmath18 ) , these frequencies agree with the rate equation . we calculate these frequencies with the current @xmath13 , and stochastically decide when and which event will occur next . in this case , by an appropriate conversion of @xmath17 , @xmath12 , and @xmath37 , we can set @xmath28 and @xmath31 to be @xmath1 without loss of the generality ( @xmath38 and @xmath39 are the only independent parameters ) . we assume that @xmath40 and @xmath41 for the purpose of further discussion . the total number of molecules in the container , @xmath42 , is approximately @xmath43 on an average . by varying @xmath12 , we can control the average number of molecules without changing the continuum limit . first , we consider the case of a large @xmath12 , i.e. , both the number of molecules in the container and flow of molecules between the container and the reservoir are large . as expected , the behavior of the system is similar to that of the rate equation with noise . as shown in fig . [ fig : a-512 ] , each @xmath13 fluctuates and vibrates around the fixed point value @xmath44 ( i.e. @xmath45 ) . this is still in the realm of stochastic differential equations . however , when @xmath12 is small , we observe novel states that do not exist in the continuum limit . as shown in fig . [ fig : a-032 ] , continuous vibrations disappear . furthermore , two chemicals are dominant and the other two are mostly extinct ( @xmath46 ) . in fig . [ fig : a-032 ] , at @xmath47 , @xmath48 and @xmath49 dominate the system and @xmath50 for the most part . we call such a state the 1 - 3 rich state . reversely , at @xmath51 , @xmath52 and @xmath53 are large and usually @xmath54 . we call this state the 2 - 4 rich state . these states appear because of the following reason . in this system , the production of @xmath8 molecules requires at least one @xmath8 molecule as a catalyst . if @xmath8 becomes extinct , the production of @xmath8 halts . @xmath13 never regains before an @xmath8 molecule flows in . in the rate equation ( eq . ( [ eqn : tr1-rate ] ) ) , the concentration @xmath19 is a continuous variable , which can be an infinitesimal but positive value . the consumption rate of @xmath19 is proportional to @xmath19 itself ; thus , @xmath19 can not reach @xmath0 exactly within finite time , even if it can go to @xmath0 asymptotically as @xmath55 . in fact , the number of molecules must be an integer . transitions from @xmath56 to @xmath46 are probabilistic and may happen in finite time . for transitions to occur , it is important that the consumption rate of @xmath19 does not converge to @xmath0 at @xmath57 . the average interval of a molecule flowing in is @xmath58 for each chemical . if @xmath17 and @xmath12 are small enough to ensure that the inflow interval is longer than the time scale of the reactions , it is likely that the state of the system @xmath56 drops to @xmath46 before an @xmath8 molecule enters . when @xmath13 reaches @xmath0 , @xmath59 is also likely to become @xmath0 . for example , if we assume that @xmath60 , then @xmath48 is likely to increase because the consumption of @xmath61 halts ; @xmath49 is likely to decrease because the production of @xmath62 halts . thus , this results in @xmath63 . when @xmath63 , the consumption rate of @xmath64 is larger than the production rate of @xmath64 ; therefore , @xmath53 starts to decrease and often reaches @xmath0 . when @xmath50 , all the reactions stop . the system stays at @xmath50 for a long time as compared with the ordinary time scale of the reactions ( @xmath65 ) . this is the 1 - 3 rich state . in the 1 - 3 rich state , the system alternately switches between @xmath63 and @xmath66 . we consider that the system is in the 1 - 3 rich state with @xmath63 . one @xmath67 molecule flowing in may resume the reactions @xmath68 and @xmath69 . generally , the former is faster because @xmath63 ; hence , @xmath52 is likely to increase . since @xmath53 = 0 , the reactions are one - way ; @xmath48 decreases and @xmath49 increases . when @xmath66 , @xmath52 starts to decrease . finally , when @xmath52 returns to @xmath0 , the reactions halt again . the system stays in the 1 - 3 rich state , until @xmath48 reaches @xmath0 . in the same manner , the inflow of @xmath64 can switch the system from @xmath66 to @xmath63 in the 1 - 3 rich state . consequently , we observe successive switching between @xmath63 and @xmath66 . in the 2 - 4 rich state , the system switches between @xmath70 and @xmath71 . in this manner , even one molecule can switch the system within the 1 - 3 or 2 - 4 rich states . we name these states `` switching states '' . now , we investigate some properties of the switching states . we introduce an index @xmath72 as a characteristic of the switching states . around the fixed point of the rate equation , @xmath73 ; in the 1 - 3 rich state , @xmath74 ; in the 2 - 4 rich state , @xmath75 . the distribution of @xmath76 is shown in fig . [ fig : a - ac - bd ] . when @xmath77 , a single peak appears around @xmath78 , which corresponds to the fixed point . by decreasing @xmath12 , the peak broadens with fluctuations . when @xmath79 , double peaks appear at @xmath80 , which correspond to the switching states . we clearly observe a symmetry - breaking transition between a continuous vibration around the fixed point with large @xmath12 and the switching states with small @xmath12 . this is a discreteness - induced transition ( dit ) that occurs with decrease of @xmath12 , which is not seen in continuum descriptions . we introduce another index @xmath81 , which represents the difference in concentrations : @xmath82 in the 1 - 3 rich state and @xmath83 in the 2 - 4 rich state . the distribution of @xmath84 is shown in fig . [ fig : a - ab - cd ] . there are double peaks around @xmath85 , which imply large imbalances , such as @xmath86 and @xmath87 , between @xmath48 and @xmath49 in the 1 - 3 rich state ( as well as the 2 - 4 rich state ) . we investigate some properties of the switching states . first , we examine how each @xmath13 changes in a switching event . we assume the 1 - 3 rich state with @xmath88 , @xmath89 , and @xmath90 . here , one @xmath67 molecule flows in ( @xmath91 ) at @xmath92 , which starts up the reactions . assuming that @xmath93 is so small that no more molecules flow in or out throughout the switching , the total number of molecules @xmath94 is conserved at @xmath95 , and @xmath53 is always @xmath0 . we can represent the state with two variables , @xmath48 and @xmath52 . in this system , only the following types of reactions * @xmath96 and * @xmath97 may change @xmath13 ; the others never take place . @xmath48 monotonously decreases . when @xmath52 reaches @xmath0 , these reactions halt completely , and the switching is completed . evidently , @xmath98 , where @xmath99 and @xmath100 are the final values of @xmath48 and @xmath49 , respectively . the frequency of the reaction @xmath96 is @xmath101 and that of @xmath97 is @xmath102 we obtain the master equation @xmath103 for @xmath104 , the probability of residence in the state @xmath105 at time @xmath37 . the initial condition is @xmath106 ; otherwise @xmath107 . we can easily follow the master equation numerically . we investigate the relationship between the initial state @xmath108 and the final state @xmath109 . if the first reaction is @xmath97 , @xmath52 instantaneously reaches @xmath0 , and as a result @xmath110 . the system fails to switch . if @xmath111 , it is more probable that the first reaction is @xmath96 . subsequently , the system carries out further reactions . in this case , it is probable that @xmath112 , i.e. , the system swaps @xmath48 and @xmath49 , as shown in fig . [ fig : a - swprob-128 ] . consequently , we observe successive switching ( as seen in fig . [ fig : a-032 ] ) . when @xmath113 , the system is likely to reach @xmath114 and break the 1 - 3 rich state . a large imbalance between @xmath48 and @xmath49 results in an unstable 1 - 3 rich state . now , we investigate the requirements for the transitions to the switching states . the rate of residence of the switching states for several @xmath17 and @xmath12 is shown in fig . [ fig : a - type1a ] . for approximately @xmath115 , we observe the switching states . if @xmath116 , the system mostly stays in the switching states . subsequently , we expect that switching states appear even for large @xmath12 if @xmath17 is very small . in fact , we observe switching states for @xmath117 as shown in fig . [ fig : a-10000 ] . furthermore , in this case also , a single molecule can induce switching . strictly speaking , if @xmath93 is the same , the rate of residence is a little smaller for larger @xmath12 . if @xmath12 is large , the system takes longer to reach @xmath46 even if the rate of reaction and the initial concentrations are the same . thus , it is less likely to exhibit switching states for the same interval of the inflow , @xmath58 . in general , some reactions are much faster than the inflow . if the number of molecules which enter within the time scale of the reactions is of the order of @xmath1 for a certain chemical , the reactions may consume all the molecules of the chemical , and the molecular discreteness of the chemical becomes significant . in other words , for the effect of the discreteness to appear in a system with several processes , it is important that the number of events of a process within the time scale of another process is of the order of @xmath1 . once the system is in the switching state , it is fairly stable and difficult to escape , especially if @xmath93 is small . to escape the 1 - 3 rich state and regain continuous vibration , at least one @xmath67 molecule and one @xmath64 molecule should flow in and it is required that after an @xmath67 molecule flows in , an @xmath64 molecule should flow in before @xmath52 returns to @xmath0 , or vice versa . we put one @xmath67 molecule into the system in the 1 - 3 rich state at @xmath92 , and one @xmath64 molecule in at @xmath118 . we assume that @xmath93 is so small that no more molecules flow in or out . then , we judge whether the system escapes from the 1 - 3 rich state . in due course , the system returns to the 1 - 3 rich ( or sometimes 2 - 4 rich ) state because there is no further flow ; thus , we should judge at the right moment . here , if @xmath119 for all @xmath27 at @xmath120 ( i.e. , waiting about 2.5 times longer than the period of the oscillation around the fixed point ) , we consider that the 1 - 3 rich state has been interrupted . we measure the probability of interruption for various initial conditions and the delay @xmath121 as shown in fig . [ fig : a - sw3b-32 ] . the system requires adequate timing of inflow to escape the switching states , which may amplify the imbalance between the stability of each state . for example , to escape the 1 - 3 rich , @xmath63 state , it is required that an @xmath67 molecule flows in , and then an @xmath64 molecule flows in with a certain delay @xmath121 , as shown in fig . [ fig : a - sw3b-32 ] . thus , the frequency of escape from the 1 - 3 rich state is approximately proportional to the product of the inflow frequencies of @xmath67 and @xmath64 . if each @xmath15 or @xmath14 is species - dependent , the stability of the 1 - 3 and 2 - 4 rich states may strongly depend on @xmath122 , the inflow frequency of @xmath8 . furthermore , if at the outset @xmath123 , it is difficult for the system to escape from the 1 - 3 rich state . in some cases where the parameters are species - dependent , the flows or the switching may lead to @xmath123 , which stabilizes the 1 - 3 rich state . these conditions are important to stabilize particular states and affect the macroscopic behavior of the system ( see section [ sect : tr2 ] ) . to close the section [ sect : tr1 ] , we briefly discuss the cases where @xmath124 . figure [ fig : a - k356 - 16 ] shows the time series of each @xmath13 for @xmath125 , @xmath126 , and @xmath127 . when @xmath128 , 1 - 3 - 5 rich and 2 - 4 - 6 rich states appear . however , these states are less stable than the 1 - 3 and 2 - 4 rich states for @xmath33 . these states collapse when any of the rich chemicals vanish ; thus , they are unstable for large @xmath7 . when @xmath7 is odd ( @xmath129 , @xmath126 , @xmath3 ) , there are no stable states where particular chemicals are extinct . however , additional reactions , or a variety of @xmath21 or @xmath14 , may stabilize or destabilize the states such that it is not always true that loops with odd-@xmath7 are unstable . the discreteness - induced transitions are not limited in the autocatalytic loop . we apply the abovementioned discussions to certain segments of a complicated reaction network with a slight modification . in the preceding section , we show that the discreteness of the molecules can induce transitions to novel `` switching '' states in autocatalytic systems . for the case where @xmath33 with uniform parameters , the 1 - 3 rich state and the 2 - 4 rich state are equivalent . in due course , the system alternates between the 1 - 3 rich and 2 - 4 rich states . the long - term averaged concentrations are still the same as the continuum limit value , @xmath130 . it will be important if macroscopic properties , such as the average concentrations , can be altered . we show that the discreteness - induced transitions may alter the long - term averaged concentrations . once again , here , we adopt the autocatalytic reaction loop @xmath9 for the @xmath33 species . now we consider the case where the parameters @xmath15 , @xmath14 , or @xmath21 are species - dependent . in the continuum limit , the concentration @xmath19 is governed by the rate equation @xmath131 the rate equation does not contain the volume @xmath12 ; hence , the average concentrations should be independent of @xmath12 . as discussed in the preceding section , for the transitions to the switching states to occur , it is necessary that the interval of the inflow is longer than the time scale of the reactions . in this model , the inflow interval of @xmath8 is @xmath132 , and the time scale of the reaction @xmath22 in order to use @xmath8 up is @xmath133 . if all the chemicals are equivalent , the discreteness of all the chemicals equally take effect , and the 1 - 3 and 2 - 4 rich states coordinately appear at @xmath134 . now , since the parameters are species - dependent , the effect of discreteness may be different for each species . for example , assuming that @xmath135 , the inflow interval of @xmath61 is longer than that of @xmath67 . thus , the discreteness of the inflow of @xmath61 may be significant for larger @xmath12 . to demonstrate a possible effect of the discreteness on the macroscopic properties of the system , we measure each average concentration @xmath136 , sampled over a long enough time to allow transitions between the 1 - 3 and 2 - 4 rich states , by gillespie s direct method . note that every @xmath136 does not depend on @xmath12 in the continuum limit . generally , in discrete simulations , the effect of the discreteness varies with @xmath12 and alters every @xmath136 . when @xmath12 is very large , the discreteness does not matter and @xmath136 is almost equal to the continuum limit value . in contrast , when @xmath12 is small , the discreteness causes @xmath136 to be very different from the continuum limit . we first investigate the case where each @xmath14 is species - dependent ( i.e. , each inflow rate is species - dependent ) , and each @xmath15 and @xmath21 are uniform ( @xmath25 , @xmath137 ) . later , we briefly discuss the case where @xmath21 is inhomogeneous . as mentioned in section [ subsect : tr1-cond ] , for the effect of the discreteness to appear , it is important that the interval of events of a process is of the order of or longer than the time scale of another process . as regards the inflow , there are two indices to determine how the discreteness appears : * the inflow interval , @xmath138 , * the number of molecules at equilibrium , @xmath139 . if the inflow interval , @xmath138 , is longer than the time scale of the reactions , the reactions may exhaust the chemical before the chemical enters , and the inflow discreteness becomes significant . furthermore , if @xmath139 is smaller than @xmath1 , @xmath13 can reach @xmath0 because of the outflow . in such cases , the relation between the inflow interval and the outflow time scale is also important . the approach time from @xmath140 to @xmath46 is of the order of @xmath141 . if the inflow interval of the chemical that causes the switching to raise @xmath13 is long enough to allow all @xmath8 molecules to flow out , the inflow discreteness may alter the stability of the states drastically . from this point of view , we classify the mechanism in cases i , i@xmath142 , and ii as follows . we start with the simplest case , @xmath143 . in this case , the rate equation has a stable fixed point with @xmath144 . when @xmath12 is large , each @xmath19 fluctuates around the fixed point , and each average concentration @xmath136 is in accordance with the fixed point value . when @xmath12 is small , @xmath136 depends on @xmath12 . [ fig : tr2-average1 ] shows each @xmath136 as a function of @xmath12 . the difference between @xmath145 and @xmath146 increases for small @xmath12 . by decreasing @xmath12 , first @xmath52 and @xmath53 reach @xmath0 and the 1 - 3 rich state appears . to reach @xmath147 or @xmath148 , the inflow interval of @xmath67 or @xmath64 should be longer than the time scale of the reactions . we set @xmath149 ; thus , the condition to achieve the 1 - 3 rich state is approximately @xmath150 , @xmath151 ; that for the 2 - 4 rich state is @xmath152 , @xmath153 . if @xmath12 satisfies @xmath152 , @xmath154 , @xmath155 , the 1 - 3 rich state appears but the 2 - 4 rich state does not . thus , @xmath145 and @xmath156 increase . we actually observed this at @xmath157 , as shown in fig . [ fig : tr2-average1 ] . for smaller @xmath12 that fulfills both the 1 - 3 and 2 - 4 rich states , the imbalance between the 1 - 3 and 2 - 4 rich states does not disappear once the system is in the 1 - 3 rich state , adequate timing for the @xmath67 and @xmath64 inflow is required to escape the state . thus , the frequency of escape from the 1 - 3 rich state is approximately proportional to @xmath158 , the product of the inflow frequencies of @xmath67 and @xmath64 . the average residence time in the 1 - 3 rich state as well as the 2 - 4 rich state is the reciprocal of the escape frequency . the ratio of the average residence time in the 1 - 3 rich state to that in the 2 - 4 rich state is @xmath159 . in addition , after escaping the switching states , the system tends to reach the 1 - 3 rich state rather than the 2 - 4 rich state because of the biased inflow . thus , the ratio of the total residence time in the 1 - 3 rich state to that of the 2 - 4 rich state is larger than @xmath160 ; hence , @xmath145 , @xmath161 , @xmath162 , even for a small difference in @xmath14 . in case i , we consider the imbalance between the @xmath61 , @xmath62 pair and the @xmath67 , @xmath64 pair . if another imbalance exists between the @xmath67 and @xmath64 inflows , the switching induced by these chemicals in the 1 - 3 rich state may be unbalanced . we consider the case where @xmath163 . in this case , the 1 - 3 rich state is more stable than the 2 - 4 rich state , which is identical to case i. in the 1 - 3 rich state , the system can be switched from @xmath63 to @xmath66 by an @xmath67 molecule ; and from @xmath66 to @xmath63 by an @xmath64 molecule . now , the inflow rate of @xmath67 is larger than @xmath64 ; thus , switching from @xmath63 to @xmath66 is more probable than vice versa , and the system tends to stay in the @xmath66 state . consequently , @xmath164 , as shown in fig . [ fig : tr2-average1 ] . this effect requires switching states . when @xmath12 is large , @xmath145 and @xmath156 are almost the same . here , we consider the case where @xmath165 . in this case also , the rate equation has a stable fixed point , @xmath166 , @xmath167 and @xmath168 , @xmath169 . ] . when @xmath12 is small , both the 1 - 3 and the 2 - 4 rich states appear . here , we consider @xmath170 , the number of @xmath64 molecules when the concentration of @xmath64 in the container and in the reservoir are at equilibrium . if @xmath171 , @xmath53 reaches @xmath0 without undergoing any reaction . the system takes @xmath172 time units to reach from @xmath173 to @xmath148 . the reaction @xmath174 also uses @xmath64 such that @xmath53 decreases faster if @xmath175 is large . if @xmath176 , the inflow of @xmath62 may switch from @xmath70 to @xmath71 and raise @xmath53 again . the inflow interval of @xmath62 is @xmath177 . if the interval is much shorter than the approach time to @xmath148 , the switching maintains @xmath176 . if the interval is longer , @xmath53 reaches @xmath0 before switching , and the 2 - 4 rich state is easily destroyed . in the 1 - 3 rich state , the system tends to maintain @xmath66 because the @xmath67 inflow is frequent . however , the @xmath61 inflow is also large enough to maintain @xmath178 . the 1 - 3 rich state retains its stability . in conclusion , the 1 - 3 rich state is more stable than the 2 - 4 rich state . in the 1 - 3 rich state , @xmath66 is preferred , and @xmath156 increases . at this stage , it is possible that @xmath179 despite the fact that @xmath180 , as shown in fig . [ fig : tr2-average2 ] . in summary , the difference in the `` extent of discreteness '' between chemical species induces novel transitions . the `` extent of discreteness '' depends on @xmath12 ; thus , we observe transitions by changing @xmath12 . the transition reported in section [ sect : tr1 ] is regarded as a _ second order _ transition involving symmetry breaking ( see figs . [ fig : a - ac - bd ] and [ fig : a - ab - cd ] ) , while the transition in this section corresponds to the _ first order _ transition without symmetry breaking ( see fig . [ fig : tr2-dist - ab - cd - casei ] ) in terms of thermodynamics . we classified the mechanism in cases i , i@xmath142 , and ii . these mechanisms can be combined . for example , we demonstrate the case where @xmath181 , @xmath182 , and @xmath183 . in this case , each @xmath136 shows a three - step change with @xmath12 , as shown in fig . [ fig : tr2-average3 ] . when @xmath12 is large , @xmath145 , @xmath184 , @xmath162 , since @xmath185 . at @xmath186 , the discreteness of the @xmath62 inflow becomes significant , and the 2 - 4 rich state appears . in the 2 - 4 rich state , the system tends to remain at @xmath70 because of the inflow imbalance between @xmath61 and @xmath62 , as observed in case i@xmath142 . figure [ fig : tr2-dist - x2 ] shows the distribution of @xmath168 . the major peak corresponds to the 2 - 4 rich , @xmath187 state for the cases when @xmath188 . on the other hand , in the 2 - 4 rich state , the outflow of @xmath64 depresses @xmath53 toward @xmath170 , as observed in fig . [ fig : tr2-ts - amp ] . by decreasing @xmath12 , the imbalance between @xmath52 and @xmath53 increases because the rate of switching , which again raises @xmath53 , decreases in proportion to @xmath12 . finally , at @xmath189 , the 2 - 4 rich state loses stability , as seen in case ii . now , the 1 - 3 rich state is preferred despite the fact that @xmath185 . @xmath156 increases to @xmath190 , which is more than @xmath191 times as large as that in the continuum limit . for extremely small @xmath12 , the 1 - 3 rich state is also unstable because @xmath48 and @xmath49 easily reach @xmath0 . in such a situation , typically only one chemical species is in the container . the system is dominated by diffusion , and @xmath146 increases again due to the large @xmath192 . note that the chemical that becomes extinct depends not only on the flows but also on the reactions . in some cases , we observe smaller @xmath156 for larger @xmath193 . it is possible that each @xmath21 varies with the species . in such cases , we can discuss the effect of discreteness in a similar way . however , the change of @xmath136 with @xmath12 is different from the case with asymmetric flows . for example , we assume that @xmath194 and @xmath195 . in the continuum limit or in the case of large @xmath12 , @xmath196 , as shown in fig . [ fig : tr2-average - r ] . in contrast , when @xmath12 is small , @xmath197 . if @xmath12 is very small , such that the total number of molecules is mostly @xmath0 or @xmath1 , reactions rarely take place . the flow of chemicals dominate the system ; thus , @xmath198 . if both the reactions and the flows are species - dependent , we simply expect the behavior to be a combination of the abovementioned cases . even this simple system can exhibit a multi - step change in concentrations along with a change in @xmath12 . it is not limited to the simple reaction loop . in fact , we observe this kind of change in concentrations with a change in the system size in randomly connected reaction networks . for a large reaction network with multiple time scales of reactions and flows , the discreteness effect may exhibit behavior that is more complicated . our discussion is largely applicable to such cases if we can define the time scales appropriately . as seen in this paper , the discreteness of molecules can alter the average concentrations . when the rates of inflow and/or the reaction are species - dependent , transitions between the discreteness - induced states are imbalanced . this may alter the average concentrations drastically from those of the continuum limit case . we demonstrated that molecular discreteness may induce transitions to novel states in autocatalytic systems , and that may result in an alteration of the macroscopic properties such as the average chemical concentrations . in biochemical pathways , it is not anomalous that the number of molecules of a chemical is of the order of @xmath199 or less in a cell . there are thousands of protein species , and the total number of protein molecules in a cell is not very large . for example , in signal transduction pathways , some chemicals work at less than @xmath200 molecules per cell . there exist only one or a few copies of genetic molecules such as dna ; furthermore , mrnas and trnas are not present in large numbers . thus , regulation mechanisms involving genes are quite stochastic . molecular discreteness naturally concerns such rare chemicals . one of the authors , kaneko , and yomo recently provided the `` minority control conjecture , '' which propounds that chemical species with a small number of molecules governs the behavior of a replicating system , which is related to the origin of heredity @xcite . matsuura et al . experimentally demonstrated that a small number of genetic molecules is essential for evolution @xcite . molecular discreteness should be significant for such chemicals , and may be relevant to characters of genetic molecules . until now , we have modeled reactions in a well - stirred medium , where only the number of molecules is taken into account while determining the behavior . however , if the system is not mixed well , we should take into account the diffusion in space . both the total number of molecules and the spatial distributions of the molecules may be significant . from a biological point of view , the diffusion in space is also important because the diffusion in cells is not always fast as compared with the time scales of the reactions . if the reactions are faster than the mixing , we should consider the system as a reaction - diffusion one , with discrete molecules diffusing in space . the relation between these time scales will be important , as indicated by mikhailov and hess @xcite . as regards these time scales , we recently found that the spatial discreteness of molecules within the so - called kuramoto length @xcite , over which a molecule diffuses in its lifetime ( lapses before it undergoes reaction ) , may yield novel steady states that are not observed in the reaction - diffusion equations @xcite . there is still room for exploration in this field , e.g. , pattern formation . our result does not depend on the details of the reaction and may be applicable to systems beyond reactions , such as ecosystems or economic systems . the inflow of chemicals in a reaction system can be seen as a model of intrusion or evolution in an ecosystem ; both systems with discrete agents ( molecules or individuals ) , which may become extinct . in this regard , our result is relevant to studies of ecosystems , e.g. , extinction dynamics with a replicator model by tokita and yasutomi @xcite . the discreteness of agents or operations might also be relevant to some economic models , e.g. , artificial markets . most mathematical methods that are applied to reaction systems can not account for the discreteness . although the utility of simulations have become convenient with the progress of computer technology , it might be useful if we could construct a theoretical formulation applicable to discrete reaction systems . on the other hand , in recent years , major advances have been made in the detection of a small number of molecules and fabrication of small reactors , which raises our hopes to demonstrate the effect of discreteness experimentally . we believe that molecular discreteness is of hidden but real importance with respect to biological mechanisms , such as pattern formation , regulation of biochemical pathways , or evolution , to be pursued in the future . this research is supported by grant - in - aid for scientific research from the ministry of education , culture , sports , science and technology of japan ( 11ce2006 , 15 - 11161 ) . is supported by a research fellowship from the japan society for the promotion of science . since the 1970s , several methods have been suggested for simulating discrete reaction systems . we briefly review some of these methods : stochsim method , gillespie s methods , and their improved versions . these methods are based on chemical master equations . in chemical master equations , we define the state of the system as the number of molecules in each chemical ; the reaction process as a series of transitions between the states . we consider each event , i.e. , a reaction event between molecules , inflow or outflow of a molecule , as a transition . they take place stochastically with a certain frequency ( probability per time unit ) determined by the current state . for the simulations used in this study , we adopted gillespie s direct method for simplicity . we also attempted a direct simulation and the next reaction method , and confirmed that our result does not depend on the simulation method . in this method , we fix the time step as @xmath201 . assuming that the frequency of the event-@xmath27 is @xmath202 , the average number of the event-@xmath27 for each step is @xmath203 . if @xmath204 , we approximately assume that at most one event occurs at each step , and the probability of the event-@xmath27 is @xmath203 ( it is possible that no event occurs in the step ) . we select an event with a random number , change the state according to the event , and recalculate each @xmath202 . the stochsim method adopts random sampling of molecules . for bimolecular reactions , we randomly choose two molecules from the system , and decide whether they react or not with certain probability . the method requires three random numbers in total for each step . in case there are some single molecular ( first - order ) reactions , we choose the second molecule from the molecules in the system and some pseudo - molecules ( dummies ) . if the second molecule is a pseudo - molecule , then we select the single molecular reaction of the first molecule , and determine whether it occurs . in the stochsim method , @xmath201 is restricted by the fastest reaction ( with the largest reaction rate per pair of molecules ) . if most of the bimolecular pairs do not react with each other , or the reaction probability varies with the species , this method may be impractical because no reaction occurs in most steps . incidentally , if the system consists of discrete molecules , it is typical that each frequency @xmath202 changes only when an event actually occurs . taking this into account , gillespie suggested two exact simulation methods : the direct method @xcite and the first reaction method @xcite . in these methods , we do not fix the time step . instead , we calculate the time lapse until the next event . in the direct method , first , we consider the total frequency of the events , @xmath205 . if @xmath202 does not change until the next event , the time lapse until the next event , @xmath121 , is exponentially distributed as @xmath206 ( @xmath207 ) . we determine the time lapse @xmath121 with an exponentially distributed random number . subsequently , the probability that the next event is @xmath27 is @xmath208 . we determine which event occurs with a random number . we then set the time @xmath37 forward by @xmath121 , update the state according to the event , and recalculate each frequency @xmath202 . iterate the above steps until the designated time elapses . the first reaction method is similar to the direct method . it is based on the fact that @xmath209 , the time lapse until the next event-@xmath27 , is exponentially distributed as @xmath210 ( @xmath211 ) . only the event with minimum @xmath209 actually occurs . we update the state , recalculate each @xmath202 , and generate all @xmath209 again with the new corresponding @xmath202 . in the first reaction method , we need as many random numbers each step as the types of events . we calculate all @xmath209 , choose the earliest , and discard the others . generally , the processor time to generate random numbers is very large ; hence , a large amount of time is wasted for several types of events . to solve this performance problem , gibson and bruck proposed a refinement of the first reaction method : next reaction method @xcite . although , in general , monte carlo simulations require independency of random numbers , they proved a safe way of recycling random numbers , which drastically promotes efficiency . in the next reaction method , we store @xmath212 , the absolute time when the next event-@xmath27 occurs , instead of @xmath209 in the first step , we have to calculate @xmath213 for every @xmath27 , according to the exponential distribution @xmath210 ( @xmath211 ) . we choose the event with the smallest @xmath212 . according to the event , we set the time @xmath37 forward to @xmath212 , change the state , and recalculate each @xmath202 . 1 . as regards the event just executed , we should recalculate @xmath214 with the exponential distribution of @xmath209 , identical to the first step . as regards other events whose frequency @xmath202 has changed , we should recalculate the corresponding @xmath212 . for such events , we convert @xmath212 as @xmath215 @xmath216 is the @xmath212 before the event and @xmath217 is that after the event . + with this conversion , the actual frequency is adjusted from @xmath218 to @xmath219 , without using random numbers . 3 . as for other events whose frequency @xmath202 has not changed , we do not need to recalculate the corresponding @xmath212 . if the event executed does not influence @xmath202 , we do not need to recalculate the concerned @xmath202 and @xmath212 ( except for those of the event just executed ) . thus , it is useful to manage the dependency of @xmath202 on each event . with this intention , we prepare a dependency graph that shows which @xmath202 should be updated after event-@xmath220 ( event-@xmath27 depends on event-@xmath220 ) . in a large reaction network , such as biochemical pathways , one chemical species can react with only a small part of the chemicals in the entire system . in such cases , recalculation is not required for irrelevant chemical species ; hence , we can accelerate simulations with help of a dependency check . it is also important to find the smallest @xmath212 quickly . for this purpose , we use a heap , a binary tree in which each node is larger than or equal to its parent . the root is the smallest at any instance ) sorting algorithm . if only one @xmath212 has changed in the step , the cost of resorting is o(@xmath221 ) . ] . x. wang , g. z. feuerstein , j. gu , p. g. lysko , and t. yue , `` interleukin-1@xmath222 induces expression of adhesion molecules in human vascular smooth muscle cells and enhances adhesion of leukocytes to smooth muscle cells '' , atherosclerosis * 115 * , 89 ( 1995 ) . for @xmath228 and @xmath227 . in this stage , @xmath13 can reach @xmath0 , and the switching states appear . in the 1 - 3 rich state , the system successively switches between the @xmath63 and @xmath66 states . the interval of switching is much longer than the period of continuous vibration ( @xmath229 ) observed in figs . [ fig : a - cont ] and [ fig : a-512 ] . around @xmath230 , a transition occurs from the 1 - 3 rich to the 2 - 4 rich state.,width=260 ] , sampled over @xmath231 time units . ( @xmath76 is actually a discrete value . here , we show the distribution as a line graph for visibility . ) when @xmath12 is large ( @xmath233 ) , there appears a single peak around @xmath78 , corresponding to the fixed point state @xmath225 . for @xmath234 , the distribution has double peaks around @xmath235 . the peak @xmath74 corresponds to the 1 - 3 rich state , with @xmath236 , @xmath90 . @xmath75 corresponds to the 2 - 4 rich state as well.,width=260 ] , sampled over @xmath231 time units . when @xmath12 is large , there appears a single peak around @xmath237 , corresponding to the fixed point . when @xmath12 is small and the system is in the switching states , the index @xmath84 shows an imbalance between the rich ( non - zero ) chemicals . for example , in the 1 - 3 rich state , @xmath84 corresponds to @xmath238 since @xmath90 for the most part . the distribution shows double peaks around @xmath239 . assuming that @xmath240 and @xmath241 in the 1 - 3 rich state , @xmath239 correspond to @xmath242 , @xmath243 . both the chemicals are likely to have a large imbalance.,width=260 ] to @xmath109 . we assume a switching event triggered by a single @xmath67 molecule in the 1 - 3 rich state , and we set the initial conditions as @xmath244 and @xmath245 . assuming that there is no further flow ( @xmath246 ) , we numerically follow the master equation ( eq . ( [ eqn : tr1-master ] ) ) until @xmath247 ( sufficiently large to ensure @xmath147 for most cases ) . for each initial condition , we show the transition probabilities of the final states @xmath109 . @xmath228 , @xmath248 . the system shows high probabilities around @xmath249 ( immediately terminated ) and @xmath250 ( switching ; @xmath111).,width=260 ] , i.e. , reactions have already stalled , as a function of @xmath37 ( in other words , the cumulative distribution of the time when @xmath52 reaches @xmath0 ) . @xmath228 , @xmath251 . in the case where @xmath252 or @xmath253 , the probability increases just after the reactions start . for such initial conditions , the reactions terminate near the initial state , as shown in fig . [ fig : a - swprob-128 ] . if @xmath113 , the probability steeply increases at @xmath254 , which corresponds to the switching . the system takes @xmath255 time units to complete the switching.,width=260 ] , the inflow frequency , sampled over @xmath256 ( @xmath257 ) , @xmath258 ( @xmath259 ) , and @xmath260 ( @xmath261 ) time units . here , we define the 1 - 3 rich state as continuation of the state in which at least one of @xmath52 or @xmath53 is @xmath0 for @xmath262 time units or longer . thus , the system may contain states with only one or no chemical , especially for small @xmath12 . we observe the switching states for @xmath263.,width=260 ] for @xmath117 and @xmath264 . for such a small @xmath17 , we observe the switching states for relatively large @xmath12 . in this case also , a single molecule can induce switching . one @xmath61 molecule flows in ( @xmath265 ) , propagates to more than @xmath266 , and then returns to @xmath0.,width=260 ] , sampled over @xmath267 times for each condition . we assume the system where @xmath90 . we inject an @xmath67 molecule at @xmath92 , then an @xmath64 molecule at @xmath118 ( no further flow ) . if @xmath268 at @xmath120 , we ascertain that the system has escaped from the 1 - 3 rich state . @xmath228 , @xmath269 . an initial large imbalance , such as @xmath270 ( @xmath271 ) , makes it easier to escape the 1 - 3 rich state . the system is unlikely to escape from the state when @xmath272 or @xmath252 . the probability is maximum at @xmath273 , which approximately corresponds to a half of the period of the vibration around the fixed point.,width=260 ] for @xmath125 , @xmath126 , @xmath127 , with @xmath274 and @xmath227 . for the case where @xmath128 , the transition occurs from the 1 - 3 - 5 rich state to the 2 - 4 - 6 rich state at @xmath275 . for the cases where @xmath125 and @xmath276 , there are no stable states such as the 1 - 3 rich state when @xmath33 or 1 - 3 - 5 rich state when @xmath128.,title="fig:",width=260 ] for @xmath125 , @xmath126 , @xmath127 , with @xmath274 and @xmath227 . for the case where @xmath128 , the transition occurs from the 1 - 3 - 5 rich state to the 2 - 4 - 6 rich state at @xmath275 . for the cases where @xmath125 and @xmath276 , there are no stable states such as the 1 - 3 rich state when @xmath33 or 1 - 3 - 5 rich state when @xmath128.,title="fig:",width=260 ] for @xmath125 , @xmath126 , @xmath127 , with @xmath274 and @xmath227 . for the case where @xmath128 , the transition occurs from the 1 - 3 - 5 rich state to the 2 - 4 - 6 rich state at @xmath275 . for the cases where @xmath125 and @xmath276 , there are no stable states such as the 1 - 3 rich state when @xmath33 or 1 - 3 - 5 rich state when @xmath128.,title="fig:",width=260 ] as a function of @xmath12 , sampled over @xmath256 ( @xmath257 ) , @xmath258 ( @xmath259 ) , and @xmath260 ( @xmath261 ) time units ( same for fig . [ fig : tr2-average2 ] ) . @xmath40 and @xmath232 . ( a ) case i : @xmath277 , @xmath278 . when @xmath12 is small , the 1 - 3 rich state appears , and the difference between @xmath145 and @xmath146 increases . ( b ) case i@xmath142 : @xmath279 , @xmath280 , @xmath281 . when @xmath12 is small , @xmath146 and @xmath162 decrease , identical to case i. in this case , the imbalance between @xmath145 and @xmath156 appears at the same time . ( figures [ fig : tr2-average1 ] , [ fig : tr2-average2 ] , [ fig : tr2-average3 ] , and [ fig : tr2-dist - x2 ] are reproduced from ref . @xcite by permission of the publisher . ) , title="fig:",width=260 ] as a function of @xmath12 , sampled over @xmath256 ( @xmath257 ) , @xmath258 ( @xmath259 ) , and @xmath260 ( @xmath261 ) time units ( same for fig . [ fig : tr2-average2 ] ) . @xmath40 and @xmath232 . ( a ) case i : @xmath277 , @xmath278 . when @xmath12 is small , the 1 - 3 rich state appears , and the difference between @xmath145 and @xmath146 increases . ( b ) case i@xmath142 : @xmath279 , @xmath280 , @xmath281 . when @xmath12 is small , @xmath146 and @xmath162 decrease , identical to case i. in this case , the imbalance between @xmath145 and @xmath156 appears at the same time . ( figures [ fig : tr2-average1 ] , [ fig : tr2-average2 ] , [ fig : tr2-average3 ] , and [ fig : tr2-dist - x2 ] are reproduced from ref . @xcite by permission of the publisher . ) , title="fig:",width=260 ] , for case i@xmath142 : @xmath279 , @xmath280 , @xmath281 , @xmath40 , and @xmath232 . there is a peak around @xmath237 for large @xmath12 . the difference in @xmath14 has a slight effect on @xmath136 . for the case where @xmath282 , there appears another peak at @xmath283 , which corresponds to the 1 - 3 rich , @xmath284 state.,width=260 ] for @xmath181 , @xmath182 , @xmath183 , @xmath40 , and @xmath286 , sampled over @xmath287 ( @xmath288 ) or @xmath289 ( @xmath79 ) time units . by decreasing @xmath12 , first , the 2 - 4 rich state appears , as seen in case i. then , the 2 - 4 rich state becomes unstable and gives way to the 1 - 3 rich state , as seen in case ii . when @xmath12 is extremely small ( @xmath290 ) , the flow of molecules governs the system , and @xmath146 increases again.,width=260 ] for @xmath181 , @xmath182 , @xmath183 , @xmath40 , and @xmath286 . ( a ) @xmath291 . the 2 - 4 rich state is dominant ( case i@xmath142 ) . inflow of @xmath62 molecules induces switching from @xmath70 to @xmath71 , which prevents @xmath53 from decreasing to @xmath0 . ( b ) @xmath228 . now , the @xmath62 inflow is rare , which allows @xmath53 to reach @xmath0 before the switching . thus , the 2 - 4 rich state is unstable ( case ii).,title="fig:",width=260 ] for @xmath181 , @xmath182 , @xmath183 , @xmath40 , and @xmath286 . ( a ) @xmath291 . the 2 - 4 rich state is dominant ( case i@xmath142 ) . inflow of @xmath62 molecules induces switching from @xmath70 to @xmath71 , which prevents @xmath53 from decreasing to @xmath0 . ( b ) @xmath228 . now , the @xmath62 inflow is rare , which allows @xmath53 to reach @xmath0 before the switching . thus , the 2 - 4 rich state is unstable ( case ii).,title="fig:",width=260 ] for @xmath181 , @xmath182 , @xmath183 , @xmath40 , and @xmath286 , sampled over @xmath287 . for large @xmath12 , a single peak around @xmath292 appears , which corresponds to the fixed point in the continuum limit . at @xmath186 , double peaks appear around @xmath293 and @xmath294 , which correspond to the 2 - 4 rich state . by decreasing @xmath12 , the two peaks spread apart . at @xmath189 , the skirt of the low - density ( left ) peak touches @xmath295 , implying that @xmath52 ( and @xmath53 ) is likely to reach @xmath0 , and thus , the 2 - 4 rich state loses stability . a peak at @xmath295 steeply grows by decreasing @xmath12 further.,width=260 ] for @xmath296 and @xmath232 with inequivalent reaction constants . for small @xmath12 , the flows of molecules dominate the system . thus , @xmath197 , which simply reflects @xmath297 ; this does not depend on how the continuum limit is imbalanced by the reactions . ( a ) @xmath298 and @xmath299 . ( b ) @xmath300 and @xmath301.,title="fig:",width=260 ] for @xmath296 and @xmath232 with inequivalent reaction constants . for small @xmath12 , the flows of molecules dominate the system . thus , @xmath197 , which simply reflects @xmath297 ; this does not depend on how the continuum limit is imbalanced by the reactions . ( a ) @xmath298 and @xmath299 . ( b ) @xmath300 and @xmath301.,title="fig:",width=260 ]

neutrons stars are born in gravitational collapse of massive , degenerate stellar cores . newly born neutron stars are hot and lepton rich objects , quite different from ordinary low temperature , lepton poor neutron stars . in view of these differences , newly born neutron stars are called _ protoneutron _ stars ; they transform into standard neutron stars on a timescale of the order of ten seconds , needed for the loss of a significant lepton number excess via emission of neutrinos trapped in the dense , hot interior . in view of the fact that the typical evolution timescale of a protoneutron star ( seconds ) is some three orders of magnitude longer , than the dynamical timescale for this objects ( milliseconds ) , one can study its evolution in the quasistatic approximation ( burrows & lattimer 1986 ) . properties of static ( non - rotating ) protoneutron stars , under various assumptions concerning composition and equation of state ( eos ) of hot , dense stellar interior were studied by numerous authors ( burrows & lattimer 1986 , takatsuka 1995 , bombaci et al . 1995 , bombaci 1996 , bombaci et al . 1996 ) . the scenario of transformation of a protoneutron star into a neutron star could be strongly influenced by a phase transition in the central region of the star . brown and bethe ( 1994 ) suggested a phase transition implied by the @xmath0 condensation at supranuclear densities . such a @xmath0 condensation could dramatically soften the equation of state of dense matter , leading to a low maximum allowable mass of neutron stars . in such a case , the massive protoneutron stars could be stabilized by the effects of high temperature and of the presence of trapped neutrinos , and this would lead to maximum baryon mass of protoneutron star larger by some @xmath1 than that of cold neutron stars . the deleptonization and cooling of protoneutron stars of baryon mass exceeding the maximum allowable baryon mass for neutron stars , would then inevitably lead to their collapse into black holes . the dynamics of such a process was recently studied by baumgarte et al . it should be mentioned , however , that the very possibility of existence of the kaon condensate ( or other exotic phases of matter , such as the pion condensate , or the quark matter ) at neutron star densities is far from being established . recently , for instance , pandharipande et al . ( 1995 ) pointed out that kaon - nucleon and nucleon - nucleon correlations in dense matter raise significantly the threshold density for kaon condensation , possibly to the densities higher than those characteristic of stable neutron stars . in view of these uncertainties , we will restrict in the present paper to a standard model of dense matter , composed of nucleons and leptons . the calculations of the static models of protoneutron stars should be considered as a first step in the studies of these objects . it is clear , in view of the dynamical scenario of their formation , that protoneutron stars are far from being static . due to the nonzero initial angular momentum of the collapsing core , protoneutron stars are expected to rotate . on the other hand , the formation scenario involves compression ( with overshoot of central density ) and a hydrodynamical bounce , so that a newborn protoneutron star begins its life in a highly excited state , pulsating around its quasistatic equilibrium . in the present paper we study the rotation of protoneutron stars ; pulsations of protoneutron stars will be discussed in a separate paper ( gondek , haensel & zdunik , in preparation ) . some aspects of rapid uniform rotation of protoneutron stars have been recently studied in ( takatsuka 1995 , hashimoto et al . however , the calculations reported by takatsuka ( 1995 ) were actually done for static ( non - rotating ) protoneutron stars , and were then used to estimate the maximum rotation frequency of uniformly rotating protoneutron stars , @xmath2 , via an `` empirical formula '' . it should be stressed , that the validity of such an `` empirical formula '' , which expresses @xmath3 in terms of the mass and radius of the extremal _ static _ configuration with maximum allowable mass , had been checked only in the restricted case of _ cold _ neutron stars ( haensel & zdunik 1989 , friedman et al . 1989 , shapiro et al . 1989 , haensel et al . 1995 , nozawa et al . . only isentropic equations of state were considered by takatsuka ( 1995 ) . hashimoto et al . ( 1995 ) calculated the structure of stationary configurations of uniformly rotating protoneutron stars , using a two - dimensional general relativistic code . these authors restricted themselves to the case with zero trapped lepton number . they assumed a constant temperature in the hot interior of the star , and used a zero temperature ( cold ) eos for @xmath4 . it should be stressed , that the assumption of @xmath5 corresponds to an isothermal state in the newtonian ( flat space - time ) theory of gravitation . in general relativity , we will define isothermal state by @xmath6 ( where @xmath7 is the lapse function and @xmath8 is the lorentz factor , see section 3.1 ) , and the effects of the space - time curvature will turn out to be rather important for massive neutron stars . also , their choice for the low density edge of the hot interior can be questioned . finally , their criterion for finding maximally rotating configuration is actually valid only for cold ( @xmath9 ) or isentropic protoneutron stars : its use in the case of the @xmath5 hot interior is unjustified ( see section 3.2 for a correct statement of the stability criterion ) . in a recent paper , lai and shapiro ( 1995 ) have studied the secular evolution , secular `` bar instability '' , and the gravitational wave emission from the newly formed , rapidly rotating neutron stars . however , these authors used unrealistic ( polytropic ) equations of state of neutron star matter . moreover , the calculations were done within newtonian theory of gravitation . in view of this , the internal structure of their models of newly born neutron stars was quite different from that characteristic of the realistic models of protoneutron stars . the problem of the secular `` bar instability '' in rapidly rotating neutron stars was also studied , using general relativity , by bonazzola et al ( 1995 ) . however , numerical calculations were done only for realistic equations of state of _ cold _ neutron star matter . in the present paper we study the properties of uniformly rotating protoneutron stars , using exact relativistic description of the rapid , stationary rotation , combined with realistic equations of state of hot dense matter , used in the whole range of temperatures and densities relevant for protoneutron stars . in particular , we calculate the maximum frequency of uniform rotation of protoneutron stars and its dependence on their baryon mass , and on the thermal state and composition of stellar interior . it is clear , that uniform rotation represents only an approximation to the actual rotational state of a newly born protoneutron star . existing numerical simulations of gravitational collapse of rotating cores of massive stars produce differentially rotating protoneutron stars ( janka & moenchmeyer 1989a , b , moenchmeyer & mueller 1989 ) . however , it should be stressed that the initial rotational state of collapsing core is unknown , and this implies uncertainty concerning the rotational state of resulting protoneutron star . it is reasonable to say , that the actual degree of nonuniformity of rotation of a protoneutron star should be considered as unknown . in the present paper we will not address the question of the physical mechanisms that could `` rigidify '' the rotational motion within the protoneutron star interior . however , we will use the approximation of uniform rotation in order to limit the number the parameter space for our numerical calculation , and also because of the relative simplicity of the stability analysis in this specific , idealized case . within our simplified model , the `` neutrinosphere '' ( which has actually a deformed , spheroidal shape ) will separate hot , neutrino - opaque interior of a protoneutron star ( hereafter referred to as `` hot interior '' ) from a significantly cooler , neutrino - transparent envelope . the actual thermal state of the hot interior of protoneutron star is determined by its formation scenario , and is expected to be influenced by the dissipative processes ( damping of pulsations , viscous damping of differential rotation , neutrino diffusion ) . for simplicity , we will restrict ourselves to two limiting cases : an isothermal ( @xmath6 , see section 3 ) , and an isentropic ( entropy per baryon @xmath10 ) hot interior . we will also consider two limiting cases of the lepton composition of the protoneutron star interior . the first case , referring to the very initial state of protoneutron star , will correspond to a fixed trapped lepton number . in the second case , neutrinos will not contribute to the lepton number of the matter , which will correspond to vanishing chemical potential of the electron neutrinos ; such a situation will take place after a deleptonization of a protononeutron star . the position of the neutrinosphere will be located using a simple prescription based on specific properties of the neutrino opacity of hot dense matter . in all cases , the equation of state of hot dense matter will be determined using one of the models of lattimer and swesty ( 1991 ) . the plan of the paper is as follows . in section 2 we describe the physical state of the interior of protoneutron star , with particular emphasis on the eos of the hot interior at various stages of evolution . we explain also our prescription for locating the `` neutrinosphere '' of a protoneutron star , and we give some details concerning the assumed temperature profile within a protoneutron star . using simple estimates of the timescales relevant for various transport processes , we justify the approximation of stationarity which is used throughout this paper . in section 3 we give a brief description of the exact equations , used for the calculation of stationary configuration of uniformly rotating protoneutron stars . we discuss also stability of rotating configurations with respect to the axially - symmetric perturbations . the numerical method , used for the calculation of rapidly rotating configurations of protoneutron star , is briefly described in section 4 , where we also discuss numerical precision of our solutions . maximum rotation frequency , for various physical conditions prevailing in the hot stellar interior , calculated as a function of the baryon ( rest ) mass of protoneutron star , is presented in section 5 . then , in section 6 we show the validity of an empirical formula , which enables one to express with a surprisingly high precision the maximum frequency of rotating protoneutron stars in terms of the mass and radius of the maximum mass configuration of static ( non - rotating ) protoneutron stars with same eos . in section 7 we study the evolutionary transformation of a rotating protoneutron star into a cold neutron star . we show that , at fixed rest mass and angular momentum , maximum rotation frequency of protoneutron stars imposes severe constraints on the rotation frequency of solitary neutron stars . finally , section 8 contains discussion of our results and conclusion . we consider a protoneutron star ( pns ) just after its formation . we assume it has a well defined `` neutrinosphere '' , which separates a hot , neutrino - opaque interior from colder , neutrino - transparent outer envelope . important parameters , which determine the local state of the matter in the hot interior are : baryon ( nucleon ) number density @xmath11 , net electron fraction @xmath12 , and the net electron - neutrino fraction @xmath13 . the calculation of the composition of hot matter and of its eos is described below . such a situation is characteristic of the very initial stage of existence of a pns . matter is composed of nucleons ( both free and bound in nuclei ) and leptons ( electrons and neutrinos ; for simplicity , we do not include muons ) . all constituents of the matter ( plus photons ) are in thermodynamic equilibrium at given values of @xmath11 , @xmath14 and @xmath15 . the composition of the matter is calculated from the condition of beta equilibrium , combined with the condition of a fixed @xmath16 , @xmath17 where @xmath18 are the chemical potentials of matter constituents . at the very initial stage we expect @xmath19 . electron neutrinos are degenerate , with @xmath20 ( in what follows we measure @xmath14 in energy units ) . the deleptonization , implying the decrease of @xmath16 , occurs due to diffusion of neutrinos outward ( driven by the @xmath21 gradient ) , on a timescale of seconds ( sawyer & soni 1979 , bombaci et al . the diffusion of highly degenerate neutrinos from the central core is a dissipative process , resulting in a significant _ heating _ of the neutrino - opaque core ( burrows & lattimer 1986 ) . this is the limiting case , reached after complete deleptonization . there is no trapped lepton number , so that @xmath23 and @xmath24 , and therefore @xmath25 . neutrinos trapped within the hot interior do not influence the beta equilibrium of nucleons , electrons and positrons , and for given @xmath11 and @xmath14 the equilibrium value of @xmath26 is determined from @xmath27 while @xmath28 . in practice , this approximation can be used as soon as electron neutrinos become non - degenerate within the opaque core , @xmath29 , which occurs after some @xmath30 seconds ( sawyer & soni 1979 , bombaci et al . the neutrino diffusion is then driven by the temperature gradient , and the corresponding timescale of the heat transport ( pns cooling ) can be estimated as @xmath31 s ( sawyer & soni 1979 ) . in principle , the temperature ( or entropy per nucleon ) profile within a pns has to be determined via evolutionary calculation , starting from some initial state , and taking into account relevant transport processes in the pns interior , as well as neutrino emission from pns . transport processes within neutrino - opaque interior occur on timescales of seconds , some three orders of magnitude longer than dynamical timescales . the very outer layer of pns becomes rapidly transparent to neutrinos , deleptonizes , and cools on a very short timescale via @xmath32 pair annihilation and plasmon decay . it seemed thus natural to model the thermal structure of the pns interior by a hot core limited by a `` neutrinosphere '' , and an outer envelope of @xmath33 mev . the transition through the `` neutrinosphere '' is accompanied by a temperature drop , which takes place over some interval of density just above the `` edge''of the hot neutrino - opaque core , situated at some @xmath34 . in view of the uncertainties in the actual temperature profiles within the hot interior of pns , we considered two extremal situations for @xmath35 , corresponding to an isentropic and an isothermal hot interior . in the first case , hot interior was characterized by a constant entropy per baryon @xmath10 . in the case of trapped lepton number , this leads to the eos of the type : pressure @xmath36)$ ] , energy density @xmath37)$ ] , and temperature @xmath38)$ ] , with fixed @xmath39 and @xmath16 . this eos will be denoted by eos[@xmath40 . in the case of an isentropic , zero trapped lepton number eos , we will have eos[@xmath41 . the condition of isothermality , which corresponds to thermal equilibrium , is more complicated . due to the curvature of the space - time within pns , the condition of isothermality corresponds to the constancy of @xmath42 ( see section 3.1 ) . the significance of the @xmath43 condition will be discussed in section 3.1 . in the static case , the isothermal state within the hot interior will be reached on a timescale corresponding to thermal equilibration , which is much longer than the lifetime of a pns . in the case of a rotating pns , the situation can be expected to be even more complicated ( see , e.g. , chapter 8 of tassoul ( 1978 ) ) . nevertheless , we considered the @xmath43 models for several reasons . first , as a limiting case so different from the @xmath10 one , it enables us to check the dependence of our results for rapidly rotating pns on the thermal state of the hot interior . moreover , for the isothermal pns we can apply the criterion of stability with respect to the axi - symmetric perturbations , and this will enable us to calculate the value of @xmath3 for the stable supramassive rotating pns models with an isothermal interior ( see section 3 ) . our calculation of the `` neutrinosphere '' within the hot pns interior was done using a simple method , described below . for a given pns model , the neutrinosphere radius , @xmath44 , has been located through the condition @xmath45 where @xmath46 is the neutrino mean free path which is calculated at the matter temperature @xmath14 , while @xmath47 is the mean energy of non - degenerate neutrinos at ( and above ) the neutrinosphere , @xmath48 . we assumed that opacity above @xmath44 is dominated by the elastic scattering off nuclei and nucleons , so that @xmath49 . then , we determined the value of the density at the neutrinosphere , @xmath34 , for a given static pns model , combining eq . ( [ r_nu ] ) with that of hydrostatic equilibrium , and readjusting accordingly the temperature profile within the outer layers of pns . this value of @xmath34 was then used in the calculations of rotating pns models . let us notice , that similar approximation , in which the `` neutrinosphere '' in rapidly rotating protoneutron star was defined as a surface of constant density , was used by janka and moenchmeyer ( 1989a , b ) . some details concerning actual calculation of the temperature profile in the vicinity of the `` neutrinosphere '' will be given in subsection 3.3 . the starting point for the construction of our eos for the pns models was the model of hot dense matter of lattimer and swesty ( 1991 ) , hereafter referred to as ls . actually , we used one specific ls model , corresponding to the incompressibility modulus at the saturation density of symmetric nuclear matter @xmath50mev . for @xmath35 we supplemented the ls model with contributions resulting from the presence of trapped neutrinos of three flavours ( electronic , muonic and tauonic ) and of the corresponding antineutrinos . in fig . 1a , b we show our eos in several cases , corresponding to various physical conditions in the hot , neutrino - opaque interior of pns . for the sake of comparison , we have shown also the eos for cold catalyzed matter , used for the calculation of the ( cold ) ns models . in fig . 1a we show eos at subnuclear densities . at these densities , both the temperature and the presence of trapped neutrinos stiffen the eos , as compared to the cold catalyzed matter one , and the stiffening is rather dramatic . the constant @xmath14 eos stiffens considerably at lower densities , which is due to the weak dependence of the thermal contribution ( photons , neutrinos ) on the baryon density of the matter ( this effect will be to some extent moderated by the factor @xmath51 in the isothermal pns , see section 3.1 ) . it is quite obvious , that @xmath5 eos becomes dominated by thermal effects below for @xmath52 . on the contrary , for isentropic eos , the effect of the trapped lepton number ( @xmath53 ) turns out to be much more important than the thermal effects . this can be see in fig . 1a , by comparing dash - dotted curve , @xmath54 $ ] , with the dotted line , which corresponds to an artificial ( unphysical ) case with small thermal effects , @xmath55 $ ] . it is clear , that the correct location of the `` neutrinosphere '' , which separates hot interior from the colder outer envelope , should be important for the determination of the radius of pns , and in consequence , of the maximum rotation frequency of a given pns model . it may be useful to compare our subnuclear eos for pns with those used by other authors . the subnuclear eos of pns , used in the papers of hashimoto et al . ( 1995 ) and bombaci et al . ( 1995 , 96 ) is very different from that used in the present paper . in particular , hashimoto et al . ( 1995 ) used a cold ( @xmath9 ) eos for the densities below @xmath56 . on the other hand , bombaci et al . ( 1995 , 96 ) stop their hot eos at the edge of the liquid interior , and use the @xmath9 ( cold catalyzed matter ) eos for the densities below @xmath57 ; in this way , they seriously underestimate thermal and neutrino trapping effects on the radius of pns . our eos above nuclear density are plotted in fig . the presence of a trapped lepton number softens the eos , while thermal effects always stiffen it with respect to that for cold catalyzed matter . the softening of the supranuclear eos at fixed @xmath16 is due to the fact , that a significant trapped lepton number increases the proton fraction , which implies the softening of the nucleon contribution to the eos . it should be stressed , that in contrast to hashimoto et al . ( 1995 ) and bombaci et al . ( 1995 , 96 ) we used a unified dense matter model , valid for both supranuclear and subnuclear densities . also , the fact that we use various assumptions about the @xmath14 and @xmath39 profiles within pns , enables us to study the relative importance of the temperature profile and that of a trapped lepton number , for the pns models . the mass - radius relation for the static pns models calculated using various versions of our eos for the hot interior is shown in fig . we assumed @xmath58 , which was consistent with our definition of the `` neutrinosphere '' . for the sake of comparison , we show also the mass - radius relation for the @xmath9 ( cold catalyzed matter ) eos , which corresponds to cold neutron star models . in the case of the isothermal hot interior with central temperature @xmath59 mev we note a very small increase of the maximum mass , as compared to the @xmath9 case ( c.f . , bombaci et al . 1995 , 1996 ) . however , the effect on the mass - radius relation is quite strong , and increases rapidly with decreasing stellar mass . in the case of the isentropic eos with a trapped lepton number , [ @xmath60 , the softening of the high - density eos due to the trapped @xmath16 leads to the decrease of @xmath61 compared to the @xmath9 case ; as far as the value of @xmath61 is concerned , the softening effect of @xmath16 prevails over that of finite @xmath39 ( this is consistent with results of takatsuka 1995 and bombaci et al . 1995 , 1996 ) . however , the thermal effect on the radius is very important even in the case of @xmath53 . this can be seen by comparing the @xmath54 $ ] curve with that corresponding to the unphysical , fictitious case of @xmath55 $ ] . the very initial state of a pns corresponds to a significant trapped lepton number . with our assumption of a `` standard '' composition of dense matter ( i , e . , excluding large amplitude @xmath0-condensate , or a very large percentage of hyperons in cold dense matter ) , the maximum baryon mass ( baryon number ) of pns is lower than that of cold ns . therefore , in our case a stable pns transforms into a stable ns , and the scenario pns@xmath62black hole , considered by baumgarte et al . ( 1996 ) is excluded . at a given mass , the radius of a pns is significantly larger than that of a cold ns . as remarked by hashimoto et al . ( 1995 ) , this should have important implications for rotating pns . it should be stressed , however , that the value of radius , especially for pns which are not close to the @xmath61 configuration , turns out to be quite sensitive to the location of the edge of the hot neutrino - opaque interior ( i.e. , to the value of @xmath34 ) . the choice of bombaci et al . ( 1996 ) would lead to a much smaller effect on @xmath63 , while that of hashimoto et al . ( 1995 ) would result in larger values of the pns radii . the eos of pns is evolving with time , due mainly to the deleptonization process , which changes the composition of matter , and also due to changes of the internal temperature of the star . however , these changes occur on the timescales @xmath641 - 10 s , which are three or more orders of magnitude longer than the dynamical timescale , governing the readjustment of pressure and gravity forces . this dynamical timescale @xmath65ms corresponds also to the characteristic periods of the pns pulsations and of their rapid rotation . in view of this , we are able to decouple pns evolution from its dynamics , and treat its rotation in the stationary approximation , with a well defined eos of the pns matter . one of the neglected dynamical processes , implied by the radiative processes and the evolution of the thermal structure of a rotating pns , is the meridional circulation of the matter . strictly uniform rotation is incompatible with an assumption of a steady thermal state , resulting from the diffusive ( radiative ) equilibrium ( see , e.g. , tassoul 1978 ) . the requirement of radiative equilibrium will necessarily imply the existence of a meridional circulation of the matter . however , the velocity of this meridional circulation will be of the order of the stellar radius divided by the thermal timescale ( the timescale of changes of the entropy of the pns interior , which is of the order of neutrino diffusion timescale ) , which is much smaller than the rotational velocity ( see , eg . , section 8 of tassoul 1978 ) . in view of this , we can neglect the effect of the meridional circulation when calculating the mechanical equilibrium of a rapidly rotating pns . finally , let us notice that in a special , idealized case of @xmath66 , the radiative flux vanishes . then , pure uniform rotation can be realized as a steady state of a pns . the problem of the calculation of the stationary state of uniform rotation of cold neutron stars , within the framework of general relativity , was considered by numerous authors ( see the review article of friedman and ipser 1992 , and references therein ) . extensive calculations for a broad set of realistic equations of state of cold dense matter were recently presented in ( cook et al . 1994 ) and ( salgado et al . 1994 ) . here , we will extend the methods used at @xmath9 to the case of hot pns . we will use the notation and formalism developed in the paper of bonazzola et al . ( 1993 ) , hereafter referred to as bgsm . one of the problems introduced by finite temperature is that , if one does not make any other assumption about the equilibrium , the equation of stationary motion does not have a first integral ( bardeen 1972 ) . in the notation of bgsm , the equation of stationary motion ( eq . 3.25 of bgsm ) reads , when the effects of temperature and rigid rotation are included , as @xmath67 where @xmath39 is the entropy per baryon , @xmath14 is the temperature , and @xmath68 $ ] is the so - called pseudo - enthalpy ( or log - enthalpy ) , @xmath7 is the lapse function appearing in the space - time metric , and @xmath8 is the lorentz factor due to rotation . the equation of the stationary motion is to be supplemented with the equations determining the metric functions ( see bsgm for the derivation and the explicit form of the complete set of equations ) . in the expression for @xmath69 , @xmath70 is the energy density ( which includes rest energy of matter constituents ) and @xmath71 is the nucleon rest mass . this equation is the general relativistic equivalent of a well - known newtonian formula ( see e.g. tassoul 1978 ) . it is straightforward to show that a _ sufficient _ condition for ( [ eqsta ] ) to be integrable is @xmath72 . in this case , one obtains a first integral of motion of the form : @xmath73 this enables us to calculate the density profile in the envelope of the pns , because we have assumed a specific @xmath74 profile within it ( see section 3.3 ) . let us stress that , due to the integral term in ( [ firstint ] ) , the pseudo - enthalphy is no more an explicit function of the metric potentials , as it was in bgsm . in fact , one must solve ( [ firstint ] ) to have @xmath69 within the star . in the two particular cases , chosen by us for the hot interior , specific first integrals of eq . ( [ eqsta ] ) can be found . first , in the case of general relativistic thermal equilibrium , @xmath75 , it is easy to show that @xmath76 where @xmath77 is the baryon chemical potential , is indeed the integral of ( [ eqsta ] ) . the constancy of @xmath78 and @xmath79 corresponds to the general - relativistic thermodynamic equilibrium , if we neglect the time dependence of the eos of pns ( i.e. , if we freeze transport phenomena ) . second , in the case of isentropic profile @xmath80 , the thermal term in ( [ eqsta ] ) vanishes and the first integral of eq . ( [ eqsta ] ) reads @xmath81 the instabilities that could develop in rapidly rotating pns , and which could limit the maximum angular frequency of these objects , are of secular and of dynamic type . let us start with the problem of secular stability with respect to the axi - symmetric perturbation of rotating configurations . we denote the total baryon number , total angular momentum , and total entropy of a pns by @xmath82 , @xmath83 , and @xmath84 , respectively . we used the _ secular _ instability criterion of friedman et al . ( 1988 ) , extended to the case of finite temperature . for the purpose of completeness , we restate here their lemma ( and correct a misprint of their paper ) . + * lemma :* consider a three - parameter family of uniformly rotating hot stellar models having an equation of state of the form @xmath85 . suppose that along a continous sequence of models labelled by a parameter @xmath86 , there is a point @xmath87 at which @xmath88 , @xmath89 and @xmath90 vanish and where @xmath91 . then the part of the sequence for which @xmath92 is unstable for @xmath86 near @xmath87 . + similarly as in the case of friedman et al . ( 1988 ) , this lemma follows directly from theorem i of sorkin ( 1982 ) , with his function @xmath84 replaced by @xmath93 , the @xmath94 quantities replaced by @xmath82 , @xmath83 , @xmath84 and the @xmath95 ones by @xmath96 , @xmath97 , @xmath98 . the conditions of the theorem are fulfilled because stellar models are configurations for which @xmath99 is minimized at fixed @xmath82 , @xmath83 and @xmath84 , _ and _ because difference in @xmath99 between two neighbouring equilibria can be expressed as ( bardeen 1970 ) @xmath100 let us stress here that this last equation requires either @xmath98 or @xmath39 to be constant through the whole star , which fixes , in each of these two cases , the temperature profile within the stellar model . we investigated the stability of our models with a specific version of this criterion , in which we choose a continous sequence of equilibria to be a sequence at fixed @xmath82 and @xmath84 . the point of the loss of stability is then simply the point of extremal @xmath83 , i.e. : @xmath101 where @xmath102 is the central density of the star . the instability discussed above is a secular one ; it will develop on the timescale needed to transport the angular momentum within the perturbed model , in order to decrease the energy of the star while changing its shape and structure . however , the timescale of the pns evolution is quite short ( seconds ) , and it is driven by the same transport processes involving neutrinos , as those which are needed to destabilize the star via the axi - symmetric perturbations . in view of this , one might expect that the secular instability described above is not efficient in disrupting the quasi - stationary configuration of rapidly rotating pns . however , we should remember that the above considerations apply to the case of infinitesimal perturbations . transport processes would be crucial for removing the energy barrier separating the initial , secularly unstable configuration , from the dynamically unstable one , which would eventually collapse into a black hole . however , newly born pns are expected to be in a highly excited state , in which various modes of stellar pulsations are excited . one may expect that the energy contained in these pulsations is sufficient to overcome the energy barrier separating the actual metastable , secularly unstable state from the dynamically unstable , collapsing one . in view of this , we expect that the secularly unstable configurations should be treated like unstable ones . therefore , the critical configuration , given by eq.(9 ) , will be thus considered as the last stable one . a rapidly rotating pns can be also susceptible to other types of secular instabilities . the instability with respect to the non axisymmetric perturbations can be driven by the gravitational radiation reaction ( grr ) . however , detailed calculations performed for hot ns suggest , that at the temperature exceeding @xmath103k these instabilities can only slightly decrease the maximum rotation frequency of uniform rotation , due to the damping effect of the matter viscosity ( cutler et al . 1990 , ipser & lindblom 1991 , lindblom 1995 , yoshida & eriguchi 1995 , zdunik 1996 ) . secular instabilities of rapidly rotating ns with respect to the non - axisymmetric `` bar '' mode were recently investigated by lai & shapiro ( 1995 ) and bonazzola et al . while lai & shapiro ( 1995 ) addressed the problem of `` bar '' instability of newly formed ns , they assumed a rather unrealistic `` ellipsoidal model '' , involving only shear viscosity as a source of viscous dissipation , and performed their calculations within the newtonian theory of gravity . on the other hand , in their relativistic calculations bonazzola et al . ( 1995 ) considered only cold ns , and found that the secular `` bar '' instability can set in before the keplerian ( mass shedding ) limit is reached only for sufficiently stiff eos of ns matter . the problem of the `` bar '' instability of pns , with realistic eos of the hot interior , will be investigated by us in the future . in any case , inclusion of possible additional secular instabilities can only decrease the maximum rotation frequency of pns below the values obtained using the simplified approach adapted in the present paper . the fact of the existence of the maximum mass of rotating pns , @xmath104 ( and maximum baryon mass , @xmath105 ) ( see section 5 ) , puts a well defined and stringent limit on @xmath106 which can be reached by pns . configurations with @xmath107 are _ dynamically unstable _ : no stationary solution exists above @xmath108 . at the same time , the angular frequency of rigid rotation can not exceed the keplerian value , @xmath106 . these two conditions , combined with the instability criterion , expressed in eq . ( 9 ) , determine the maximum frequency ( or , strictly speaking , an upper bound on the frequency ) of rigid rotation of pns . practical implementation of these criteria will be described in section 5 . a consistent study of pns would require an exact treatment of the thermal transport in the frame of a non - stationary spacetime . unfortunately , such a study is beyond the scope of the present work , and we have thus chosen to impose `` by hand '' the temperature profile within the star , following prescription described in the subsection 2.3 . we divided the interior of pns into the hot interior ( @xmath109 ) , a layer corresponding to the temperature drop within the `` neutrinosphere '' ( @xmath110 ) , and the low temperature , neutrino - transparent outer envelope with @xmath111 . we have chosen two types of temperature profiles within the hot , neutrino - opaque core @xmath112 : * the isothermal profile : @xmath113 , with @xmath114 , * the isentropic profile : @xmath115 , with @xmath116 . for the transition region and the low temperature envelope , we used a suitable profile @xmath74 : @xmath117 the function @xmath118 was chosen for computational convenience as a suitable combination of an exponential and a gaussian function , selected to lead to @xmath74 of class @xmath119 through the transition ( temperature drop ) region . our typical choice was @xmath120 ; increasing this value up @xmath121 led to a very small increase of the stellar radius . as the mass of a massive pns is almost entirely contained in the hot neutrino - opaque core ( a rough estimate for a typical star gives less than @xmath122 of the total mass for the cool envelope mass ) , we supposed that our stability criterion remained valid also in the case of the presence of the low temperature envelope . we used a code based on the @xmath123 formulation of the einstein equations in stationary axisymmetric spacetimes ( bgsm ) . the four elliptic equations obtained were solved by means of a spectral method , in which the functions are expanded in different polynomial bases ( chebyshev for @xmath124 , legendre for @xmath125 and fourier for @xmath126 ) . we refer the reader to bgsm for a detailed description of the code , including the description of the `` virial '' indicator used for monitoring the convergence and the precision reached . let us just say that the code was modified to take into account thermal effects and , contrary to salgado et al . ( 1994 ) , to converge to the solution with various quantities being held fixed ( for example @xmath127 or @xmath83 or @xmath84 ) . here we briefly outline some of the numerical checks we made . the two - parameters eos was interpolated using bicubic splines subroutine from the nag library . in this way , the thermodynamic functions are of class @xmath128 , but the thermodynamic consistency is not conserved . the global relative error , evaluated by the means of the virial check ( see bgsm ) , is @xmath129 . this relatively `` low '' precision is due to the thermodynamical inconsistencies . we used two grids in @xmath124 for the star and one in @xmath130 for the exterior with @xmath131 in each , and one grid in @xmath125 with @xmath132 . we checked that a greater @xmath133 or @xmath134 does not change the results by more than a few @xmath135 at most , which stays within the global precision reached . let us stress that such a low number of grid points is sufficient to reach high accuracy within a spectral method ( that would not be the case with a finite difference scheme ) . the temperature drop at the `` neutrinosphere '' is not always located on the border of the internal grid in @xmath124 , which could influence the precision ( remember that the spectral methods are very sensitive to discontinuities ) . we checked that , in fact , even when the `` neutrinosphere '' is far from the border of the grid , the global physical quantities of the star did not change much ( we found also relative variations of a few @xmath135 at most ) . for a `` simple '' model where @xmath136 were held fixed , convergence required no more than @xmath137 iterations and @xmath138 minute of cpu time on a silicon graphix indigo@xmath139 workstation . however , the number of iterations can reach @xmath140 if one needs to converge to , e.g. , a keplerian configuration at fixed supra - massive baryon mass . finally , we compared our results at @xmath9 and @xmath141 with the results of another code used by one of us ( p. haensel ) , and found that the relative differences in the global properties of the stellar models are of the order of @xmath142 , which can be imputed to the different interpolation procedures . for a given eos , rapidly rotating pns models can be divided into two families : normal and supramassive one . normal models are those with baryon mass which does not exceed the maximum allowable baryon mass of static ( non - rotating ) configurations , @xmath143 . a _ normal _ rotating model can be transformed into a static configuration of the same baryon mass , through a continuous decrease of @xmath97 . rotation of pns increases their maximum allowable mass with respect to the static case , up to @xmath144 . rotating pns models with @xmath145 are called _ supramassive_. such a supramassive rotating model can not transform into a static model , because in the process of decreasing angular velocity it will collapse into a black hole . the maximum angular velocity of the normal and supramassive rotating models results from different stability conditions . in the case of normal rotating configurations the value of @xmath97 is bound by the mass shedding limit , which corresponds to the keplerian velocity at the stellar equator , @xmath146 . the value of @xmath106 turns out to be rather sensitive to the location of the `` neutrinosphere '' within the pns . this sensitivity is particularly large in the case of the isothermal profile of the hot neutrino - opaque core . this is visualized in fig . 3 , where we show the dependence of the mass shedding limit @xmath106 for pns with baryon mass @xmath147 on the value of @xmath34 , for the central temperatures ranging between 10 and 20 mev . the dependence on the value of @xmath34 weakens with increasing @xmath127 . also , this effect is much less important in the case of isentropic pns . llllll & & & & & + eos & @xmath148 & @xmath149 & @xmath104 & @xmath150 & @xmath151 + & @xmath152 $ ] & [ km ] & @xmath152 $ ] & [ km ] & [ hz ] + & & & & & + & & & & & + @xmath9 , @xmath153 & 2.048 & 10.59 & 2.430 & 14.34 & 1625 + @xmath154 , @xmath153 & 2.053 & 11.17 & 2.322 & 14.79 & 1521 + @xmath155 , @xmath53 & 1.957 & 10.85 & 2.180 & 14.40 & 1522 + @xmath156 , @xmath53 & 1.977 & 11.53 & 2.172 & 15.32 & 1388 + & & & & & + & & & & & + for supramassive rotating pns , the limit on @xmath97 has been set by the condition of stability of rotating models with respect to the axi - symmetric perturbations , which was combined with the mass shedding stability condition . our calculations have been performed for both isothermal ( @xmath43 ) and isentropic ( @xmath10 ) hot interiors of pns . in both cases , the absolute maximum of keplerian frequency for rotating models , which were stable with respect to the axi - symmetric perturbations , was obtained for a rotating configuration with a maximum baryonic mass ( and gravitational mass ) , @xmath157 [ @xmath104 ] . actually , rotating configuration with @xmath104 and that with @xmath2 do not generally coincide ( see , e.g. , cook et al . 1994 , stergioulas & friedman 1995 ) . however , the difference is very small , and it could not be detected within the precision of our numerical code . the value @xmath104 depends on the value of @xmath78 in the case of isothermal hot interior , but we preferred to parametrize it in terms of central temperature , @xmath159 . the dependence of @xmath104 on the value of @xmath160 is displayed in fig . 4 a. the effect of temperature on @xmath104 is opposite to that seen for @xmath148 : thermal effects lower the value of @xmath104 as compared to that for cold neutron stars . this is due to the thermal increase of the equatorial radius , which prevails over the thermal stiffening of the central core of pns . the dependence of @xmath161 on the value of @xmath39 in the case of isentropic hot interior of pns is represented in fig . 4 b , in the case of @xmath162 ( trapped lepton number ) and for @xmath53 . the decrease of @xmath161 is there smaller than in the case of the isothermal pns models . in fig . 5 a , 5 b we show our results for the maximum rotation frequency of stable pns models , reached for supramassive configurations . in the case of the isothermal pns , thermal effects tend to decrease the value of @xmath2 , but even in the case of @xmath163mev the relative effect does not exceed ten percent . irregularities at lower @xmath160 result from the limited accuracy of determination of the critical configuration with an isothermal core . maximum rotation frequency in the case of the isentropic hot neutrino - trapped cores of pns is shown , for several values of the central entropy per baryon , in fig . 5 b. the value of @xmath2 is also decreasing with increasing @xmath39 . in general , we find that the decrease of @xmath2 , as compared to cold neutron stars , is in our case smaller than that found by hashimoto et al . ( 1995 ) ; this may be due to the lower value of the density at the edge of the hot core , assumed by these authors . some differences may also be due to different supranuclear eos , and to a different treatment of the thermal effects , in particular , to their assumption of @xmath5 . in order to visualize the importance of the relativistic effects on @xmath164 in the isothermal interior of a pns , we show in fig . 6 the temperatures @xmath164 and @xmath78 , for @xmath165 @xmath166 , and a uniform keplerian rotation . as one can see , in the central part of the pns , @xmath14 is about @xmath167 percent greater than @xmath168 . a similar effect is seen also in the case of an isentropic hot interior , which in fig . 6 corresponds to @xmath169 . the calculation of the rotating pns ( and ns ) models is incomparably more difficult than that of the static models . in the case of cold ns , one finds a surprisingly precise universal formula , which relates the maximum rotation frequency to the mass and radius of the maximum mass configuration for the static models ( haensel & zdunik 1989 , friedman at al . 1989 , shapiro et al . 1989 , haensel et al . 1995 , nozawa et al . 1996 ) , @xmath170 where the most recent value of @xmath171 , based on calculations performed for a broad set of realistic _ cold _ eos of dense matter , is @xmath172 ( haensel et al . 1995 ) . the `` empirical formula '' , eq . ( 10 ) , reproduces the values of @xmath2 for cold ns models with typical precision better than @xmath173 ( although in some specific cases the deviations can reach nearly @xmath174 , see nozawa et al . 1996 ) . the validity of the empirical formula for @xmath2 is of great practical importance . for cold eos of dense matter , it enables one to get immediately a rather precise estimate of the value of @xmath2 , using easily calculated static neutron star models , and avoiding in this way incomparably more difficult 2-d calculations of rotating ns models and the analysis of their stability . it is of interest to check , whether `` empirical formula '' is valid also in the case of hot pns . let us notice , that the subnuclear eos of pns , with possible lepton number trapping , is very different from that of cold catalyzed matter ( see fig . 1 ) . in order to check the validity of the empirical formula , we define the eos depending parameter @xmath175 , @xmath176 where the right - hand - side is calculated using exact results for a specific eos . the values of @xmath175 for our models of pns are displayed in fig . 7 a , 7 b. let us notice , that for isentropic hot cores with trapped lepton number the value of @xmath175 shows a decreasing trend with increasing @xmath39 . in the case of isothermal hot cores with @xmath153 we see the opposite trend : the value of @xmath175 tends to increase with central temperature . the value of @xmath177 will lead to a very good empirical formula for pns , with precison comparable to that used for cold ns . within the precision of the empirical formulae , the values of @xmath178 and @xmath179 can be considered as being equal . maximum rotation frequencies and maximum baryon masses of pns with various eos , and corresponding frequencies of cold ns configurations , obtained via cooling at constant @xmath83 and @xmath127 ( @xmath83-losses due to neutrino emission are neglected ) . [ cols= " < , < , < , < " , ] rapidly rotating hot , neutrino - opaque pns evolves eventually into a cold , rotating ns , which under favourable circumstances can be observed as a solitary pulsar . let us assume , that such a transformation took place at constant baryon mass : this assumption is valid if mass accreted after the formation of a pns is negligibly small . ( notice , that we can _ define _ the moment of formation of a pns as that at which the accretion ends ) . second assumption refers to the angular momentum of the star . let us neglect for the time being the angular momentum loss due to emission of neutrinos ( mainly during deleptonization ) , as well as that due to gravitational radiation ( if rotating star deviated from the axial symmetry ) . inclusion of both effects could only _ decrease _ the final angular momentum of the ns . under these two assumptions , the transformation of a pns into a solitary radio pulsar takes place at constant baryon mass , @xmath127 , and angular momentum , @xmath83 . the maximum angular frequency of a solitary radio pulsar of a given baryon mass would be then determined by the parameters of a maximally rotating pns of the same baryon mass . in the present paper we restrict ourselves , similarly as hashimoto et al . ( 1995 ) , to the case of rigid rotation of pns . we will first consider `` typical '' range of baryon masses expected for the ns born in a gravitational collapse of a massive stellar core , @xmath180 ( for our eos of dense matter this would correspond to @xmath181 ) . all stars considered are normal , and the maximum initial frequency of pns will be then @xmath182 $ ] , and @xmath183 $ ] , for the limiting case of the zero trapped lepton number , and maximum trapped lepton number neutrino - opaque cores , respectively . the values of these limiting angular frequencies , for the isothermal and isentropic pns cores , are plotted versus @xmath127 in fig . 8 ( thin dashed and dash - dotted lines , respectively ) . our results displayed in fig . 8 lead to some conclusions , relevant to those neutron stars which remained solitary after their birth in the gravitational collapse of a massive stellar core . if the starting configuration ( at the end of significant accretion , after the revival of a stagnated shock ) was that with trapped lepton number and isentropic , then the maximum frequency of such a neutron star , in the gravitational mass range of 1.3 - 1.8 @xmath184 , can not exceed 600 - 900 hz , the lower frequency limit corresponding to the lower mass limit . the corresponding range of minimum periods would be from 1.6 ms to 1.1 ms , respectively . if the `` initial isolated configuration '' ( i.e. , that with no subsequent accretion ) was a little older , and therefore deleptonized , and if we assume that it was isothermal , then the resulting constraints would be less stringent : for the range of masses 1.3 - 1.8 @xmath184 we get minimum rotation periods of 1.25 - 1.1 ms , respectively . in both cases cooling resulted in speeding up of the rotation of the star : such situation is characteristic of `` normal rotating models '' . similar evolutionary considerations can be applied to the `` maximally rotating configurations '' : isentropic one with @xmath169 and isothermal ones with @xmath185mev , 25 mev , respectively . our results are shown in table 2 . the initial `` hot '' configurations are supramassive , and evolve into more compact cool ones ( at constant @xmath186 and @xmath83 ) _ decreasing _ their angular frequency ( slowing down ) . this purely relativistic effect was recently pointed out by hashimoto et al . the `` relativistic slowing down '' during cooling of the maximally rotating isentropic configuration with @xmath169 corresponds to the increase of the period from @xmath187=0.72 $ ] ms up to @xmath188 ms . if such a scenario of formation of solitary pulsars is valid , an absolute limit on their period would be @xmath189 , and not @xmath190 ms , obtained for the @xmath9 eos . the slowing down factor of @xmath191 coincides with that obtained by hashimoto et al . ( 1995 ) for a different eos and a different scenario of formation of solitary cold pulsars . actually , they assumed that the initial configuration is that with @xmath192 . their initial model would to some extent correspond to our isothermal models ( remember however the factor @xmath193 in our temperature profiles , which is absent in their calculations ) . using tables 1 , 2 we obtain @xmath194/p_{\rm min}[t=0]=0.92 $ ] , which is quite close to 1 . in the present paper we studied rapid rotation of protoneutron stars , using a specific model of hot , dense matter . our models of rapidly rotating protoneutron stars were based on several simplifications , which were necessary in order to make the problem tractable . in order to avoid difficulties and/or ambiguities of the largely unknown `` real situation '' , we restricted ourselves to studying idealized , limiting cases . in many places we introduced approximations , which were crucial for making numerical calculations feasible . we assumed rigid rotation within protoneutron star . this simplified greatly our considerations , and reduced dramatically the number of stellar models . actually , our formalism allows calculation of the quasi - stationary differentially rotating configurations , after introducing an additional @xmath195 term appearing on the right hand side of the equation of stationary motion , eq . ( 4 ) ( see section 5.1 of bgsm ) . we plan to perform studies of differentially rotating protoneutron stars in the near future , as the next step in our investigations of dynamics of protoneutron stars . while existing numerical simulations of gravitational collapse of rotating cores of massive stars yield a differentially rotating protoneutron star , the calculations stop too early after bounce . in contrast to our models of protoneutron stars , the object which comes out from the simulations of moenchmeyer and mueller has an extended , very hot envelope , produced by the shock wave immediately after bounce ( janka & moenchmeyer 1989a , b , moenchmeyer & mueller 1989 ) . also , the lack of knowledge of the initial angular velocity distribution within the collapsing core results in the uncertainty in the rotational state of produced protoneutron star . clearly , the study of quasistationary differential rotation of protoneutron stars should take into account all these uncertainties . deleptonization of protoneutron star is connected with energy and angular momentum losses . however , angular momentum taken away by neutrinos is expected to constitute at most a few percent of the total stellar angular momentum ( kazanas 1977 ) . inclusion of this effect would slightly decrease some of our `` evolutionary limits '' of section 7 . our treatment of the thermal state of the protoneutron star interior should be considered as very crude . the temperature profile might be affected by convection . also , our method of locating the `` neutrinosphere '' was very approximate . clearly , the treatment of thermal effects can be refined , but we do not think this will change our main results . our calculations were performed for only one model of the nucleon component of dense hot matter . the model was realistic , and enabled us to treat in a unified way the whole interior ( core as well as the envelope ) of the protoneutron star . however , in view of the uncertainties in the eos of dense matter at supranuclear densities , one should of course study the whole range of theoretical possibilities , for a broad set - from soft to stiff - of supranuclear , high temperature eos . an example of such an investigation , in the case of the _ static _ protoneutron stars , is the study of bombaci et al . ( 1996 ) . in view of the possible importance of the protoneutron star - neutron star connection for the properties of solitary pulsars , similar studies should also be done for rotating proto - neutron stars . the calculations of the present paper were done under the assumption , that the frequency of uniform rotation of protoneutron stars is limited only by the mass shedding and the secular instability with respect to the axi - symmetric perturbations . in view of this , our results can be considered only as upper bounds to the maximum rotation frequency . our main conclusion is that the minimum rotation period of solitary neutron stars , born as rapidly rotating protoneutron stars , is significantly larger , than the corresponding limit for cold neutron stars . inclusion of additional secular instabilities in rapidly , uniformly rotating protoneutron stars can only strengthen this conclusion . we are very grateful to e. gourgoulhon for his help at the initial stage of this project . we are also very grateful to w. dziembowski for introducing us into the difficult subject of meridional circulation . this research was partially supported by the jumelage program `` astronomie france - pologne '' of cnrs / pan , by the kbn grant no . p304 014 07 , and by the mesr grant 94 - 3 - 1544 . the numerical computations have been performed on the silicon graphics workstations , purchased thanks to the support of the spm department of the cnrs and the institut national des sciences de lunivers . bardeen j.m . , 1970 , apj , 162 , 71 bardeen j.m . , 1972 , in `` black holes - les astres occlus '' , les houches , ed . de witt c. & de witt b.s . , gordon and breach sci . pub . , new york baumgarte t.w . shapiro , s.a . teukolsky , 1996 , apj 458 , 680 bombaci i. , prakash m. , prakash m. , ellis p.j . , lattimer j.m . , brown g.e . , 1995 , a , 583 , 623 bombaci i. , 1996 , a&a 305 , 871 bombaci i. , prakash m. , prakash m. , ellis p.j . , lattimer j.m . , brown g.e . , 1996 , reports , in press bonazzola s. , gourgoulhon e. , salgado m. , marck j.a . , 1993 , a&a 278 , 421 bonazzola s. , frieben j. , gourgoulhon e. , 1995 , apj , 460 , 379 brown g.e . , bethe h.a . , 1994 , apj 423 , 659 burrows a. , lattimer j.m . , 1986 , apj , 307 , 178 cook g.b . , shapiro s.l . , teukolsky s.a . , 1994 , apj 424 , 823 cutler c. , lindblom l. , splinter r.j . , 1990 , apj 363 , 603 friedman j.l . , ipser j.r . , sorkin r.d . , 1988 , apj 325 , 722 friedman j.l . , ipser j.r . , parker l. , 1989 , phys . 62 , 3015 friedman j.l . , ipser j.r . , 1992 , phil . trans . lond . a , 340 , 391 haensel p. , zdunik j.l . , 1989 , nature 340 , 617 haensel p. , salgado m. , bonazzola s. , 1995 , a&a 296 , 745 hashimoto m - a , oyamatsu k. , eriguchi y. , 1995 , apj 436 , 257 ipser j.r . , lindblom l. , 1991 , apj 373 , 213 janka h .- t . , moenchmeyer r. , 1989a , a&a , 209 , l5 janka h .- t . , moenchmeyer r. , 1989b , a&a , 226 , 69 kazanas d. , 1977 , nature 267 , 501 lai d. , shapiro s.l . , 1995 , apj , 442 , 259 lattimer j.m . , swesty f.d , 1991 , nucl . a535 , 331 lindblom l. , 1995 , apj 438 , 265 moenchmeyer r. , mueller e. , 1989 , in `` timing neutron stars '' , ed . oegelman h. , van den heuvel e.p.j , kluwer , dordrecht nozawa t. , hashimoto m. , oyamatsu k. , eriguchi y. , 1996 , phys . rev . d 53 , 1845 pandharipande v.r . , pethick c.j . , thorsson v. , 1995 , phys.rev.lett . 75 , 4567 salgado m. , bonazzola s. , gourgoulhon e. , haensel p. , 1994 , a&a 291 , 155 sawyer r.f . , 1980 , apj 237 , 187 sawyer r.f . , soni a. , 1979 , apj 230 , 859 shapiro s.l . , teukolsky s.a . , wasserman i. , 1989 , nature 340 , 451 sorkin r.d . , 1982 , apj 257 , 847 stergioulas n. , friedman j.l . , 1995 , apj 444 , 306 takatsuka t. , 1995 , nucl . a588 , 365 tassoul j.l . , 1978 , `` theory of rotating stars '' , princeton university press , princeton yoshida s. , eriguchi y. , 1995 , apj 438 , 830 zdunik j.l . , 1996 , a&a 308 , 828

recently there has been a considerable interest in trying to obtain atomic decompositions of the space @xmath6 . these decompositions are usually obtained in terms of frames generated by a family of functions translated on a regular grid , and dilated by powers of a dilation matrix . the uniformity of the grid and the structure of the dilations can be exploited to obtain very sharp results . for irregular grids and unstructured dilations or if dilations are replaced by other transformations the situation is more complex and requires different techniques . one method is to use the regular case and try to obtain perturbations of the grid that preserve the frame structure . another possibility is to obtain irregular samples of the continuous transform , that have the required properties . in this article we study frame decompositions of the space @xmath4 using translations of a family of functions on irregular grids , and arbitrary dilations , and we even replace dilations by other transformations . our approach is different and very general , allowing quite general constructions . we prove the existence of smooth time - frequency frame atoms in several variables . the setting includes as particular cases , wavelet frames on irregular lattices and with a set of dilations or transformations that do not have a group structure . another particular case are non - harmonic gabor frames with non - uniform covering of the euclidean space . it also leads to new constructions of wavelet and gabor frames with regular lattice translates . one of the nice features of the proposed method is that it unifies different atomic decompositions . for the case of regular lattices guido weiss and his group @xcite developed a very fundamental program to characterize a large class of decompositions of @xmath4 through certain equations that the generators must satisfy . this is an important attempt to unify gabor and wavelets decompositions . other fundamental construction of mra wavelet frames on regular lattices can also be found in @xcite , @xcite , @xcite . our methods can be used to produce a substantial part of these systems . a set @xmath7 is a wavelet set if the inverse fourier transform of the characteristic function of the set is a wavelet . wavelet sets , frame wavelet sets and methods for constructing such sets have been studied recently @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite @xcite , @xcite . our methods give constructions of wavelet sets with translations on irregular grids . let @xmath8 and @xmath9 be countable index sets . we consider families of functions @xmath10 and discrete sets @xmath11 such that the collection @xmath12 form a frame for @xmath4 . the wavelet case is obtained when @xmath13 with a an expansive matrix and @xmath14 a fixed atom . we want to stress here that our constructions are much more general , allowing for example a different invertible ( not necessarily expansive ) matrix @xmath15 for each @xmath16 for the case of orthogonal wavelets , yang wang @xcite has recently considered wavelet sets associated with arbitrary families of invertible matrices and irregular sets of translates . he gave conditions for the existence of such wavelet sets and related them to spectral pairs . irregular wavelet and gabor frames also have been studied as perturbations of uniform ( lattice translate ) frames and also as sampling of the continuous wavelet/ gabor transform . see @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite,@xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . the approach in this article can be considered in the spirit of the classic construction in 1 dimension of smooth regular tight frames done by daubechies , grossmann and meyer in @xcite . they found , for the case of uniform lattices , general conditions on a compactly supported smooth function @xmath17 , in order that it generates a tight gabor frame of @xmath18 . in the affine case they found necessary and sufficient conditions for a band limited function in order that it generates a smooth wavelet frame . see also @xcite . there were other related attempts to obtain atomic decompositions of functional spaces using very general systems . see for example @xcite , @xcite in the context of locally compact groups . this paper is organized as follows : section [ notation ] introduces the notation and some preliminaries . section [ wc ] presents a theorem on wavelet construction on arbitrary , sufficiently dense , but otherwise irregular grids and with arbitrary dilation or even invertible transformation matrices . specific constructions of such wavelets are obtained in section [ ew ] , first in the 1-d case and then in the multidimensional case . a general theory of frame atomic decomposition of @xmath4 is obtained in section [ gen - res ] . using the concept of outer frame , reconstruction formulas for these atomic decompositions are obtained in section [ rf ] . throughout the paper @xmath8 and @xmath9 will denote countable index sets , and @xmath19 will stand for the function @xmath20 we will use @xmath21 to denote the lebesgue measure of a measurable set @xmath22 . a set @xmath23 of measurable functions on @xmath24 is called a _ riesz partition of unity _ ( * rpu * ) , if there exist constants @xmath25 such that @xmath26 let @xmath27 be a family of measurable subsets of @xmath5 . a _ riesz partition of unity * associated to @xmath28 * _ , is a set @xmath23 of measurable functions , such that 1 . @xmath29 2 . there exist constants @xmath25 such that @xmath30 * if @xmath31 is a rpu , then @xmath32 is also a rpu . * if @xmath33 , we will say that @xmath31 is a _ regular partition of unity_. * if the sets in @xmath28 are essentially disjoint ( i.e. @xmath34 ) , the family @xmath35 will yield a regular partition of unity associated to @xmath28 . * every rpu @xmath31 can be normalized to obtain a regular partition of unity by considering @xmath36 * given a family @xmath37 of measurable sets on @xmath5 , define @xmath38 where @xmath39 is the cardinal of the set @xmath40 . the value @xmath41 is called the _ covering index _ of @xmath28 . we now recall the definition of frame for a given close subspace @xmath42 of @xmath4 . a set of functions @xmath43 is a _ frame _ for @xmath42 if @xmath44 and there exist constants @xmath45 , such that @xmath46 for a measurable set @xmath47 we will denote by @xmath48 the functions that have support in @xmath49 , and by @xmath50 the functions whose fourier transform has support in @xmath49 , i.e. @xmath51 we will need also the following definition . let @xmath52 be a hilbert space . a collection of vectors @xmath53 is an _ outer frame _ for a closed subspace @xmath42 of @xmath52 , if @xmath54 is a frame for @xmath42 , where @xmath55 is the orthogonal projection onto @xmath42 , or equivalently , there exist constants @xmath45 , such that @xmath56 related definitions to the concept of outer frames appear in @xcite , @xcite . throughout the paper , we will use the following immediate and very useful fact about frames : + if @xmath57 is a frame for @xmath48 , and @xmath58 , then @xmath57 is an outer frame for @xmath59 . our first results concerns the construction of wavelets @xmath60 with translates on the arbitrary irregular grid @xmath61 and with an arbitrary countable family of invertible @xmath2 matrices @xmath3 . [ wavelets ] [ wavelet ] let @xmath62 be a set of finite measure , @xmath17 a function in @xmath4 and @xmath63 a family of invertible matrices . for each @xmath64 set @xmath65 and let @xmath66 . assume that @xmath28 is a covering of @xmath5 , @xmath67 is a rpu with bounds @xmath68 and @xmath69 and that @xmath70 . consider @xmath71 such that for each @xmath64 , the set @xmath72 forms a frame for @xmath73 with lower and upper frame bounds @xmath74 and @xmath75 respectively . if @xmath76 and @xmath77 , then the collection @xmath78 is a wavelet frame of @xmath4 with bounds @xmath79 and @xmath80 , generated by a single function @xmath81 , where @xmath81 is the inverse fourier transform of @xmath17 . since for each @xmath64 we have that @xmath82 forms a frame for @xmath73 with lower and upper frame bounds @xmath74 and @xmath75 respectively , an application of part [ coro-2 ] of corollary [ transl ] for the matrix @xmath83 shows that @xmath84 forms a frame of @xmath85 with the same bounds . from the definition of @xmath86 , @xmath87 is then a frame for @xmath85 with frame bounds @xmath88 and @xmath89 . on the other side , if @xmath90 then @xmath67 is associated to @xmath91 . so , we can apply proposition [ genexpdil ] to @xmath92 and @xmath93 , to conclude that , @xmath94 forms a frame of @xmath4 with lower frame bound @xmath95 and upper frame bound @xmath96 . this gives that , @xmath97 forms a frame of @xmath4 with frame bounds @xmath79 ( or @xmath98 ) and @xmath80 . the theorem now follows from an application of the inverse fourier transform . 1 . the set of matrices @xmath99 can be arbitrary and need not have a group structure . the set @xmath100 can also be chosen to have a simple structure . for example , @xmath101 , @xmath102 where @xmath103 is a rotation and @xmath104 a dilation matrix , will be used to construct directional wavelets . an even simpler example is @xmath105 , @xmath106 , where @xmath107 is an invertible matrix which gives a construction of wavelet frames on @xmath5 . 3 . note that @xmath17 does not need to be compactly supported . we will use the theorem above to construct specific examples of wavelets , e.g. , directional wavelets , isotropic wavelets , etc . * interesting particular cases of theorem [ wavelet ] . * 1 . @xmath108 . + let @xmath109 be such that @xmath110 is a frame for @xmath73 with frame bounds @xmath111 and @xmath112 . then @xmath113 forms a wavelet frame of @xmath4 with bounds @xmath79 and @xmath80 . 2 . @xmath114 , with @xmath115 . + each of the following sets are wavelet frames of @xmath4 with bounds @xmath79 and @xmath80 : @xmath116 1 . if the set @xmath28 is a tiling of @xmath5 then the wavelets constructed above are shannon - like wavelets , thus not well localized in space . to obtain well localized space - frequency wavelets , @xmath117 must be constructed to be a smooth partition of unity , e.g. , at least @xmath118 as demonstrated in the examples in section [ ew ] , below reconstruction formulas for such wavelet frames are developed in section [ rf ] . to be able to use theorem [ wavelet ] to construct concrete examples of wavelet frames on irregular grids , we first need to construct exponential frames @xmath119 ( also called fourier frames ) for @xmath120 . exponential frames play a central role in sampling theory for paley - wiener spaces ( also known as spaces of band - limited functions ) . the density of a set @xmath121 and separateness of the points in @xmath122 play a fundamental role for finding exponential frames @xmath119 for @xmath120 . a sequence @xmath123 is separated if @xmath124 there are many notions for the density of a set @xmath122 . we start with three definitions that are due to beurling . 1 . a lower uniform density @xmath125 of a separated sequence @xmath126 is defined as @xmath127 where @xmath128^d}\right)$ ] , where @xmath129 denotes the cardinal of the set @xmath130 . an upper uniform density @xmath131 of a separated sequence @xmath122 is defined as @xmath132 where @xmath133^d}\right)$ ] . 3 . if @xmath134 , then @xmath122 is said to have uniform beurling density @xmath135 . the limits in the definitions of @xmath125 and @xmath131 exist ( see @xcite ) . as an example , let @xmath136 be separated and assume that there exists @xmath137 such that @xmath138 , for all @xmath139 . then @xmath140 . for the one dimensional case , beurling proved the following theorem . ( beurling ) [ b1 ] let @xmath141 be separated , @xmath142 and @xmath143 $ ] . if @xmath144 then @xmath145 is a frame for @xmath146 . this previous result however is only valid in one dimension . for higher dimensions , beurling introduced the following notion : the gap @xmath147 of the set @xmath148 is defined as @xmath149 equivalently , the gap @xmath147 can be defined as @xmath150 it is not difficult to show that if @xmath122 has gap @xmath147 , then @xmath151 . for a separated set @xmath122 , and for the case where @xmath152 is the ball @xmath153 of radius @xmath154 centered at the origin , beurling @xcite proved the following result : [ beurlingthm ] let @xmath126 be separated , and @xmath155 . if @xmath156 , then @xmath145 is a frame for @xmath146 . for a very clear exposition of some of the beurling density results see @xcite . the construction in the following theorem ( which is a particular case of theorem [ wavelet ] ) , generalizes a similar result of @xcite to the irregular case . see also @xcite . [ 1dwvlt ] let @xmath157\cup [ 1/2,1]$ ] , @xmath158 , and let @xmath159 be a real valued function such that @xmath160 $ ] , @xmath161 , and @xmath162 on @xmath163 $ ] . assume that for each @xmath164 , the sequence @xmath165 is separated and that @xmath166 . then for each @xmath167 , the set @xmath168 is a frame of @xmath169 , where @xmath170 . if furthermore the sets @xmath171 are chosen such that the frame bounds @xmath74 and @xmath75 satisfy @xmath172 and @xmath173 , then the set @xmath174 where @xmath175 is a wavelet frame for @xmath18 . the wavelet frame constructed in the theorem above is of the form + @xmath174 which is slightly different form than the one constructed in theorem [ wavelet ] . this discrepancy is due to a convenient choice of the irregular set @xmath176 that we have adopted in the statement of the theorem above . note that the wavelets constructed in the theorem above are real and symmetric . actually , if one wants @xmath81 with good decay , @xmath177 can be easily constructed to be @xmath178 , @xmath179 , even @xmath180 . as a corollary , if we choose the sampling sets @xmath181 to be nested , i.e. , @xmath182 , we get [ 1dwvltc1 ] let @xmath183 , @xmath184 , and @xmath177 be as in theorem [ 1dwvlt ] . assume that the sequences @xmath185 are separated and such that @xmath186 . if @xmath187 , then for each @xmath167 , the set @xmath188 is a frame for @xmath189 . if furthermore the frame bounds @xmath74 and @xmath75 satisfy @xmath172 and @xmath190 , then the set @xmath191 where @xmath192 is a wavelet frame for @xmath18 . since @xmath193 we have that @xmath194)\ge 2 \#(x_{j}\cap [ -r , r]-1 ) $ ] . thus @xmath195 . but @xmath187 , therefore @xmath196 . the corollary then follows directly from theorem [ 1dwvlt ] . from theorems [ 1dwvlt ] and [ wavelet ] , we immediately get the following corollary . [ 1dwvltc2 ] let @xmath183 , @xmath184 , and @xmath197 be as in theorem [ 1dwvlt ] . assume that the set @xmath198 is separated and that @xmath199 . then , the set @xmath200 is a frame of @xmath169 , and the set of functions @xmath201 where @xmath192 is a wavelet frame for @xmath18 . 1 . _ shannon - type wavelet frames _ : we use corollary [ 1dwvltc1 ] , with @xmath202 and @xmath203}$ ] to get wavelet frames of the form @xmath204 these wavelets are not well localized since the decay at @xmath205 is @xmath206 . shannon - type wavelet bases _ : if we choose @xmath181 such that @xmath207 , then by kadec s @xmath208-theorem , we immediately get that @xmath209 constructed above form a wavelet riesz basis for @xmath18 . 3 . _ well localized wavelet frames _ : for faster decay of the wavelet frames , we choose @xmath177 to be a smoother function . let @xmath210}\ast\cdots\ast\chi_{[0,1]}$ ] be the b - spline of degree @xmath211 ( note that @xmath212 $ ] ) . let @xmath213 , and @xmath214 . then we get a wavelet frame of the form @xmath215 for this case the wavelets decay as @xmath216 . 1 . for @xmath217 , let @xmath218 , and let @xmath219 . if @xmath220 and @xmath221 , then using proposition [ rdframes ] below for product frames , the set @xmath222 form a frame for @xmath223 ^ 2}$ ] . let @xmath224 , and @xmath225 , then @xmath226 . we can then use theorem [ wavelet ] to construct wavelet frames for @xmath227 : * _ shannon - type radial wavelets _ : let @xmath228 , then @xmath17 is radial . thus the function @xmath81 defined as @xmath229 satisfies @xmath230 , where @xmath231 . we then construct the shannon - like wavelet frame for @xmath227 , as in theorem [ wavelet ] . a related construction of non - separable radial shannon - type frame wavelets and multiwavelets can be found in @xcite , and @xcite . * _ well localized radial wavelets _ : to construct wavelet frames with polynomial decay in space , we let @xmath232 , and construct the wavelet frames using theorem [ wavelet ] ( see figure [ fig2 ] ) . + radial wavelet frames that are well localized in space . , width=188 ] 2 . the points @xmath233 lie on an irregular grid of the form @xmath234 . however , we may be interested in points @xmath235 that do not lie on irregular grids of the form @xmath234 . for this case , the same constructions above can be used to form wavelets frame for @xmath227 , as long as the gap @xmath236 . 3 . as in the 1-d examples above , we can also use corollary [ 1dwvltc1 ] to construct wavelets on irregular grids satisfying @xmath237 . directional wavelet frames _ : [ examples-4 ] we can easily construct directional wavelet frames as follows : let @xmath238 be a region defined by @xmath239 , and define @xmath240 . let @xmath241 , and @xmath103 be the matrix of a rotation by an angle @xmath242 . let @xmath81 be such that @xmath243 , then we obtain the wavelet frame for @xmath227 of the form @xmath244 . the index @xmath245 codes for the resolution of the wavelet , while the index @xmath246 codes for four possible directions . thus the wavelet frame coefficients encode time scale as well as directional information . clearly one can choose any number of directions and adapt the previous construction . an obvious modification as shown in figure [ fig1 ] , yields wavelet frames with polynomial decay . + well - localized directional wavelet : the regions @xmath247 and @xmath248 that can be used to construct well localized directional wavelets . , width=188 ] + very nice constructions of smooth directional wavelet frames on regular grids were obtained before in @xcite . _ spiral _ in this example we will define a dilation covering by spiral annulus sectors . + let @xmath249 , and @xmath250 the spiral curve defined by @xmath251 for @xmath252 define @xmath253 to be the rotation of angle @xmath254 : @xmath255 $ ] . the curve @xmath250 satisfies : @xmath256 note that for positive @xmath254 the matrix @xmath257 is expansive . + now we are ready to define the covering elements . set @xmath258 and @xmath259 , for some integer @xmath260 so that @xmath261 . define the spiral annulus sector @xmath262 ( see figure [ fig3 ] ) . so @xmath263 is compact and @xmath264 is a disjoint covering of @xmath265 . + choose @xmath266 sufficiently small and @xmath17 a smooth function that does not vanish in @xmath263 and with support in @xmath267 . define @xmath268 . select a separated set @xmath269 such that @xmath270 . + the set @xmath271 form a wavelet frame of @xmath227 generated by a single wavelet @xmath81 that is band - limited , with good decay and directional in frequency . + spiral wavelet frames . , width=188 ] 6 . obviously , all the constructions above can be generalized to @xmath5 for any dimension @xmath272 . some of the wavelet frames may be associated with mras . for example , the so called shannon wavelet frame constructed above is associated with the shannon mra @xmath273\ } $ ] , @xmath274 . in general however , the precise relation needs further investigation . in this section we will develop a general method for constructing time - frequency frame atoms in several variables . this construction allows us to construct the previously introduced wavelet frames on irregular grids and with arbitrary dilation matrices or other types of transformations . it also allows us to construct non - harmonic gabor frames on non - uniform coverings of @xmath5 as described in section [ ctfa ] below . let @xmath275 and @xmath276 be non - empty open subsets of @xmath5 , @xmath277 is an invertible @xmath278 map , with @xmath278 inverse @xmath279 , i.e. , @xmath280 is a @xmath278 homeomorphism with @xmath278 inverse @xmath279 . define @xmath281 and @xmath282 , where @xmath283 denotes the derivative of @xmath280 at @xmath284 , @xmath285 $ ] . we have the following proposition : [ chvar ] with the above notation , assume that @xmath254 is positive and @xmath286 is finite , then if @xmath57 is a frame for @xmath59 with frame bounds @xmath111 and @xmath112 , then @xmath287 is a frame for @xmath288 with frame bounds @xmath289 and @xmath290 . assume that @xmath291 . then for @xmath64 , by an application of the change of variables formula @xmath292 since @xmath293 is finite and bounded away from zero , @xmath294 is in @xmath59 . using that @xmath295 is a frame of @xmath296 with bounds @xmath297 we have @xmath298 now , applying again the change of variables theorem , we obtain @xmath299 we have that @xmath300 = \left(\det[t^{\prime}(y)]\right)^{-1}$ ] for all @xmath301 . thus @xmath302 that is , from , and we have @xmath303 this proves the upper inequality for @xmath304 . the lower inequality is obtained in the same way , with the obvious modifications . proposition [ chvar ] establishes that @xmath280 defines an isomorphism @xmath305 defined by @xmath306 , thereby transforming frames into frames . by taking @xmath280 to be a translation or a dilation , we get the well known result : [ transl ] let @xmath307 be an open subset of @xmath5 . let @xmath308 be any point , and @xmath115 an invertible matrix . we have , 1 . * translation * [ coro-1 ] @xmath57 is a frame for @xmath48 with bounds @xmath111 and @xmath112 , if and only if @xmath309 is a frame for @xmath310 with the same bounds . 2 . * dilation * [ coro-2 ] @xmath57 is a frame for @xmath48 with bounds @xmath111 and @xmath112 , if and only if @xmath311 is a frame for @xmath312 with the same bounds . in fact the corollary remains true if we only assume that @xmath183 is measurable . part [ coro-1 ] is a direct application of the proposition , for the case that @xmath313 . for part [ coro-2 ] , and the transformation @xmath314 , the proposition tells us that @xmath315 is a frame with frame bounds @xmath316 and @xmath317 , therefore dividing each function by @xmath318 we obtain the result . for the next theorem , we need to introduce some definitions . let @xmath319 be a rpu with bounds @xmath68 and @xmath69 . for each @xmath320 set @xmath321 let @xmath322 , and define @xmath323 for a given @xmath324 , we discard all those @xmath167 such that @xmath325 has measure zero . note that if @xmath326 , then we can only claim that @xmath327 is a rpu associated to @xmath328 with constants @xmath324 and @xmath69 . [ unionframe ] let @xmath329 . 1 . [ unionframe-1 ] assume that @xmath330 is a frame for @xmath331 with lower and upper frame bounds @xmath74 and @xmath75 respectively . if @xmath76 and @xmath332 , then @xmath333 is a frame for @xmath4 , with frame bounds @xmath334 and @xmath335 . [ unionframe-2 ] assume that for each @xmath336 , @xmath330 is a frame for @xmath337 with lower and upper frame bounds @xmath74 and @xmath75 respectively . if @xmath338 and @xmath339 , then @xmath340 is a frame for @xmath341 , with frame bounds @xmath342 and @xmath335 . 1 . given @xmath343 and denoting by @xmath344 we will first show that @xmath345 for @xmath346 where we used dominated convergence for the interchange of the integral with the sum . the other inequality is analogous . + for each @xmath64 and @xmath343 , we use the fact that @xmath347 is a frame for @xmath331 , and that @xmath348 to obtain @xmath349 so summing over @xmath167 @xmath350 2 . for the second case we observe that the set @xmath351 forms a rpu associated to @xmath352 with bounds @xmath324 and @xmath69 , for if @xmath353 , @xmath354 furthermore , by hypothesis , @xmath355 these subspaces @xmath337 correspond to the subspaces @xmath331 defined in for the rpu @xmath356 , and therefore we can use the previous result ( applied to @xmath357 instead of @xmath4 ) to conclude that @xmath358 is a frame for @xmath359 . the proof is complete by noting that @xmath360 . * note that in the previous theorem , instead of choosing a frame for the subspaces @xmath331 , we could have chosen any collection of functions of @xmath4 that form an outer frame for @xmath331 . * if @xmath17 is a bounded function and @xmath361 , then it is easy to see that @xmath362 if and only if @xmath363 . so the spaces @xmath331 defined in will coincide in most of the cases with @xmath364 . as a very important particular case of the previous theorem , we have the following corollaries . [ unionframe - cor ] let @xmath365 be a family of subsets of @xmath5 , not necessarily disjoint , and let @xmath366 be a rpu with constants @xmath367 and @xmath69 not necessarily associated to @xmath28 . assume that @xmath330 is a frame for @xmath368 with lower and upper frame bounds @xmath74 and @xmath75 respectively . if @xmath76 and @xmath332 , then 1 . [ unionframe - cor-1 ] if @xmath67 is associated to @xmath28 ( i.e. @xmath369 ) , then @xmath333 is a frame for @xmath370 , with frame bounds @xmath334 and @xmath335 . [ unionframe - cor-2 ] if instead @xmath371 , then also @xmath333 is a frame for @xmath372 , with frame bounds @xmath342 and @xmath335 . denote by @xmath373 the orthogonal projection on the subspace @xmath331 defined in . since @xmath374 forms a frame for @xmath368 with bounds @xmath74 and @xmath75 , then @xmath375 forms a frame for @xmath331 with the same bounds . so , by theorem [ unionframe ] , @xmath376 forms a frame of @xmath377 with frame bounds @xmath79 and @xmath80 . we obtain part @xmath378 of the corollary observing that : @xmath379 the second part is a consequence of the fact that @xmath380 is a rpu associated to @xmath28 with bounds @xmath324 and @xmath69 and that @xmath381 . therefore we can apply the second part of theorem [ unionframe ] . * for the first case , the set @xmath382 is not necessarily a frame for @xmath383 or even for @xmath384 , even though @xmath385 is a frame for @xmath386 . * in the second case , the subspace @xmath331 is contained in @xmath387 . * note that if @xmath67 is a regular partition of unity , then the frame bounds for the frame constructed in the corollary , are @xmath111 and @xmath112 . [ new - cor ] let @xmath388 be a rpu with constants @xmath367 and @xmath69 and @xmath389 and @xmath390be coverings of @xmath5 such that @xmath391 and @xmath392 for all @xmath64 and some constant @xmath393 . assume that @xmath330 is a frame for @xmath368 with lower and upper frame bounds @xmath74 and @xmath75 respectively . if @xmath76 and @xmath332 , then @xmath333 is a frame for @xmath6 , with frame bounds @xmath342 and @xmath335 . as in the proof of the theorem , we have the inequalities @xmath394 on the other side , @xmath395 and then @xmath396 and @xmath397 the following proposition is a direct application of fubini s theorem , and allows us to construct frames in a product space . by induction the theorem can be generalized to hold for any finite number of factors . [ rdframes ] let @xmath398 and @xmath399 be measurable subsets of @xmath400 and @xmath401 respectively , and let @xmath402 and @xmath403 be frames for @xmath404 and @xmath405 with frame bounds @xmath406 and @xmath407 . then @xmath408 is a frame of @xmath409 with frame bounds @xmath410 . for any function @xmath411 in @xmath409 , @xmath412 which yields the lower frame bound @xmath413 . the upper frame bound @xmath414 can be obtained in a similar fashion . we now particularize our previous results to frames of the form @xmath415 , where @xmath17 is a fixed function . using the fourier transform , these types of frames allow us to construct wavelets @xmath416 with translates on the arbitrary irregular grid @xmath122 and with an arbitrary countable family of invertible @xmath2 matrices @xmath3 ( cf . theorem [ wavelet ] ) . first as a particular case of corollary [ unionframe - cor ] , we obtain the following proposition . [ genexpdil ] assume that @xmath417 forms a covering of @xmath5 , and let @xmath418 be a rpu with bounds @xmath68 and @xmath69 associated to @xmath28 . assume also that @xmath419 is a frame for @xmath368 with lower and upper frame bounds @xmath74 and @xmath75 respectively . if @xmath76 and @xmath77 , then @xmath420 is a frame for @xmath4 with frame bounds @xmath79 and @xmath80 . if @xmath421 , @xmath86 , and @xmath422 are chosen such that @xmath423 , @xmath252 , @xmath424 and @xmath425 , where @xmath250 is a lattice in @xmath5 , then we obtain the standard gabor or weyl - heisenberg frames . thus in general , the construction above can be viewed as non - harmonic gabor frames with variable windows @xmath422 . the following wavelet frame construction is a direct application of the previous proposition . taking @xmath426 , the set @xmath427 is a wavelet frame for @xmath4 with frame bounds @xmath79 and @xmath80 . therefore , there exists a dual frame @xmath428 such that @xmath429 we can also use proposition [ genexpdil ] and beurling theorem [ b1 ] , to obtain a non - harmonic gabor frame as the following example shows : [ gf ] [ non - harmonic gabor frames ] let @xmath430 $ ] and let @xmath431}\ast\chi_{[0,1]}\ast\chi_{[0,1]}\ast\chi_{[0,1]}$ ] be the b - spline of degree @xmath432 . clearly , @xmath433 . let @xmath185 be sets in @xmath434 chosen such that for each @xmath167 , @xmath435 and such that the frame bounds @xmath74 and @xmath75 satisfy @xmath172 and @xmath173 . then using proposition [ genexpdil ] and beurling theorem [ b1 ] , we obtain a non - harmonic gabor frame of the form @xmath436 . obviously , we can also use a non - uniform partition of @xmath434 and get generalized non - harmonic gabor frames as discussed earlier . if @xmath62 be a measurable subset of @xmath5 , and if the family of sets @xmath28 is defined by means of expanding or contracting @xmath183 , then we obtain the theorems of section [ wc ] as particular cases of proposition [ genexpdil ] . by corollaries [ transl ] and [ unionframe - cor ] , we can construct a frame for @xmath437 starting from a frame for @xmath438 where @xmath104 is _ any _ subset of @xmath5 with nonempty interior . hence to build a frame for @xmath439 it is enough to start with a frame for @xmath440 , where @xmath275 is an open disk . specifically , since any bounded measurable set can be covered by a finite number of translates of @xmath275 , we can use corollary [ transl](1 ) to find a frame for @xmath439 . we can also expand @xmath275 until it covers @xmath183 and use corollary [ transl](2 ) . this shows that there are many ways to construct frames for @xmath439 starting from a frame for @xmath440 . obviously , the particular construction will depend on the application . we will describe now two particular constructions . 1 . given a frame for @xmath441 where @xmath104 is a measurable subset of @xmath5 , we will construct a frame for @xmath442 . let @xmath250 be a regular lattice in @xmath5 ( i.e. @xmath443 , where @xmath103 is an invertible @xmath2 matrix with real entries ) , and let @xmath104 be a measurable subset of @xmath5 such that @xmath444 with a finite covering index . let @xmath183 be a measurable subset of @xmath5 and define @xmath445 for @xmath446 . let @xmath447 . by corollary [ transl ] , if @xmath448 is a frame for @xmath449 , then @xmath450 is a frame for @xmath451 . hence , @xmath452 is also a frame for @xmath453 . therefore , by corollary [ unionframe - cor ] , @xmath454 is a frame for @xmath73 . as an example , when @xmath183 is a measurable subset of @xmath5 , @xmath455^d$ ] , and @xmath456 , we have that if @xmath457^d } : k \in k\}$ ] is a frame for @xmath458^d)$ ] , then @xmath459 is a frame for @xmath73 ( recall that @xmath460^d + \gamma)$ ] ) . + it is easy to see that the covering requirement @xmath461 in the previous construction is not restrictive . specifically , we only need @xmath462 , where @xmath254 is any positive real number . furthermore , the construction remains valid if for each @xmath463 we choose a _ different _ set @xmath464 such that @xmath465 is a frame for @xmath449 , and @xmath466 and @xmath467 , where @xmath468 and @xmath469 are the lower and upper frame bounds of @xmath470 . finally , note that if @xmath471 we obtain a frame for @xmath4 . 2 . if @xmath183 is bounded we can construct frames for @xmath442 using theorem [ beurlingthm ] . let @xmath472 , and @xmath473 such that @xmath474 . let @xmath475 be such that @xmath476 . then using beurlings theorem ( [ beurlingthm ] ) we obtain that @xmath477 is a frame of @xmath478 . so , using corollary [ transl](1 ) , @xmath479 is an outer frame of @xmath442 , and @xmath480 is a frame for @xmath442 . in view of the previous results , we will be interested in constructing particular kinds of riesz partitions of unity associated to special coverings of the space . the next results provide the necessary tools to accomplish this task . if @xmath107 is a @xmath481 matrix , we will say that @xmath107 is _ expansive _ , if @xmath482 for every eigenvalue @xmath356 of @xmath107 . we will use the following known result ( see for example @xcite , pg . let @xmath40 be in @xmath483 and @xmath266 . there exists a matrix norm @xmath484 such that @xmath485 where @xmath486 is the spectral radius of the matrix @xmath40 , and there exists a norm @xmath487 in @xmath488 such that @xmath489 as a consequence of this lemma , if @xmath107 is an expansive matrix , then there exists a norm @xmath490 in @xmath488 such that @xmath491 and therefore @xmath492 in particular for every @xmath493 , @xmath494 [ prop * * ] let @xmath107 be a @xmath2 expansive matrix and @xmath495 a bounded set such that 1 . there exists @xmath266 such that @xmath496 . [ prop**1 ] 2 . @xmath497.[prop**2 ] then @xmath498 , the covering index of the family @xmath499 , is finite , i.e. there exists an integer @xmath500 such that @xmath501 . for @xmath502 define @xmath503 since @xmath504 , it is enough to prove that @xmath505 and @xmath506 are uniformly bounded in @xmath5 . we will see that @xmath505 is uniformly bounded . a similar argument proves the claim for @xmath506 . it is easy to see that @xmath507 thus , if @xmath505 is bounded on @xmath276 , then , by ( [ prop**2 ] ) @xmath505 is bounded in @xmath265 with the same bound . now , using ( [ prop**1 ] ) , we see that there exist @xmath508 such that @xmath509 , for all @xmath510 . fix @xmath510 . if @xmath511 , then there exists @xmath512 such that @xmath513 and @xmath514 and therefore @xmath515 . but since @xmath516 , necessarily there exists @xmath517 such that for every @xmath518 . hence for every @xmath510 , @xmath519 . the next proposition shows the construction of a large class of riesz partitions of unity , for families of sets obtained by dilation of a compact set . [ prop2 ] let @xmath62 be a compact set and @xmath107 a @xmath2 expansive matrix such that 1 . @xmath520 [ prop2 - 1 ] 2 . [ prop2 - 3]@xmath521 . let @xmath17 be any measurable function , and @xmath522 some constants such that 1 . @xmath523 2 . [ prop2 - 5 ] @xmath524 on @xmath183 3 . @xmath525 on @xmath526 , where @xmath527 and @xmath528 then the family of functions @xmath529 is a rpu associated to @xmath530 . if @xmath531 , by ( [ prop2 - 3 ] ) there exists @xmath164 and @xmath532 such that @xmath533 , so by ( [ prop2 - 5 ] ) , @xmath534 now by proposition [ prop * * ] , the covering index @xmath535 of the family @xmath536 is finite . using ( [ prop2 - 1 ] ) and ( [ prop2 - 3 ] ) , we see that @xmath537 and since @xmath538 which proves the proposition . this proposition generalizes easily to the case where we replace for each @xmath167 , @xmath539 by an invertible matrix @xmath15 in such a way that @xmath540 is a covering of @xmath265 with finite index . the next lemma shows , that the assumption of having a compact set that covers @xmath5 by dilations is actually necessary , if one wants to construct a rpu associated to a covering of @xmath5 of the form @xmath541 , where @xmath276 is a bounded open set . if @xmath107 is invertible , and @xmath276 is a bounded open set such that 1 . [ lem - i ] there exists @xmath266 with @xmath542 2 . [ lem - ii ] @xmath497 then there exists a compact set @xmath543 such that @xmath544 . it is clearly enough to find @xmath183 such that @xmath545 . let @xmath546 we will prove ( by the contradiction ) that for some @xmath547 , @xmath548 covers @xmath276 by dilations by @xmath107 . assume that for each @xmath547 , there exists @xmath549 such that @xmath550 . since @xmath551 is compact , and @xmath552 , there exists a subsequence @xmath553 and @xmath554 such that @xmath555 . by ( [ lem - i ] ) and ( [ lem - ii ] ) there exists @xmath517 such that @xmath556 , and since @xmath557 is open , @xmath558 for @xmath559 . set @xmath560 and @xmath561 , then @xmath562 for @xmath563 . choose @xmath564 small enough such that @xmath565 , and so @xmath566 . let @xmath567 be such that if @xmath568 , then @xmath569 . then for @xmath570 , and @xmath571 we have @xmath572 and thus @xmath573 so @xmath574 for all @xmath211 such that @xmath575 . this contradicts our assumption that @xmath576 , since , if @xmath568 and @xmath577 , we have @xmath578 , and so @xmath579 . we show next that it is not difficult to construct bounded sets that cover @xmath5 by dilations . let @xmath495 be a bounded set such that @xmath580 , and @xmath107 an expansive @xmath481 matrix . set @xmath581 , then @xmath582 is a covering of @xmath265 with finite covering index . furthermore , if @xmath583 then the sets @xmath584 are disjoint . choose @xmath266 such that @xmath585 . let @xmath586 be an arbitrary point . by equation @xmath587 , for some norm @xmath487 in @xmath488 . so there exists a positive integer @xmath211 such that @xmath588 , @xmath589 , i.e. , @xmath590 . since @xmath276 is bounded , and @xmath591 , there exists @xmath592 such that @xmath593 and @xmath594 , i.e. , @xmath595 . the finiteness of the covering index follows from proposition [ prop * * ] , and if @xmath583 the disjointness property follows immediately . * if we want to obtain coverings of @xmath596 with compact sets , we can choose the set @xmath49 where @xmath183 is the set constructed in the previous lemma . clearly the sets @xmath597 cover @xmath596 , and if @xmath598 , then @xmath597 are almost disjoint , i.e. @xmath599 . * even though for _ any _ set @xmath276 it is true that @xmath600 , in general it is not true that the family @xmath601 is disjoint , as the following example shows : let @xmath602 $ ] and @xmath276 the rectangle @xmath603\times[-1,1]$ ] . we can combine the results of this section with theorem [ wavelet ] , to obtain the following recipe to construct smooth , wavelet frames of @xmath5 associated to a single dilation matrix @xmath107 and an irregular grid . the wavelets obtained by this method can be constructed to have polynomial decay of any degree , as exemplified in section [ wf1d ] . * recipe : * * select a bounded set @xmath495 such that @xmath580 and @xmath604 . * select a function @xmath17 of class @xmath605 such that @xmath606 on @xmath607 and @xmath608 for some small @xmath266 . * consider a set @xmath609 , such that @xmath122 is separated and @xmath610 , where @xmath611 . then the following collection is a frame of @xmath4 : @xmath612 where @xmath81 is the inverse fourier transform of @xmath17 . we first note that although the set @xmath613 in theorem [ wavelet ] is a wavelet frame for @xmath4 , it is not in general true that for a fixed @xmath167 the set @xmath614 is a frame , unless the wavelet frame is of shannon - type , hence , not well - localized . thus , for well - localized wavelets , it appears that the reconstruction of a function @xmath343 from the wavelet coefficients @xmath615 can not be obtained in a stable way by first reconstructing at each level @xmath167 and then obtaining @xmath411 by summing over all levels @xmath167 . however what follows shows that this is still possible : @xmath618 this , together with the assumptions that @xmath619 is a frame for @xmath620 for each @xmath167 , and that @xmath621 , permits to reconstruct @xmath622 using a dual frame @xmath623 of @xmath72 . note that , since @xmath624 is unitary then @xmath625 is a dual frame of @xmath626 for @xmath368 . thus , it is always possible to reconstruct each @xmath627 in a stable way and then obtain @xmath411 by summing up over all levels @xmath167 . one drawback is that the dual frame may not be well - localized , since each @xmath628 may be discontinuous at the boundary of @xmath183 . to treat this problem we simply note that @xmath629 multiplying both sides by @xmath422 we obtain @xmath630 . if we choose @xmath17 to be in @xmath631 , and if @xmath628 is in @xmath632 , then @xmath633 will be in @xmath631 and therefore decays polynomially in space . hence the partial sums of the series ( [ recfor ] ) will have good convergence properties . we can then sum equation over @xmath64 to obtain @xmath634 , and then divide by @xmath635 to obtain @xmath636 . c. k. chui , w. he , j. stckler and q. sun , _ compactly supported tight affine frames with integer dilations and maximum vanishing momemnts _ , advances computational math . , special issue on frames , * 18 * ( 2003 ) , no . 2 , 159187 . m. papadakis , g. gogoshin , i. a. kakadiaris , d. j. kouri , and d. k. hoffman , _ non - separable radial frame multiresolution analysis in multidimensions and isotropic fast wavelet algorithms _ , spie - wavelet x , to appear .

our current hypothesis is that the galaxies and black holes observed today originated over 13 gyr ago , growing from seeds of primordial density perturbations . one can test this hypothesis by studying the star formation rate ( sfr ) history , from the epoch of reionization ( eor ) at redshifts @xmath21 , through the peak era at @xmath22 ( hopkins & beacom 2006 ) down to the present epoch . the feedback of ionizing radiation , kinetic energy , and heavy elements leaves imprints on early stars , supernovae , and galaxies , providing a fossil record " that can be detected through abundances in galactic halo stars and the intergalactic medium ( igm ) and in the distributions of mass , metallicity , and luminosity of galaxies . determining when and how the universe was reionized by these early sources have been important questions for decades ( gunn & peterson 1965 ; sunyaev 1977 ; robertson et al . it has been suggested that igm reionization was complete by @xmath23 ( fan et al . 2001 ; gnedin & fan 2006 ; fan et al . 2006 ; hu & cowie 2006 ) , based on strong ly@xmath2 absorption from neutral hydrogen along lines of sight to qsos at @xmath24 . becker et al . ( 2007 ) and songaila ( 2004 ) used transmission of the ly@xmath2(and ly@xmath25 ) forest out to @xmath26 and @xmath27 , respectively , to suggest a smoothly decreasing ionization rate toward higher redshifts . recent surveys of high - redshift galaxies and ly@xmath2 emitters ( bouwens et al . 2011a ; ouchi et al . 2010 ; kashikawa et al . 2011 ; ono et al . 2012 ; schenker et al . 2012 ) infer an increasing igm neutral fraction from the declining populations between @xmath3 . further evidence for an increasing neutral fraction comes from the decreasing sizes of ionized near zones " associated with quasars between @xmath28 and @xmath29 ( carilli et al . 2010 ) and from the ly@xmath2 damping wing in the transmission profile toward the newly discovered quasar at @xmath30 ( mortlock et al . these studies all suggest that the igm is becoming increasingly neutral between @xmath31 , marking the end of cosmic reionization when ionized regions overlap and percolate . whether the epoch of full reionization occurs at @xmath0 is still not ascertained . a contrasting estimate of the eor comes from the measured optical depth , @xmath32 , to electron scattering of the cosmic microwave background ( cmb ) in _ wmap-7 _ observations ( larson et al . 2011 ; komatsu et al . the error bars come from the central 68% in the marginalized cumulative distribution . using additional cosmological parameter constraints , they infer single - epoch reionization at @xmath33 . although such a high redshift could be explained with @xmath34cdm simulations and modeled sfr histories ( choudhury & ferrara 2005 ; trac & cen 2007 ; shull & venkatesan 2008 ) , the cmb observations are at variance with optical surveys that suggest late reionization , unless reionization is a process that extends to higher redshifts . for a fully ionized igm , including both h and he , the optical depth @xmath14 for @xmath35 ( see section 2.1 ) . in the cmb analysis ( larson et al . 2011 ) , marginalization of @xmath36 with other cosmological parameters allows the possibility of lower optical depth , with a reionization epoch as low as @xmath37 at 95% c.l . one can also invoke a partially ionized igm at @xmath38 , as discussed by many groups ( cen 2003 ; venkatesan et al . 2003 ; ricotti & ostriker 2004 ; benson et al . 2006 ; shull & venkatesan 2008 ) . however , even with the recent progress in finding high-@xmath12 galaxies , we still do not know whether galaxies are the sole agents of reionization . current observations of high-@xmath12 galaxies leave open several ionization scenarios , some involving simple hydrogen reionization at @xmath39 and reionization at @xmath40 , and others with more complex ionization histories ( bolton & haehnelt 2007 ; venkatesan , tumlinson & shull 2003 ; cen 2003 ) that depend on sfrs at @xmath41 . shull & venkatesan ( 2008 ) demonstrated how the cmb optical depth constrains the sfr and igm metallicity history at @xmath17 , and trenti & shull ( 2010 ) quantified the metallicity - driven transition from population iii ( metal - free stars ) to population ii ( stars formed from metal - enriched gas ) . several recent observations provide valuable constraints on the luminosity function of high-@xmath12 galaxies . the number density of galaxies appears to drop rapidly at @xmath17 ( bouwens et al . 2009 , 2010a , b , 2011a , b , c ) . with a comoving sfr density @xmath42 at @xmath43 ( gonzlez et al . 2010 ; bouwens et al . 2011a ) , the observed galaxies do not produce enough ionizing photons in the lyman continuum ( lyc ) to maintain a photoionized igm against recombinations . however , the luminosity function is steep , and the total lyc budget requires extrapolation to low - luminosity galaxies ( trenti et al . 2010 ; bouwens et al . moreover , the conversion from sfr to lyc production rate relies on insecure calibrations from theoretical models and comparison with high - mass , low - metallicity stars . we revisit the calculation of lyc photon production and assess the high-@xmath12 galaxy contribution to reionization . we also analyze several factors , such as the photon escape fraction ( @xmath11 ) , igm clumping factor ( @xmath44 ) , and electron temperature ( @xmath45 ) , which enter the calculation of the critical star formation rate " ( @xmath46 ) necessary to maintain a photoionized igm . in section 2 , we calculate @xmath46 ( @xmath47 mpc@xmath48 ) in a filamentary igm , equating the production rate of lyman continuum ( lyc ) photons with the hydrogen recombination rate . the photoionization rate depends on the mass function of stellar populations , their evolutionary tracks and stellar atmospheres , and the escape fraction , @xmath11 of lyc photons away from their galactic sources . the recombination rate depends on the density and temperature of the igm , properties we explore with cosmological simulations . in section 3 , we give our results for the critical sfr at @xmath49 and present our new sfr simulator , a user - friendly interface for calculating @xmath50 and @xmath51 . in section 4 , we discuss the implications for the hydrogen eor . consistency between high - redshift galaxies and cmb optical depth appears to require @xmath0 and a partially ionized igm at @xmath17 . the peak signal from redshifted 21-cm emission would likely occur during the heating period between @xmath52 ( 145163 mhz ) when the hydrogen neutral fraction @xmath53 ( pritchard et al . 2010 ; lidz et al . we denote by @xmath1 ( @xmath54 ) the global star formation rate per co - moving volume . using a simple argument ( madau et al . 1999 ) , balancing photoionization with radiative recombination , we estimate the critical sfr to maintain igm photoionization at @xmath17 , assuming that the lyc photons are produced by populations of massive ( ob - type ) stars . because the mass in collapsed objects ( clusters , groups , galaxies ) is still small at high redshift , the igm contains most of the cosmological baryons , at mean density @xmath55 for a hubble constant denoted @xmath56 , we adopt the wmap-7 ( plus bao + @xmath57 ) parameters , @xmath58 and @xmath59 ( komatsu et al . 2011 ) relative to a critical density @xmath60 g cm@xmath48 . from the corresponding helium mass fraction @xmath62 ( peimbert et al . 2007 ) , we adopt a mean hydrogen number density , @xmath63 in a fully ionized igm , the cmb optical depth back to @xmath64 can be written as the integral of @xmath65 , the electron density times the thomson cross section along proper length , @xmath66 \ ; dz \ ; , \ ] ] for a standard @xmath34cdm cosmology ( @xmath67 ) with @xmath68^{1/2}$ ] . this integral can be done analytically ( shull & venkatesan 2008 ) , @xmath69 \left [ \ { \omega_m ( 1+z_{\rm rei})^3 + \omega_{\lambda } \}^{1/2 } - 1 \right ] \ ; .\ ] ] in the high - redshift limit , when @xmath70 , this expression simplifies to @xmath71 ( 1+z_{\rm rei})^{3/2 } \approx ( 0.0521 ) \left [ \frac { ( 1+z_{\rm rei})}{8 } \right ] ^{3/2}\ ] ] independent of @xmath72 to lowest order . the helium and electron densities are written @xmath73 and @xmath74 for singly ionized helium , where @xmath75 by number . to these formulae , we add @xmath76 , from electrons donated by reionized at @xmath77 ( shull et al . helium therefore contributes @xmath788% to @xmath36 , and a fully ionized igm produces @xmath14 , 0.060 , and 0.070 back to redshifts @xmath79 , 8 , and 9 , respectively . a comoving volume of 1 mpc@xmath80 contains @xmath81 hydrogen atoms . our simple ionization criterion requires a sfr density that produces a number of lyc photons equal to @xmath82 over a hydrogen recombination time , @xmath83^{-1}$ ] . the hydrogen case - b recombination rate coefficient ( osterbrock & ferland 2006 ) is @xmath84 , scaled to an igm temperature @xmath85 . for typical igm ionization histories and photoelectric heating rates , numerical simulations predict that diffuse photoionized filaments of hydrogen have temperatures ranging from 5000 k to 20,000 k ( dav et al . 2001 ; smith et al . these are consistent with temperatures inferred from observations of the ly@xmath2 forest at @xmath86 ( becker et al . 2011 ) . owing to gravitational instability , a realistic igm is inhomogenous and filamentary . semi - analytical models of the reionization of the universe often adopt a clumping factor " , @xmath87 , to account for inhomogeneity in estimates of the enhanced recombination rate in denser igm filaments . the clumping factor therefore plays an important role in computing the critical sfr density needed to maintain the reionization of the universe . the clumping factor is also used in numerical simulations to implement `` sub - grid physics '' , in which changes in the density field occur on scales below the resolution of the simulation and are also approximated by the factor @xmath44 ( gnedin & ostriker 1997 ; madau et al . 1999 ; miralda - escud et al . 2000 ; miralda - escud 2003 ; kohler et al . 2007 ) . we correct the recombination time for density variations scaled to a fiducial @xmath88 , found in the simulations described below . at @xmath43 , the igm filaments have electron density @xmath89 ^ 3 c_h$ ] , and the characteristic times for hydrogen recombination and hubble expansion are , @xmath90^{-3 } \ ; , \\ t_{\rm h } & \approx & [ h_0 \omega_m^{1/2 } ( 1+z)^{3/2}]^{-1 } \approx ( 1.18~{\rm gyr } ) \left [ \frac { ( 1+z)}{8 } \right]^{-3/2 } \ ; . \end{aligned}\ ] ] in our calculations , we express the reionization criterion as @xmath91 , where @xmath46 is the critical sfr density ( @xmath54 ) and @xmath92 is the conversion factor from @xmath1 to the lyc production rate ( see section 2.2 ) . we define @xmath11 as the fraction of lyc photons that escape from their galactic sources into the igm ( dove & shull 1994 ) . recent statistical estimates ( nestor et al . 2011 ) suggest that @xmath93 for an ensemble of 26 lyman - break galaxies and 130 ly@xmath2 emitters at @xmath94 , and it could be higher for the lower - mass galaxies that likely dominate the escaping lyc at @xmath24 ( fernandez & shull 2011 ) . the lyc production efficiency , @xmath92 , is expressed in units @xmath95 photons per @xmath96 of star formation , since typical massive stars emit @xmath97 lyc photons over their lifetime . to evaluate @xmath92 , we convert the sfr ( by mass ) into numbers of ob - stars and compute the total number of ionizing photons produced by a star of mass @xmath98 over its lifetime . we then integrate over an imf , @xmath99 , with a range @xmath100 . the standard mass range is 0.1 @xmath96 to 100 @xmath96 , but changes in the mass range and imf slope will affect the lyc production substantially . for example , differences in the imf have been associated with higher jeans masses in low - metallicity gas in the high - redshift igm ( abel et al . 2002 ; bromm & loeb 2003 ) , and tumlinson ( 2007 ) and smith et al . ( 2009 ) noted the potential influence of cmb temperature on modes of high - redshift star formation . consequently , most cosmological simulations or calculations include a metallicity - induced imf transition ( trenti & shull 2010 ) between low - metallicity population iii star formation and metal - enhanced population ii . the number of lyc photons produced per total mass in star formation depends on the imf of the stellar population and is given by the conversion factor @xmath92 . we calculate this conversion by integrating the total number of lyc photons produced over the entire mass in star formation . @xmath101 here , @xmath102 is the lifetime - integrated number of lyc photons as a function of mass calculated from stellar atmosphere models and evolutionary tracks . the imf , @xmath103 , is integrated over the mass range @xmath100 , where @xmath98 is expressed in solar units . in the sfr simulator discussed in section 3.3 , the user can choose between a normal and broken imf power law . the lower integration limit in the numerator , @xmath104 , is the mass at which stars no longer produce significant amounts of lyc photons . in our calculator , we use and compare two models that calculate @xmath102 . first , using stellar atmospheres and evolutionary tracks ( r. s. sutherland & j. m. shull , unpublished ) , we find that , over its main - sequence and post - main - sequence lifetime , an ob star of mass @xmath98 produces a total number @xmath105 of ionizing photons , where @xmath106 over the mass range @xmath107 and for metallicities @xmath108 . we have fitted our results to the form @xmath109 , for @xmath110 ( the mass @xmath104 defines the effective lower limit for stars that produce significant numbers of lyc photons ) . for metallicities in the range @xmath111 ( where @xmath112 is the solar metallicity ) and for masses @xmath113 , the fitted coefficients are @xmath114 and @xmath115 . thus , for @xmath116 , we have @xmath117 and @xmath118 , while for @xmath119 we have @xmath120 and @xmath121 . inserting these values for @xmath102 into equation ( 8) , we integrate these lyc photon yields over the imf , to derive the conversion coefficient , @xmath92 , from _ total mass _ in star formation to _ total number _ of lyc photons produced ( in units of @xmath95 photons per @xmath96 ) . the integrals in eq . ( 8) can be done analytically as functions of the imf parameters and metallicity . for a salpeter imf ( @xmath122 , @xmath123 ) we find : @xmath124 ( for @xmath125 ) , @xmath126 ( for @xmath127 ) , and @xmath128 ( for @xmath129 ) . adopting a typical ( low-@xmath130 ) value @xmath131 ( @xmath132 photons per @xmath96 ) , we can solve for the critical sfr for reionization , scaled to clumping factor @xmath6 , escape fraction @xmath7 , and gas temperature @xmath133 : @xmath134^{3 } \left [ \frac { c_h/3 } { f_{\rm esc}/0.2 } \right ] \left [ \frac { 0.004 } { q_{\rm lyc } } \right ] t_4^{-0.845 } \ ; .\ ] ] our chosen value of @xmath88 is consistent with recent downward revisions ( pawlik et al . 2009 ) and with our numerical simulations discussed in section 2.3 . escape fractions @xmath135 have been inferred from observations at @xmath40 ( shapley et al . 2006 ; nestor et al . 2011 ) and theoretical expectations ( fernandez & shull 2011 ) . equation ( 9 ) agrees with prior estimates ( madau et al . 1999 ) when adjusted for our new scaling factors , in particular the ratio @xmath136 ) . their earlier paper assumed @xmath137 , @xmath138 , @xmath139 , @xmath140 , and @xmath141 . our fiducial redshift has increased from @xmath141 to @xmath142 , appropriate for the new discoveries of high - redshift galaxy candidates for reionization . expressed with the same coefficients in eq . ( 9 ) , their coefficient would be nearly the same , @xmath143 at @xmath142 . one of the improvements in our formulation is to better identify the dependences on the physical parameters ( @xmath44 , @xmath11 , @xmath45 ) and the sfr - to - lyc conversion factor ( @xmath144 , which can change with different imfs and atmospheres . a related calculation is the production rate of ionizing ( lyc ) photons per unit volume , needed to balance hydrogen recombinations . with the same assumptions as above , this is @xmath145 ^ 3 t_{4}^{-0.845 } \left ( \frac { c_h}{3 } \right ) \ ; .\ ] ] for standard high - mass stars ( o7 v , solar metallicity ) each with lyc photon luminosity @xmath146 s@xmath147 , this rate corresponds to an o - star space density @xmath148 at @xmath142 . schaerer ( 2002 , 2003 ) also computed models of population iii and low - metallicity stars based on non - lte model atmospheres and new stellar evolution tracks and evolutionary synthesis models . the lifetime total number of lyc photons produced per star of mass @xmath98 is @xmath149 . for stars of mass parameter @xmath150 , the number of ionizing photons , @xmath151 , emitted per second per star , is given by : @xmath152 = \left\ { \begin{array}{lll } 43.61 + 4.90x - 0.83x^2 & z = 0 , & 9 - 500 m_\odot , \\ 39.29 + 8.55x & z=0 , & 5 - 9 m_\odot,\\ 27.80 + 30.68x-14.80x^2 + 2.50x^3 & z = 0.02 z_\odot , & 7 - 150 m_\odot , \\ 27.89 + 27.75x-11.87x^2 + 1.73x^3 & z= z_\odot , & 7 - 120 m_\odot \\ \end{array } \right . \label{eq : qvalues}\ ] ] and the star s lifetime @xmath153 is given by : @xmath154 = \left\ { \begin{array}{ll } 9.785 - 3.759x+1.413x^2 - 0.186x^3 & z = 0 , \\ 9.59 - 2.79x+0.63x^2 & z = 0.02 z_\odot , \\ 9.986 - 3.497x+0.894x^2 & z = z_\odot \end{array } \right . \label{eq : lifetimes}\ ] ] sample values of @xmath92 and @xmath46 for different imfs are shown in table 1 , for both model atmospheres . one can obtain different values of @xmath92 for power - law imfs by varying their high - mass slope @xmath25 and the minimum and maximum masses . increases in @xmath92 translate into _ decreases _ in critical sfr . for example , at @xmath155 , if the minimum mass is raised to @xmath156 with a salpeter slope ( @xmath157 ) , the lyc production factor rises to @xmath158 , some 2.5 times higher than the equivalent model , @xmath128 for @xmath159 . if the imf is flatter , @xmath160 instead of 2.35 , with minimum mass fixed at @xmath159 , @xmath161 ( 4.6 times higher ) . at the upper end of the imf , if one increases @xmath162 from 100 to 200 @xmath96 , keeping @xmath157 and @xmath163 , one finds @xmath164 ( 1.4 times higher ) . all of these variations can be explored with our reionization / sfr simulator , described in section 3.3 . the critical sfr of @xmath165 can be understood from simple arithmetic . over the 386 myr recombination time ( eq . [ 6 ] ) with @xmath6 , @xmath7 , and @xmath142 , approximately 20 million stars are formed per comoving mpc@xmath80 in a salpeter imf ( @xmath166 ) with mean stellar mass @xmath167 . of these stars , a small fraction , @xmath168 , are massive ob stars ( @xmath169 . at an average of @xmath170 lyc photons and escape fraction @xmath7 , these @xmath7814,000 ob stars will produce a net ( escaping ) @xmath171 lyc photons over their lifetime . these photons are sufficient to ionize @xmath81 hydrogen atoms mpc@xmath48 . the simulations used in this study were performed with ` enzo ` , an eulerian adaptive mesh - refinement ( amr ) , hydrodynamical + n - body code ( bryan & norman 1997 ; oshea et al . 2004 , 2005 ) . smith et al . ( 2011 ) enhanced this code by adding new modules for star formation , primordial chemistry , and cooling rates consistent with ionizing radiation , metal transport , and feedback . the ionizing background is spatially constant and optically thin , but variable in redshift . at this stage , we have not implemented radiative transfer or spectral filtering by the igm . for the clumping factor calculation , we ran a simulation on a 50@xmath172 mpc static grid ( unigrid " ) cube with @xmath173 cells , denoted as run 50_1024_2 in table 1 of smith et al . the radiative heating from the ionizing background plays an important role in determining the properties of the filamentary structure . as a filament is ionized and heated , its density drops and its temperature rises ; both effects reduce the recombination rate . to study these effects , we ran four moderate - resolution ( 50@xmath172 mpc unigrid cube with @xmath174 cells ) simulations with varying ionizing backgrounds , summarized in table 2 . after initial submission of this paper , we ran a @xmath175 simulation , to check convergence and assess the cosmic variance " among eight sub - volumes of the @xmath173 and @xmath175 simulations . the standard uv background was taken from haardt & madau ( 2001 ) , although we also explore new computations of high-@xmath12 sfrs by trenti et al . ( 2010 ) and haardt & madau ( 2012 ) . these four simulations were : ( 1 ) no photoionizing background ; ( 2 ) uv background ramped up from @xmath176 to @xmath177 ( run 50_512_2 from smith et al . 2011 ) ; ( 3 ) uv background ramped up from @xmath178 to @xmath179 ; and ( 4 ) uv background ramped up from @xmath178 to @xmath179 , but twice as strong as in ( 3 ) . post - processing of the simulations was performed using the data analysis and visualization package , ` yt ` , documented by turk et al . ( 2011 ) . for regions of ionized hydrogen of density @xmath180 , we calculate the clumping factor , @xmath44 . using two different methods . the first calculation , which has been used in some earlier studies , uses density weighting . in this density field " ( df ) method we define @xmath181 and average the density fields ( quantities @xmath182 , denoting either @xmath180 or @xmath183 ) where parameter averages are computed by summing over grid cells ( @xmath184 ) , subject to various `` cuts '' on the igm overdensity and gas temperature and weighted by factors , @xmath185 , @xmath186 in the second calculation , we compare the local recombination rate to the global average recombination rate , averaged over density and temperature in cells , @xmath187 where again @xmath188 denotes a weighted average and @xmath189 is the case - b radiative recombination rate coefficient for hydrogen , as tabulated by osterbrock & ferland ( 2006 ) . we refer to this as the recombination rate " ( rr ) method . because the purpose of the clumping factor is to correct for an enhanced recombination rate , we believe @xmath190 to be a more appropriate representation . the recombinations that are important to removing ionizing photons occur in the filamentary structure of the igm , and the clumping factor should only be calculated in these structures . to assess the critical sfr necessary to maintain an ionized medium , we focus on grid cells that are significantly ionized . because a negligible amount of recombination occurs in cells containing mostly neutral gas , these cells should not contribute to the clumping factor . if we do not exclude neutral gas , the clumping factor is extremely high ( @xmath191 ) before the ionizing background is turned on . we find large density gradients between the neutral gas ( uniformly distributed over the simulation ) and the ionized gas . because the clumping factor is a measure of the inhomogeneity of the medium , a large density gradient yields a large value of @xmath44 . if low - density voids are included in the calculation , the larger density gradient leads to an overestimate of the clumping . by setting both upper and lower density thresholds , we can exclude collapsed halos and low - density voids from our calculations . previous studies ( miralda - escud 2000 ; miralda - escud et al . 2003 ; pawlik et al . 2009 ) addressed these issues by setting a density threshold that excludes collapsed halos , but they did not set a lower limit to exclude voids from their calculations . to explore these effects in a filamentary igm , we make various cuts of our data in baryon overdensity ( @xmath192 ) and in temperature , metallicity , and hydrogen ionization fraction ( @xmath193 ) . in our standard formulation , we include only those cells that meet the following criteria : @xmath194 , @xmath195 k , @xmath196 , and @xmath197 . we believe this data cut " adequately represents unenriched igm lying on an adiabat ( see figure 19 of smith et al . we also explore the clumping factor with no lower density threshold ( total range @xmath198 ) . our results , presented in section 3.1 , show a small increase in @xmath199 from the wider range in densities by including low - density cells with @xmath200 . however , these low - density voids do not contribute substantially to the recombination rate . in summary , our prescriptions for calculating the clumping factor yield a more physical representation of the enhanced recombination rate that is an important component to many reionization models . by not assuming a fully ionized medium and by specifically following @xmath180 , we are able to exclude denser neutral gas that does not contribute appreciably to the recombination rate . we make simple assumptions on the reionization process and reionization history ( section 3.2 ) , turning on the ionizing radiation field at redshifts @xmath142 or @xmath201 and following the thermal history arising from photoelectric heating , radiative cooling , and pressure smoothing ( pawlik et al . the metal - line and molecular cooling , metal transport , and feedback included in our simulations also allow us to accurately represent the thermodynamics of the gas , which has been shown to have a significant effect on the evolution of the clumping factor . future models will include discrete sources and radiative transfer , accounting for temperature increases arising from photo - heating with spectral hardening ( abel & haehnelt 1999 ) . because our current simulations employ a spatially constant ionizing radiation field , we anticipate carrying out these more realistic situations . early studies of igm clumping adopted high values , @xmath202 at @xmath86 ( gnedin & ostriker 1997 ) . as discussed earlier , we believe these values are too high for the ionized igm filaments , which have expanded as a result of the heat deposited by lyc photons . the differences between observations and inferred critical sfr densities can largely be attributed to this high clumping factor ( sawicki & thompson 2006 ; bouwens et al . more recent studies have trended towards less clumping . raievi & theuns ( 2011 ) argue that using a global clumping factor overestimates the recombination rate , and that local values should be used instead . in this study , we calculate the clumping factor for a series of high - resolution cosmological simulations for ionized hydrogen and helium that explore how the photoheating of an ionizing background affects the igm thermodynamics ( density and temperature ) . we impose criteria in overdensity , temperature , metallicity , and ionization fraction to constrain our calculations to igm filaments where recombinations and ionization fronts are most important . the clumping in simulation @xmath203 is calculated as a function of redshift from @xmath204 to @xmath205 for both @xmath206 and @xmath190 , weighted by overdensity and by volume . note that the unigrid simulations automatically yield volume weighting , when averaged over cells . for the remainder of our simulations , we find a power - law overdensity distribution , with an average form @xmath207 . weighting these cells by overdensity gives undue emphasis on small numbers of high - density cells . this method is also biased , since it does not calculate the clumping factor where most of the recombinations are occurring . figure 1 compares the df and rr methods and shows the difference between the different weights used in our calculations . in both methods , weighting by overdensity results in higher clumping factors . the recombination - rate method ( eq . [ 15 ] ) results in a lower clumping factor because it accounts for both density and temperature effects from the ionizing background on the clumping averages . photoelectric heating during photoionization causes the filaments to expand , lowering the density . the increased electron temperature also reduces the radiative recombination rate coefficient . the df method ( eq . [ 13 ] ) includes no dependence on the recombination rate coefficient , and therefore only counts the effect of photoheating on the filament density . see section 3.2 for further discussion . in our current simulations , we turn on a spatially constant uv background at @xmath142 or @xmath201 . the effects of different density cuts " in the summation over cells ( see section 2.3 ) are shown in figure 2 . we find little difference at @xmath208 ( turn - on of ionizing radiation ) , and see a small increase in @xmath44 at @xmath209 when we include the lower density cells with @xmath200 . in future , more refined simulations with radiative transfer , we expect photoionization to be initiated in high - density regions , where the stars are located . once the uv background is turned on , lower - density regions are easily photoionized . the photoelectric heating of the uv background causes the filaments to expand and become diffuse , resulting in a lower clumping factor . the elevated temperature also reduces the recombination rate coefficient , which in turn lowers @xmath44 when calculated by the rr method . when calculating a mean recombination rate , only the recombining ionized gas is relevant . by @xmath210 , the igm is nearly completely ionized and only small regions of remain . as noted earlier , in more refined simulations , the remaining would be largely self - shielded from the uv background and likely to reside in clumps of high density yet unreached by ionization fronts . figure 3 explores the convergence and cosmic variance among different simulations . in two panels , we compare results from our @xmath174 simulations with larger simulations with @xmath173 and @xmath175 cells . in each panel , we show the average clumping factors , together with those in eight sub - volumes . the average values of @xmath44 agree in the @xmath173 and @xmath175 simulations , and the small ( @xmath211% ) variance among these sub - volumes suggests that the @xmath173 and @xmath175 simulations are converged . over the redshift range @xmath212 in the @xmath175 simulation , the clumping factor is well fitted by a power - law , @xmath213 ^{-1.1 } \ ; , \ ] ] representing a slow rise in clumping to lower redshift , after the turn - on of ionizing radiation . to study the effect of radiative heating from the ionizing background on filamentary structure , we ran a suite of moderate - resolution simulations with different ionizing backgrounds ( see table 2 for details ) . the redshift at which the uv background is turned on affects the clumping factor , causing it to drop and then recover at lower redshift . pawlik et al . ( 2009 ) attribute this effect to _ jeans filtering _ , where the photo - heating raises the cosmological jeans mass , preventing further accretion onto low - mass halos and smoothing out small - scale density fluctuations . the photoheating also heats the filaments we are focusing on , which lowers the hydrogen recombination rate coefficient , @xmath214 , and causes the filaments to expand ; both of these effects reduce the clumping factor . without a uv background , the clumping continues to rise as the filaments gravitationally collapse . pawlik et al . ( 2009 ) also claim that the clumping factor at @xmath177 is insensitive to when the background is turned on , as long as it is turned on at @xmath215 . when an ionizing background is introduced , photoheating acts as a positive feedback to reionization by lowering the clumping factor and making it easier to stay ionized . this same photoheating mechanism also suppresses star formation in low - mass halos , which in turn lowers the ionizing photon production rate by star - forming galaxies and acts as a negative feedback to reionization ( pawlik et al . 2009 ) . these thermodynamic processes affect clumping and structure and emphasize the importance of carefully modeling the strength of feedback processes and their effects on jeans mass . we have explored what happens when the ionizing radiation turned on earlier , especially during the interval @xmath3 marking the transition from a neutral to fully ionized igm . in a series of @xmath174 simulations , we explore the influence of turn - on of photoionizing radiation between redshifts @xmath142 and @xmath201 . figure 4 shows @xmath44 , computed for the moderate - resolution simulations via the density - field method and weighted by volume . we do not plot simulation @xmath216 , since it is identical to simulation @xmath217 . as in the results of pawlik et al . ( 2009 ) , we find that photoheating from the ionizing background results in a decrease in the clumping factor . the clumping factors fall along two tracks : a higher track at redshifts above the turn - on of the uv background , and a lower one after turn - on . by @xmath141 , we find nearly identical clumping factors for both backgrounds . after a substantial recovery time , the redshift when the background is turned on is not important . before this recovery , the clumping factor of the earlier background is lower by a factor of @xmath782 , resulting in earlier reionization . one can compare the strengths of feedback to reionization , where positive feedback " lowers the clumping factor and negative feedback " suppresses star formation . at @xmath218 the total stellar mass ( sfr density ) of simulation @xmath219 is 1.19 ( 1.13 ) times lower than that of simulation @xmath220 , while the clumping factor is 1.64 times lower . for simulation @xmath217 , the total stellar mass ( sfr density ) and clumping factor are 1.52 ( 1.49 ) and 1.66 , respectively , times lower than those of simulation @xmath220 . this suggests that the positive feedback introduced by the background is greater than the negative feedback . however , the stellar mass ( sfr density ) does not recover in the same manner as the clumping factor . once photoheating suppresses the formation of small - mass halos , the hubble flow takes over and prevents them from collapsing and forming stars . therefore , the redshift at which the background is turned on determines whether the positive or negative feedback dominates and whether the ionizing background will cause reionization to be accelerated or delayed . in connection with this project , we have developed a user interface for calculating the critical sfr density @xmath221 needed to maintain the igm ionization at a given redshift . the software computes the effects of variations in the stellar imf ( slope , mass - range ) and model atmospheres , and the redshift evolution of metallicity and gas thermodynamics ( density , temperature , coupling to the cmb ) . the clumping factor and lyc escape fraction are free parameters in the calculator . our simulations find ranges of @xmath222 depending on redshift , overdensity , and thermal phase of the ( photoionized or shock - heated ) igm . for the new @xmath175 simuation , the global mean clumping factor is @xmath223 , with a power - law fit @xmath224^{-1.1}$ ] for redshifts between @xmath212 . this calculator is a useful tool for determining the population of galaxies responsible for reionization . the critical sfr per co - moving volume ( eq . [ 9 ] ) is obtained by balancing the production rate of lyc photons with the number of hydrogen recombinations . here , @xmath225 is the clumping factor of ionized hydrogen , @xmath11 is the escape fraction of lyc photons from their host galaxies , and @xmath226 is the temperature scaled to @xmath227 k. the conversion factor , @xmath92 , from stellar mass to total number of lyc photons produced , is regulated by the imf and model atmospheres . the user has the option of controlling these parameters to determine the resulting critical sfr density , subject to several observational constraints . this simulator can be accessed at http://casa.colorado.edu/~harnessa/sfrcalculator with an ` html ` interface for easy use . two standard tests of the sfr use the simulator to calculate the ionization histories of and and compare them to the ionized volume filling factor , @xmath228 , and the cmb optical depth , @xmath51 . the calculator derives @xmath51 by integrating the differential form of eq . ( 3 ) for various sfr histories and parameters . the average evolution of @xmath229 is found by numerical integration of the rate equation ( madau , haardt , & rees 1999 ) expressing the sources and sinks of ionized zones , @xmath230 the source term , @xmath231 , represents the net ionizing photon production rate , computed from selected models of the sfr density , @xmath1 . here , @xmath232 is the mean hydrogen number density , and @xmath233 is the hydrogen recombination timescale , which depends on @xmath44 as shown in eq . the igm is assumed to be fully ionized when @xmath234 . we integrate a similar equation for @xmath235 to follow photoionization in the qso 4-ryd continuum . we use the qso emissivities at 1 ryd ( haardt & madau 2012 ) , extrapolated to 4 ryd assuming a spectrum with specific flux @xmath236 . figure 5 shows ionization histories , quantified by @xmath228 and @xmath51 , for various values of clumping factor , escape fraction , igm temperature , and lyc - production efficiencies ( @xmath92 ) . we adopt the sfr history from trenti et al . ( 2010 ) with an evolving luminosity function . our standard model adopts @xmath6 , @xmath237 , @xmath238 , and @xmath131 ( @xmath239 photons / m@xmath240 ) . different curves show the effects of changing @xmath44 and @xmath11 , including two models in which these parameters evolve with redshift . with these parameters , we are typically able to complete reionization by @xmath43 , consistent with observations of gunn - peterson troughs , redshift evolution in ly@xmath2 emitters , and igm neutral fraction . the model with a constant @xmath241 does not complete the ionization until @xmath242 , which is far too late . thus , we be believe that lyc escape fractions must be considerably larger ( @xmath243% ) perhaps evolving to higher values at @xmath24 shown by green and magenta curves . figure 6 compares the dependence of @xmath228 and @xmath51 on sfr histories , computed with an evolving luminosity function ( trenti et al . 2010 ) and new calculations ( haardt & madau 2012 ) of the star formation rate density , @xmath244 . figure 7 compares the effects of two choices of model atmospheres , with a fixed sfr history from trenti et al . the on - line calculator provides ionization fractions , @xmath228 and @xmath245 , together with cmb optical depth , @xmath51 . in the simulator , users can select other igm parameters and sfr histories . the main difference between the two sfr models lies in the assumptions at @xmath246 . haardt & madau ( 2012 ) rely on an empirical extrapolation of the star formation rate as a function of redshift , while trenti et al . ( 2010 ) adopt a physically motivated model based on the evolution of the dark - matter halo mass function . the two approaches are similar at @xmath247 , but differ significantly at higher redshift , where an empirical extrapolation does not capture the sharp drop in the number density of galaxies observed at @xmath248 ( see figure 8 in oesch et al . 2011 ) . as a consequence , the reionization history from the haardt & madau ( 2012 ) model is more extended at high @xmath12 , especially when the efficiency of reionization is increased because of evolving clumping factor and escape fraction ( our preferred models , green and magenta curves in fig . the two models yield quite different predictions for the duration of reionization , defined as the redshift interval , @xmath249 , over which @xmath229 evolves from 20% to 80% ionized . our preferred sfr models ( green and magenta lines in fig . 5 ) have @xmath19 ( from @xmath250 to @xmath251 ) , whereas the haardt - madau sfr histories exhibit a more extended interval , @xmath252 ( from @xmath253 to @xmath43 ) . this difference in @xmath249 could be tested by upcoming 21-cm experiments ; see bowman & rodgers ( 2010 ) for an initial constraint , @xmath254 . figure 8 illustrates a third constraint on the reionization epoch ( pritchard et al . 2010 ; lidz et al . 2011 ) comparing sfr histories with estimates of the ionizing background at @xmath255 ( fan et al . 2006 ; bolton & haehnelt 2007 ; haardt & madau 2012 ) . the lyc co - moving emissivity ( in photons s@xmath147 mpc@xmath48 ) , defined as @xmath231 , can be related to the ionizing background at @xmath256 , using recent estimates of the hydrogen photoionization rate , @xmath257 , from haardt & madau ( 2012 ) and the lyc mean free path , @xmath258 , from songaila & cowie ( 2010 ) . the lyc emissivity is proportional to the star formation rate density , computed from our halo mass function model ( trenti et al . 2010 ) , integrated down to absolute magnitudes @xmath259 ( bouwens et al . 2011a ) or to @xmath260 , the faint limit suggested by trenti et al . we adopt a fiducial lyc production parameter @xmath131 , corresponding to @xmath239 lyc photons produced per @xmath96 of star formation , and we use two different models for lyc escape fraction , @xmath11 ( constant at 20% and varying with redshift ) . for quantitative values , we assume an ionizing background with specific intensity , @xmath261 ( in units erg @xmath262 s@xmath147 sr@xmath147 hz@xmath147 ) with power - law index @xmath263 at energies above @xmath264 ryd . the hydrogen photoionization rate is @xmath265 $ ] for a hydrogen photoionization cross section @xmath266 with @xmath267 @xmath268 . for this spectrum , the frequency - integrated ionizing intensity is @xmath269 , and the lyc photon flux ( photons @xmath262 s@xmath147 ) integrated over all solid angles is @xmath270 . because we analyze the _ photon _ mean - free path , we normalize @xmath271 to @xmath272 and @xmath273 , @xmath274 by approximating @xmath272 as the product of the lyc emissivity and mean - free path , we arrive at the calibrations : @xmath275 \\ \phi_{\rm lyc } & = & \frac { ( \alpha+3)}{\alpha } \frac { \gamma_{\rm hi } } { \sigma_0 } = ( 1.98 \times 10 ^ 5~{\rm photons~cm}^{-2}~{\rm s}^{-1 } ) \left [ \frac { \gamma_{\rm hi } } { 5 \times 10^{-13}~{\rm s}^{-1 } } \right ] \ ; . \end{aligned}\ ] ] finally , we relate @xmath272 to the star - formation rate @xmath1 and ionization rate @xmath273 , @xmath276 \left [ \frac { 0.1 } { f_{\rm esc } } \right ] \left [ \frac { 0.004}{q_{\rm lyc } } \right ] \left [ \frac { 9.8~{\rm pmpc } } { \lambda_{\rm hi } } \right ] \left [ \frac { 6.5}{1+z } \right]^{3 } \ ; .\ ] ] here , we have scaled the parameters to the same values assumed by lidz et al . ( 2011 ) , namely @xmath277 , @xmath278 , an ionizing spectral index @xmath279 , and redshift @xmath280 , at which songaila & cowie ( 2010 ) fit @xmath281 proper mpc . our parameter @xmath131 corresponds to the lidz et al . ( 2011 ) lyc photon production calibration , @xmath282 per @xmath47 of star formation . however , our coefficient , @xmath283 , is slightly larger than their value , @xmath284 , an effect that may arise from our different method of relating @xmath273 to the ionizing radiation field and sfr . more careful examination of these parameters suggests that , at @xmath285 , the hydrogen ionization rate @xmath286 , given in table 3 of haardt & madau ( 2012 ) , and the observed mean free path , @xmath258 , at @xmath280 may be closer to 6 proper mpc ( see figure 10 of songaila & cowie 2010 ) . rescaling to those two parameters , we find a similar coefficient of @xmath287 . this sfr density at @xmath255 ( see also figure 8) is comparable to the critical value needed to maintain reionization at @xmath142 , but the observed rates and mean free paths are declining rapidly with redshift . all three constraints on sfr suggest that full reionization is more likely to occur at redshift @xmath0 than at @xmath288 . ( uv background turned on at @xmath176 ) with different weights . red lines : clumping calculated via density field ( df ) method . blue lines : clumping via recombination rate ( rr ) method ( see section 2.2 ) . solid lines correspond to weighting by volume , and dashed lines correspond to weighting by baryon overdensity . we believe the rr method , with volume weighting , is a more accurate measure of clumping . ] simulation , using the rr method in equation ( 15 ) . solid line : clumping factor summed over cells with overdensity between @xmath289 , excluding low - density voids with @xmath200 . dashed line : including all cells with @xmath198 . ] agree for @xmath173 and @xmath175 simulations , with variance computed over eight sub - volumes in our simulations . top panel : eight @xmath290 sub - volumes in @xmath175 run . bottom panel : eight @xmath174 sub - volumes in @xmath173 run . in each panel , our original @xmath174 run is shown as dotted black line , and values for @xmath175 and @xmath173 runs as heavy black lines . large variations in @xmath44 at high redshifts arise from small numbers of cells prior to turn - on of the ionizing background at @xmath201 ( top ) and @xmath142 ( bottom ) . for the @xmath175 simulation ( top panel ) , the clumping factor is well - fitted by @xmath224^{-1.1}$ ] between @xmath212 . , title="fig : " ] agree for @xmath173 and @xmath175 simulations , with variance computed over eight sub - volumes in our simulations . top panel : eight @xmath290 sub - volumes in @xmath175 run . bottom panel : eight @xmath174 sub - volumes in @xmath173 run . in each panel , our original @xmath174 run is shown as dotted black line , and values for @xmath175 and @xmath173 runs as heavy black lines . large variations in @xmath44 at high redshifts arise from small numbers of cells prior to turn - on of the ionizing background at @xmath201 ( top ) and @xmath142 ( bottom ) . for the @xmath175 simulation ( top panel ) , the clumping factor is well - fitted by @xmath224^{-1.1}$ ] between @xmath212 . , title="fig : " ] igm simulations , with sums weighted by volume ( see eq . [ 14 ] ) . red solid line has uv background ( haardt & madau 2001 ) turned on at @xmath178 ( simulation @xmath217 ) , blue dashed line at @xmath176 ( simulation @xmath219 ) , and green dot - dashed line has no background ( simulation @xmath220 ) . we observe two tracks for @xmath199 : a high - track for redshifts above turn - on of the uv background , and a lower track after turn - on . during most of the reionization epoch , from @xmath21 , @xmath44 lies between 1.5 and 3 . ] and cmb optical depth @xmath51 versus redshift ( eqs . [ 3 ] and [ 17 ] ) computed with @xmath291 k , @xmath131 ( @xmath132 photons per @xmath96 ) and using sfr history and evolving luminosity function from trenti et al . ( 2010 ) integrated down to @xmath260 . the igm is assumed to be fully ionized when @xmath234 . blue line : constant @xmath6 and @xmath241 ( this model reionizes too late to be viable ) . red line : constant @xmath6 and @xmath7 . green line : constant @xmath6 but redshift - dependent @xmath292 ( haardt & madau 2012 ) . magenta line : redshift - dependent @xmath293 ( pawlik et al . 2009 ) and @xmath294 . our two preferred models with variable @xmath44 or @xmath11 ( green , magenta ) naturally produce @xmath0 . solid black circles indicate redshifts of 20% , 50% , and 80% ionization ; the duration @xmath19 is defined between 20% and 80% points . ] , with same color codes as fig . 5 , comparing two models for sfr history . solid lines show sfr history from trenti et al . ( 2010 ) , and dashed lines show sfr from equation ( 53 ) in haardt & madau ( 2012 ) , which is larger and more extended at higher redshifts , particularly for @xmath295 . bottom : corresponding cmb optical depths @xmath51 . additional optical depth may arise from sources at @xmath17 . , title="fig : " ] , with same color codes as fig . 5 , comparing two models for sfr history . solid lines show sfr history from trenti et al . ( 2010 ) , and dashed lines show sfr from equation ( 53 ) in haardt & madau ( 2012 ) , which is larger and more extended at higher redshifts , particularly for @xmath295 . bottom : corresponding cmb optical depths @xmath51 . additional optical depth may arise from sources at @xmath17 . , title="fig : " ] , @xmath245 , and @xmath51 from our on - line sfr / reionization simulator . solid lines show model atmospheres labeled as sutherland & shull ( s&s ) , while dashed lines are for schaerer ( 2002 , 2003 ) , assuming @xmath296 . models assume sfr history of trenti et al . ( 2010 ) and parameters with constant @xmath6 and redshift - dependent @xmath292 ( haardt & madau 2012 ) . dot - dashed line shows ionization history , @xmath245 , computed for qso ( 4-ryd continuum ) ionization as described in section 3.3 . in these models , is fully ionized by @xmath43 and by @xmath297 . additional optical depth may arise from sources at @xmath17 . ] ) from halo mass function model ( trenti et al . 2010 ) integrated for galaxies down to @xmath259 ( solid blue line ) and to @xmath298 ( dotted green line ) . red data points are from bouwens et al . ( 2011a ) for galaxies down to @xmath259 . solid data point at @xmath285 is from ionizing background constraint ( eq . 21 ) , using haaradt & madau ( 2012 ) model of @xmath299 and lyc mean free paths from ( songaila & cowie 2010 ) . bottom : co - moving emissivity , @xmath300 , of lyc photons , corresponding to above sfrs , integrated to @xmath259 and @xmath298 . we use a lyc production parameter @xmath131 ( @xmath301 lyc photons s@xmath147 per @xmath96 yr@xmath147 ) and two different escape - fraction models : @xmath7 ( blue curves and data points ) and @xmath302 ( red curves and data points ) . , title="fig : " ] ) from halo mass function model ( trenti et al . 2010 ) integrated for galaxies down to @xmath259 ( solid blue line ) and to @xmath298 ( dotted green line ) . red data points are from bouwens et al . ( 2011a ) for galaxies down to @xmath259 . solid data point at @xmath285 is from ionizing background constraint ( eq . 21 ) , using haaradt & madau ( 2012 ) model of @xmath299 and lyc mean free paths from ( songaila & cowie 2010 ) . bottom : co - moving emissivity , @xmath300 , of lyc photons , corresponding to above sfrs , integrated to @xmath259 and @xmath298 . we use a lyc production parameter @xmath131 ( @xmath301 lyc photons s@xmath147 per @xmath96 yr@xmath147 ) and two different escape - fraction models : @xmath7 ( blue curves and data points ) and @xmath302 ( red curves and data points ) . , title="fig : " ] the major results of our study can be summarized with the following points : 1 . we calculated the critical star formation rate required to maintain a photoionized igm , incorporating four free parameters ( @xmath44 , @xmath11 , @xmath45 , @xmath303 ) that control the rates of lyc photon production and radiative recombination . our best estimate at @xmath142 is @xmath304 ^ 3 ( c_h/3)(0.2/f_{\rm esc } ) t_4^{-0.845}$ ] for fiducial values of igm clumping factor @xmath305 , lyc escape fraction @xmath7 , temperature @xmath8 k , and standard imfs and low - metallicity stellar atmosphere ( @xmath131 ) . an epoch of full reionization at @xmath0 is consistent with recent optical / ir measurements of sfr history and a rising igm neutral fraction at @xmath3 , marking the tail end of reionization . these observations include the decrease in numbers of high-@xmath12 galaxies and ly@xmath2 emitters and igm damping - wing intrusion into the ly@xmath2 transmission profiles in high-@xmath12 qsos . our newly developed sfr and reionization calculator , now available on - line at + http://casa.colorado.edu/~harnessa/sfrcalculator , allows users to calculate the ionization history , critical sfr , and cmb optical depth , and to assess whether observed sfrs , imfs , and other parameters are consistent with igm reionization . 4 . reconciling late reionization at @xmath0 with @xmath13 of the cmb likely requires an epoch of partial ionization . a fully ionized igm back to @xmath79 produces @xmath306 , and an additional optical depth , @xmath16 , may arise from early sources of uv / x - ray photons at @xmath17 . alternatively , one can appeal to the likelihood contours from wmap , which allow optical depths as low as @xmath307 ( 95% c.l . ) the _ planck _ experiment may clarify the situation in several years . if the eor is more complex , as we suggest , then redshifted 21-cm experiments should focus on the interval @xmath308 , corresponding to frequencies 124167 mhz , for the maximum signal of igm heating ( @xmath309 mk ) produced when the hydrogen neutral fraction @xmath53 ( pritchard et al . 2010 ) . we conclude by speculating about which future observations can best constrain the eor . additional data from from the _ planck _ mission should provide confirmation of the cmb optical depth with smaller error bars . this information will constrain the additional amount of ionization at @xmath17 and narrow the range @xmath310 produced by high - redshift sources in the partially ionized igm . ongoing surveys for high-@xmath12 galaxies , ly@xmath2 emitters , and qso near - zone sizes will better quantify the rise of neutral fraction , @xmath311 at @xmath312 . on the theoretical front , we are running larger igm simulations on @xmath175 and @xmath313 unigrids and will add discrete sources of ionizing radiation and radiative transfer in order to capture the heating and clumping more accurately . we will also include source turn - on at @xmath208 to test the decrease and recovery of @xmath44 , as seen in figures 3 and 4 . finally , we plan to carry out more detailed modeling of the ( 21-cm ) signal , coupled to the kinetic and spin temperatures driven by heating at @xmath17 . as described earlier , the duration of the reionization transition and the 21-cm emission during the heating phase could provide discriminants of various sfr histories at @xmath24 . this work was supported by grants to the astrophysical theory program ( nnx07-ag77 g from nasa and ast07 - 07474 from nsf ) at the university of colorado boulder . we thank joanna dunkley , chris carilli , richard ellis , and piero madau for useful discussions on reionization and cmb optical depth statistics . we are grateful to the referee for suggesting additional model comparisons with the ionizing background at @xmath256 . cccccccc 0.1 & 100 & 2.35 & @xmath314 & 0.00236 & 0.00286 & 0.0306 & 0.0253 + 0.1 & 100 & 2.35 & @xmath315 & 0.00397 & 0.00558 & 0.0181 & 0.0129 + 0.1 & 100 & 2.35 & @xmath316 & 0.00401 & 0.00752 & 0.0180 & 0.0096 + 0.1 & 100 & 2.35 & @xmath317 & 0.00365 & @xmath318 & 0.0197 & @xmath318 + 0.1 & 100 & 2.35 & @xmath319 & 0.00383 & @xmath318 & 0.0188 & @xmath318 + 1.0 & 100 & 2.35 & @xmath319 & 0.00976 & @xmath318 & 0.0074 & @xmath318 + 0.1 & 100 & 2.00 & @xmath319 & 0.01267 & @xmath318 & 0.0057 & @xmath318 + 0.1 & 200 & 2.35 & @xmath319 & 0.00543 & @xmath318 & 0.0133 & @xmath318 + lcccc [ tbl : sims ] @xmath203 & 50 & 1024 & 7 & 1 + @xmath219 & 50 & 512 & 7 & 1 + @xmath217 & 50 & 512 & 9 & 1 + @xmath216 & 50 & 512 & 9 & 2 + @xmath220 & 50 & 512 & n / a & 0 + @xmath320 & 50 & 1536 & 9 & 1 +

the use of general principles to investigate systems whose microscopic makeup is unclear can be very rewarding . sometimes this method gives information on a whole class of systems . among such principles , thermodynamics , believed to be of universal applicability , stands out . an instructive example of its use is the application of the second law of thermodynamics to the problem of viscous flow , for which it permits the inference that the two viscosity coefficients must be positive without need to resort to microscopic expressions for the latter @xcite . in this work we use the generalization of the second law of thermodynamics holding in the presence of black holes ( the generalized second law gsl ) to reach further conclusions about the shear viscosity coefficient of an arbitrary fluid . we do this by describing a new paradox for ideal fluid flow in the presence of a black hole . this indicates that the correlation length of a real fluid can not be arbitrarily small . by implication the energy - momentum tensors describing systems which display macroscopic fluid behavior must be subject to a restriction : the shear viscosity , a function of the thermodynamic state of the system , can not be arbitrarily small . thereby the gsl opens an alternative macroscopic approach to the recently proposed lower bound on viscosity @xcite . the gsl is a unique law of physics that bridges thermodynamics and gravity . it is rooted in the understanding that a black hole , basically a pure gravity entity , is endowed with well defined entropy @xcite proportional to its surface area . the gsl @xcite then claims that the sum of the entropy of all black holes and the total ordinary entropy in the black holes exterior never decreases . while this formulation is reminiscent of the ordinary second law , the gsl is exceptional in that it relates ordinary entropy a rather elusive object from the mechanical viewpoint and the surface area of the black hole ( formally the area of its horizon ) whose evolution is quite mechanical in nature . from this point of view it is little surprise that the gsl has provided unexpected information on entropy . an example is the upper bound on the entropy of weakly self - gravitating thermodynamic systems ( the universal bound on entropy ube @xcite ) . while in particular cases the bound can be verified directly , it is the gsl which really makes understanding of the bound in generic situations easy . however , the gsl gives more than just a simpler way to see some results derivable by other means . at the microscopic level the gsl represents a piece of the yet to be established theory of quantum gravity . in particular , this law permits , in principle , to draw conclusions that from the microscopic viewpoint would only be derivable from a fundamental theory combining quantum mechanics with gravity . for example , the gsl gives an indication of the number of particle species in nature @xcite . in another example , the gsl reveals an _ a priori _ bound on the strength of the electromagnetic interaction @xcite . let us now describe the ideal fluid paradox revealed by invoking the gsl . we assume the existence of physical fluids with arbitrarily small shear viscosity @xmath0 at fixed values of some two thermodynamic variables , say the entropy and the energy densities , @xmath1 and @xmath2 , respectively . ( in this work we consider only simple fluids , so the values of @xmath1 and @xmath2 completely determine the thermodynamic state of the system . ) here physical " means , among other things , that the fluid satisfies the gsl ; this could , in principle , be checked by considering the fluid at the microscopic level , but we shall not go into such detail . the assumption that the fluid has arbitrarily small viscosity allows us to describe its flow as ideal fluid flow , possibly with shocks ( the zero viscosity limit of a given flow can be non - trivial ; see refs . it turns out , as we shall show , that for sufficiently slow accretion of the fluid onto the black hole the overall entropy decreases and the gsl is violated . the realizability , in principle , of such slow accretion flows will be demonstrated , so that the small viscosity assumption engenders a paradox . of course , ideal fluid paradoxa exist already in non - relativistic physics , for example , the famous dalembert paradox . this maintains that an ideal fluid with no boundaries exerts no force whatsoever on a body moving through it with constant velocity . in particular , there is no lift force , so swimming or flying would be impossible in such a fluid . thus the established fact that swimming is possible in any fluid implies that every real fluid must have nonzero viscosity . there is however an essential difference between the dalembert paradox and the paradox described in this work . dalembert s paradox says nothing about the actual magnitude of @xmath0 . in contrast , our paradox implies , as we shall see , the existence of a lower bound on @xmath0 for given @xmath1 and @xmath2 . this conclusion is in concord with the recent ingenious conjecture of kovtun , starinets and son ( kss ) @xcite ( see also ref . @xcite ) . based on holographic calculations of the viscosity coefficient for certain strongly coupled quantum field theories with gravity duals , they suggested that the viscosity @xmath0 of a general , possibly non - relativistic , fluid is subject to the universal restriction @xmath3 currently this bound is considered a conjecture well supported for a certain class of field theories see the detailed discussion in ref . @xcite and the references therein . in the sequel we discuss the relation between the ideal fluid paradox presented in the next section and the kss bound . we also argue that a frequent objection to the validity of the kss bound is likely to be ruled out by the gsl . our paper is structured as follows . in sec . [ paradx ] we show explicitly that the slow accretion of a truly ideal fluid onto a schwarzschild black hole leads to a contradiction with the gsl . one escapes from the paradox by recognizing that every fluid must have a nonvanishing correlation length which restricts the range of applicability of the ideal fluid paradigm . in sec . [ kssp ] we obtain a lower bound on the correlation length and a generic estimate of the viscosity of real fluids , which together bound the viscosity to entropy density ratio from below . although this is not yet the kss bound , we consider there the connection between it and the ube , and argue that the gsl provides a natural frame for elucidation of the origin of the former . in the sec . [ summary ] we summarize our results and arguments . the realizability , in principle , of the slow accretion flow assumed in sec . [ paradx ] is demonstrated in the appendix . unless otherwise stated , we work in units with @xmath4 , where @xmath5 is speed of light and @xmath6 is boltzmann s constant . our metric signature is @xmath7 . in the present section we consider ideal fluid flow into a spherical black hole . for some flows , we demonstrate that it is possible for the gsl to be violated so that the total entropy of the system decreases . it follows that the assumption of a perfect continuum down to an arbitrarily small scale is not consistent with the gsl . consider a flow , not necessarily spherically symmetric , in which fluid is absorbed by the black hole . the rate of change of the total entropy @xmath8 of the system is the sum of the rate of change of the entropy of the black hole exterior , @xmath9 , and that of the black hole entropy @xmath10 : @xmath11 here and below we use schwarzschild coordinates @xmath12 , with @xmath13 where @xmath14 is the schwarzschild radius . we first calculate @xmath15 . let the fluid s proper entropy density be @xmath1 . since the fluid is assumed ideal , there is no dissipative contribution to the entropy current density which is thus purely convective , and must take the form @xmath16 , where @xmath17 is the fluid four - velocity . the fluid can carry entropy into the hole leading to a decrease of @xmath9 . the explicit expression for this comes from the entropy balance equation @xcite , @xmath18 where @xmath19 stands for the determinant of the schwarzschild metric @xmath20 . @xmath15 of the black hole exterior ( @xmath21 ) is thus @xmath22 we have not included a contribution from the outer boundary of the domain of integration because we intend to specialize to stationary flows . in any such situation the entropy flow into the hole per unit @xmath23-time is given by the r.h.s . of eq . ( [ exterior ] ) . the expression is non - positive because @xmath24 for infalling matter . we have assumed that the flow is differentiable , and that it contains no shocks ; otherwise there exists an additional contribution to @xmath15 associated with the entropy generation in the shocks @xcite . let us now consider the second term of the r.h.s . in eq . ( [ total ] ) . the absorption of the fluid by the black hole increases the latter s mass @xmath25 , producing an increase in the black hole entropy given through energy conservation by @xmath26 where @xmath27 is the black hole temperature , and @xmath25 is its mass . we shall now write down the flux of energy into the black hole . let @xmath28 be the killing vector associated with the stationarity of the black hole . then the energy - momentum tensor of the fluid , @xmath29 , must obey @xmath30 or equivalently @xmath31 since @xmath32 is _ minus _ the energy density of the fluid , @xmath33 the energy - momentum tensor of the ideal fluid is given by @xmath34 where @xmath2 is the energy density and @xmath35 is the pressure in the comoving frame . consider now the normalization condition @xmath36 , written as @xmath37.\ ] ] since @xmath38 and @xmath39 are physical velocity components , they should be bounded at @xmath40 . hence since @xmath41 is future and inwardly pointed , and @xmath42 as @xmath43 , we can infer from the last equation that @xmath44 at the horizon . combining all the above we find @xmath45 thus @xmath46 in harmony with hawking s area theorem @xcite . we observe from eqs . ( [ exterior ] ) and ( [ black ] ) that while the ( negative ) rate of change of @xmath9 is proportional to the first power of @xmath47 , the ( positive ) rate of change of the black hole entropy is proportional to the _ second _ power of @xmath47 . thus for sufficiently small @xmath48 , the total entropy of the system will _ decrease _ , in violation of the gsl . explicitly we have @xmath49,\ ] ] where we stress that at the horizon @xmath47 is never positive . we observe that when the accretion velocity @xmath50 ( the suitable mean value of @xmath51 over the horizon ) obeys @xmath52 the total entropy decreases and the gsl is broken . we now proceed to search for such flows . we first reconsider known explicit solutions . the above formulae apply for a generic , not necessarily spherically symmetric , accretion flow onto the black hole ; in particular it may have non - vanishing @xmath53 and @xmath54 . as an example , consider the well known bondi flow ( see ref . @xcite and references therein ) . bondi flow starts from rest at infinity and is spherically symmetric . for a polytropic equation of state , @xmath55 where @xmath56 is the adiabatic exponent , the accretion velocity is given by @xmath57 for @xmath58 and @xmath59 for @xmath60 @xcite . it follows that bondi flow will obey the gsl if the following condition holds : @xmath61 we have used the usual expression @xmath62 for the schwarzschild black hole . the above conditions would seem to obtain for any fluid with positive pressure thanks to the ube @xcite . this bound states that the entropy @xmath63 of a generic , weakly self - gravitating , thermodynamic system satisfies @xmath64 where @xmath65 is the body s total energy , while @xmath66 is its linear size . our experience is that the inequality here is usually a strong one . now the minimal size of a parcel of fluid consistent with the fluid description is its correlation length @xmath67 . for a gas , @xmath67 is the mean free path , while for a liquid it is typically the intermolecular distance . applied to such a parcel the ube tells us that ( see the next section for more details ) @xmath68 where we have passed from the total entropy and energy to their densities by dividing by the parcel s proper volume . again , this bound will in most cases be a strong inequality . since the continuum description down to accretion at the black hole makes sense only if @xmath69 , we may conclude that the inequalities ( [ bondi ] ) will always hold with at least one order of magnitude difference between the r.h.s . and the l.h.s . of the equations in the above argument we have tacitly assumed that @xmath70 . negative pressure is often discussed in cosmology ( dark energy ) ; the functioning of the gsl in the face of dark energy is fraught with subtleties @xcite . thus bondi flow satisfies the gsl by virtue of the ube . clearly , this happens because the accretion velocity approaches the speed of light , @xmath71 . we now turn to examples of slow ideal fluid accretion flows that do violate the gsl . it is clear from the previous analysis that the gsl holds for any accretion flow of ideal fluid with @xmath71 . we expect such flows whenever spherical accretion starts from a distance @xmath72 from the black hole large compared to @xmath14 ( for bondi flow @xmath73 ) . but what is @xmath50 in the limit @xmath74 ? we now show that @xmath50 vanishes in that limit , thereby giving an example of ideal fluid flow that violates the gsl . we study steady , spherically symmetric accretion flow which starts at @xmath75 with the initial flow and sound velocities , @xmath76 and @xmath77 , specified there . we assume that the relevant initial conditions are physically realizable ; see the appendix for the discussion of this assumption . following ref . @xcite we introduce @xmath78 . the continuity and the euler equations can be written as @xmath79 where a prime denotes @xmath80 , @xmath81 is the baryon number density , and @xmath82,\\ & & d_2=\frac{2u^2/r - m / r^2}{u},\end{aligned}\ ] ] with @xmath83 the local sound velocity . obviously @xmath84 is always positive due to the causality constraint @xmath85 ( sound velocity is smaller than velocity of light ) @xcite . here we consider only flows with @xmath86 so @xmath87 at the horizon . in bondi flow @xmath88 becomes negative at large @xmath89 which signifies that there exists the so - called sonic radius @xmath90 such that @xmath91 . as is clear from eqs . ( [ derivs ] ) , where @xmath88 stands in the denominator , the sonic radius is a special though regular point of the bondi flow , which in other situations could signify the presence a shock . by contrast , here we shall choose a range of initial parameters for the flow which ensure that @xmath92 , so that there is no sonic point . we assume initial parameters for the flow satisfying @xmath93 it is easy to see from eq . ( [ d ] ) that @xmath94 is indeed positive . we do not need to derive the explicit form of the flow in order to find @xmath50 . it suffices to consider the conservation of the bernoulli integral along the streamlines . the relativistic version of the bernoulli equation reads @xcite @xmath95 assuming for simplicity the polytropic equation of state ( [ poly ] ) one finds the following relation @xcite @xmath96,\ ] ] where @xmath97 is the baryon mass . the above relation allows us to rewrite the bernoulli equation as @xmath98 evaluating the above equation first at @xmath40 and then at @xmath75 , equating the results , using inequalities ( [ in1])-([in2 ] ) and assuming that at the horizon @xmath99 remains much smaller than unity ( which we have verified numerically ) , we find that @xmath100 where the ellipsis stands for subleading terms . the above expression holds for any @xmath101 and @xmath56 , the latter assumed to be not too close to unity . actually , the use of the polytropic equation is not essential . using eq . ( [ bern ] ) directly we would find @xmath102}{n(r_i)\left[\rho(r_h)+p(r_h)\right ] } \sqrt{1-\frac{2m}{r_i}}.\ ] ] if we may assume that there are no abrupt changes in @xmath103 throughout the flow , we recover a result of form eq . ( [ accrvel ] ) . we may conclude that an ideal fluid accretion flow which starts close to the horizon with initial parameters complying with inequalities ( [ in1])-([in2 ] ) can have arbitrarily small @xmath50 . according to the criterion ( [ ineq0 ] ) it will thus incur a violation of the gsl . of course for the above argument to be convincing , we must still demonstrate that it is possible for a flow to start near the horizon with sufficiently small velocity ( and sound speed ) . in the appendix we show in detail that a cord which respects fundamental physical constraints can be used to bring objects to rest arbitrarily near the horizon . one can envision the setting up of the initial conditions for the fluid we require at @xmath75 by this means . the above violation of the gsl raises a paradox . in the real world the gsl can not be violated . this law has been shown to follow from fundamental concepts in quantum theory and classical gravity @xcite . can the paradox be removed without calling on new physics just for the occasion ? an obvious solution would be to call on hawking s radiance to generate entropy that would at least compensate for the decrement of total entropy pointed out in sec . [ paradox ] . after all , in free hawking emission the thermal radiation entropy creation rate exceeds the associated rate of decrease of the black hole entropy @xcite . but there exist reasons to reject this as the resolution of our problem . the hawking radiation is capable of preventing a violation of the gsl that would arise if high entropy radiation of the same nature , i.e. , electromagnetic , were injected into a black hole with @xmath104 above the effective radiation temperature @xcite . its efficacy here is related to the principle of detailed balance in equilibrium ( in fact , for incoming thermal radiation at temperature @xmath104 the hawking radiation exactly balances the entropy decrement ) . now detailed balance refers to modes of the same physical system . there is no detailed balance between radiation and fluid modes . thus if entropic radiation were injected alongside the fluid accretion , the hawking radiation might prove incapable of compensating for the entropy decreases of the two kinds . the above discussion also suggests we should look for a way out of the paradox that hinges on the physics of the fluid itself . in our discussion in sec . [ paradox ] we assume that one can deposit the fluid at an arbitrarily small distance from the horizon at an arbitrary velocity . since the physical paradigm used is that of fluids , one should , as a matter of principle , demand that the said distance is still larger than the correlation length @xmath67 at which the hydrodynamic description first becomes applicable . to be in the schwarzschild spacetime a small proper length @xmath105 away from the horizon at @xmath106 corresponds to the schwarzschild coordinate @xmath107 ; here @xmath108 and is given explicitly by @xmath109 let us thus substitute @xmath110 for @xmath72 in our expression ( [ accrvel ] ) : latexmath:[\[\label{u } correlation length limits the smallness of the accretion velocity @xmath50 . with eq . ( [ u ] ) the expression in square brackets in eq . ( [ entr2 ] ) takes the form @xmath112\,,\ ] ] where we used the expression for the black hole temperature . in order for the gsl to be obeyed the above factor must be nonpositive . this is actually guaranteed by the fluid version of the ube , eq . ( [ bekensteinforfluid ] ) , as long as the pressure @xmath35 is nonnegative ( see the next section for more details ) . this removes the paradox . the moral of the discussion is that a paradox arises if one relies on the continuum description of a fluid down to an arbitrarily small scale , that is if one takes the notion of ideal fluid literally . to be rid of paradoxa one must take cognizance of the finite correlation length of any physical fluid , _ and _ must accept that the entropy capacity of fluid matter is limited according to bound ( [ bekensteinforfluid ] ) . the paradox uncovered in sec . [ paradox ] and its resolution in sec . [ remove ] clearly show that the ideal fluid paradigm is inconsistent . in particular , the picture of a fluid as a perfect continuum is shown to be physically unacceptable : the fluid in question must have a nonvanishing correlation length . the medium is a fluid only over scales exceeding the correlation length . it turns out that finiteness of the correlation length is incompatible with the vanishing of various transport coefficients like shear viscosity and heat conductivity , another feature of the ideal fluid paradigm . for a gas there is a simple way to see this . the usual estimates of the mentioned transport coefficients @xcite have them proportional to the mean free path of the gas molecules . but in a gas the mean free path and correlation length are the same thing . thus the finite correlation length forces the transport coefficients of the gas , in particular the shear viscosity , to be nonvanishing . we shall see in the sequel that the same conclusion applies generally to any liquid as well . thus a fluid can be fully compatible with the gsl only if it is dissipative to some extent ( and thus not ideal ) . another way of putting this is that a fluid with arbitrarily small shear viscosity is unphysical . how large must the correlation length @xmath67 be ? microscopically speaking it must obviously be at least as large as the intermolecular distance . but can we say something without delving into the structure of the fluid ? let us approach this question in the spirit of wilson s work on the renormalization group @xcite . we consider some thermodynamic system with the typical linear size @xmath113 . because the system is macroscopic ( by definition ) , one can represent any extensive thermodynamic variable as some quantity times the system s volume @xmath114 . for example , one can write the entropy as @xmath115 and the energy as @xmath116 . now decrease @xmath113 . for sufficiently small @xmath113 , already comparable with the system correlation length @xmath67 , the system ceases to be macroscopic , and extensivity is generally lost . this means that the expressions for the entropy and energy densities themselves become dependent on @xmath113 . the system is no longer a continuum . can one set a lower bound on the correlation radius @xmath67 purely from macroscopic considerations ? the standard answer to the above question would be in the negative . but , surprisingly , the true answer is yes . as we have seen , for a macroscopic system the ube can be restated in the form ( [ bekensteinforfluid ] ) , which immediately leads to the inequality @xmath117 the main point here is that a macroscopic system with size much smaller than @xmath118 would violate the ube ; thus the correlation length can not be much smaller . we emphasize , again , that @xmath67 may easily be much larger than the minimal scale ( [ bekenstein11 ] ) . the above result answers the following question : given a macroscopic system with given entropy and energy densities , what is its minimal possible correlation length ? it shows that macroscopic quantities do know " about the minimal correlation length of the system . we mentioned earlier that a nonvanishing correlation length @xmath67 ensures that the transport coefficients of a gas ( which are proportional to @xmath67 ) do not vanish . for example , for shear viscosity one has the order of magnitude estimate @xcite @xmath119 where the speed of sound @xmath83 is of the order of molecular speed @xcite . the above estimate together with eq . ( [ bekenstein11 ] ) give that @xmath120 is subject to the inequality @xmath121 let us show that the estimate ( [ estimate ] ) , and thus the bound ( [ new1 ] ) , must hold also for liquids sufficiently far from a critical point . in this preliminary treatment we neglect heat conductivity and bulk viscosity . we note that , for any fluid , density perturbations at scale much larger than @xmath67 generate sound waves . on the other hand , perturbations with scale much smaller than @xmath67 are not coherent and produce no sound . furthermore , one can still use hydrodynamics asymptotically to describe the evolution of perturbations with scale @xmath67 . then the demand that there is no well - defined sound at smaller scales gives the asymptotic condition that at scale @xmath67 the wave decay time @xmath122 , as found from hydrodynamics @xcite , should be comparable with the wave period @xmath123 . this produces the estimate ( [ estimate ] ) for liquids . alternatively , the results ( [ estimate])-([new1 ] ) can be recovered by considering a sound wave already propagating through the liquid . its fourier components with reduced wavelengths near or below @xmath67 should decay over a distance comparable to @xmath67 since we can not have macroscopic flow at those smaller scales . now from the _ macroscopic _ point of view , the decay must be caused by transport processes controlled by the viscosities or heat conductivity . according to fluid theory @xcite , a sound wave of wavelength @xmath124 penetrates a distance of order @xmath125 into the liquid before damping out . according to the above argument , for @xmath126 this penetration length should be @xmath67 , which gives eq . ( [ estimate ] ) again . for relativistic fluids the above consideration should be modified slightly . here the wave decay time includes @xmath127 instead of @xmath2 @xcite , which leads to @xmath128 for most realistic fluids @xmath129 , see @xcite , and @xmath130 , so this modification is not essential for the order of magnitude estimate of @xmath0 , see the next subsection for an example . the above estimates , however , must fail sufficiently close to a critical point . both @xmath0 and @xmath67 diverge at the critical temperature with the power law @xmath131 holding in the vicinity of the critical point . here @xmath132 is much less than @xmath133 @xcite . this is incompatible with eq . ( [ estimate ] ) because both @xmath2 and @xmath83 are finite at the critical point . the reason for the failure of the estimate is complications in the asymptotic matching procedure above , related to the difference of the critical exponents of the various quantities involved . another issue is that near a critical point the static correlation length @xmath67 becomes different from the scale beyond which hydrodynamics applies . of experimental viscosities of eleven pure liquids ( data taken mostly from ref . @xcite ) vs log@xmath134 of the estimates from eq . ( [ estimate ] ) with @xmath67 identified with the average intermolecular separation . both viscosities and estimates are in si units ( mpa s ) . nonstandard symbols are `` m '' for methane , `` p '' for propane , `` a '' for acetone , `` e '' for ethanol , `` u '' for undecane ( c@xmath135h@xmath136 ) , `` ni '' for nitrobenzene ( phenil - no@xmath137 ) and `` g '' for glycerol ( c@xmath138h@xmath139o@xmath138 ) . a repeated symbol correspond to viscosities at several pressures and temperatures . the solid line is the locus of @xmath140 ; the dotted lines demarcate the region where the viscosity lies within a factor of 3 of @xmath141 . ] nevertheless , sufficiently far from the critical point the estimate ( [ estimate ] ) seems to produce remarkably good results , as shown by fig . [ fig : fig1 ] for a number of pure liquids . ( the only resounding failure is glycerol , one of the most viscous fluids known . the problem with it may be that @xmath67 is much larger than the intermolecular separation , which we routinely used as estimator for @xmath67 . ) this fact is remarkable because it allows to reach conclusions on the magnitude of viscosity quite possibly of practical value even for liquids for which no microscopic theory is available . let us now turn to the lower bound on the viscosity - entropy density ratio ( [ new1 ] ) . it evidently holds if the estimate ( [ estimate ] ) works . note that eq . ( [ new1 ] ) will most often be a strong inequality , particularly for nonrelativistic systems , for which the ube is known to be a very liberal bound . thus the bound ( [ new1 ] ) is generally valid far from a critical point . on the other hand , near the critical point the bound ( [ new1 ] ) holds trivially because its l.h.s . diverges while its r.h.s . remains finite . thus the divergence of shear viscosity at the critical point @xcite only strengthens the bound . the considerations in this subsection provide a strong argument in favor of the existence of a generic lower bound on the viscosity - entropy density ratio . the reformulation of the arguments with regard to the other transport coefficients , including thermal conductivity , is left for future work . it was the challenge presented by the quark - gluon plasma ( qgp ) which motivated the activity leading to the formulation of the kss bound . as a practical use of our estimate ( [ estimate ] ) , let us apply it to the qgp . quantum chromodynamics ( qcd ) predicts the existence of such a form of matter where the usually confined quarks and gluons are essentially free forming a fireball with a typical size @xmath142 fm . the transition from the hadronic state of matter to the qgp occurs at densities of @xmath143 gev/@xmath144 . qcd lattice calculations predict that the transition should take place at a critical temperature @xmath145 mev , and that the speed of sound for a qgp above @xmath146 mev is about @xmath147 ( see ref . @xcite and further references therein ) . measurements on the qgp at brookhaven s relativistic heavy ion collider ( rhic ) indicate that the ratio of energy density @xmath148 to entropy density @xmath1 is roughly proportional to the qgp temperature @xmath149 ( see , e.g. , fig . 3 in ref . specifically , for the high energy results of rhic one has @xmath150 using this and estimate ( [ estimate ] ) we have @xmath151 where in the last step we assumed @xmath147 at @xmath152 mev , @xmath153 fm corresponding to a number density @xmath154 /@xmath144 , and employed @xmath155 mev fm . our estimate for the qgp viscosity is thus in harmony with the kss bound ( [ kss ] ) . moreover , it is in quite reasonable agreement with the experimental data which require a shear viscosity to entropy density ratio as low as @xmath156 @xcite . we draw attention to the similarity between inequality ( [ new1 ] ) with @xmath5 and @xmath157 restored , namely @xmath158 and the conjectured kss bound ( [ kss ] ) . at first sight this inequality seems to fall well below the kss bound since in many cases @xmath159 . however , we must keep in mind that , especially in such nonrelativistic circumstances , we expect the inequality to be a strong one . the minimum @xmath120 may thus be well above the literal bound ( [ new2 ] ) , and not necessarily at variance with eq . ( [ kss ] ) . for relativistic media @xmath160 , and there is no difference to speak of between bounds ( [ kss ] ) and ( [ new2 ] ) . thus arguments based on the ube , and ultimately on the gsl , seem to suggest a _ raison dtre _ for the mysterious kss bound . before proceeding we wish to provide an alternative viewpoint . the kss bound eq . ( [ kss ] ) is usually interpreted as saying that a fluid can not be too perfect . however , in the form @xmath161 the bound is really an entropy bound , specifically an upper bound on the entropy density of an arbitrary fluid ( we return to the use of units with @xmath162 as in the previous sections ) . clearly the above bound is reminiscent of the ube , eq . ( [ bekensteinentropy ] ) , and its fluids version , eq . ( [ bekensteinforfluid ] ) . but at least for non - relativistic fluids , the kss bound is a tighter entropy bound by orders of magnitude than the ube . restoring the speed of light @xmath5 we introduce the variable @xmath163 ( @xmath67 being , again , the correlation length , either the mean free path for a gas , or , typically , the intermolecular separation for a liquid ) which has the dimensions of viscosity . thus eq . ( [ bekensteinforfluid ] ) gives @xmath164 . using the estimate ( [ estimate ] ) we have @xmath165 . it follows that for nonrelativistic fluids eq . ( [ kssentropy ] ) is indeed a much tighter entropy bound than eq . ( [ bekensteinforfluid ] ) . thus the kss bound can be viewed as a tightened version of the ube in guise of eq . ( [ bekensteinforfluid ] ) which is available for the class of systems exhibiting macroscopic fluid behavior . both the ideal fluid paradox of sec . [ paradox ] and the interpretation of the kss bound as an entropy bound for fluids suggest the gsl as a natural frame for investigating relations between entropy bounds . because the ube is obtained in the most transparent way via the gsl , one may hope that the kss bound should likewise be obtainable most simply with help of the gsl . we turn now to considerations in this direction . first we remark that it is not clear how to obtain the kss bound directly from microscopic physics . inspection of the green - kubo formula @xcite for the viscosity in terms of fluctuations shows no apparent connection of the viscosity and the entropy . such consideration affords no special status to the ratio @xmath120 , at least not from the viewpoint of nongravitational physics . on the other hand , the kss conjecture emerged from holographic type arguments that connect quantum field theory with gravity . it might thus turn out that a derivation of the kss bound requires use of the still nonexistent theory of quantum gravity . but even if this were true , one need not loose heart . there is general agreement that black hole entropy , and the gsl which hinges on it , reflect some aspect of quantum gravity . the kss is not universally accepted . many objections to it rely on scenarios where very many particle species are supposed to exist in nature ( see ref . @xcite for an example and references ) . these are the same objections raised against the validity of the ube ( see ref . @xcite for references ) . in fact , any entropy bound whatsoever invites attacks of this sort , for if the number of species that may show up is unlimited , the entropy can be made as large as desired while keeping parameters like energy or total particle number fixed . although it is formally true that with many species available , the kss bound must fail , this does not detract from its heuristic usefulness ; one is often interested on the entropy of a specific system , one whose particle content is fixed ahead of time . it should also be mentioned that many particle scenarios , if not arbitrarily legislating particle proliferation , conjure up fine - tuned or baroque setups to beget the required large number of species of quasiparticles or excitations . this aspect considerably decreases the appeal of the species proliferation arguments . both of the above considerations make it clear that a ground up approach to deriving the kss bound is unlikely to succeed . alternative , indirect approaches are needed . a point in favor of employing the gsl to investigate the origin of the kss bound is that the gsl knows " the actual number of species in nature @xcite . for example , black hole entropy , which plays a crucial role in the gsl , should in principle depend on the number of elementary fields , yet all derivations endow it with a fixed coefficient which may be thought as determined by the actual number of species . this feature would act to neutralize the above mentioned argument against the validity of the kss bounds . we have worked out the expression for the rate of change of the total entropy of a system consisting of a schwarzschild black hole and ideal fluid which can accrete onto the hole . the well known bondi flow is the case of fast accretion with velocity near to the speed of light ; for it the gsl is always obeyed due to the ube . for sufficiently slow accretion velocity the flow violates the gsl . the question of whether flow with the required small accretion velocity is a realistic option is answered in the affirmative by a critical revision , in the appendix , of the venerable argument @xcite claiming that it is impossible to adiabatically lower mass to near the black hole horizon . since it is known from microscopic considerations @xcite that any physical system should comply with the gsl , the above result would constitute a serious paradox if the ideal fluid is a continuum , as usually considered . we find , however , that the paradox can be defused if one takes into account that the continuum picture must break down at some level by virtue of the fluid having a nonvanishing correlation length . an auxiliary role in the nullification of the paradox is played by the universal entropy bound . it gives a lower bound on the correlation length in relation to the fluid s entropy density . the lower bound on the correlation length of an arbitrary thermodynamic system in terms of its ( macroscopic ) entropy and energy densities is an unexpected and thought - provoking aspect of the ube . it makes it clear that the gsl , though a macroscopic law , `` knows '' about the microscopic structure of matter . the lower bound must rank as one of the most impressive consequences of the extension of the second law of thermodynamics to include black holes . the breakdown of the continuum description of a fluid has momentous consequences : discreteness of matter and thermal fluctuations necessarily engender nonideal behavior parameterized by the viscosity and heat conduction coefficients . we have shown that one can expect the existence of a universal lower bound on the viscosity of an arbitrary fluid . our argument yields a seemingly novel estimate for the viscosity of a fluid far from the critical region ; this estimate is shown to be better than an order of magnitude estimate for a range of pure liquids . in addition , the estimate works reasonably well for the quark - gluon plasma , which viscosity was a subject of much study lately @xcite . together with the lower bound on the correlation length , the viscosity estimate yields a lower bound on viscosity per unit entropy density which may be expressed in terms of the sound velocity of the fluid . the appearance of the speed of sound here , in contrast to the ( implicit ) speed of light in the original kss arguments , may signify that the kss argument can be improved , or may perhaps disclose that the field theories considered by kss as representatives of strongly coupled fluids are not sufficiently generic in all these theories the sound velocity is comparable with the speed of light . an additional issue is whether the bound can be extended to other transport coefficients . if there is indeed a lower bound on the ratio of viscosity to entropy density for an arbitrary fluid , then it is likely to be rooted in some very basic principle of physics . we have shown that the generalized second law of thermodynamics , combining as it does gravity with thermodynamics , can be such a principle . starting from it , one can anticipate that the viscosity to entropy ratio of an arbitrary fluid obeys a certain lower bound . in particular , the gsl incorporates physics that weakens the recent objections @xcite to the bound s very existence . on the basis of the above , we propose that the use of the gsl may lead to a clarification of the kss bound s origin . although we have made no concrete progress towards reaching a final sharp inequality , we have given for the first time physical arguments indicating that this elusive bound , whose existence is sometimes disputed , indeed exists . i. f. thanks v. lebedev for an illuminating discussion . g. b. thanks the hebrew university for a golda meir fellowship . j. d. b.s research is supported by grant 694/04 of the israel science foundation , established by the israel academy of sciences . gibbons @xcite claimed that one can not lower a rope to near a black hole at an arbitrarily small velocity , without it being torn at some finite distance from the horizon . the conclusion was based upon the assertion that the stresses that arise in the rope become so strong that they violate the weak energy condition @xcite which must be obeyed by the energy - momentum tensor of physical matter . here we reexamine the problem and show that , in fact , the energy condition need not be violated and thus that adiabatic lowering down to the horizon is , in principle , possible . in the stationary state of some matter in the vicinity of a spherical black hole , the energy momentum tensor obeys @xmath166 where we used the symmetry of @xmath167 @xcite . below we concentrate on configurations for which @xmath168 is diagonal . using the same coordinates as in sec . [ balance ] , the @xmath169 component of the above equations is satisfied automatically , while the radial component gives @xmath170 the above equation gives a linear relation between the different components of the stress tensor which need not , _ a priori _ , respect the weak energy condition . as an example , consider a thin spherical shell in equilibrium . for the shell the contribution of @xmath171 in the above equation is negligible . this can be seen by noting that @xmath171 vanishes at the shell boundaries , and as a result its values within the shell vanish together with the ratio of shell thickness to radius . far from the hole , where the shell is describable by the classical linear elasticity theory , the maximal value of @xmath171 is found to be proportional to the square of the above ratio . so the thin shell is basically supported by the tangential stresses @xmath172 and @xmath173 , which must be equal because of spherical symmetry . ( [ radial ] ) gives @xmath174 . now the weak energy condition would demand @xmath175 . it is thus clear that a stationary thin shell would violate the condition at @xmath176 . thus a physical thin shell , i.e. one obeying the energy condition , can not support itself arbitrarily close to the horizon ( see ref . @xcite for the more detailed discussion ) . ref . @xcite argued that the rope can not be in equilibrium with its lower end arbitrarily close to the black hole , similarly to the thin shell above . here we wish to correct this conclusion . we shall assume that the rope fibers can be considered radial , such as in the case of a conical rope filling the portion of space defined by some solid angle . since the rope can only support stresses along its fibers , the general form of its energy - momentum tensor is @xmath177 $ ] , and eq . ( [ radial ] ) gives @xmath178 the above linear equation can be used to express the stress @xmath179 in terms of the rope density @xmath2 and the boundary condition at the lower end of the rope , @xmath180 , which is defined by the load . the solution can be written as a sum @xmath181 , where @xmath182 describes the stresses caused by the load , while @xmath183 describes the stresses caused by the rope s own weight . the expression for @xmath182 ( the solution for a weightless rope ) expresses the `` constancy of the tension '' along the rope : @xmath184 where @xmath185 is determined by the weight of the load ; for a point mass @xmath97 we have @xmath186 . note that @xmath182 is negative , as befitting a tension . the `` constancy of the tension '' gives monotonically decreasing @xmath187 as one moves out because the rope cross - section increases with @xmath89 by the assumed symmetry of the fibers . the expression for @xmath183 ( the solution without the load ) describes how the force caused by the rope above some @xmath89 balances the weight of the rope below : @xmath188 for definiteness we shall consider below the case of the rope with a constant density @xmath189 where integration gives @xmath190 . \nonumber\end{aligned}\ ] ] we now show that the above solution does not violate the weak energy condition @xmath191 , and in particular , that no violation occurs even when the rope s end is arbitrarily close to the horizon . by choosing sufficiently large @xmath189 we may always disregard @xmath182 , so it is sufficient to show that @xmath192 . it is easy to see from eq . ( [ a1 ] ) that the maximum of @xmath193 , at given @xmath189 and @xmath25 , grows as @xmath194 decreases . this just means that the maximal stress is the bigger the closer the lower end of the rope is from the horizon . therefore , it is enough to show that @xmath192 for @xmath194 which is infinitesimally close to @xmath106 . in general , @xmath195 , vanishing as it is at both endpoints , @xmath180 and @xmath196 , has a unique maximum at @xmath197 . it is easy to see numerically that in the limit @xmath198 the point @xmath199 also tends to @xmath14 . defining @xmath200 we find analytically @xmath201 where @xmath202 and the dots stand for subleading terms . it follows from the above that the weak energy condition is obeyed for any @xmath203 . thus , at least from the viewpoint of the energy conditions , it is possible in principle to lower a body adiabatically all the way down to the horizon by means of a suitably constructed rope . our result is at variance with that of gibbons @xcite . the discrepancy arises because he missed a term when calculating a certain 4-divergence ( eq . ( 5 ) in ref . @xcite ) . by taking this term into account , the main result , eq . ( 8) in ref . @xcite , now reads @xmath204 here @xmath205 is the tension , @xmath206 the rope s cross section , @xmath207 the energy per unit proper length of the rope , and @xmath208 is the norm of the timelike killing field @xmath209 ( the spacetime is taken to be stationary ) . in the original derivation , the term @xmath149 in the denominator of the l.h.s . ( [ gibb ] ) is absent . the corrected eq . ( [ gibb ] ) is indeed in accord with our eq . ( [ drs ] ) . observing that in the schwarzschild geometry @xmath210 and further @xmath211 for the case of a conical rope , one easily checks that eq . ( [ gibb ] ) reduces to our eq . ( [ drs ] ) . our analysis demonstrates that no problem of principle militates against the creation of the boundary conditions required for the slow accretion flow considered in sec . [ paradox ] here . a suitably designed rope could be used to deposit parcels of fluid at rest near the horizon . j. zhou , b. wang , y. gong , and e. abdalla , preprint , arxiv:0705.1264v2 ; g. izquierdo and d. pavon , phys . b * 639 * , 1 ( 2006 ) and preprint arxiv : gr - qc/0612092 ; j. a. de freitas pacheco and j. e. horvath , preprint arxiv:0709.1240 .

central to a complete picture of galaxy evolution is the distribution of the interstellar matter ( ism ) within each galaxy and how that ism forms stars . given that the ism mass on galactic scales is dominated by molecular and atomic gas , observing the tracers of these gas components is necessary for measuring the ism distribution within the disks of spiral galaxies . accordingly , observations of the hi 21-cm line and the co @xmath9 2.6-mm line are often used as tracers of the atomic and molecular gas , respectively ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? while conversion of the velocity - integrated brightness temperature of the hi line , or @xmath10 , to atomic gas column density , @xmath11 , is usually straightforward ( though not always , e.g. , * ? ? ? * ) , the conversion of i(co ) to molecular gas column density @xmath12 is not quite so certain ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , especially given that the co @xmath9 line is known to be optically thick ( e.g. , see * ? ? ? * ; * ? ? ? * ; * ? ? ? recent observations @xcite suggest that the @xmath13 conversion factor , or x - factor , @xmath14 , is roughly constant within the disk of our galaxy , with @xmath15 , to be abbreviated as @xmath16 ; this or a similar value of @xmath14 is often called the `` standard '' value . this uniform x - factor value for our galaxy s disk now applies to the disks of external galaxies , where @xmath17 is inferred and , on average , is _ radially _ non - varying from the inner disk to a galactocentric radius of @xmath18 @xcite . the evidence for co - dark gas , both theoretically and observationally , is a further complication ( see , for example * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? it is thus advantageous to employ tracers other than co @xmath9 as independent checks on ism surface density variations to test recent physical models of @xmath14 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) observed gas column and surface densities can then provide insights into large - scale star formation in galaxies . the schmidt - kennicutt ( s - k ) law , for example , states that star formation rate surface density , @xmath19 , is related to the gas surface density , @xmath20 , by @xmath21 with @xmath22 to @xmath23 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the index @xmath24 is appealing because the gas depletion time ( @xmath25 ) is constant thoughout the spiral disks . while there is some evidence that @xmath26 on size scales of @xmath27 ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , there is also evidence of non - linear and even non - universal slopes on such size scales ( see , e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , especially on the scales of individual giant molecular clouds ( gmcs ) @xcite . there is also strong evidence for an inside - out formation of galaxies ( e.g. * ? ? ? * ) , which is at odds with a constant gas depletion time and , therefore , with having @xmath28 . comparison between those results on the large ( i.e. , galactic ) scales with those on gmc scales is problematic . @xcite suggest that the schmidt law on the scales of gmcs are fundamentally different from the s - k law apparent on larger ( i.e. galactic ) scales ; the latter are not the `` result of an underlying physical law of star formation . '' many of the abovementioned results used observations of the optically thick co @xmath9 line and adopted a spatially constant @xmath14 . in contrast , millimetre ( mm ) , submillimetre ( submm ) , and far - infrared ( far - ir ) continuum observations sample optically thin continuum emission from the dust grains that pervade both the atomic and molecular gas . recently , there have been many papers of the far - ir / submm continuum emission of external galaxies from the _ planck _ and _ herschel _ missions ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? these papers find , for example , that the dust and stellar masses of galaxies are correlated @xcite and that spiral galaxies and dusty early type galaxies have @xmath29 to @xmath30 of dust @xcite . a major stumbling block to determining accurate surface densities from dust continuum emission is the unknown mass absorption coefficient , @xmath31 , at millimetre wavelengths . millimetre continuum emission is less temperature sensitive than that at submillimetre and far - ir wavelengths ; this provides an important constraint on dust mass and sometimes the spectral emissivity index , @xmath32 , can be constrained as well . observationally , the relevant quantity determined is the dust optical depth to gas column density ratio , @xmath33 . @xcite and @xcite have found that @xmath34 at 857@xmath0ghz ( wavelength of 350@xmath35 ) in the hi gas in the solar neighbourhood and @xmath36 at 250@xmath35 ( corresponding to @xmath37 at 857@xmath0ghz for @xmath38 ) in the hi gas in the taurus molecular complex . given that @xmath38 applies to the dust in our galaxy ( see , e.g. , * ? ? ? * ) and that the dust to _ hydrogen _ gas mass is about 0.01 , those observed @xmath33 correspond to @xmath39 to 0.5@xmath40 in the dust associated with hi . the dust associated with h@xmath41 , however , has @xmath33 double that in hi @xcite . consequently , estimating @xmath33 and @xmath31 from comparing the observed dust continuum emission against the hi gas emission alone , while useful , must be viewed with caution . the various estimates of @xmath31 suggest that determining the exact _ total _ mass of dust within a galaxy is uncertain by a factor of a few . in spite of the uncertain dust mass absorption coefficient , dust continuum emission can provide estimates of @xmath14 in our galaxy as well as in external galaxies ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? such observations have shown that while @xmath14 can be more or less spatially constant in some cases , like in the disk of our galaxy and other external galaxies ( see * ? ? ? * ; * ? ? ? * ) , there can be regions of `` dark '' gas , h@xmath41 with no co emission , both in our galaxy and other galaxies ( e.g. , see * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? therefore , observations of dust continuum emission provide a vital check on results inferred from co @xmath9 observations . even with the many recent advances mentioned above , there are many questions left unanswered . for example , do the inferred x - factor values ( i.e. * ? ? ? * ; * ? ? ? * ) apply to the outer disks of all galaxies ? also , are there systematic differences of the x - factor between arm and inter - arm regions ? can previous methods of observationally inferring the dust mass absorption coefficient at millimetre wavelengths be refined ? how do the answers to those questions influence the specific form of the observed s - k law in a given galaxy ? to address these questions and to better understand the gas and dust in spiral galaxies and their relationship to star formation , we observed the grand - design , face - on spiral galaxies m@xmath051 and m@xmath083with the bolometer - array camera , _ aztec _ ( aztronomical thermal emission camera ) , mounted on the 15-m jcmt in hawaii at a wavelength of 1.1@xmath0 mm . both of these galaxies are nearby with distances of less than 10@xmath0mpc ( see table [ tab1 ] for details ) and , hence , the spiral arms in both galaxies are resolved across the arms in the _ jcmt / aztec _ observations , which have a spatial resolution of 20@xmath42 . both galaxies have been studied extensively at numerous wavelengths ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? recent _ herschel _ observations of m@xmath051 and m@xmath083 at 70 , 160 , 250 , 350 , and 500@xmath35 have provided maps of the dust temperature , surface density , and even spectral emissivity index , @xmath32 @xcite , but with spatial coverage that is slightly more limited than those of the _ aztec _ 1.1@xmath0 mm continuum maps presented here ; the _ aztec _ images cover a few more kiloparsecs at the adopted distances given in table [ tab1 ] . as a result , these _ aztec _ 1.1@xmath0 mm continuum images extend both the spatial and the wavelength coverage of the dust emission in both m@xmath051 and m@xmath083 . this greater spatial coverage placed restrictions on the other wavelengths available for comparison with the _ aztec _ 1.1@xmath0 mm data ; there were no 250 , 350 , and 500@xmath35 data towards the outer disks of m@xmath051 and m@xmath083 . there are , however , _ spitzer _ 160@xmath35 data covering both of these galaxies . this wavelength is the longest of the _ spitzer _ data and , along with the _ aztec _ 1.1@xmath0 mm data , is the one most likely associated with the dust component(s ) that dominate the mass of the dust . in addition , the surface densities in the current work were estimated from the observational data . specifically , @xmath33 at @xmath43 was inferred by comparison with the hi column densities and removing upper outliers , because these upper outliers were assumed to represent positions with co - dark gas . this approach has the advantage that final gas masses inferred were not dependent on dust models . .adopted parameter values [ cols="<,<,<",options="header " , ] @xmath44 for galactocentric radii of 1.8 - 5.5@xmath0kpc and 1.0 - 2.9@xmath0kpc for m@xmath051 and m@xmath083 , respectively . @xmath45 in red light for m@xmath051 and @xmath46-band for m@xmath083 . @xmath47 the star formation surface density map normalized to the molecular gas surface density determined from the continuum and the hi line . @xmath48 the star formation surface density map normalized to the molecular gas surface density determined from the co @xmath9 line and the x - factor . maps of m@xmath051 and m@xmath083 were made in the 1.1@xmath0 mm continuum from observations with the instrument _ aztec _ with the _ jcmt_. combining with these maps with the corresponding _ spitzer _ 160@xmath35 ( or , more properly , 155.9@xmath35 ) maps gave estimates of the gas surface densities in these two galaxies ( see appendix [ appsurf ] and [ appgsdm ] for a detailed discussion ) . with these gas surface density maps , spatial variations of the x - factor were estimated . in addition , we investigated the relationship between the gas surface density and that of the star formation rate . these are dealt with in more detail below . the most important results of this work regarding the x - factor are the following : 1 . the average x - factor for each galaxy can be estimated from the current observations , even if crudely . those average values are @xmath49 and @xmath50 for m@xmath051 and m@xmath083 , respectively . 2 . the x - factor is higher in the interarm regions than in the arms . 3 . there seems to be co - dark gas that resides mostly in the outer disks of both m@xmath051 and m@xmath083 . the latter two results are robust to a range of adopted @xmath33 values . the variation of the x - factor spatially and from source to source could be due , in part , to variations in metallicity . theoretical work using the observational data also support a dependence of @xmath14 on metallicity ( e.g. , * ? ? ? * ; * ? ? ? in contrast , @xcite do not find a strong correlation of @xmath14 with metallicity . however , their sample only had a metallicity range of 0.5 - 0.8 dex within factors of 3 of solar . the irregular galaxies observed by israel and others ( e.g. , see * ? ? ? * ; * ? ? ? * ; * ? ? ? ? * ; * ? ? ? * ; * ? ? ? * ) typically had metallicities much less than solar , sometimes only a few percent of solar , and found x - factors an order - of - magnitude or more higher than the standard value . so a strong @xmath14-metallicity relation may exist for galaxies with strongly sub - solar metallicities . for galaxies with roughly solar metallicities , while metallicity alone is apparently insufficient in constraining @xmath14 , it is still relevant . for example , using the data points for ngc4321 from the left panel of figure 10 in @xcite yields a correlation coeffient of @xmath510.6 . this suggests that each galaxy has its own @xmath14-metallicity relationship . given that sub - solar metallicities imply larger x - factors , perhaps the interarm regions of m@xmath051 and m@xmath083 have sub - solar metallicities , while being at solar - level in their arms.51 . whether this implies the same metallicity in both regions is unclear . ] the models of @xcite , as well as their figure 1 , suggest that the metallicity in the interarm regions would be systematically lower by a factor of @xmath522 - 3 in order to increase @xmath14 by a factor of 2 with respect to that in the spiral arms ( see their equation # 8) . while observations of metallicity in the ism of m@xmath051 and m@xmath083 apparently do not support such a systemically lower metallicity between their spiral arms ( see * ? ? ? * ; * ? ? ? * ) , they also do not rule it out : such observations are toward hii regions and are heavily biased toward the spiral arms . this could , in turn , affect the results of studies of the effects of the spiral arms on star formation . @xcite , for example , looked at whether the sfr normalized to the molecular gas surface density is higher in the spiral arms of three galaxies . they adopted a spatially constant x - factor , which , to within a factor of 2 , is likely correct . this will be discussed further in section [ sfrgsd ] . the modelling by @xcite mentioned above suggests that the x - factor depends mainly on the metallicity of the gas in a galaxy and , to a lesser extent , on the average co surface brightness . using the roughly solar metallicities of m@xmath051 and m@xmath083 ( see * ? ? ? * ; * ? ? ? * ) , the observed co brightnesses from the data used here , and applying expression ( 8) of @xcite yields @xmath53 for m@xmath051 and @xmath54 to 6@xmath55 for m@xmath083 ; this is much higher than the observed values found in the current work . admittedly , our estimates are uncertain by factors of about 2 . nevertheless , our estimate of @xmath56 for m@xmath051 agrees with the result of @xcite who find @xmath57 from using the observed extinction in hii regions . accordingly , the theoretical models need further adjustments . as mentioned previously , there is evidence that adopting a spatially constant x - factor does not account for co - dark gas . for example , table [ tab3 ] suggests that about half the total gas mass is unaccounted for when using the co @xmath9 and hi 21-cm spectral lines ( see section [ xfac ] and appendix [ appgsdm ] ) . if we adopt a threshold for co - dark gas that corresponds to an @xmath14 that is a factor of 4 higher than the average for each galaxy , then some of the edges of the shaded regions of figures [ fig13 ] and [ fig14 ] indicate such gas . it is worth noting that these edges are well within the boundaries of the co maps . of course , possible alternative interpretations for the high x - factor or its high lower limits are not entirely ruled out . these include the following : * an extended low surface brightness artifact in the 1.1@xmath0 mm continuum maps of m@xmath051 and m@xmath083 . * insufficient mapping of co @xmath9 in the outer disks of these galaxies . * dust with unusual properties such as unusually high dust - mass absorption coefficient ( i.e. , @xmath31 ) and/or a high dust - to - gas mass ratio . * optically thick hi 21-cm emission each of the above could mimic the presence of co - dark gas . the first alternative is unlikely given that the simulations of the _ aztec _ 1.1@xmath0 mm observations have accurately accounted for any constant offsets in the m@xmath051 and m@xmath083 maps . the second alternative is unlikely because there is evidence for such co - dark gas seen at radii well within the boundaries of the existing co maps , as mentioned previously . the third alternative is unlikely because having a combination of high @xmath31 and high @xmath58 would not be sufficient for positions with @xmath14 two orders of magnitude larger . the fourth alternative is a partly valid explanation for co - dark gas in our galaxy according to the @xcite . they estimated that up to half of the dark gas could be due to optically thick hi 21-cm emission . even if that were the case for m@xmath051 and m@xmath083 , it would not account for lower limits to @xmath14 that are one or two orders of magnitude higher than the average inner disk value . in short , none of the alternatives mentioned above are likely to entirely rule out co - dark gas . nevertheless , these alternatives themselves are _ not _ entirely ruled out either and could partly account for some of the high x - factor values inferred . in any event , _ more and deeper mapping of co and other gas tracers of the outer disks of these galaxies is essential for understanding the nature of the dust and gas at these large galactocentric radii . _ the existence of co - dark gas in our galaxy has been known for a while ( see * ? ? ? * and references therein ) and has been confirmed recently by the @xcite . they find @xmath59 for our galaxy . also , as mentioned previously there is much additional evidence for such gas in the galaxy and external galaxies ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? so it is quite likely that there is co - dark gas in both m@xmath051 and m@xmath083 . indeed , the current observations suggest that such gas is present in the outer disks of both m@xmath051 and m@xmath083 . the co - dark gas could account for as much as half of the total gas mass , although the alternative interpretations given above could reduce that fraction by roughly 50% . in section [ sfvgsd ] , we examined how the sfr varied with the gas surface density in a number of ways . comparing between the continuum and spectral line tracers given in figure [ fig18 ] , all show higher slopes for the continuum tracers than for the spectral - line tracers . figures [ fig5a ] and [ fig6a ] illustrate that the continuum - derived radial profile of the gas surface density is flatter than that for the spectral - line derived gas surface for large galactocentric radii . having a smaller range of gas surface densities will naturally increase the slope in the sfr versus gas surface density plots . even though @xcite used far - uv and 24@xmath35 data to estimate the sfr , whereas we used h@xmath60 and 24@xmath35 , that is unlikely to account for the difference between their fitted slopes and ours ; their figure 9 makes such a comparison and the difference in fitted slopes between the two sfr tracers is 10% or less . so if differences in sfr tracers can not explain the difference between the slope obtained by @xcite and those obtained in the current work , then it must be the difference in gas surface density tracer . the continuum tracer used here gives slopes of @xmath61 , which is consistent with the eyeball inspection of the far upper right panel of figure 8 of @xcite for the inner @xmath62 radius of m@xmath083 . estimation of gas surface or column densities is problematic because , as exemplified in the current work , different tracers can yield different results . there are systematics that affect the column density estimates using co @xmath9 and different systematics when using infrared and millimetre continuum . these systematics yield the differing slopes and also the observed scatter , which is sometimes quite large with _ reduced _ chi - square as high as @xmath63 to 70 . this scatter is partly intrinsic , because the sfr depends on more physical conditions than on just column density . but systematic errors in estimating surface densities also play a role . corrections applied systematically to those errors would represent a smooth gradient with galactocentric radius and not simply corrections to a few individual points . for example , observations have inferred x - factor values that are factors of about 5 or more lower than the standard value ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) in the centres of external galaxies as well as in the central region of our own galaxy @xcite . but these centres only represent the central few hundred parsec radii and would thus represent very few points in the plots of figure [ fig18 ] . only a large , smooth gradient in parameters like the x - factor or @xmath33 would change the slope uniformly in those plots , and there is no evidence for such . nevertheless , there is evidence of a weak dependence of the x - factor on the sfr . @xcite explored this possibility with their physical models of molecular clouds , finding that @xmath64 with @xmath65 . the systematically different power - law indices for the dust tracers from those for the gas tracers are reconcilable by adopting that relationship between @xmath14 and @xmath19 . the computation of appendix [ appxfsfr ] yields @xmath66 and @xmath67 for m@xmath051 and m@xmath083 , respectively . these have uncertainties of 3 - 4% and are thus significantly different from that of the models of @xcite . these are more consistent with the models of @xcite(see * ? ? ? * ) . that the m@xmath051 and m@xmath083 @xmath68 values are significantly different from each other argues that there is no universal relationship between @xmath14 and @xmath19 . this is not surprising given that both of those quantities have complex dependences on the physical structure within the molecular gas . the value of the slope of the surface densities of sfr versus those of the molecular gas is an indication of the large - scale evolution of a galaxy . as briefly alluded to previously , following the `` stream '' of points in each of the panels of figure [ fig18 ] from low to high surface densities is equivalent to travelling from large to small galactocentric radii . the @xmath69 ( or alternatively @xmath70 $ ] ) ratio is really the inverse depletion time of the molecular gas . for slope @xmath71 , @xmath72 and the gas is depleted uniformly throughout a galaxy , as pointed out by @xcite and found by them and @xcite . however , _ all _ the panels of figure [ fig18 ] have slopes @xmath73 , strongly supporting an inside - out depletion of molecular gas in both m@xmath051 and m@xmath083 . the panels of figures [ fig19 ] and [ fig20 ] yield molecular gas depletion times from their centres out to radii of about 8@xmath0kpc for m@xmath051 and 6@xmath0kpc for m@xmath083 . for m@xmath051 , the continuum tracers give molecular gas depletion times of about 1.2@xmath0gyr in the centre to 20@xmath0gyr in the outer disk and , with the co line tracer , these times are 0.8 to 2.5@xmath0gyr . for m@xmath083 , the continuum tracers suggest depletion times of about 0.7@xmath0gyr in the centre to 10@xmath0gyr in the outer disk and , again for co , these times are about 0.4 to 5@xmath0gyr . this inside - out evolution of the star formation in the disks of galaxies is supported by the visible - light observations of @xcite . at radii from about 8@xmath0kpc to 11@xmath0kpc for m@xmath051 , the molecular gas depletion times at these radii extend by nearly two orders of magnitude higher than the 20@xmath0gyr estimate for galactocentric radius of 8@xmath0kpc . for m@xmath083 , this extension of the gas depletion time occurs for radii slightly beyond 6@xmath0kpc and is by about 1.5 orders of magnitude . figures [ fig19 ] and [ fig20 ] are equivalent to figure 5 of @xcite , displaying the inverse depletion times of the molecular gas in the form of images of entire galaxies . those images in the current paper ( and , to some extent , those of * ? ? ? * ) are suggestive of spiral structure . such spiral structure implies an enhancement of the sfr due to the spiral arms beyond that of arms simply collecting and compressing gas and dust . accordingly , we subjected these images to spiral arm fourier analysis and found only a weak spiral structure with arm / interarm ratios usually within factors of 2 of unity . the right panels were found to have a spiral structure with high _ interarm _ values of @xmath69 and lower arm values , which is corrected when accounting for the higher interarm @xmath14 ( see section [ xfac ] ) . in contrast , the left panels were found to have a spiral structure with high @xmath74 $ ] on the arms and lower values between the arms ; the arm / interarm ratios are 1.3 for m@xmath051 and 2.0 for m@xmath083 . these arm / interarm ratios are at least partly explained by the higher @xmath33 between the arms than in the arms ( see appendix [ appgsdm]) 18% higher for m@xmath051 and 8% higher for m@xmath083 . so , conservatively speaking , even the continuum tracer in our work _ confirms the work of @xcite that spiral arms only enhance the star formation rate because of increasing the surface density of gas and dust with _ no _ additional enhancement . _ the uncertainties in the current work do not permit completely ruling out such an enhancement , but suggest that any such enhancement would be small ( i.e. , a factor of @xmath75 ) . that enhancement , should it be real , could be accounted for by orbit - crowding in the spiral arms raising the inverse depletion time of the gas in those arms ( e.g. * ? ? one key question is whether the conclusions are still valid if the diffuse emission is removed @xcite . but the lower spatial resolution of the gas tracer observations impede determination of the surface densities associated only with the star - forming regions . so a surface - density versus surface - density plot is difficult to create ( see * ? ? ? * ) , making comparison in the context of much previous work difficult . removing the diffuse emission _ might _ still result in the slope of the @xmath76 versus @xmath77 still being higher when a dust - continuum tracer is used in place of a gas - line tracer ( see figure [ fig18 ] ) , due to the co - dark gas not traced by spectral lines or maybe due to @xmath14 varying with the sfr . also , the inside - out galactic - scale evolution of star formation , as indicated by the superlinear slopes , is likely still valid due to the support of independent work @xcite . @xcite use hierarchical bayesian linear regression on the observational data , finding that no one s - k relation holds for all galaxies . this is consistent with the current work where no one power - law applies to either m@xmath051 or m@xmath083 . indeed , any simple power - law fit is inapplicable given the poor quality of fits in figure [ fig18 ] . not only are there different slopes in different galaxies , but there are also different offsets , particularly in post - starburst galaxies ( see * ? ? ? , if no one such relation holds for all galaxies , then the s - k relation does not represent a universal physical law of star formation ( see * ? ? ? perhaps the s - k relation for a disk galaxy is a measure of how the inverse depletion time of the molecular gas varies radially in that disk galaxy ( e.g. , inside - out or outside - in star formation ) , which , in turn , is affected by the many properties of , and processes in , the disk of that particular galaxy ( see , for example * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the spiral galaxies m@xmath051 and m@xmath083 were observed with the bolometer array _ aztec _ on the _ jcmt _ in the 1.1@xmath0 mm continuum at 20@xmath42 spatial resolution . the extended emission , including the interarm emission and exponential disks , was faithfully recovered in the final maps out to galactocentric radii of more than 12@xmath0kpc for both galaxies . this was verified by simulations that show that only small corrections are necessary for the m@xmath051 image and negligible corrections for that of m@xmath083 . the 1.1@xmath0mm - continuum fluxes are @xmath2 and @xmath3 for m@xmath051 and m@xmath083 , respectively . the uncertainties are largely due to that of the calibration . these images were combined with the 160@xmath35 image of _ spitzer _ to obtain dust temperatures and column densities . this approach was adopted , rather than using dust models , for two reasons . one reason was to have an independent test of the models ( see * ? ? ? * ) . the other reason is that multi - wavelength far - ir data at the long wavelengths ( i.e. @xmath78 ) at which the bulk of the dust mass radiates is not available for the outer disks of m@xmath051 and m@xmath083 , other than the 160@xmath35 data of _ spitzer_. another model - independent approach was to constrain , at least roughly , the @xmath33 at 1.1@xmath0 mm by the observations , rather than simply adopting a dust mass absorption coefficient , @xmath79 . gas column densities were estimated by calibrating against hi - dominant positions to estimate the dust optical depth to gas column density ratio @xmath33 . the method of calibrating against the hi - dominant positions was improved by crudely estimating the effects of the co - dark gas ( see below ) . neither galaxy has a strong radial variation in the gas surface density beyond galactocentric radii of about 3@xmath0kpc . out to a galactocentric radius of 14@xmath0kpc , the best estimate of the mass of gas in m@xmath051 is @xmath4 . out to 12@xmath0kpc in m@xmath083 , this best estimate is @xmath5 . ( see adopted distances in table [ tab1 ] . ) pre - existing maps of co @xmath9 permitted the creation of maps of the @xmath6 or x - factor for both m@xmath051 and m@xmath083out to galactocentric radii of 6 - 8@xmath0kpc . both galaxies have x - factor values that are higher in the _ inter_arm than in the arms by a factor of @xmath7 - 2 . in the central few kiloparsecs of m@xmath051 , this interarm / arm ratio rises to @xmath80 . in m@xmath051 , there is no significant radial variation of @xmath14 . in m@xmath083 , however , there is evidence at the many-@xmath81 level of radial variation of the x - factor by factors of 2 to 3 , where the central 2@xmath0kpc radius has a roughly flat x - factor which declines to a minimum at about 4@xmath0kpc . within galactocentric radii @xmath82 - 8@xmath0kpc , the spatially averaged x - factor is about 1@xmath55 . beyond the outer radius of the x - factor map for each galaxy , comparison of the radial profile of the gas surface density derived from the continuum with those of the surface brightnesses of co and hi permits estimates of lower limits of the x - factor that reach one to two orders of magnitude higher than the inner disk values . this suggests the existence of co - dark molecular gas in the outer disks of m@xmath051 and m@xmath083 , although alternative explanations are only partly ruled out . nevertheless , these alternatives do not entirely account for the high x - factor values in the outer disk . a two - dimensional fourier analysis of the spiral structure at 1.1@xmath0 mm and at visible ( or near visible ) wavelengths revealed that the spiral structure in red light and that in the 1.1@xmath0 mm continuum and were the same in m@xmath051 . for m@xmath083 , the spiral structure in @xmath46-band compared with that in the 1.1@xmath0 mm continuum showed that the bar s effect in m@xmath083 is conspicuous in @xmath46-band and not at 1.1@xmath0 mm . these results suggest that the spiral density wave in m@xmath051 is influencing the interstellar medium and stars similarly , while the bar potential in m@xmath083 has a different influence on the interstellar medium from that on the stars . log - log plots of the star formation rate surface densities against those of the gas traced by spectral lines ( i.e. , of hi and co ) had slopes of @xmath7 whether total gas or just molecular gas surface density . for the plots with gas surface densities traced by the continuum emission , the slopes were @xmath61 whether total gas ( using continuum only ) or just molecular ( using continuum with hi subtracted ) gas surface densities . these plots , especially with the continuum tracers , show a threshold gas surface density at which the sfr rises by two or more orders of magnitude . the existence of this threshold gas surface density is insensitive to within a factor of @xmath523 for the adopted @xmath33 . the value of this threshold density is @xmath83 . this threshold is somewhat less conspicuous in the spectral line tracers than for the continuum tracers . the fitted slopes suggest that the depletion of the molecular gas occurs first at small galactocentric radii and then at increasing radii in an inside - out galactic evolution . this is seen more clearly in maps of the ratio of the surface densities of the sfr to that of the molecular gas . for both these galaxies and the continuum tracer , the molecular gas depletion time in the centres is about 1@xmath0gyr , rising at radii of 6 - 8@xmath0kpc to around 10 - 20@xmath0gyr . further out , the depletion times rise by one or two orders of magnitude . the spectral line tracer , i.e. co @xmath9 , suggests molecular gas depletion times in the outer disks that are appreciably less than 10 - 20@xmath0gyr . the images of the inverse depletion time show signs of spiral structure . superficially , this suggests that spiral arms effect the sfr beyond just heightening the gas surface density . however , correcting for the x - factor spatial variation and for the spatial variation of the @xmath33 removes or nearly removes such spiral structure . this apparently confirms the result of @xcite that the arms merely heighten the sfr in the same proportion as they heighten the gas surface density . greater spatial resolution is required for confirming this result . in the future , we need deeper mapping of molecular tracers in the outer disks of these spiral galaxies . as well , a better method of calibrating the @xmath33 at millimetre wavelengths is needed . so far , the method employed in the current work is functional , but only crudely . either a new method or refinement of the method described here is necessary . consultations with rich rand , divakara mayya , daniel rosa are greatly appreciated . we also thank the anonymous referee , whose comments noticeably improved the manuscript . aniano , g. et al . 2012 , apj , 756 , 138 baes , m , allaert , f. , sarzi , m. , de looze , i. , fritz , j. , gentile , g. , hughes , t. m. , puerari , i. , smith , m. w. l. , and viaene , s. 2014 , mnras , 444 , l90 baillard , a. , bertin , e. , de lapparent , v. , fouqu , p. , arnouts , s. , mellie , y. , pell , r. , leborgne , j .- f . , prugniel , p. , makarov , d. , makarova , l. , mccracken , h. j. , bijaoui , a. , and tasca , l. 2011 , a&a , 532 , 74 bendo , g. j. , boselli , a. , dariush , a. , pohlen , m. , roussel , h. , sauvage , m. , smith , m. w. l. , wilson , c. d. , baes , m. , cooray , a. , clements , d. l. , cortese , l. , foyle , k. , galametz , m. , gomez , h. l. , lebouteiller , v. , lu , n. , madden , s. c. , mentuch , e. , o halloran , b. , page , m. j. , remy , a. , schulz , b. , and spinoglio , l. , 2012 , mnras , 419 , 1833 bigiel , f. , leroy , a. , walter , f. , brinks , e. , de blok , w. j. g. , madore , b. , and thornley , m. d. 2008 , aj , 136 , 2846 blasco - 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brightness maps of figures [ fig1 ] and [ fig2 ] were ratioed with the _ spitzer / mips _ 160@xmath35 , effectively 155.9@xmath35 , maps of m@xmath051 and m@xmath083 @xcite to produce temperature maps under the assumption that the dust emissivity index , @xmath32 , was 2.0 . as stated in the introduction , this was the most appropriate value for @xmath32 as found from the observations of @xmath86 by @xcite . _ in practice , the adopted value for @xmath32 will have only a small effect on the derived gas column densities , provided that those derived column densities are calibrated against observed gas column densities . _ by using those positions where atomic gas dominated the column densities to calibrate the @xmath33 for the continuum - derived column densities , it was found that the column densities derived from the continuum data for @xmath87 to 2.5 did not deviate by more than 30% from those for @xmath88 . consequently , observations at only two wavelengths are necessary for a reasonable approximation of the gas surface density map . the @xmath33 derived from the hi comparisons are 2.2-@xmath89 for m@xmath051 and m@xmath083 , respectively . these values lead to gas mass estimates that appear to be too low ( see sections [ xfac ] , [ xfsv ] , and below in this appendix for details ) . @xcite simply adopted the dust mass - absorption coefficient of @xcite and @xcite , which is a factor of @xmath90 smaller than the values determined here . @xcite also determined a higher dust - to - gas mass ratio than that adopted here . this is equivalent to adopting a @xmath33 that is factors of 5.0 - 5.3 smaller than that derived from the hi comparisons . because we adopt a constant dust - to - gas ratio , the two @xmath33 cases are referred to as the high-@xmath31 ( from hi comparisons ) and low-@xmath31 @xcite cases . given the low masses that result from the high-@xmath31 case , the hi comparisons must be appropriately modified . the hi comparisons were repeated after removing upper outliers that are apparently unreliable due to possible undetected gas . this results in an intermediate case , where a @xmath33 that is a factor of two lower than that from the simple atomic gas calibration method is found . unless otherwise stated , all masses and surface densities quoted are for this intermediate-@xmath33 ( most realistic ) case . this approach yields a roughly 50% uncertainty in column density and mass estimates , using @xmath91 . the surface brightness at frequency @xmath92 , @xmath93 , is related to the dust - derived gas column density , @xmath94 , by @xmath95 where @xmath96 is the mean atomic weight per hydrogen atom , @xmath97 is the mass of a hydrogen atom , @xmath58 is the dust - to - gas mass ratio , and @xmath31 is the dust mass absorption coefficient . the mass - absorption coefficient varies with frequency as follows : @xmath98 in which @xmath99 is dust mass - absorption coefficient at a reference frequency @xmath100 . the dust temperature , @xmath101 , is of course estimated by evaluating expression ( [ eq1 ] ) at frequency @xmath102 , corresponding to the wavelength of 1.1@xmath0 mm for the _ jcmt / aztec _ observations , and again at frequency @xmath103 , corresponding to the wavelength of 155.9@xmath35 for the _ spitzer / mips _ observations , and taking the ratio of the two . as well as using the maps at the two wavelengths ( i.e. the _ jcmt / aztec _ at 1.1@xmath0 mm and the _ spitzer / mips _ map at 155.9@xmath35 ) , there are many additional details involved in estimating the column densities , @xmath94 . these include estimating the value of @xmath33 ( or the product of @xmath58 and @xmath31 ) , convolving the _ aztec _ 1.1@xmath0 mm map to the resolution of the _ spitzer / mips _ 155.9@xmath35 map , estimating the noise levels in the maps , characterizing any systematic effects introduced into the @xmath101 and @xmath94 maps due to the observations and processing of the _ aztec _ data , and any relevant colour corrections to the continuum data . in particular , a constant offset correction of @xmath104-@xmath105 must be added to the column densities for m@xmath051 as dictated by the simulations to correct for inadequately recovering the large - scale emission ( but was not applied to the figures ) . there are a couple issues that should be addressed in discussing the reliability of gas surface densities and masses as inferred from dust emission in the current work . one issue is to what level calibration differences between that of _ spitzer / mips _ and that of _ herschel / pacs _ for their 160@xmath35 data would affect the results and conclusions of the current work ( see * ? ? ? another issue is the validity of the simple approach used here instead of using a more detailed dust model ( e.g. , * ? ? ? as for the former issue , @xcite compared the 160@xmath35 data for both instruments and characterized their systematic differences . depending on the calibration scheme used , the _ herschel / pacs _ 160@xmath35 intensities were either 1.6% or 25% systematically higher than those for _ spitzer / mips_. scaling up the 160@xmath35 intensities used in the current work by 25% would increase the computed dust temperatures , but by differing amounts depending on initial computed temperature . most of the those temperatures were determined to be between 10@xmath0k and 23@xmath0k . after the hypothetical correction , those would be 10.3@xmath0k and 25@xmath0k . if there were no calibration of the column densities against other data , those corrected temperatures would require the column densities to be corrected downwards by 5% and 11% , respectively . the warmer positions in both galaxies tend to have higher column densities . so the higher column densities would be corrected downwards by more than the lower column densities , on average . this reduces the dynamic range of the determined column densities by roughly 6% . the overall scaling of those column densities would remain unchanged because of the calibration against hi column densities . so , such a correction , were it necessary , would not appreciably change the current results . as for the second issue , the models of @xcite combined with data at mid - ir and far - ir wavelengths can provide estimates of a number of parameters . for the current work , however , there are two reasons why such an approach was not used . one reason is that , as mentioned in the introduction , the _ aztec _ 1.1@xmath0 mm map has more spatial coverage than the _ herschel _ data for both m@xmath051 and m@xmath083 . _ spitzer _ does have sufficient spatial coverage for comparison with the _ aztec _ data , but does not have the long - wavelength data that probes the bulk of the dust mass , other than at 160@xmath35 . so , to consistently treat each entire galaxy , the _ aztec _ 1.1@xmath0 mm data were combined with the _ spitzer _ 160@xmath35 data for both m@xmath051 and m@xmath083 . another reason that the more model - dependent approach was not used is that we need to have model - independent tests that can confirm or refute the validity of the models , as stated recently by @xcite . the current work , for example , finds a dust opacity to gas column density ratio @xmath33 at 1.1@xmath0 mm that is about double that inferred from @xcite . but temporarily adopting the @xmath33 of @xcite and adopting the appropriate distance yields nearly identical masses for both the more sophistocated approach ( i.e. , in * ? ? ? * ) and the simpler approach ( i.e. , the current work ) for the ngc@xmath05194 field . this strongly suggests that the simpler approach of the current work is reasonable at recovering some basic dust properties . the dust optical depth to gas column density ratio , @xmath33 , and especially the dust mass absorption coeffient , @xmath31 , can be quite uncertain , ranging from @xmath106 to @xmath107 . ( see the following for examples of disparate values : * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . accordingly , attempting to measure @xmath33 observationally could narrow the uncertainty . this is most easily observed in gas dominated by atomic hydrogen because the gas column densities are easy to determine . another approach was followed by @xcite for observations of the galaxies m@xmath099 and m@xmath0100 in which they plotted the star formation rate surface density , @xmath19 , against gas surface density and against that for dust , adjusting the @xmath31 until both surface densities agree at large @xmath19 . this resulted in a @xmath31 value corresponding to the unusually low @xmath108 mentioned above . the large @xmath33 initially derived in the current work was from comparing the observed optical depths at 1.1@xmath0 mm with the observed gas column densities in regions dominated by atomic gas . the @xcite observed @xmath34 at 857@xmath0ghz and @xmath36 at 250@xmath35 in the dust associated with the hi gas within the galaxy . the @xcite also observed @xmath109 at 250@xmath35 for the dust associated with molecular gas , which is about double that observed for dust associated with atomic gas , remembering that the @xmath110 is the column density of hydrogen _ nuclei_. at a wavelength of 1.1@xmath0 mm , these numbers correspond to @xmath111 and @xmath112 for dust associated with hi and @xmath113 for that associated with h@xmath41 ( using a spectral emissivity index of @xmath38 for the galaxy , e.g. , * ? ? ? * ; * ? ? ? the current work initially finds @xmath114 for m@xmath051 and @xmath115 for m@xmath083 for dust along lines of sight with gas column densities dominated by @xmath116 . this is more than double the values observed in our galaxy and are even 40% larger than the dust associated with @xmath117 in our galaxy . this large @xmath33 is suspect and requires a re - evaluation of the calibration of @xmath33 at 1.1@xmath0 mm . one explanation for the observed large @xmath33 would be co - dark gas . the presence of such gas can be crudely tested . this is done by starting with dust - derived gas column densities , @xmath94 , that are only roughly calibrated i.e. , to within a factor of a few . the difference @xmath118 then provides a rough measure of molecular gas column density . the ratio @xmath119 ( where the @xmath14 is the average used for each galaxy ) is a crude measure of the amount of dark gas a _ dark _ ratio . then @xmath33 can be recomputed using only those positions with dark ratios less than some upper limit . choosing a dark ratio upper limit as high as 100 reduces the derived @xmath33 by a few percent , while choosing unity yields a @xmath33 that is a factor of about 4 lower . the latter is risky given that this could unnecessarily exclude too many positions with very little dark gas . accordingly , a more conservative correction factor would be around 2 . reducing the @xmath33 value by this factor scales up the column densities by the same factor and the x - factor scales in proportion to @xmath118 , or up by a factor of @xmath523 for both m@xmath051 and m@xmath083 . the best estimates of these values are hence @xmath120 , corresponding to @xmath121 for @xmath122 for both m@xmath051 and m@xmath083 . these are well within a factor of 2 of the values found by the @xcite for our galaxy . the best estimates for the x - factors for m@xmath051 and m@xmath083 are around 1@xmath55 . table [ tab3 ] displays the best estimated values for the gas masses and the x - factors . that table shows that roughly half of the total mass of the gas is unaccounted for using spectral lines . this is discussed further in section [ xfsv ] . possible spatial variations in @xmath33 should also be considered . both m@xmath051 and m@xmath083 show _ de_creases in @xmath33 by about 15 - 20% in going from galactocentric radii of 1 or 2@xmath0kpc to about 6 or 7@xmath0kpc . correcting for the co - dark gas changes this result . the problem is that the numbers of points used in the comparisons becomes very small ( i.e. @xmath123 ) when the upper limit to the dark ratio is reduced . such small numbers do not give reliable estimates of @xmath33 . so the radial spatial variation of @xmath33 is uncertain , although there is no clear evidence that it has a strong radial variation . in going from interarm positions to spiral arm positions , the @xmath33 in m@xmath051 decreases by 18% . in m@xmath083 , this decrease is 8% . again , correcting for the co - dark gas changes these results to 25% for m@xmath051 and an 11% _ in_crease for m@xmath083 , although the number of points available for the estimation is not really sufficient . the x - factor , @xmath14 , can be straightforwardly computed from @xmath124 given that @xmath94 , @xmath11 , and @xmath125 are all maps , @xmath14 is also a map , but not a very extended map because of the mounting uncertainties that result from the numerous computations with pre - existing data required to reach @xmath14 . both the m@xmath051 and m@xmath083 @xmath14 maps do not extend beyond about 7@xmath0kpc from the centre of each map , mostly because the co maps do not extend further than that . figures [ fig13 ] , [ fig13a ] , [ fig14 ] , and [ fig14a ] show the x - factor maps and radial profiles for m@xmath051 and m@xmath083 . the mean values of the x - factor listed in table [ tab5 ] , which we call @xmath126 , are the @xmath127-weighted means of each @xmath14 map and only really applies to the central 7@xmath0kpc radius in each galaxy . given that the co @xmath9 emission falls to the noise level at a radius of about 7 to 8@xmath0kpc , we are unable to compute @xmath14 beyond that in the outer disks . nevertheless , we can use the radial curves of figures [ fig5a ] and [ fig6a ] to infer rough lower limits on @xmath14 in the outer disks . the radial variation of @xmath14 with respect to the mean value in the inner disk , @xmath126 , is a simple variation on expression ( [ eq5 ] ) : @xmath128 where @xmath129 is the molecular gas column density estimated from a spatially constant x - factor : @xmath130 . this measure of the column density is very noisy in these observations of the outer disks and a useful proxy is necessary for inferring some crude lower limit on @xmath131 for radii beyond 8@xmath0kpc . the proxy adopted for @xmath132 was a pessimistic estimate of the 3-sigma uncertainties of the gas column density , i.e. , @xmath133 $ ] . this gives the most conservative ( i.e. least extreme ) estimates for @xmath131 beyond a radius of 8@xmath0kpc . for m@xmath051 , even in the high-@xmath31 ( low-@xmath31 ) case , the above proxy yields lower limits to @xmath14 of roughly @xmath134 to @xmath135 ( @xmath136 to @xmath137 ) depending on the radius in the outer disk . for m@xmath083 , even in the high-@xmath31 ( low-@xmath31 ) case , these rough lower limits to @xmath14 are @xmath138 to @xmath139 ( @xmath140 to @xmath141 ) . the logarithm of the sfr surface density is plotted against that of the gas surface density for different tracers of that gas surface density . the plots of the surface densities of sfr versus gas are displayed in figure [ fig18 ] . the uncertainty of each fitted slope is the formal error of the fit and was scaled by the square - root of the reduced chi - square in order to give a more conservative and more realistic estimate of this formal error . one important consideration in these fits is that the spatial resolution of the m@xmath051 data is limited to 38@xmath42 , because of the _ spitzer _ 160@xmath35 resolution . for m@xmath083 this resolution is 55@xmath42 resolution due to the co observations . given that each point in these plots represents a single pixel , the effective number of independent points in each plot is the number of pixels divided by 20 for m@xmath051 and by 55 for m@xmath083 . nevertheless , the linear resolution on each galaxy is about the same , with 1.5@xmath0kpc for m@xmath051 and 1.2@xmath0kpc for m@xmath083 . the sfr surface density maps were created from images in the h@xmath60 line and in the 24@xmath35 continuum . according to @xcite , the sfr , @xmath142 , within a source is given by @xmath143\qquad .\hfill \label{eqb1}\ ] ] @xmath144 is the integrated luminosity of the h@xmath60 line in @xmath145 ; @xmath92 is the frequency corresponding to the wavelength of 24@xmath35 ; @xmath146 is the luminosity of the 24@xmath35 continuum per unit frequency bandwidth in @xmath147 ; @xmath148 ; @xmath149 ; @xmath142 is in units of @xmath150 . if @xmath19 is the sfr surface density , then @xmath151 , where @xmath152 is the source solid angle and d is the source distance . using @xmath153 , where @xmath154 is the source flux , and also using @xmath155 , with @xmath46 as the source surface brightness , results in @xmath156\qquad .\hfill \label{eqb2}\ ] ] for @xmath19 in units of @xmath157 , @xmath158 in units of @xmath159 , and @xmath160 in units of @xmath161 , we have @xmath162 and @xmath163 . the numerical value of @xmath164 remains unchanged . specifically for m@xmath051 and m@xmath083 , the sfr surface density maps were created by applying expression ( [ eqb2 ] ) to the h@xmath60 ( i.e. , the line - integrated images of * ? ? ? * ; * ? ? ? * ) and the spitzer 24@xmath35 images @xcite of both galaxies . the images were convolved to the appropriate resolution and rebinned to 9@xmath42 pixels . the @xmath158 image was converted to units of @xmath165 . the final spatial resolution of the sfr surface density map for m@xmath051 is 38@xmath42 , which is that of the _ spitzer _ 160@xmath35 image . for m@xmath083 this is 55@xmath42 , which is that of the co @xmath9 map . the negative tails of the histograms of pixel values provided estimates of the noise levels . the physical models of @xcite explore the possible dependence of the x - factor on the the star formation rate . in their section 5.1 , their results can be parameterized in the form @xmath166 where they find @xmath65 . with the results of the fits illustrated in figure [ fig18 ] , we can test this result as well as specify the parameter @xmath167 . we repeat the derivation in section 5.1 of @xcite in more detail and adopt their nomenclature where applicable . if co ( 1 - 0 ) is used to estimate the h@xmath41 surface density , @xmath168 , then we let @xmath169 represent this surface density as estimated by co. then @xmath170^{n_{obs } } = k_{sco } \left[{x_{fm}}i(co)\right]^{n_{obs}}\ , \hfill \label{eqxs2}\ ] ] where @xmath171 is the power - law index that is observed when using co - derived molecular gas surface densities and @xmath126 is the mean x - factor value chosen for the galaxy observed . is unimportant because it is the @xmath172 combination that matters . ] using a tracer that supposedly gives the `` true '' @xmath168 , would have a similar relation : @xmath173^{n_{act } } = k_{sco } \left[{x_f}i(co)\right]^{n_{act}}\qquad , \hfill \label{eqxs3}\ ] ] where @xmath174 is the actual correct power - law index and @xmath14 is the correct point - by - point value of the x - factor . combining expressions ( [ eqxs2 ] ) and ( [ eqxs3 ] ) so as to eliminate @xmath125 and comparing with ( [ eqxs1 ] ) yields @xmath175 and also , @xmath176 in the context of the current work , we assume that the dust tracers of the molecular gas give @xmath177 ( i.e. , ) . so the fits in the second panels of figure [ fig18 ] yield the parameters @xmath178 and @xmath174 for m@xmath051 and m@xmath083 . the fits in the fourth panels provide @xmath179 and @xmath171 . the uncertainties of these quantities are likely dominated by systematics . this means that @xmath167 has a roughly 50% uncertainty . the effects of systematics on @xmath68 are difficult to estimate . the formal uncertainties of @xmath68 using the current data are 3 - 4% . with the above in mind , expressions ( [ eqxs4 ] ) and ( [ eqxs5 ] ) applied to the current data yield @xmath180 and @xmath66 for m@xmath051 and 0.79 and @xmath67 for m@xmath083 . note that these values for @xmath167 give @xmath14 in @xmath55 units .

general relativity and ordinary differential geometry should be replaced by non - commutative geometry at some point between the currently accessible energies of about 1 - 10 tev ( after starting the large hadron collider ( lhc ) at cern ) and the planck scale , which is @xmath2 times higher , where space - time and gravity should be quantized . this could occur either at the planck scale or below . quantum field theory on a non - commutative space - time ( ncqf ) could very well be an intermediate theory relevant for physics at energies between the lhc and the planck scale . it certainly looks intermediate in structure between ordinary quantum field theory on commutative @xmath3 and string theory , the current leading candidate for a more fundamental theory including quantized gravity . ncqft in fact appears as an effective model for certain limits of string theory @xcite . in joint work with r. gurau , j. magnen and f. vignes - tourneret @xcite , using direct space methods , we provided recently a new proof that the grosse - wulkenhaar scalar @xmath4 theory on the moyal space @xmath5 is renormalisable to all orders in perturbation theory . the grosse - wulkenhaar breakthrough @xcite was to realize that the right propagator in non - commutative field theory is not the ordinary commutative propagator , but has to be modified to obey langmann - szabo duality @xcite . grosse and wulkenhaar were able to compute the corresponding propagator in the so called `` matrix base '' which transforms the moyal product into a matrix product . this is a real _ tour de force _ ! they use this representation to prove perturbative renormalisability of the theory up to some estimates which were finally proven in @xcite . our direct space method builds upon the previous works of filk and chepelev - roiban @xcite . these works however remained inconclusive @xcite , since these authors used the right interaction but not the right propagator , hence the problem of ultraviolet / infrared mixing prevented them from obtaining a finite renormalised perturbation series . we also extend the grosse - wulkenhaar results to more general models with covariant derivatives in a fixed magnetic field @xcite . our proof relies on a multiscale analysis analogous to @xcite but in direct space . non - commutative field theories ( for a general review see @xcite ) deserve a thorough and systematic investigation , not only because they may be relevant for physics beyond the standard model , but also ( although this is often less emphasized ) because they can describe effective physics in our ordinary standard world but with non - local interactions . in this case there is an interesting reversal of the initial grosse - wulkenhaar problematic . in the @xmath4 theory on the moyal space @xmath5 , the vertex is sort of god - given by the moyal structure , and it is ls invariant . the challenge was to overcome uv / ir mixing and to find the right propagator which makes the theory renormalisable . this propagator turned out to have ls duality . the harmonic potential introduced by grosse and wulkenhaar can be interpreted as a piece of covariant derivatives in a constant magnetic field . now to explain the ( fractional ) quantum hall effect , which is a bulk effect whose understanding requires electron interactions , we can almost invert this logic . the propagator is known since it corresponds to non - relativistic electrons in two dimensions in a constant magnetic field . it has ls duality . but the interaction is unclear , and can not be local since at strong magnetic field the spins should align with the magnetic field , hence by pauli principle local interactions among electrons in the first landau level should vanish . we can argue that among all possible non - local interactions , a few renormalisation group steps should select the only ones which form a renormalisable theory with the corresponding propagator . in the commutative case ( i.e. zero magnetic field ) local interactions such as those of the hubbard model are just renormalisable in any dimension because of the extended nature of the fermi - surface singularity . since the non - commutative electron propagator ( i.e. in non zero magnetic field ) looks very similar to the grosse - wulkenhaar propagator ( it is in fact a generalization of the langmann - szabo - zarembo propagator ) we can conjecture that the renormalisable interaction corresponding to this propagator should be given by a moyal product . that s why we hope that non - commutative field theory is the correct framework for a microscopic _ ab initio _ understanding of the fractional quantum hall effect which is currently lacking . even for regular commutative field theory such as non - abelian gauge theory , the strong coupling or non - perturbative regimes may be studied fruitfully through their non - commutative ( i.e. non local ) counterparts . this point of view is forcefully suggested in @xcite , where a mapping is proposed between ordinary and non - commutative gauge fields which do not preserve the gauge groups but preserve the gauge equivalent classes . we can at least remark that the effective physics of confinement should be governed by a non - local interaction , as is the case in effective strings or bags models . in other words we propose to base physics upon the renormalisability principle , more than any other axiom . renormalisability means genericity ; only renormalisable interactions survive a few rg steps , hence only them should be used to describe generic effective physics of any kind . the search for renormalisabilty could be the powerful principle on which to orient ourselves in the jungle of all possible non - local interactions . renormalisability has also attracted considerable interest in the recent years as a pure mathematical structure . the work of kreimer and connes @xcite recasts the recursive bphz forest formula of perturbative renormalisation in a nice hopf algebra structure . the renormalisation group ambiguity reminds mathematicians of the galois group ambiguity for roots of algebraic equations . finding new renormalisable theories may therefore be important for the future of pure mathematics as well as for physics . that was forcefully argued during the luminy workshop `` renormalisation and galois theory '' . main open conjectures in pure mathematics such as the riemann hypothesis @xcite or the jacobian conjecture @xcite may benefit from the quantum field theory and renormalisation group approach . considering that most of the connes - kreimer works uses dimensional regularization and the minimal dimensional renormalisation scheme , it is interesting to develop the parametric representation which generalize schwinger s parametric representation of feynman amplitudes to the non commutative context . it involves hyperbolic generalizations of the ordinary topological polynomials , which mathematicians call kirchoff polynomials , and physicist call symanzik polynomials in the quantum field theory context @xcite . we plan also to work out the corresponding regularization and minimal dimensional renormalisation scheme and to recast it in a hopf algebra structure . the corresponding structures seem richer than in ordinary field theory since they involve ribbon graphs and invariants which contain information about the genus of the surface on which these graphs live . a critical goal to enlarge the class of renormalisable non - commutative field theories and to attack the quantum hall effect problem is to extend the results of grosse - wulkenhaar to fermionic theories . the simplest theory , the two - dimensional gross - neveu model can be shown renormalisable to all orders in their langmann - szabo covariant versions , using either the matrix basis @xcite or the direct space version developed here @xcite . however the @xmath6-space version seems the most promising for a complete non - perturbative construction , using pauli s principle to controll the apparent ( fake ) divergences of perturbation theory . in the case of @xmath7 , recall that although the commutative version is until now fatally flawed due to the famous landau ghost , there is hope that the non - commutative field theory treated at the perturbative level in this paper may also exist at the constructive level . indeed a non trivial fixed point of the renormalization group develops at high energy , where the grosse - wulkenhaar parameter @xmath8 tends to 1 , so that langmann - szabo duality become exact , and the beta function vanishes . this scenario has been checked explicitly to all orders of perturbation theory @xcite . this was done using the matrix version of the theory ; again an @xmath6-space version of renormalisation might be better for a future rigorous non - perturbative investigation of this fixed point and a full constructive version of the model . finally let us conclude this short introduction by reminding that a very important and difficult goal is to also extend the grosse - wulkenhaar breakthrough to gauge theories . one considers free electrons : @xmath9 where @xmath10 is the canonical conjugate of @xmath11 . the moment and position @xmath12 and @xmath13 have commutators @xmath14=0,\ \ [ r_i , r_j]=0,\ \ [ p_i , r_j]=\imath\hbar\delta_{ij } .\end{aligned}\ ] ] the moments @xmath15 have commutators @xmath16=-\imath\hbar\epsilon_{ij}eb,\ \ [ r_i , r_j]=0,\ \ [ \pi_i , r_j]=\imath\hbar\delta_{ij } .\end{aligned}\ ] ] one can also introduce coordinates @xmath17 , @xmath18 corresponding to the centers of the classical trajectories @xmath19 which _ do not commute _ : @xmath20=\imath\hbar\epsilon_{ij}\frac{1}{e b}},\ \ [ \pi_i , r_j]=0 . \end{aligned}\ ] ] this means that there exist heisenberg - like relations between quantum positions . one considers the string action in a generalized background @xmath21 where @xmath22 is the string worldsheet , @xmath23 is a tangential derivative along the worldsheet boundary @xmath24 and @xmath25 is an antisymmetric background tensor . the equations of motion determine the boundary conditions : @xmath26 boundary conditions for coordinates can be neumann ( @xmath27 ) or dirichlet ( @xmath28 , corresponding to branes ) . after conformal mapping of the string worldsheet onto the upper half - plane , the string propagator in background field is @xmath29 \ ; .\end{aligned}\ ] ] for some constant symmetric and antisymmetric tensors @xmath30 and @xmath31 . evaluated at boundary points on the worldsheet , this propagator is @xmath32 where the @xmath31 term simply comes from the discontinuity of the logarithm across its cut . interpreting @xmath33 as time , one finds @xmath34= i \theta^{\mu \nu } , \end{aligned}\ ] ] which means that string coordinates lie in a non - commutative moyal space with parameter @xmath31 . there is an equivalent argument inspired by @xmath35 theory : a rotation sandwiched between two @xmath36 dualities generates the same constant commutator for string coordinates . the recent progresses concerning the renormalisation of non - commutative field theory have been obtained on a very simple non - commutative space namely the moyal space . from the point of view of quantum field theory , it is certainly the most studied space . let us start with its precise definition . let us define @xmath37 and @xmath38 the free algebra generated by @xmath39 . let @xmath40 a @xmath41 non - degenerate skew - symmetric matrix ( wich requires @xmath42 even ) and @xmath43 the ideal of @xmath38 generated by the elements @xmath44 . the moyal algebra @xmath45 is the quotient @xmath46 . each element in @xmath45 is a formal power series in the @xmath47 s for which the relation @xmath48 holds . usually , one puts the matrix @xmath40 into its canonical form : @xmath49 sometimes one even set @xmath50 . the preceeding algebraic definition whereas short and precise may be too abstract to perform real computations . one then needs a more analytical definition . a representation of the algebra @xmath45 is given by some set of functions on @xmath51 equipped with a non - commutative product : the _ groenwald - moyal _ product . what follows is based on @xcite . [ [ sec : lalgebre - ca_theta ] ] the algebra @xmath45 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the moyal algebra @xmath45 is the linear space of smooth and rapidly decreasing functions @xmath52 equipped with the non - commutative product defined by : @xmath53 , @xmath54 this algebra may be considered as the `` functions on the moyal space @xmath55 '' . in the following we will write @xmath56 instead of @xmath57 and use : @xmath58 , @xmath59 , @xmath60 for the fourier transform and @xmath61 for the twisted convolution . as on @xmath62 , the fourier transform exchange product and convolution : @xmath63 one also shows that the moyal product and the twisted convolution are * associative * : @xmath64 using , we show the associativity of the @xmath65-produit . the complex conjugation is * involutive * in @xmath45 @xmath66 one also have @xmath67 [ prop : trace ] for all @xmath68 , @xmath69 @xmath70 where @xmath71 is the ordinary convolution . in the following sections , we will need lemma [ lem : moyal - prods ] to compute the interaction terms for the @xmath72 and gross - neveu models . we write @xmath73 . [ lem : moyal - prods ] for all @xmath74 , let @xmath75 . then @xmath76 [ cor : int - moyal ] for all @xmath74 , let @xmath75 . then @xmath77 the cyclicity of the product , inherited from proposition [ prop : trace ] implies : @xmath78 , @xmath79 and allows to extend the moyal algebra by duality into an algebra of tempered distributions . [ [ sec : extens - par - dual ] ] extension by duality + + + + + + + + + + + + + + + + + + + + let us first consider the product of a tempered distribution with a schwartz - class function . let @xmath80 and @xmath81 . we define @xmath82 and @xmath83 . [ defn : tf ] let @xmath84 , @xmath85 , we define @xmath86 and @xmath87 by @xmath88 for example , the identity @xmath89 as an element of @xmath90 is the unity for the @xmath65-produit : @xmath91 , @xmath92 we are now ready to define the linear space @xmath93 as the intersection of two sub - spaces @xmath94 and @xmath95 of @xmath90 . [ defn : m ] @xmath96 one can show that @xmath93 is an associative @xmath71-algebra . it contains , among others , the identity , the polynomials , the @xmath97 distribution and its derivatives . then the relation @xmath98 often given as a definition of the moyal space , holds in @xmath93 ( but not in @xmath45 ) . the simplest non - commutative model one may consider is the @xmath99-theory on the four - dimensional moyal space . its lagrangian is the usual ( commutative ) one where the pointwise product is replaced by the moyal one : @xmath100 = & \int d^4x \big ( -\frac{1}{2 } \partial_\mu \phi \star \partial^\mu \phi + \frac{1}{2 } m^2 \,\phi \star \phi + \frac{\lambda}{4 } \phi \star \phi \star \phi \star \phi\big)(x).\label{eq : phi4naif}\end{aligned}\ ] ] thanks to the formula , this action can be explicitly computed . the interaction part is given by the corollary [ cor : int - moyal ] : @xmath101 the main characteristic of the moyal product is its non - locality . but its non - commutativity implies that the vertex of the model is only invariant under cyclic permutation of the fields . this restricted invariance incites to represent the associated feynman graphs with ribbon graphs . one can then make a clear distinction between planar and non - planar graphs . this will be detailed in section [ sec : multi - scale - analysismatrix ] . thanks to the delta function in , the oscillation may be written in different ways : @xmath102 the interaction is real and positive . ] : @xmath103 it is also translation invariant as shows equation . the property [ prop : trace ] implies that the propagator is the usual one : @xmath104 . the non - locality of the @xmath65-product allows to understand the discovery of minwalla , van raamsdonk and seiberg @xcite . they showed that not only the model is nt finite in the uv but also it exhibits a new type of divergences making it non - renormalisable . in the article @xcite , filk computed the feynman rules corresponding to . he showed that the planar amplitudes equal the commutative ones whereas the non - planar ones give rise to oscillations coupling the internal and external legs . a typical example is the the non - planar tadpole : @xmath105 if @xmath106 , this amplitude is finite but , for small @xmath107 , it diverges like @xmath108 . in other words , if we put an ultraviolet cut - off @xmath109 to the @xmath110-integral , the two limits @xmath111 and @xmath112 do not commute . this is the uv / ir mixing phenomena . a chain of non - planar tadpoles , inserted in bigger graphs , makes divergent any function ( with six points or more for example ) . but this divergence is not local and ca nt be absorbed in a mass redefinition . this is what makes the model non - renormalisable . we will see in sections [ sec : prop - et - renorm ] and [ sec : direct - space ] that the uv / ir mixing results in a coupling of the different scales of the theory . we will also note that we should distinguish different types of mixing . the uv / ir mixing was studied by several groups . first , chepelev and roiban @xcite gave a power counting for different scalr models . they were able to identify the divergent graphs and to classify the divergences of the theories thanks to the topological data of the graphs . then v. gayral @xcite showed that uv / ir mixing is present on all isospectral deformations ( they consist in curved generalisations of the moyal space and of the non - commutative torus ) . for this , he considered a scalar model and discovered contributions to the effective action which diverge when the external momenta vanish . the uv / ir mixing is then a general characteristic of the non - commutative theories , at least on the deformations . the situation remained so until h. grosse and r. wulkenhaar discovered a way to define a renormalisable non - commutative model . we will detail their result in section [ sec : multi - scale - analysismatrix ] but the main message is the following . by adding an harmonic term to the lagrangian , @xmath113 = & \int d^4x \big ( -\frac{1}{2 } \partial_\mu \phi \star \partial^\mu \phi + \frac{\omega^2}{2 } ( \tilde{x}_\mu \phi ) \star ( \tilde{x}^\mu \phi ) + \frac{1}{2 } m^2 \,\phi \star \phi + \frac{\lambda}{4 } \phi \star \phi \star \phi \star \phi\big)(x)\label{eq : phi4omega}\end{aligned}\ ] ] where @xmath114 and the metric is euclidean , the model , in four dimensions , is renormalisable at all orders of perturbation @xcite . we will see in section [ sec : direct - space ] that this additional term give rise to an infrared cut - off and allows to decouple the different scales of the theory . the new model ( [ eq : phi4omega ] ) , we call it @xmath72 , do not exhibit any mixing . this result is very important because it opens the way towards other non - commutative field theories . in the following , we will call _ _ vulcanisation__. ] the procedure consisting in adding a new term to a lagrangian of a non - commutative theory in order to make it renormalisable . + the propagator @xmath115 of this @xmath116 theory is the kernel of the inverse operator @xmath117 . it is known as the mehler kernel @xcite @xmath118 langmann and szabo remarked that the quartic interaction with moyal product is invariant under a duality transformation . it is a symmetry between momentum and direct space . the interaction part of the model is ( see equation ) @xmath119=&\int d^{4}x\,\frac{\lambda}{4}(\phi\star\phi\star\phi\star\phi)(x)\\ = & \int\prod_{a=1}^{4}d^{4}x_{a}\,\phi(x_{a})\,v(x_{1},x_{2},x_{3},x_{4})\label{eq : vx}\\ = & \int\prod_{a=1}^{4}\frac{d^{4}p_{a}}{(2\pi)^{4}}\,\hat{\phi}(p_{a})\,\hat{v}(p_{1},p_{2},p_{3},p_{4})\label { eq : vp } \intertext{with } v(x_{1},x_{2},x_{3},x_{4})=&\frac{\lambda}{4}\frac{1}{\pi^{4}\det\theta}\delta(x_{1}-x_{2}+x_{3}-x_{4 } ) \cos(2(\theta^{-1})_{\mu\nu}(x_{1}^{\mu}x_{2}^{\nu}+x_{3}^{\mu}x_{4}^{\nu}))\notag\\ \hat{v}(p_{1},p_{2},p_{3},p_{4})=&\frac{\lambda}{4}(2\pi)^{4}\delta(p_{1}-p_{2}+p_{3}-p_{4})\cos(\frac 12 \theta^{\mu\nu}(p_{1,\mu}p_{2,\nu}+p_{3,\mu}p_{4,\nu}))\notag\end{aligned}\ ] ] where we used a _ cyclic _ fourier transform : @xmath120 . the transformation @xmath121 exchanges and . in addition , the free part of the model is nt covariant under this duality . the vulcanisation adds a term to the lagrangian which restores the symmetry . the theory is then covariant under the langmann - szabo duality : @xmath122\mapsto&\omega^{2}\,s[\phi;\frac{m}{\omega},\frac{\lambda}{\omega^{2 } } , \frac{1}{\omega } ] . \end{aligned}\ ] ] by symmetry , the parameter @xmath8 is confined in @xmath123 . let us note that for @xmath124 , the model is invariant . [ [ parag : interpharm ] ] the interpretation of that harmonic term is not yet clear . but the vulcanisation procedure already allowed to prove the renormalisability of several other models on moyal spaces such that @xmath125 @xcite , @xmath126 @xcite and the lsz models @xcite . these last are of the type @xmath113 = & \int d^nx \big ( \frac{1}{2 } \bar{\phi}\star(-\partial_\mu+\xt_{\mu}+m)^{2}\phi + \frac{\lambda}{4 } \bar{\phi } \star \phi \star \bar{\phi } \star\phi\big)(x).\label{eq : lszintro}\end{aligned}\ ] ] by comparison with , one notes that here the additional term is formally equivalent to a fixed magnetic background . deep is the temptation to interpret it as such . this model is invariant under the above duality and is exactly soluble . let us remark that the complex interaction in makes the langmann - szabo duality more natural . it does nt need a cyclic fourier transform . the @xmath127 have been studied at @xmath124 where they also exhibit a soluble structure . apart from the @xmath72 , the modified bosonic lsz model @xcite and supersymmetric theories , we now know several renormalizable non - commutative field theories . nevertheless they either are super - renormalizable ( @xmath128 @xcite ) or ( and ) studied at a special point in the parameter space where they are solvable ( @xmath129 @xcite , the lsz models @xcite ) . although only logarithmically divergent for parity reasons , the non - commutative gross - neveu model is a just renormalizable quantum field theory as @xmath72 . one of its main interesting features is that it can be interpreted as a non - local fermionic field theory in a constant magnetic background . then apart from strengthening the `` vulcanization '' procedure to get renormalizable non - commutative field theories , the gross - neveu model may also be useful for the study of the quantum hall effect . it is also a good first candidate for a constructive study @xcite of a non - commutative field theory as fermionic models are usually easier to construct . moreover its commutative counterpart being asymptotically free and exhibiting dynamical mass generation @xcite , a study of the physics of this model would be interesting . + the non - commutative gross - neveu model ( @xmath130 ) is a fermionic quartically interacting quantum field theory on the moyal plane @xmath131 . the skew - symmetric matrix @xmath40 is @xmath132 the action is @xmath133=&\int dx\lbt\psib\lbt-\imath\slashed{\partial}+\omega\xts+m+\mu\,\gamma_{5}\rbt\psi+v_{\text{o}}(\psib,\psi ) + v_{\text{no}}(\psib,\psi)\rbt(x)\end{aligned}\ ] ] where @xmath134 , @xmath135 and @xmath136 is the interaction part given hereafter . the @xmath137-term appears at two - loop order . we use a euclidean metric and the feynman convention @xmath138 . the @xmath139 and @xmath140 matrices form a two - dimensional representation of the clifford algebra @xmath141 . let us remark that the @xmath142 s are then skew- hermitian : @xmath143 . [ [ propagator ] ] propagator + + + + + + + + + + the propagator corresponding to the action is given by the following lemma : [ xpropa1gn ] the propagator of the gross - neveu model is @xmath144 with @xmath145 et @xmath146 . + we also have @xmath147 . if we want to study a @xmath148-_color _ model , we can consider a propagator diagonal in these color indices . [ [ interactions ] ] interactions + + + + + + + + + + + + concerning the interaction part @xmath149 , recall that ( see corollary [ cor : int - moyal ] ) for any @xmath150 , @xmath151 , @xmath152 , @xmath153 in @xmath45 , @xmath154 this product is non - local and only invraiant under cyclic permutations of the fields . then , contrary to the commutative gross - neveu model , for which there exits only one spinorial interaction , the @xmath130 model has , at least , six different interacitons : the _ orientable _ ones [ eq : int - orient ] @xmath155 where @xmath156 s and @xmath157 s alternate and the _ non - orientable _ ones [ eq : int - nonorient ] @xmath158 all these interactions have the same @xmath6 kernel thanks to the equation . the reason for which we call these interactions orientable or not will be clear in section [ sec : direct - space ] . the matrix basis is a basis for schwartz - class functions . in this basis , the moyal product becomes a simple matrix product . each field is then represented by an infinite matrix @xcite . in the matrix basis , the action takes the form : @xmath100=&(2\pi)^{d/2}\sqrt{\det\theta}\big(\frac 12\phi\delta\phi+\frac{\lambda}{4}\operatorname*{tr}\phi^{4}\big)\label { eq : sphi4matrix}\end{aligned}\ ] ] where @xmath159 and @xmath160 the ( four - dimensional ) matrix @xmath161 represents the quadratic part of the lagragian . the first difficulty to study the matrix model is the computation of its propagator @xmath30 defined as the inverse of @xmath161 : @xmath162 fortunately , the action is invariant under @xmath163 thanks to the form of the @xmath40 matrix . it implies a conservation law @xmath164 the result is @xcite @xmath165 where @xmath166 and @xmath115 is a function in @xmath8 : @xmath167 . the main advantage of the matrix basis is that it simplifies the interaction part : @xmath168 becomes @xmath169 . but the propagator becomes very compllicated . + let us remark that the matrix model is _ dynamical _ : its quadratic part is not trivial . usually , matrix models are _ local_. a matrix model is called * local * if @xmath170 and * non - local * otherwise . in the matrix theories , the feynman graphs are ribbon graphs . the propagator @xmath171 is then represented by the figure [ fig : propamatrix ] . in a local matrix model , the propagator preserves the index values along the trajectories ( simple lines ) . the power counting of a matrix model depends on the topological data of its graphs . the figure [ fig : ribbon - examples ] gives two examples of ribbon graphs . each ribbon graph may be drawn on a two - dimensional manifold . actually each graph defines the surface on which it is drawn . let a graph @xmath30 with @xmath149 vertices , @xmath43 internal propagators ( double lines ) and @xmath172 faces ( made of simple lines ) . the euler characteristic @xmath173 gives the genus @xmath174 of the manifold . one can make this clear by passing to the * dual graph*. the dual of a given graph @xmath30 is obtained by exchanging faces and vertices . the dual graphs of the @xmath175 theory are tesselations of the surfaces on which they are drawn . moreover each direct face broken by exernal legs becomes , in the dual graph , a * puncture*. if among the @xmath172 faces of a graph , @xmath176 are broken , this graph may be drawn on a surface of genus @xmath177 with @xmath176 punctures . the figure [ fig : topo - ruban ] gives the topological data of the graphs of the figure [ fig : ribbon - examples ] . @xmath178 @xmath179v=3@xmath180i=3@xmath180f=2@xmath180b=2@xmath181 + @xmath182 @xmath183v=2@xmath184i=3@xmath184f=1@xmath184b=1@xmath185 in @xcite , v. r. , r. wulkenhaar and f. v .- t . used the multi - scale analysis to reprove the power counting of the non - commutative @xmath116 theory . let @xmath30 a ribbon graph of the @xmath186 theory with @xmath148 external legs , @xmath149 vertices , @xmath43 internal lines and @xmath172 faces . its genus is then @xmath187 . four indices @xmath188 are associated to each internal line of the graph and two indices to each external line , that is to say @xmath189 indices . but , at each vertex , the left index of a ribbon equals the right one of the neighbour ribbon . this gives rise to @xmath190 independant identifications which allows to write each index in terms of a set @xmath191 made of @xmath190 indices , four per vertex , for example the left index of each half - ribbon . + the graph amplitude is then @xmath192 where the four indices of the propagator @xmath30 of the line @xmath97 are function of @xmath193 and written + @xmath194 . we decompose each propagator , given by : @xmath195 we have an associated decomposition for each amplitude @xmath196 where @xmath197 runs over the all possible assignements of a positive integer @xmath198 to each line @xmath97 . we proved the following four propositions : [ thm - th1 ] for @xmath35 large enough , there exists a constant @xmath199 such that , for @xmath200 $ ] , we have the uniform bound @xmath201 [ thm - th2 ] for @xmath35 large enough , there exists two constants @xmath199 and @xmath202 such that , for @xmath203 $ ] , we have the uniform bound @xmath204 this bound allows to prove that the only diverging graphs have either a constant index along the trajectories or a total jump of @xmath205 . [ prop : bound3 ] for @xmath35 large enough , there exists a constant @xmath199 such that , for @xmath206 $ ] , we have the uniform bound @xmath207 this bound shows that the propagator is almost local in the following sense : with @xmath208 fixed , the sum over @xmath209 does nt cost anything ( see figure [ fig : propamatrix ] ) . nevertheless the sums we ll have to perform are entangled ( a given index may enter different propagators ) so that we need the following proposition . [ prop : bound4 ] for @xmath35 large enough , there exists a constant @xmath199 such that , for @xmath206 $ ] , we have the uniform bound @xmath210 we refer to @xcite for the proofs of these four propositions . about half of the @xmath190 indices initially associated to a graph is determined by the external indices and the delta functions in ( [ ig ] ) . the other indices are summation indices . the power counting consists in finding which sums cost @xmath211 and which cost @xmath212 thanks to ( [ thsum ] ) . the @xmath211 factor comes from ( [ th1 ] ) after a summation over an index . ] @xmath213 , @xmath214 we first use the delta functions as much as possible to reduce the set @xmath193 to a true minimal set @xmath215 of independant indices . for this , it is convenient to use the dual graphs where the resolution of the delta functions is equivalent to a usual momentum routing . the dual graph is made of the same propagators than the direct graph except the position of their indices . whereas in the original graph we have @xmath216 { oberfig-16.eps}}$ ] , the position of the indices in a dual propagator is @xmath217{oberfig-17.eps}}\;. \label{dual - assign}\end{aligned}\ ] ] the conservation @xmath218 in ( [ ig ] ) implies that the difference @xmath219 is conserved along the propagator . these differences behave like an _ angular momentum _ and the conservation of the differences @xmath220 and @xmath221 is nothing else than the conservation of the angular momentum thanks to the symmetry @xmath222 of the action : @xmath223{oberfig-8.eps}}&\qquad l = m+\ell\;,~~ n = k+(-\ell ) . \label{angmom}\end{aligned}\ ] ] the cyclicity of the vertices implies the vanishing of the sum of the angular momenta entering a vertex . thus the angular momentum in the dual graph behaves exactly like the usual momentum in ordinary feynman graphs . + we know that the number of independent momenta is exactly the number @xmath224 ( @xmath225 for a connected graph ) of loops in the dual graph . each index at a ( dual ) vertex is then given by a unique _ reference index _ and a sum of momenta . if the dual vertex under consideration is an external one , we choose an external index for the reference index . the reference indices in the dual graph correspond to the loop indices in the direct graph . the number of summation indices is then @xmath226 where @xmath227 is the number of broken faces of the direct graph or the number of external vertices in the dual graph . + by using a well - chosen order on the lines , an optimized tree and a @xmath228 bound , one can prove that the summation over the angular momenta does not cost anything thanks to ( [ thsum ] ) . recall that a connected component is a subgraph for which all internal lines have indices greater than all its external ones . the power counting is then : @xmath229 and @xmath230 , @xmath231 , @xmath232 , @xmath233 and @xmath234 are respectively the numbers of external legs , of vertices , of ( internal ) propagators , of faces and broken faces of the connected component @xmath235 ; @xmath236 is its genus . we have [ pc - slice ] the sum over the scales attributions @xmath137 converges if @xmath237 . we recover the power counting obtained in @xcite . from this point on , renormalisability of @xmath7 can proceed ( however remark that it remains limited to @xmath200 $ ] by the technical estimates such as ( [ th1 ] ) ; this limitation is overcome in the direct space method below ) . the multiscale analysis allows to define the so - called effective expansion , in between the bare and the renormalized expansion , which is optimal , both for physical and for constructive purposes @xcite . in this effective expansion only the subcontributions with all _ internal _ scales higher than all _ external _ scales have to be renormalised by counterterms of the form of the initial lagrangian . in fact only planar such subcontributions with a single external face must be renormalised by such counterterms . this follows simply from the the grosse - wulkenhaar moves defined in @xcite . these moves translate the external legs along the outer border of the planar graph , up to irrelevant corrections , until they all merge together into a term of the proper moyal form , which is then absorbed in the effective constants definition . this requires only the estimates ( [ th1])-([thsummax ] ) , which were checked numerically in @xcite . in this way the relevant and marginal counterterms can be shown to be of the moyal type , namely renormalise the parameters @xmath238 , @xmath208 and @xmath8 term can be absorbed in a rescaling of the field , called `` field strength renromalization . '' ] . notice that in the multiscale analysis there is no need for the relatively complicated use of polchinski s equation @xcite made in @xcite . polchinski s method , although undoubtedly very elegant for proving perturbative renormalisability does not seem directly suited to constructive purposes , even in the case of simple fermionic models such as the commutative gross neveu model , see e.g. @xcite . the bphz theorem itself for the renormalised expansion follows from finiteness of the effective expansion by developing the counterterms still hidden in the effective couplings . its own finiteness can be checked e.g. through the standard classification of forests @xcite . let us however recall once again that in our opinion the effective expansion , not the renormalised one is the more fundamental object , both to describe the physics and to attack deeper mathematical problems , such as those of constructive theory @xcite . the matrix base simplfies very much at @xmath239 , where the matrix propagator becomes diagonal , i.e. conserves exactly indices . this property has been used for the general proof that the beta function of the theory vanishes in the ultraviolet regime @xcite , leading to the exciting perspective of a full non - perturbative construction of the model . we give here the results we get in @xcite . in this article , we computed the @xmath6-space and matrix basis kernels of operators which generalize the mehler kernel . then we proceeded to a study of the scaling behaviours of these kernels in the matrix basis . this work is useful to study the non - commutative gross - neveu model in the matrix basis . the following lemma generalizes the mehler kernel @xcite : [ hinxspace]let @xmath240 the operator : @xmath241 the @xmath6-space kernel of @xmath242 is : @xmath243 @xmath244 the mehler kernel corresponds to @xmath245 . the limit @xmath246 gives the usual heat kernel . let @xmath240 be given by with @xmath247 . its inverse in the matrix basis is : @xmath248 where @xmath166 and @xmath115 is a function of @xmath8 : @xmath167 . on the moyal space , we modified the commutative gross - neveu model by adding a @xmath249 term ( see lemma [ xpropa1gn ] ) . we have @xmath250 it will be useful to express @xmath30 in terms of commutators : @xmath251 where @xmath252 with @xmath145 and @xmath253 . + we now give the expression of the fermionic kernel in the matrix basis . the inverse of the quadratic form @xmath254 is given by in the preceeding section : @xmath255 the fermionic propagator @xmath30 ( [ xfullprop ] ) in the matrix basis may be deduced from the kernel . we just set @xmath256 , add the missing term with @xmath257 and compute the action of @xmath258 on @xmath259 . we must then evaluate @xmath260 in the matrix basis : @xmath261 this allows to prove : let @xmath262 the kernel , in the matrix basis , of the operator + @xmath263 . we have : @xmath264 where @xmath265 is given by ( [ eq : propinit - b ] ) and the commutators bu the formulas ( [ x0gamma ] ) and ( [ x1gamma ] ) . the first two terms in the equation ( [ eq : matrixfullprop ] ) contain commutators and will be gathered under the name @xmath266 . the last term will be called @xmath267 : @xmath268 we use the multi - scale analysis to study the behaviour of the propagator and revisit more finely the bounds to . in a slice @xmath269 , the propagator is @xmath270 @xmath271 let @xmath272 and @xmath273 . without loss of generality , we assume @xmath274 and @xmath275 . then the smallest index among @xmath276 is @xmath208 and the biggest is @xmath277 . we have : [ maintheorem ] under the assumptions @xmath278 and @xmath279 , there exists @xmath280 ( @xmath281 depends on @xmath8 ) such that the propagator of the non - commutative gross - neveu model in a slice @xmath269 obeys the bound @xmath282 the mass term is slightly different : @xmath283 we can redo the same analysis for the @xmath116 propagator and get @xmath284 which allows to recover the bounds to . let us consider the propagator of the non - commutative gross - neveu model . we saw in section [ sec : bornes ] that there exists two regions in the space of indices where the propagator behaves very differently . in one of them it behaves as the @xmath116 propagator and leads then to the same power counting . in the critical region , we have @xmath285 the point is that such a propagator does not allow to sum two reference indices with a unique line . this fact was useful in the proof of the power counting of the @xmath116 model . this leads to a _ uv / ir mixing . let us consider the graph in figure [ fig : sunsetj ] where the two external lines bear an index @xmath286 and the internal one an index @xmath287 . the propagator obeys the bound in prop . which means that it is almost local . we only have to sum over one index per internal face . on the graph of the figure [ fig : sunseti ] , if the two lines inside are true external ones , the graph has two broken faces and there is no index to sum over . then by using prop . we get @xmath288 . the sum over @xmath269 converges and we have the same behaviour as the @xmath116 theory , that is to say the graphs with @xmath289 broken faces are finite . but if these two lines belongs to a line of scale @xmath287 ( see figure [ fig : sunsetj ] ) , the result is different . indeed , at scale @xmath269 , we recover the graph of figure [ fig : sunseti ] . to maintain the previous result ( @xmath290 ) , we should sum the two indices corresponding to the internal faces with the propagator of scale @xmath291 . this is not possible . instead we have : @xmath292 the sum over @xmath269 diverges logarithmically . the graph of figure [ fig : sunseti ] converges if it is linked to true exernal legs et diverges if it is a subgraph of a graph at a lower scale . the power counting depends on the scales lower than the lowest scale of the graph . it ca nt then be factorized into the connected components : this is uv / ir mixing . + let s remark that the graph of figure [ fig : sunseti ] is not renormalisable by a counter - term in the lagrangian . its logarithmic divergence ca nt be absorbed in a redefinition of a coupling constant . fortunately the renormalisation of the two - point graph of figure [ fig : sunsetj ] makes the four - point subdivergence finite @xcite . this makes the non - commutative gross - neveu model renormalisable . we want now to explain how the power counting analysis can be performed in direct space , and the `` moyality '' of the necessary counterterms can be checked by a taylor expansion which is a generalization of the one used in direct commutative space . in the commutative case there is translation invariance , hence each propagator depends on a single difference variable which is short in the ultraviolet regime ; in the non - commutative case the propagator depends both of the difference of end positions , which is again short in the uv regime , but also of the sum which is long in the uv regime , considering the explicit form ( [ eq : mehler ] ) of the mehler kernel . let @xmath30 be an arbitrary connected graph . the amplitude associated with this graph is in direct space ( with hopefully self - explaining notations ) : @xmath293 \prod_l c_l \ ; , \nonumber \\ c_l= & \frac{\omega^2}{[2\pi\sinh(\omega t_l)]^2}e^{-\frac{\omega}{2}\coth(\omega t_l)(x_{v , i(l)}^{2}+x_{v',i'(l)}^{2 } ) + \frac{\omega}{\sinh(\omega t_l)}x_{v , i(l ) } . x_{v',i'(l ) } - \mu_0 ^ 2 t_l}\ ; .\nonumber \label{amplitude1}\end{aligned}\ ] ] with these notations , defining @xmath298 , the propagators in our graph can be written as : @xmath299 ^ 2 } e^{-\frac{\omega}{4}\coth(\frac{\alpha_l}{2 } ) { u_l^2}- \frac{\omega}{4}\tanh(\frac{\alpha_l}{2 } ) { v_l^2 } - \frac{\mu_0 ^ 2}{\omega } \alpha_l}\ ; .\label{tanhyp}\ ] ] @xmath301\nonumber\\ c^i ( u , v ) = & \int_{m^{-2i}}^{m^{-2(i-1 ) } } \frac{\omega d\alpha}{[2\pi\sinh(\alpha)]^2 } e^{-\frac{\omega}{4}\coth(\frac{\alpha}{2 } ) { u^2}- \frac{\omega}{4}\tanh(\frac{\alpha}{2 } ) { v^2 } - \frac{\mu_0 ^ 2}{\omega } \alpha}\ ; , \end{aligned}\ ] ] we pick a tree @xmath36 of lines of the graph , hence connecting all vertices , pick with a root vertex and build an _ orientation _ of all the lines of the graph in an inductive way . starting from an arbitrary orientation of a field at the root of the tree , we climb in the tree and at each vertex of the tree we impose cyclic order to alternate entering and exiting tree lines and loop half - lines , as in figure [ otree ] . then we look at the loop lines . if every loop lines consist in the contraction of an entering and an exiting line , the graph is called orientable . otherwise we call it non - orientable as in figure [ nono ] . there are @xmath306 @xmath97 functions in an amplitude with @xmath306 vertices , hence @xmath306 linear equations for the @xmath307 positions , one for each vertex . the _ position routing _ associated to the tree @xmath36 solves this system by passing to another equivalent system of @xmath306 linear equations , one for each branch of the tree . this is a triangular change of variables , of jacobian 1 . this equivalent system is obtained by summing the arguments of the @xmath97 functions of the vertices in each branch . this change of variables is exactly the @xmath6-space analog of the resolution of momentum conservation called _ momentum routing _ in the standard physics literature of commutative field theory , except that one should now take care of the additional @xmath308 cyclic signs . we have , calling @xmath310 the remaining integrand in ( [ amplitude2 ] ) : @xmath311 \ , \big]\ ; i_g(\{x_{v , i } \ } ) \\ = & \int \prod_{b } \delta \left ( \sum_{l\in t_b \cup l_b } u_{l } + \sum_{l\in l_{b,+}}v_{l}-\sum_{l\in l_{b,-}}v_{l } + \sum_{f\in x_b}\epsilon(f ) x_f \right ) i_g(\{x_{v , i } \ } ) , \nonumber \end{aligned}\ ] ] where @xmath312 is @xmath313 depending on whether the field @xmath314 enters or exits the branch . we can now use the system of delta functions to eliminate variables . it is of course better to eliminate long variables as their integration costs a factor @xmath315 whereas the integration of a short variable brings @xmath316 . rough power counting , neglecting all oscillations of the vertices leads therefore , in the case of an orientable graph with @xmath148 external fields , @xmath306 internal vertices and @xmath317 internal lines at scale @xmath269 to : * a factor @xmath318 coming from the @xmath211 factors for each line of scale @xmath269 in ( [ eq : propbound - phi4 ] ) , * a factor @xmath319 for the @xmath320 short variables integrations , * a factor @xmath321 for the long variables after eliminating @xmath309 of them using the delta functions . in the non - orientable case , we can eliminate one additional long variable since the rank of the system of delta functions is larger by one unit ! therefore we get a power counting bound @xmath324 , which proves that only _ orientable _ graphs may diverge . in fact we of course know that not all _ orientable _ two and four point subgraphs diverge but only the planar ones with a single external face . ( it is easy to check that all such planar graphs are indeed orientable ) . since only these planar subgraphs with a single external face can be renormalised by moyal counterterms , we need to prove that orientable , non - planar graphs or orientable planar graphs with several external faces have in fact a better power counting than this crude estimate . this can be done only by exploiting their vertices oscillations . we explain now how to do this with minimal effort . following filk @xcite , we can contract all lines of a spanning tree @xmath36 and reduce @xmath30 to a single vertex with `` tadpole loops '' called a `` rosette graph '' . this rosette is a cycle ( which is the border of the former tree ) bearing loops lines on it ( see figure [ fig : rosette ] ) : remark that the rosette can also be considered as a big vertex , with @xmath325 fields , on which @xmath148 are external fields with external variables @xmath6 and @xmath326 are loop fields for the corresponding @xmath327 loops . when the graph is orientable , the rosette is also orientable , which means that turning around the rosette the lines alternatively enter and exit . these lines correspond to the contraction of the fields on the border of the tree @xmath36 before the filk contraction , also called the `` first filk move '' . the rosette contribution after a complete first filk reduction is exactly : @xmath330 where the @xmath331 variables are the long or external variables of the rosette , counted with their signs , and the quadratic oscillations for these variables is @xmath332 we have now to analyze in detail this quadratic oscillation of the remaining long loop variables since it is essential to improve power counting . we can neglect the secondary oscillations @xmath333 and @xmath334 which imply short variables . the second filk reduction @xcite further simplifies the rosette factor by erasing the loops of the rosette which do not cross any other loops or arch over external fields . it can be shown that the loops which disappear in this operation correspond to those long variables who do not appear in the quadratic form @xmath335 . the basic mechanism to improve the power counting of a single non - planar subgraph is the following : @xmath336 in these equations we used for simplicity @xmath290 instead of the correct but more complicated factor @xmath337 ( of course this does not change the argument ) and we performed a unitary linear change of variables @xmath338 , @xmath339 to compute the oscillating @xmath340 integral . the gain in ( [ gainoscill ] ) is @xmath341 , which is the difference between @xmath342 and the normal factor @xmath343 that the @xmath344 integral would have cost if we had done it with the regular @xmath345 factor for long variables . to maximize this gain we can assume @xmath346 . finally it remains to consider the case of subgraphs which are planar orientable but with more than one external face . in that case there are no crossing loops in the rosette but there must be at least one loop line arching over a non trivial subset of external legs ( see e.g. line @xmath347 in figure [ fig : rosette ] ) . we have then a non trivial integration over at least one external variable , called @xmath6 , of at least one long loop variable called @xmath348 . this `` external '' @xmath6 variable without the oscillation improvement would be integrated with a test function of scale 1 ( if it is a true external line of scale @xmath349 ) or better ( if it is a higher long loop variable ) can not be the full combination of external variables in the `` root '' @xmath97 function . ] . but we get now @xmath350 so that a factor @xmath315 in the former bound becomes @xmath351 hence is improved by @xmath316 . in this way we can reduce the convergence of the multiscale analysis to the problem of renormalisation of planar two- and four - point subgraphs with a single external face , which we treat in the next section . remark that the power counting obtained in this way is still not optimal . to get the same level of precision than with the matrix base requires e.g. to display @xmath174 independent improvements of the type ( [ gainoscill ] ) for a graph of genus @xmath174 . this is doable but basically requires a reduction of the quadratic form @xmath335 for single - faced rosette ( also called `` hyperrosette '' ) into @xmath174 standard symplectic blocks through the so - called `` third filk move '' introduced in @xcite . we return to this question in section [ hyperbo ] . the idea is that one should compare its amplitude to a similar amplitude with a `` moyal factor '' @xmath352 factorized in front , where @xmath353 . but precisely because the graph is planar with a single external face we understand that the external positions @xmath6 only couple to _ short variables _ @xmath354 of the internal amplitudes through the global delta function and the oscillations . hence we can break this coupling by a systematic taylor expansion to first order . this separates a piece proportional to `` moyal factor '' , then absorbed into the effective coupling constant , and a remainder which has at least one additional small factor which gives him improved power counting . this is done by expressing the amplitude for a graph with @xmath355 , @xmath356 and @xmath357 as : @xmath358 \ e^{\imath uru+\imath usv } \\ & \hspace{-2 cm } \bigg\ { { \delta(\delta ) } + \int_{0}^{1}dt\bigg [ \mathfrak{u}\cdot \nabla \delta(\delta+t\mathfrak{u } ) + \delta(\delta+t\mathfrak{u } ) [ \imath xqu + { \mathfrak r } ' ( t ) ] \bigg ] e^{\imath txqu + { \mathfrak r}(t ) } \bigg\ } \ .\nonumber\end{aligned}\ ] ] where @xmath359 is the propagator taken at @xmath360 , @xmath361 and @xmath362 is a correcting term involving @xmath363 $ ] . following the same strategy we have to taylor - expand the coupling between external variables and @xmath354 factors in two point planar graphs with a single external face to _ third order _ and some non - trivial symmetrization of the terms acording to the two external arguments to cancel some odd contributions . the corresponding factorized relevant and marginal contributions can be then shown to give rise only to again the bphz theorem itself for the renormalised expansion follows by developing the counterterms still hidden in the effective couplings and its finiteness follows from the standard classification of forests . see however the remarks at the end of section [ sec : vari - indep ] . since the bound ( [ eq : propbound - phi4 ] ) works for any @xmath366 , an additional bonus of the @xmath6-space method is that it proves renormalisability of the model for any @xmath8 in @xmath3670,1]$ ] in @xmath368 is irrelevant since it can be rewritten by ls duality as an equivalent model with @xmath8 in @xmath3670,1]$ ] . ] , whether the matrix method proved it only for @xmath8 in @xmath3670.5,1]$ ] . the interaction @xmath371 ensures that perturbation theory contains only orientable graphs . for @xmath372 the @xmath6-space propagator still decays as in the ordinary @xmath7 case and the model has been shown renormalisable by an easy extension of the methods of the previous section @xcite . consider the @xmath6-kernel of the operator @xmath374 the gross - neveu model or the critical langmann - szabo - zarembo models correspond to the case @xmath375 . in these models there is no longer any confining decay for the `` long variables '' but only an oscillation : @xmath376 this kind of models are called critical . their construction is more difficult , since sufficiently many oscillations must be proven independent before power counting can be established . the prototype paper which solved this problem is @xcite , which we briefly summarize now . the main technical difficulty of the critical models is the absence of decreasing functions for the long @xmath331 variables in the propagator replaced by an oscillation , see ( [ eq : criticalpropa ] ) . note that these decreasing functions are in principle created by integration over the @xmath377 variables . ] : @xmath378 but to perform all these gaussian integrations for a general graph is a difficult task ( see @xcite ) and is in fact not necessary for a bphz theorem . we can instead exploit the vertices and propagators oscillations to get rationnal decreasing functions in some linear combinations of the long @xmath331 variables . the difficulty is then to prove that all these linear combinations are independant and hence allow to integrate over all the @xmath331 variables . to solve this problem we need the exact expression of the total oscillation in terms of the short and long variables . this consists in a generalization of the filk s work @xcite . this has been done in @xcite . once the oscillations are proven independant , one can just use the same arguments than in the @xmath116 case ( see section [ sec : routing - filk - moves ] ) to compute an upper bound for the power counting : [ lem : compt - puissgn ] let @xmath30 a connected orientable graph . for all @xmath379 , there exists @xmath380 such that its amputated amplitude @xmath381 integrated over test functions is bounded by @xmath382 as in the non - commutative @xmath116 case , only the planar graphs are divergent . but the behaviour of the graphs with more than one broken face is different . note that we already discussed such a feature in the matrix basis ( see section [ sec : prop - et - renorm ] ) . in the multiscale framework , the feynamn diagrams are endowed with a scale attribution which gives each line a scale index . the only subgraphs we meet in this setting have all their internal scales higher than their external ones . then a subgraph @xmath383 of scale @xmath269 is called _ critical _ if it has @xmath384 and that the two `` external '' points in the second broken face are only linked by a single line of scale @xmath287 . the typical example is the graph of figure [ fig : sunseti ] . in this case , the subgrah is logarithmically divergent whereas it is convergent in the @xmath116 model . let us now show roughly how it happens in the case of figure [ fig : sunseti ] but now in @xmath6-space . the same arguments than in the @xmath116 model prove that the integrations over the internal points of the graph [ fig : sunseti ] lead to a logarithmical divergence which means that @xmath385 in the multiscale framework . but remind that there is a remaining oscillation between a long variable of this graph and the external points in the second broken face of the form @xmath386 . but @xmath331 is of order @xmath387 which leads to a decreasing function implementing @xmath388 of order @xmath389 . if these points are true external ones , they are integrated over test functions of norm @xmath349 . then thanks to the additional decreasing function for @xmath388 we gain a factor @xmath290 which makes the graph convergent . but if @xmath6 and @xmath390 are linked by a single line of scale @xmath287 ( as in figure [ fig : sunsetj ] ) , instead of test functions we have a propagator between @xmath6 and @xmath390 . this one behaves like ( see ): @xmath391 the integration over @xmath388 instead of giving @xmath392 gives @xmath290 thanks to the oscillation @xmath393 . then we have gained a good factor @xmath394 . but the oscillation in the propagator @xmath395 now gives @xmath396 instead of @xmath397 and the integration over @xmath398 cancels the preceeding gain . the critical component of figure [ fig : sunseti ] is logarithmically divergent . this kind of argument can be repeated and refined for more general graphs to prove that this problem appears only when the extrernal points of the auxiliary broken faces are linked only by a _ single _ lower line @xcite . this phenomenon can be seen as a mixing between scales . indeed the power counting of a given subgraph now depends on the graphs at lower scales . this was not the case in the commutative realm . fortunately this mixing does nt prevent renormalisation . note that whereas the critical subgraphs are not renormalisable by a vertex - like counterterm , they are regularised by the renormalisation of the two - point function at scale @xmath291 . the proof of this point relies heavily on the fact that there is only one line of lower scale . let us conclude this section by mentionning the flows of the critical models . one very interesting feature of the non - commutative @xmath116 model is the boundedness of its flows and even the vanishing of its beta function for a special value of its bare parameters @xcite . note that its commutative counterpart ( the usual @xmath399 model on @xmath400 ) is asymptotically free in the infrared and has then an unbounded flow . it turns out that the flow of the critical models are not regularized by the non - commutativity . the one - loop computation of the beta functions of the non - commutative gross - neveu model @xcite shows that it is asymptotically free in the ultraviolet region as in the commutative case . in ordinary commutative field theory , symanzik s polynomials are obtained after integration over internal position variables . the amplitude of an amputated graph @xmath30 with external momenta @xmath107 is , up to a normalization , in space - time dimension @xmath42 : @xmath401 the first and second symanzik polynomials @xmath402 and @xmath403 are where the first sum is over spanning trees @xmath36 of @xmath30 and the second sum is over two trees @xmath405 , i.e. forests separating the graph in exactly two connected components @xmath406 and @xmath407 ; the corresponding euclidean invariant @xmath408 is , by momentum conservation , also equal to @xmath409 . since the mehler kernel is still quadratic in position space it is possible to also integrate explicitly all positions to reduce feynman amplitudes of e.g. non - commutative @xmath7 purely to parametric formulas , but of course the analogs of symanzik polynomials are now hyperbolic polynomials which encode the richer information about ribbon graphs . the reference for these polynomials is @xcite , which treats the ordinary @xmath7 case . in @xcite , these polynomials are also computed in the more complicated case of critical models . defining the antisymmetric matrix @xmath410 as @xmath411 the @xmath412functions appearing in the vertex contribution can be rewritten as an integral over some new variables @xmath413 . we refer to these variables as to _ hypermomenta_. note that one associates such a hypermomenta @xmath413 to any vertex @xmath149 _ via _ the relation @xmath414 consider a particular ribbon graph @xmath30 . specializing to dimension 4 and choosing a particular root vertex @xmath415 of the graph , one can write the feynman amplitude for @xmath30 in the condensed way @xmath416^{2 } d\alpha_{\ell } \int d x d p e^{-\frac { \omega}{2 } x g x^t}\end{aligned}\ ] ] where @xmath417 , @xmath418 summarizes all positions and hyermomenta and @xmath30 is a certain quadratic form . if we call @xmath419 and @xmath420 the external variables we can decompose @xmath30 according to an internal quadratic form @xmath335 , an external one @xmath35 and a coupling part @xmath421 so that @xmath422 performing the gaussian integration over all internal variables one obtains : @xmath423^{2 } d\alpha\frac{1}{\sqrt{\det q } } e^{-\frac{\ot}{2 } \begin{pmatrix } x_e & \bar{p } \\ \end{pmatrix } [ m - p q^{-1}p^{t } ] \begin{pmatrix } x_e \\ \bar{p } \\ \end{pmatrix}}\ .\end{aligned}\ ] ] this form allows to define the polynomials @xmath424 and @xmath425 , analogs of the symanzik polynomials @xmath354 and @xmath149 of the commutative case ( see ) . they are defined by @xmath426 hu_{g , \bar{v } } ( t ) ^{-2 } e^{- \frac { hv_{g , \bar{v } } ( t , x_e , p_{\bar v})}{hu_{g , \bar{v } } ( t ) } } .\end{aligned}\ ] ] they are polynomials in the set of variables @xmath427 ( @xmath428 ) , the hyperbolic tangent of the half - angle of the parameters @xmath429 . * the polynomials @xmath424 and @xmath425 have a strong positivity property . roughly speaking they are sums of monomials with positive integer coefficients . this positive integer property comes from the fact that each such coefficient is the square of a pfaffian with integer entries , * leading terms can be identified in a given `` hepp sector '' , at least for _ orientable graphs_. a hepp sector is a complete ordering of the @xmath432 parameters . these leading terms which can be shown strictly positive in @xmath424 correspond to super - trees which are the disjoint union of a tree in the direct graph and a tree in the dual graph . hypertrees in a graph with @xmath306 vertices and @xmath172 faces have therefore @xmath433 lines . ( any connected graph has hypertrees , and under reduction of the hypertree , the graph becomes a hyperrosette ) . similarly one can identify `` super - two - trees '' @xmath425 which govern the leading behavior of @xmath425 in any hepp sector . for the bubble graph of figure [ figex1 ] : @xmath435}^2+t_1t_2\big{[}2p_2 ^ 2+(1 + 16s^4)(x_1-x_4)^2 \big{]}\,,\nonumber\\ & + t_1 ^ 2\big{[}p_2 + 2s(x_1-x_4)\big{]}^2\nonumber\\\end{aligned}\ ] ] @xmath437}+t_1 ^ 2\big{[}p_2 + 2s(x_3-y_4 ) \big{]}^2\,,\nonumber\\ & + t_1t_2\big{[}8s^2y_2 ^ 2 + 2(p_2 - 2sy_4)^2+(x_1+x_3)^2 + 16s^4(x_1-x_3)^2\big{]}\nonumber\\ & + t_1 ^ 2t_2 ^ 24s^2(x_1-y_2)^2\,,\nonumber\end{aligned}\ ] ] finally , for the half - eye graph ( see fig . [ figeye ] ) , we start by defining : @xmath440 the @xmath438 polynomial with fixed hypermomentum corresponding to the vertex with two external legs is : @xmath441 whereas with another fixed hypermomentum we get : @xmath442 note that the leading terms are identical and the choice of the root perturbs only the non - leading ones . moreover note the presence of the @xmath443 term . its presence can be understood by the fact that in the sector @xmath444 the subgraph formed by the lines @xmath445 has two broken faces . this is the sign of a power counting improvement due to the additional broken face in that sector . to exploit it , we have just to integrate over the variables of line @xmath446 in that sector , using the second polynomial @xmath447 for the triangle subgraph @xmath448 made of lines @xmath445 . non - commutative qft seemed initially to have non - renormalisable divergencies , due to uv / ir mixing . but following the grosse - wulkenhaar breakthrough , there has been recent rapid progress in our understanding of renormalisable qft on moyal spaces . we can already propose a preliminary classification of these models into different categories , according to the behavior of their propagators : * ordinary models at @xmath449 such as @xmath7 ( which has non - orientable graphs ) or @xmath450 models ( which has none ) . their propagator , roughly @xmath451 is ls covariant and has good decay both in matrix space ( [ th1]-[thsummax ] ) and direct space ( [ tanhyp ] ) . they have non - logarithmic mass divergencies and definitely require `` vulcanization '' i.e. the @xmath8 term . * `` supermodels '' , namely ordinary models but at @xmath452 in which the propagator is ls invariant . their propagator is even better . in the matrix base it is diagonal , e.g. of the form @xmath453 , where @xmath454 is a constant . the supermodels seem generically ultraviolet fixed points of the ordinary models , at which non - trivial ward identities force the vanishing of the beta function . the flow of @xmath8 to the @xmath452 fixed point is very fast ( exponentially fast in rg steps ) . * `` critical models '' such as orientable versions of lsz or gross - neveu ( and presumably orientable gauge theories of various kind : yang - mills , chern - simons ... ) . they may have only logarithmic divergencies and apparently no perturbative uv / ir mixing . however the vulcanized version still appears the most generic framework for their treatment . the propagator is then roughly @xmath455 . in matrix space this propagator shows definitely a weaker decay ( [ mainbound1 ] ) than for the ordinary models , because of the presence of a non - trivial saddle point . in direct space the propagator no longer decays with respect to the long variables , but only oscillates . nevertheless the main lesson is that in matrix space the weaker decay can still be used ; and in @xmath6 space the oscillations can never be completely killed by the vertices oscillations . hence these models retain therefore essentially the power counting of the ordinary models , up to some nasty details concerning the four - point subgraphs with two external faces . ultimately , thanks to a little conspiration in which the four - point subgraphs with two external faces are renormalised by the mass renormalisation , the critical models remain renormalisable . this is the main message of @xcite . * `` hypercritical models '' which are of the previous type but at @xmath452 . their propagator in the matrix base is diagonal and depends only on one index @xmath208 ( e.g. always the left side of the ribbon ) . it is of the form @xmath456 . in @xmath6 space the propagator oscillates in a way that often exactly compensates the vertices oscillations . these models have definitely worse power counting than in the ordinary case , with e.g. quadratically divergent four point - graphs ( if sharp cut - offs are used ) . nevertheless ward identities can presumably still be used to show that they can still be renormalised . this probably requires a much larger conspiration to generalize the ward identities of the supermodels . parametric representation can be derived in the non - commutative case . it implies hyper - analogs of symanzik polynomials which condense the information about the rich topological structure of a ribbon graph . using this representation , dimensional regularization and dimensional renormalisation should extend to the non - commutative framework . remark that trees , which are the building blocks of the symanzik polynomials , are also at the heart of ( commutative ) constructive theory , whose philosophy could be roughly summarized as `` you shall use trees , but you shall _ not _ develop their loops or else you shall diverge '' . it is quite natural to conjecture that hypertrees , which are the natural non - commutative objects intrinsic to a ribbon graph , should play a key combinatoric role in the yet to develop non - 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we study surfaces in space @xmath0 such that through each point of the surface one can draw two circles fully contained in the surface . hereafter by a _ circle _ we mean either an ordinary circle in @xmath0 or a straight line . in this paper we reduce finding all such surfaces to the algebraic problem of finding all pythagorean @xmath1-tuples of polynomials . in a subsequent publication we are going to solve the latter problem . the problem of finding such surfaces traces back to the works of darboux from xixth century . basic examples a one - sheeted hyperboloid and a nonrotational ellipsoid are discussed in hilbert cohn - vossen s `` anschauliche geometrie '' . there ( and respectively , in a recent paper @xcite by nilov and the first author ) it is also proved that a smooth surface containing two lines ( respectively , a line and a circle ) through each point is a quadric or a plane . a torus contains @xmath2 circles through each point : a meridian , a parallel , and two villarceau circles . [ cols="^ " , ] all these examples are particular cases of a _ darboux cyclide _ , i.e. , a subset of @xmath0 given by the equation @xmath3 where @xmath4 and @xmath5 $ ] of degree @xmath6 do not vanish simultaneously ; see figure [ movie ] to the left . equivalently , a _ darboux cyclide _ is the stereographic projection of the intersection of the sphere @xmath7 with another @xmath8-dimensional quadric ( * ? ? ? * section 2.2 ) . almost each darboux cyclide contains at least @xmath9 circles through each point ( and there is an effective algorithm to count their actual number @xcite ) . conversely , darboux has shown that @xmath10 circles through each point guarantee that an analytic surface is a darboux cyclide . this result has been improved over the years : in fact already @xmath8 , or @xmath9 orthogonal , or @xmath9 cospheric circles are sufficient for the same conclusion ( * ? ? ? * theorem 3 ) , ( * ? ? ? * theorem 1 ) , ( * ? ? ? * theorem 20 in p. 296 ) , cf . hereafter two circles are called _ cospheric _ , if they are contained in one @xmath9-dimensional sphere or plane . recently there has been a renewed interest to surfaces containing @xmath9 circles through each point due to pottmann who considered their potential applications to architecture @xcite . pottmann noticed that the _ euclidean translational surface _ @xmath11 , where @xmath12 are two fixed generic circles in @xmath0 , contains @xmath9 circles through each point but is not a darboux cyclide ( * ? ? ? * example 3.9 ) . another example with similar properties was given by zub in 2011 : the stereographic projection of a _ clifford translational surface _ @xmath13 , where @xmath12 are now circles in the sphere @xmath14 identified with the set of unit quaternions . the projection itself is called a _ clifford translational surface _ as well ; see figure [ movie ] to the right . it may have degree up to @xmath15 . each degree @xmath15 surface in @xmath7 containing a _ great _ circle and another circle through each point is clifford translational ( * ? ? ? * corollary 2c ) . more examples can be obtained from these translational surfaces by _ mbius transformations _ , i.e. , compositions of inversions . euclidean and clifford translational surfaces are not mbius transformations of each other ( * ? ? ? * theorem 2b ) . for related transformations taking lines to circles see @xcite . we conjecture that the above ones are the only possible surfaces containing @xmath9 circles through each point . let us make this statement precise . we switch to a local problem involving a piece of a surface instead of a closed one and circular arcs instead of circles . by an _ analytic surface _ in @xmath16 we mean the image of an injective real analytic map of a planar domain into @xmath16 with nondegenerate differential at each point . we use the same notation for the map and the surface ; no confusion arises from this . a circle ( or circular arc ) _ analytically depending _ on a point is a real analytic map of an analytic surface into the variety of all circles ( or circular arcs ) in @xmath16 . analyticity is not really a restriction @xcite . [ mainthm ] if through each point of an analytic surface in @xmath0 one can draw two transversal circular arcs fully contained in the surface ( and analytically depending on the point ) then the surface is a mbius transformation of a subset of either a darboux cyclide , or euclidean or clifford translational surface . we hope to deduce main conjecture [ mainthm ] from the following 4-dimensional counterpart . the @xmath2-dimensional problem seems to be more accessible than the @xmath8-dimensional one because of nice approach using quaternions . in what follows identify @xmath17 with the skew field @xmath18 of quaternions , and @xmath0 with the set @xmath19 of purely imaginary quaternions . mbius transformations in @xmath17 are precisely the nondegenerate maps of the form @xmath20 and @xmath21 , where @xmath22 ; see @xcite for an exposition . circles in @xmath17 are precisely the nondegenerate curves having a parametrization of the form @xmath23 ( outside one point ) , where @xmath22 are fixed and @xmath24 runs . denote by @xmath25 $ ] the set of polynomials with quaternionic coefficients of degree at most @xmath26 in the variable @xmath27 and at most @xmath28 in the variable @xmath29 ( the variables commute with each other and the coefficients ) . denote @xmath30 . define @xmath31 , @xmath32 , and @xmath33 analogously . for each @xmath34 and real numbers @xmath35 ( but not quaternions ) the value @xmath36 is well - defined . thus the polynomial @xmath37 or a rational expression in such polynomials defines a surface in @xmath17 . [ 4dbconj ] assume that through each point of an analytic surface in @xmath17 one can draw two noncospheric circular arcs fully contained in the surface ( and analytically depending on the point ) . assume that for each point in some dense subset of the surface the number of circular arcs passing through the point and fully contained in the surface is finite . then some mbius transformation of the surface has a parametrization @xmath38 for some polynomials @xmath39 such that @xmath40 . conversely , almost each surface contains two circular arcs @xmath41 and @xmath42 through each point because the curves @xmath23 and @xmath43 are circular arcs for almost each @xmath22 . e.g. , @xmath44 is a clifford translational surface , and @xmath45 is a torus . the first result of this paper is the following assertions reducing the @xmath8-dimensional problem to the @xmath2-dimensional one . they are proved in section [ sec : proofs ] . [ prop21][prop:21-split ] if surface is contained in @xmath0 ( respectively , in @xmath7 ) then it is a subset of either an euclidean ( respectively , clifford ) translational surface or a darboux cyclide ( respectively , an intersection of @xmath7 with another @xmath8-dimensional quadric ) . [ cor-3d ] if a surface in @xmath0 is a mbius transformation of surface in @xmath17 then the former surface is a mbius transformation of a subset of either a darboux cyclide , or euclidean or clifford translational surface . surfaces containing two circles through each point are particular cases of surfaces containing two conic sections or lines through each point . the latter surfaces have been classified by brauner , schicho , and lubbes @xcite . in the particular case of so - called supercyclides a classification was given by degen @xcite . the schicho classification is up to a week equivalence relation , thus it does not allow automatically to find all surfaces containing two circles through each point . however the following parametrization result by schicho provides the first step toward the solution of our problem . the notions used in the statement are defined analogously to the above ; see section [ sec : proofs2 ] for details . [ schicho ] assume that through each point of an analytic surface in a domain in @xmath28-dimensional complex projective space one can draw two transversal conic sections intersecting each other only at this point ( and analytically depending on the point ) such that their intersections with the domain are contained in the surface . assume that through each point in some dense subset of the surface one can draw only finitely many conic sections such that their intersections with the domain are contained in the surface . then the surface ( possibly besides a one - dimensional subset ) has a parametrization @xmath46 for some @xmath47 such that the conic sections are the curves @xmath41 and @xmath42 . conversely , one can see immediately that almost each surface contains two conic sections @xmath41 and @xmath42 through each point . theorem [ schicho ] in particular implies that each surface containing two conic sections through each point is contained in an projective subspace of dimension at most @xmath15 and has the degree at most @xmath15 ( by standard elimination of variables ) . theorem [ schicho ] is proved completely analogously to ( * ? ? ? * theorem 11 ) , where the case when @xmath48 is considered ; see also ( * ? ? ? * theorem 17 ) . for convenience of the reader we give the proof in section [ sec : proofs2 ] . in comparison to @xcite we have added some details to make it accessible to nonspecialists . the main result of the paper is the following corollary also proved in section [ sec : proofs2 ] . [ haupt][param ] assume that through each point of an analytic surface in @xmath49 ( respectively , in @xmath16 ) one can draw two noncospheric circular arcs fully contained in the surface ( and analytically depending on the point ) . assume that through each point in some dense subset of the surface one can draw only finitely many circular arcs fully contained in the surface . then the surface ( possibly besides a one - dimensional subset ) has a parametrization @xmath50 where @xmath51 satisfy the equation @xmath52 ( respectively , the equation @xmath53 for some @xmath54 ) . this result reduces the above conjectures to solving the equation @xmath55 for `` pythagorean @xmath28-tuples '' in real polynomials . the resulting problem is hard but seems more accessible because of possible induction over the involved parameters ( @xmath28 , number of variables , degree ) . the problem has been solved for @xmath48 and @xmath2 in ( * ? ? ? * theorem 2.2 ) using that @xmath56 $ ] is a unique factorization domain ( ufd ) . in case of one variable a similar argument works for @xmath57 and @xmath1 because @xmath58 $ ] is still a ufd in a sense ( * ? ? ? * theorem 1 in chapter 2 ) , cf . @xcite , @xcite and conjecture [ conj-1-var ] below . the main difficulty of passing to two variables is that @xmath59 $ ] is _ not _ a ufd any more ; see example [ ex - beauregard ] below taken from @xcite . the case of two variables and @xmath60 arising in our geometric problem seems to be the simplest case not accessible by the methods available before . so far we give only one result in this direction , which is nice and interesting in itself , useful for the proof of the above conjectures and also for theorem [ prop21 ] above . the result is proved in section [ sec : final ] . [ l - splitting - basic ] if @xmath61 for some @xmath62 and @xmath63 , @xmath64 of degree @xmath9 then @xmath65 is reducible . [ cor - splitting ] if @xmath66 for some @xmath67 , @xmath68 such that @xmath69 and @xmath70 have degree @xmath9 then for some @xmath71 and @xmath72 we have @xmath73 to summarize , the results of this paper reduce main conjecture [ mainthm ] to the following completely algebraic conjecture . by a _ mbius transformation _ of @xmath74 we mean a linear transformation @xmath75 which preserves the homogeneous equation @xmath76 of @xmath74 . [ conj - alg ] polynomials @xmath77 satisfy equation @xmath66 if and only if up to mbius transformation of @xmath74 not depending on @xmath78 we have @xmath79 for some @xmath39 , @xmath80 such that @xmath81 . the parametrizations of conjectures [ conj - alg ] and [ 4dbconj ] are related to each other via the stereographic projection @xmath82 . we plan to prove all the above conjectures in a subsequent publication . this is reasonable because the remaining algebraic part of the conjectures is proved by very different methods . in this section we prove the results on the classification of surfaces parametrized by quaternionic rational functions of small degree up to mbius transformation : theorem [ prop21 ] and corollary [ cor-3d ] . let us prove several lemmas required for the proof of theorem [ prop21 ] . these lemmas are independent in the sense that the proof of each one uses the statements but not the proofs of the other ones . in what follows @xmath83 are arbitrary polynomials not vanishing identically . linear homogeneous polynomials @xmath84 $ ] are defined by the formulae @xmath85 and @xmath86 . by a _ possibly degenerate surface _ we mean an analytic map of a planar domain into @xmath16 not necessarily with nondegenerate differential , and also the image of the map , if no confusion arises . a _ possibly degenerate hypersurface _ is defined analogously . first we consider the ( possibly degenerate ) surface @xmath87 being a particular case of surface . we are going to estimate its degree . for that we estimate the degree of the ( possibly degenerate ) hypersurface @xmath88 it has the rational parametrization @xmath89 of degree at most @xmath9 in each variable , and by elimination of variables ( not used in the paper ) it has degree at most @xmath15 . we prove a much sharper estimate . [ l-3folddeg4 ] the ( possibly degenerate ) hypersurface @xmath90 is contained in an algebraic hypersurface of degree at most @xmath2 . if a point @xmath91 is contained in the hypersurface @xmath90 then @xmath92 for some @xmath93 not vanishing simultaneously . the latter quaternionic equation can be considered as a system of @xmath2 real linear homogeneous equations in the variables @xmath94 with the coefficients linearly depending on the parameters @xmath95 . the system has a nonzero solution if and only if the determinant of the system vanishes , which gives an algebraic equation in @xmath95 of degree at most @xmath2 . the algebraic equation defines the required algebraic hypersurface unless the determinant vanishes identically . assume that the determinant vanishes identically . then the system has a nonzero solution for each point @xmath96 . but the set of values of the fraction @xmath90 is at most three - dimensional . one can have a nonzero solution for each point in @xmath97 only if @xmath98 for some nonzero @xmath99 . assume that @xmath100 without loss of generality . then by the linearity @xmath101 performing a linear change of the parameters @xmath94 we may assume that both @xmath102 and @xmath103 do not depend on @xmath104 . further denote by @xmath105 and @xmath106 the resulting linear polynomials , defining the same hyperfurface @xmath90 as the initial ones . then the above equation takes the form @xmath107 . consider the equation as a system of @xmath2 real linear homogeneous equations in the variables @xmath108 . the system has a nonzero solution if and only if all the @xmath109 minors of the system vanish , which gives four algebraic equations in @xmath95 of degree at most @xmath8 . if at least one of the @xmath109 minors does not vanish identically then it defines the required algebraic surface . if all the @xmath109 minors vanish identically then repeat the argument of the previous paragraph to get linear homogeneous @xmath110 and @xmath111 depending only on @xmath9 variables again , we either get the required algebraic surface or proceed to the case when @xmath112 and @xmath113 depend only on @xmath114 variable . in the latter case any hyperplane passing through the point @xmath115 is the required algebraic hypersurface . we consider the case when the constructed algebraic hypersurface degenerates to @xmath19 separately . [ l-3fold - r3 ] assume that the ( possibly degenerate ) hypersurface @xmath90 is contained in the hyperplane @xmath19 . then the map @xmath90 is a composition of a map of the form either @xmath116 or @xmath117 with a mbius transformation of @xmath19 ( with constant coefficients ) , where @xmath118 $ ] , @xmath119 $ ] , @xmath120 $ ] are linear homogeneous , and @xmath121 . the inverse stereographic projection @xmath122 maps the hypersurface @xmath90 contained in @xmath123 to the hypersurface @xmath124 contained in @xmath7 . thus @xmath125 identically . in particular , for each @xmath126 the condition @xmath127 implies the condition @xmath128 . denote @xmath129 and @xmath130 . define a real linear map from the linear span of @xmath131 into the linear span of @xmath132 by the formula @xmath133 . the map is well - defined because the condition @xmath134 implies @xmath135 . the map is isometric because @xmath125 . extend it to an isometry @xmath136 . each isometry @xmath136 has one of the forms @xmath137 or @xmath138 for some @xmath139 ( * ? ? ? * theorem 3.2 ) . therefore either @xmath140 or @xmath141 . in the former case set @xmath142 and @xmath143 we have @xmath144 . up to a mbius transformation this is @xmath145 , which projects stereographically to @xmath146 . in the latter case set @xmath147 and @xmath148 . we have @xmath149 . up to a mbius transformation this is @xmath150 , which projects stereographically to @xmath151 . [ l - quadrics ] the ( possibly degenerate ) surfaces @xmath152 and @xmath153 , where @xmath154 , @xmath155 , @xmath156 , and @xmath157 , are contained in certain quadrics in @xmath19 . the surface @xmath158 is a subset of the sphere @xmath159 . the surface @xmath153 is a rational surface in @xmath0 of degree at most @xmath114 in each variable , hence a subset of a quadric . [ l - deg4 ] if the ( possibly degenerate ) surface @xmath160 is contained in @xmath19 then it is contained in an irreducible algebraic surface of degree at most @xmath2 . by lemma [ l-3folddeg4 ] the hypersurface @xmath90 , and hence @xmath160 , is contained in an irreducible algebraic hypersurface of degree @xmath161 . one of the components of the intersection of the algebraic hypersurface with the hyperplane @xmath19 is the required surface unless the algebraic hypersurface coincides with the hyperplane . in the latter case the assumptions of lemma [ l-3fold - r3 ] are satisfied . by lemmas [ l-3fold - r3 ] and [ l - quadrics ] the surface @xmath160 is a mbius transformation of a quadric , hence has degree @xmath161 . now the following folklore lemma allows us to prove a particular case of theorem [ prop21 ] . the lemma is a particular case of ( * ? ? ? * theorem 11 ) . we give the proof for convenience of the reader . [ l - cyclide ] if two irreducible algebraic surfaces of degree at most @xmath2 in @xmath0 are symmetric with respect to the unit sphere then they both are darboux cyclides . denote @xmath162 . let @xmath163 and @xmath164 be equations of our surfaces of minimal degrees . since the surfaces are symmetric with respect to the unit sphere it follows that @xmath165 for some @xmath166 and @xmath167 . expand @xmath168 and @xmath169 , where @xmath170 and @xmath171 are homogeneous of degree @xmath172 . we get @xmath173 . hence @xmath174 . assume that @xmath175 ( otherwise the proof is analogous ) . the degree of the left- and right - hand sides equal @xmath176 and @xmath15 respectively . thus @xmath177 . comparing the highest - degree terms we get @xmath178 , hence @xmath179 is divisible by @xmath180 . comparing the degree @xmath181 terms we get @xmath182 , hence @xmath183 is divisible by @xmath184 . therefore @xmath164 is a darboux cyclide . analogously @xmath163 is a darboux cyclide . [ prop11 ] if the surface @xmath87 , where @xmath83 , is contained in @xmath0 ( respectively , in @xmath7 ) then it is a subset of a darboux cyclide ( respectively , an intersection of @xmath7 with another @xmath8-dimensional quadric ) . for a surface in @xmath185 the result follows from lemmas [ l - deg4 ] and [ l - cyclide ] applied to the surfaces @xmath160 and @xmath186 . now consider a surface in @xmath7 . project it stereographically to @xmath0 . the resulting surface has the form @xmath187 . by the lemma for @xmath0 it is a darboux cyclide . by ( * section 2.2 ) the initial surface is an intersection of @xmath7 with another quadric . let us proceed to the general case of theorem [ prop21 ] . we start with a folklore lemma . [ l - axial ] the curve @xmath188 , where @xmath22 , @xmath167 , @xmath189 , are fixed and @xmath24 runs , is a circle ( possibly without one point ) . if @xmath190 is contained in @xmath191 then the plane of the circle is orthogonal to the vector @xmath192 . we have @xmath193 for some @xmath194 , where @xmath195 . this is a composition of translations , rotations , and an inversion applied to the line @xmath196 , once @xmath167 and @xmath189 . thus @xmath190 is a circle ( possibly without one point ) . now assume that @xmath197 . then @xmath198 . since the coefficients of the polynomial in the right - hand side vanish it follows that @xmath199 and @xmath200 , hence @xmath201 is orthogonal to @xmath202 . assume further that @xmath203 , otherwise @xmath190 is a line and there is nothing to prove . then the circle @xmath204 is contained in the plane spaned by @xmath201 and @xmath205 . so @xmath206 is contained in a plane orthogonal to @xmath202 as well . a similar result holds for the curve @xmath207 , where @xmath208 . if @xmath209 then by lemma [ prop11 ] @xmath210 is a subset of a darboux cyclide . analogously , if @xmath211 then @xmath212 is a darboux cyclide . assume further that @xmath213 . since @xmath214 it follows that either @xmath215 and @xmath216 or vice versa . assume the former without loss of generality . by left division of both @xmath217 and @xmath218 by the leading coefficient of @xmath217 we may achieve @xmath219 for some @xmath220 . analogously , assume that @xmath221 for some @xmath222 . performing an appropriate linear change of variables @xmath27 and @xmath29 we may achieve @xmath223 . if @xmath224 then @xmath225 for appropriate @xmath226 , hence @xmath227 is a darboux cyclide by the previous paragraph . the same holds for @xmath228 . assume further that @xmath229 . then we have @xmath230 for some @xmath231 because the left - hand sides vanishes identically for @xmath232 or @xmath233 . if @xmath234 then @xmath235 is a subset of a euclidean translational surface because by lemma [ l - axial ] the curves @xmath236 and @xmath237 are circles ( not degenerating to points because @xmath227 is a nondegenerate surface ) . assume further that @xmath238 . by the above @xmath239 for each @xmath240 . thus @xmath241 , hence @xmath242 , @xmath243 , @xmath244 , and @xmath245 . since @xmath246 this implies that @xmath247 . by lemma [ l - axial ] the curves @xmath41 and @xmath42 are circles ( or points ) whose planes are orthogonal to the vector @xmath247 . thus all these circles and hence the surface @xmath248 are contained in one plane . as in the previous proof , we may assume that @xmath215 , @xmath216 and @xmath213 . since @xmath249 it follows that @xmath250 and @xmath251 . by splitting lemma [ l - splitting - basic ] there exist @xmath252 and @xmath253 such that @xmath254 splits : @xmath255 or @xmath256 . since @xmath257 , without loss of generality we may assume that @xmath258 , @xmath259 . in the case when @xmath260 we have @xmath261 is a product of two circles @xmath262 and @xmath263 in @xmath7 because @xmath258 , @xmath259 . thus @xmath227 is a subset of a clifford translational surface . in the case when @xmath264 we have @xmath265 because @xmath266 . since @xmath267 it follows by lemma [ prop11 ] that @xmath227 is contained in the intersection of @xmath7 with another @xmath8-dimensional quadric . assume that a surface @xmath268 is a mbius transformation of surface . since a mbius transformation takes a hyperplane to either a hyperplane or a @xmath8-dimensional sphere it follows that the latter surface is contained either in a hyperplane or a @xmath8-dimensional sphere . perform a mbius transformation @xmath269 ( a similarity ) , where @xmath270 , taking the obtained hyperplane ( respectively , the @xmath8-dimensional sphere ) to @xmath191 ( respectively , to @xmath7 ) . it takes the surface @xmath271 to the surface @xmath272 again of form . by theorem [ prop21 ] the resulting surface either a darboux cyclide , or euclidean translational surface , or intersection of @xmath7 with another quadric , or clifford translational surface . in the latter two cases perform the inversion with the center at the point @xmath273 and the radius @xmath274 projecting the surface stereographically from @xmath7 to @xmath191 . this gives either a darboux cyclide ( by ( * ? ? ? * section 2.2 ) ) or a clifford translational surface . in all cases the resulting surface in @xmath191 is a mbius transformation of the initial surface @xmath268 . we conclude the section by an open problem : find a short proof that euclidean and clifford translational surfaces are not mbius transformations of each other ; this is ( * ? ? ? * theorem 2b ) . in this section we prove the results on parametrization of surfaces containing two conics or circles through each point : theorem [ schicho ] and corollary [ haupt ] . the proof uses well - known methods and goes along the lines of @xcite . we use the following notions . let @xmath275 be the @xmath28-dimensional complex projective space with the homogeneous coordinates @xmath276 . throughout we use the standard topology in @xmath275 ( not the zariski one ) . a _ analytic surface _ in @xmath275 is the image of an injective complex analytic map from a domain in @xmath277 into @xmath275 with nondegenerate differential at each point . an _ algebraic subset _ @xmath278 is the solution set of some system of algebraic equations . algebraic subsets of dimension @xmath114 and @xmath9 are called _ projective algebraic surfaces _ and _ algebraic curves _ , respectively . recall that the set of all conics in @xmath275 including the ones degenerating into lines and pairs of lines is naturally identified with an algebraic subset of @xmath279 for some large @xmath280 ( depending on @xmath28 ) . the latter subset is called _ the variety of all conics in @xmath275_. a conic _ analytically depending _ on a point is a complex analytic map of an analytic surface in @xmath275 into the variety of all conics in @xmath275 . an _ analytic family _ of conics is a complex analytic map @xmath281 of a domain in @xmath282 into the variety of all conics in @xmath275 . if no confusion arises , the image of this map is also called a _ family _ of conics . let us prove several lemmas required for the proof of theorem [ schicho ] . these lemmas are independent in the sense that the proof of each one uses the statements but not the proofs of the other ones . in what follows @xmath227 is a surface satisfying the assumptions of theorem [ schicho ] unless otherwise indicated . denote by @xmath283 and @xmath284 the two conics drawn through a point @xmath285 . [ l - cover ] there are two analytic families of conics @xmath286 , @xmath287 , and a domain @xmath288 such that @xmath289 , each pair @xmath286 , @xmath287 intersects transversely at a unique point @xmath290 , and @xmath291 , @xmath292 . take a point @xmath293 . draw the two conics @xmath294 and @xmath295 in the surface through the point . through each point @xmath296 draw another conic @xmath297 in the surface . we get an analytic family of conics @xmath287 . analogously we get an analytic family of conics @xmath286 . by the assumptions of theorem [ schicho ] the conics @xmath298 and @xmath299 intersect transversely at a unique point . by continuity there is @xmath300 such that for @xmath301 the conics @xmath286 and @xmath287 intersect transversely at a unique point @xmath290 , and @xmath302 . take a sufficiently small neighborhood @xmath303 of the point @xmath304 in @xmath275 . then @xmath289 . it remains to show that @xmath291 and @xmath292 . indeed , otherwise the image of the analytic map @xmath305 or @xmath306 in the variety of all conics in @xmath275 is @xmath9-dimensional . then there are infinitely many conics @xmath307 through each point in an open subset of @xmath227 such that @xmath308 . this contradicts to one of the assumptions of theorem [ schicho ] . thus the families @xmath286 and @xmath287 are the required . [ l - algebraic ] the surface @xmath227 is contained in an irreducible algebraic surface @xmath309 . the family of conics @xmath286 is contained in an irreducible algebraic curve in the variety of all conics . the conics @xmath287 given by lemma [ l - cover ] have at most @xmath2 common points because they do not all coincide . thus we may assume that they do not have common points inside the domain @xmath303 ( one can restrict the surface to a smaller domain , if necessary ) . then by the analyticity each point of @xmath303 belongs to at most countable number of conics @xmath287 . take a sufficiently small @xmath300 such that @xmath310 for each @xmath301 . consider the set @xmath311 of all conics in @xmath275 intersecting each conic @xmath287 , where @xmath312 , transversely at a unique point , which belongs to the domain @xmath303 . let us show that the set @xmath311 is an open subset of an algebraic subvariety of the variety of all conics in @xmath275 . indeed , the set of all conics intersecting a fixed conic @xmath287 is clearly an algebraic subvariety . the set of all conics intersecting each conic @xmath287 , where @xmath312 , is the intersection of infinitely many such algebraic subvarieties , and hence also an algebraic subvariety . the set @xmath311 is open in the latter subvariety because each conic sufficiently close to a conic lying in @xmath311 can only intersect a conic @xmath287 , where @xmath312 , transversely at a unique point inside the domain @xmath303 ( or not intersect at all ) . let us prove that the dimension of the set @xmath311 is @xmath114 . the dimension is at least @xmath114 because @xmath311 contains all the conics @xmath286 , where @xmath313 . to estimate the dimension from above , take an arbitrary conic @xmath314 . since each point of @xmath303 belongs to at most countable number of conics @xmath287 it follows that the conic @xmath307 has uncountably many intersection points with @xmath315 . hence @xmath316 , because @xmath227 is analytic . if the dimension of @xmath311 were at least @xmath9 then there would be infinitely many conics @xmath307 through each point in an open subset of @xmath227 such that @xmath308 . this would contradict to one of the assumptions of theorem [ schicho ] . thus the dimension of @xmath311 is exactly @xmath114 . the irreducible component of the algebraic closure of the set @xmath311 containing all the conics @xmath286 is the required algebraic curve in the variety of all conics . ( since the family @xmath286 is analytic , it can not `` jump '' from one irreducible component to another . ) the union of all the conics of the irreducible component ( including the ones degenerating to lines and pairs of lines ) is the required irreducible algebraic surface @xmath309 . the algebraic surface @xmath309 contains the analytic surface @xmath227 because @xmath309 contains the open subset @xmath317 of @xmath227 . [ rem - infinity ] if we drop the assumption that the number of conic through certain points is finite in theorem [ schicho ] then a similar argument shows that @xmath227 is contained in an algebraic surface @xmath318 and @xmath286 is contained in an algebraic subvariety @xmath311 of the variety of all conics in @xmath275 such that @xmath319 . let @xmath320 be algebraic subsets . a _ rational map _ @xmath321 is a map of an open dense subset of @xmath322 into the set @xmath323 given by polynomials in homogeneous coordinates of @xmath275 . ( dashes in the notation remind that a rational map may not be defined everywhere in @xmath322 . ) if the restriction of a rational map @xmath324 to certain open dense subsets of @xmath322 and @xmath323 is bijective , and the inverse map is rational as well , then @xmath324 is called a _ birational map_. a rational map @xmath325 is called a _ rational function_. a projective algebraic surface @xmath326 is _ unirationally ruled _ ( or simply _ uniruled _ ) , if for some algebraic curve @xmath311 there is a rational map @xmath327 with dense image . a surface @xmath326 is _ birationally ruled _ ( or simply _ ruled _ ) , if there is a birational map @xmath327 . a curve @xmath311 is _ rational _ , if there is a birational map @xmath328 . a surface @xmath326 is _ rational _ , if there is a birational map @xmath329 . [ l - uniruled ] the surface @xmath309 is unirationally ruled . let @xmath330 be the irreducible curve in the variety of conics given by lemma [ l - algebraic ] . consider the algebraic set @xmath331 the second projection @xmath332 is a rational map such that a generic fiber is a conic ( and hence a rational curve ) . by the noether enriques theorem ( * ? ? ? * theorem iii.4 ) @xmath333 is birationally ruled . in particular there is a rational map @xmath334 with dense image . compose the map with the first projection @xmath335 , which is surjective because the surfaces @xmath333 and @xmath309 are compact and the image contains the open subset @xmath336 by lemma [ l - cover ] . we get a rational map @xmath337 with dense image , i.e. , @xmath309 is unirationally ruled . the following folklore lemma is the most technical part of our proof . [ l - ruled ] each unirationally ruled surface is birationally ruled . let @xmath309 be a unirationally ruled surface and @xmath338 be a rational map with dense image . by the hironaka theorem ( or by an earlier zariski theorem sufficient in our situation ) the surface @xmath309 has a _ desingularization _ @xmath339 , i.e. , a proper birational map from a smooth projective algebraic surface @xmath340 to the surface @xmath309 . let @xmath341 be the inverse rational map of the desingularization . consider the rational map @xmath342 . by the theorem on eliminating indeterminacy ( * theorem ii.7 ) this rational map equals to a composition @xmath343 , where @xmath344 is a smooth projective algebraic surface , the first map is birational , and the second map is rational and defined everywhere . since @xmath345 has dense image and the surfaces are compact it follows that @xmath346 has dense image and @xmath347 is surjective . in particular , @xmath344 is birationally ruled . by the enriques theorem ( * ? ? ? * theorem vi.17 and proposition iii.21 ) , a smooth projective algebraic surface is birationally ruled if and only if for each @xmath348 the @xmath349-th tensor power of the exterior square of the cotangent bundle has no sections except identical zero ( in other terminology , _ the surface has kodaira dimension _ @xmath350 , or _ all plurigeni vanish _ ) . assume , to the contrary , that @xmath340 has such a section ( _ pluricanonical section _ ) . then the pullback under the surjective rational map @xmath351 is such a section for the birationally ruled surface @xmath344 , a contradiction . thus @xmath340 , and hence @xmath309 , is birationally ruled . [ l - rational ] the surface @xmath309 is rational . by lemmas [ l - uniruled ] and [ l - ruled ] the surface @xmath309 is birationally ruled . thus there is a birational map @xmath352 . consider the two conics through a generic point of the surface @xmath309 . their images are two distinct rational curves through a point of @xmath353 . since there is only one @xmath354-fiber through each point , at least one of the rational curves is nonconstantly projected to the curve @xmath311 . by the lroth theorem ( * ? ? ? * theorem v.4 ) the curve @xmath311 must be rational , and hence @xmath309 is rational . we conjecture that the following generalization of lemma [ l - rational ] is true : _ an algebraic surface containing two rational curves through almost each point is rational_. see ( * ? ? ? * definition iv.3.2 , theorems iv.5.4 , iv.3.10.3 , corollary iv.5.2.1 , exercise iv.3.12.2 ) for a sketch of the proof . [ l - linear - pencil ] each of the families @xmath286 and @xmath287 consists of level sets of some rational function @xmath355 . we use the following notions ; see @xcite for details . in order to apply intersection theory , @xmath339 ; see the first paragraph of the proof of lemma [ l - ruled ] . a _ divisor _ on @xmath340 is a formal linear combination of irreducible algebraic curves on @xmath340 with integer coefficients . closure of the preimage of an algebraic hypersurface under a rational map from @xmath340 to a projective space can be considered as a divisor , once the preimage is one - dimensional and one counts the irreducible components with their multiplicities . the collection of preimages of all the hyperplanes under a rational map is called a _ linear family _ of divisors . in particular , a one - dimensional _ linear _ _ family _ of divisors is the collection of level sets of a rational function . two divisors are _ linear equivalent _ , if they are contained in a one - dimensional linear family . @xmath356 of two divisors @xmath357 and @xmath358 on @xmath340 is the number of their intersection points counted with multiplicities ( once the number of intersection points is finite ) . two divisors are _ numerically equivalent _ , if their intersection with each algebraic curve on @xmath340 are equal , once the number of intersection points is finite . the _ degree _ of a divisor is the sum of the degrees of the irreducible components counted with multiplicities . the degree of a divisor equals the intersection of the divisor with a generic hyperplane section of @xmath359 . assume to the contrary that one of the analytic families @xmath286 and @xmath287 , say , the first one , is not linear . by the hironaka theorem the inverse map @xmath360 is defined everywhere except a finite set . thus the pullback @xmath361 is an analytic family of algebraic curves . we use the notation @xmath361 for the family of ( closed ) algebraic curves ( not to be confused with the set of preimages @xmath362 being algebraic curves possibly with a finite number of points removed ) . for an algebraic curve @xmath363 distinct from each @xmath364 the intersection @xmath365 continuously depends on @xmath366 , and hence is constant . thus each two curves of the family @xmath364 are numerically equivalent . on a smooth rational surface , numerical equivalence implies linear equivalence @xcite . since @xmath286 is nonlinear it follows that @xmath364 is nonlinear and hence must be contained in a linear family of divisors in @xmath340 of dimension at least @xmath9 . the image of the latter family under the projection @xmath367 is at least two - dimensional algebraic family @xmath311 of divisors on @xmath309 . the divisors of the family @xmath311 are linear combinations of curves with positive coefficients , because they arise from a linear family ( and negative numbers can not occur as multiplicities of preimages ) . all the divisors of the family @xmath311 have the same degree because their pullbacks are numerically equivalent . ( indeed , for two divisors @xmath368 , and a general position hyperplane @xmath369 we have @xmath370 . ) since the curves @xmath286 are conics it follows that the degrees of all these divisors are @xmath9 . therefore the divisors of the family @xmath311 are either conics or pairs of lines or lines of multiplicity @xmath9 . this is possible only if there are infinitely many conic sections or lines through each point in an open subset of the surface , which contradicts to the assumptions of theorem [ schicho ] . thus both families @xmath286 and @xmath287 must be linear . [ l - parametrization ] there is a rational map @xmath371 taking the families @xmath286 and @xmath287 to the sets of curves @xmath372 and @xmath373 respectively , such that the restriction of the map to some dense sets is bijective . consider the pair of rational functions given by lemma [ l - linear - pencil ] whose level sets are the two families of conics @xmath286 and @xmath287 . the pair of rational functions defines a rational map @xmath371 . since for sufficiently small @xmath374 each pair @xmath286 and @xmath287 has an intersection point it follows that the image of this rational map contains a neighborhood of @xmath375 , and thus is dense . since the intersection point is unique it follows that the point @xmath376 has exactly one preimage for sufficiently small @xmath374 , and hence for almost all @xmath377 . thus the restriction of the rational map @xmath371 to appropriate dense subsets is bijective . now we apply the following well - known result for which we could not find any direct reference . ( we have found several more general results in the literature but each time the proof that the inverse map is rational , the only assertion we need , was omitted . ) [ l - postulate ] if a rational map between open dense subsets of rational surfaces is bijective then the inverse map is rational as well , and hence birational . ( s. orevkov , private communication ) it suffices to prove the lemma for a map @xmath324 between open dense subsets of @xmath378 . for a generic @xmath379 the preimage @xmath380 is an open dense subset of the algebraic curve @xmath381 defined by the algebraic equation @xmath382 in the variables @xmath383 , where @xmath384 is the first projection . restrict the map @xmath324 to the preimage . we get a bijective rational map between open dense subsets of the curves @xmath381 and @xmath373 . clearly , it extends to a rational homeomorphism @xmath385 . then by the classification of algebraic curves the curve @xmath381 is rational . identify @xmath381 and @xmath373 with @xmath354 . consider the graph of the rational map @xmath385 as a subset of @xmath378 . by elimination of variables it follows that the graph is the zero set of some polynomial @xmath386 $ ] . since the map @xmath385 is bijective it follows that the polynomial @xmath37 has degree @xmath114 in each variable and hence the inverse map @xmath387 is also rational . this implies that the inverse map @xmath388 is rational in the variable @xmath366 for fixed generic @xmath104 . analogously , @xmath388 is rational in @xmath104 for fixed generic @xmath366 . thus @xmath388 is rational . [ l - final ] assume that a birational map @xmath389 takes the sets of curves @xmath372 and @xmath373 to conics or lines . write the birational map as @xmath390 for some coprime @xmath391 $ ] . then @xmath392 . for a birational map between surfaces , by elimination of indeterminacy ( * ? ? ? * theorem ii.7 ) there is always an algebraic curve @xmath393 such that outside the curve @xmath393 the map is injective and has nondegenerate differential . fix a generic value of @xmath27 . denote @xmath394 . the curve @xmath395 is a conic or a line . cut it by a generic hyperplane @xmath396 in @xmath275 . the intersection consists of at most @xmath9 points . by general position they are not contained in the image of @xmath393 . since the @xmath397 are coprime it follows that @xmath398 have no common roots . since the birational map is injective outside @xmath393 it follows that the equation @xmath399 has at most @xmath9 solutions . these solutions have multiplicity @xmath114 because the birational map has nondegenerate differential outside @xmath393 . this implies that the polynomials @xmath400 have degree at most @xmath9 . analogously , @xmath401 have degree at most @xmath9 in @xmath27 , and the lemma follows . it follows directly by lemmas [ l - cover ] , [ l - algebraic ] , [ l - parametrization ] , [ l - postulate ] , [ l - final ] . theorem [ schicho ] remains true ( with almost the same proof ) without the assumption that the number of conic sections through certain points is finite except that then one can not conclude that the two drawn conic sections @xmath283 and @xmath284 are necessarily the curves @xmath41 and @xmath42 . for the proof of corollary [ haupt ] we need the following lemmas . in the rest of this section @xmath402 is a surface satisfying the assumptions of corollary [ haupt ] . [ l - reduction ] the surface @xmath227 ( possibly besides a one - dimensional subset ) has parametrization , where @xmath47 satisfy equation , and @xmath403 runs through some ( not open ) subset of @xmath277 . since the circular arcs through each point of @xmath227 are noncospheric , their respective circles are transversal and intersect at a unique point . extend @xmath404 analytically to a complex analytic surface @xmath309 in a sufficiently small neighborhood of @xmath227 in @xmath275 modulo the boundary ( so that the boundaries of @xmath227 and @xmath309 are contained in the boundary of the neighbourhood ) . extend the two real analytic families of circular arcs in @xmath227 to complex analytic families of ( complex ) conics in @xmath275 . by analyticity @xmath309 satisfies the assumptions of theorem [ schicho ] . by theorem [ schicho ] the surface @xmath309 ( possibly besides a one - dimensional subset ) has parametrization by polynomials @xmath392 such that the circular arcs have the form @xmath41 and @xmath42 . ( however , the converse is not true : most of the curves @xmath41 and @xmath42 are not circular arcs but parts of complex conics in @xmath309 . ) since the surface @xmath309 is contained in the complex quadric extending the sphere @xmath49 , it follows that the polynomials satisfy equation . let us reparametrize the surface to make the polynomials @xmath405 real . [ l - make - real ] the surface @xmath227 ( possibly besides a one - dimensional subset ) has a parametrization , where @xmath47 satisfy equation , and @xmath403 runs through certain open subset of @xmath406 . start with the parametrization given by lemma [ l - reduction ] . draw two circular arcs of the form @xmath41 and @xmath42 through a point of the surface @xmath227 . through another pair of points of the first circular arc , draw two more circular arcs of the form @xmath42 . perform a complex fractional - linear transformation of the parameter @xmath29 so that the second , the third , and the fourth circular arcs obtain the form @xmath407 , respectively ( we consider the part of the surface where the denominator of the transformation does not vanish ) . perform an analogous transformation of the parameter @xmath27 so that @xmath408 become circular arcs intersecting the circular arc @xmath233 . after performing the transformations and clearing denominators we get a parametrization @xmath409 of the surface @xmath404 , where still @xmath47 and @xmath403 runs through a subset @xmath410 . let us prove that actually @xmath411 . take @xmath412 sufficiently close to @xmath413 . then @xmath414 is a point of the surface @xmath227 sufficiently close to @xmath415 . draw the two circular arcs @xmath41 and @xmath42 through the point @xmath414 . by continuity it follows that the circular arc @xmath42 intersects the circular arc @xmath232 in @xmath227 . the intersection point can only be @xmath416 . in particular , we get @xmath417 . in a quadratically parametrized conic , the cross - ratio of any four points equals the cross - ratio of their parameters . since three ( real ) points @xmath418 of the circular arc @xmath232 have real @xmath29-parameters it follows that all ( but one ) points of the circular arc have real @xmath29-parameters . in particular , @xmath419 . analogously , @xmath420 . we have proved that all @xmath412 sufficiently close to the origin are real . by the analyticity @xmath326 is an open subset of @xmath406 . [ l - real ] let @xmath421 $ ] . assume that for all the points @xmath403 from some open subset of @xmath406 the point @xmath422 is real . then @xmath423 for some real @xmath424 $ ] and complex @xmath425 $ ] . without loss of generality assume that @xmath426 is not identically zero . take @xmath427 such that @xmath428 is not identically zero and take generic real @xmath78 from the open set in question . by the assumption of the lemma @xmath422 hence @xmath429 is real , therefore @xmath430 . since this holds for generic real @xmath78 it follows that @xmath431 as polynomials . take decompositions @xmath432 into coprime reduced irreducible factors @xmath433-\mathbb{r}[u , v]$ ] , @xmath434 $ ] with the powers @xmath435 , and constant factors @xmath436 , @xmath437 . since @xmath56 $ ] is a unique factorization domain , the relation @xmath431 implies that @xmath438 and @xmath439 for each @xmath440 . without loss of generality assume that @xmath441 for each @xmath440 ( otherwise replace @xmath442 by @xmath443 and vice versa ) . it remains to set @xmath444\end{aligned}\ ] ] for a surface @xmath227 in @xmath49 the corollary follows from lemmas [ l - reduction][l - real ] . now consider a surface @xmath227 in @xmath445 . project it stereographically to @xmath49 . the resulting surface has a parametrization as granted by the theorem for @xmath49 . thus the initial surface has the parametrization @xmath446 with @xmath447 , as required . the following result is useful for applications of corollary [ haupt ] . if through each point of an analytic surface in @xmath0 one can draw infinitely many circular arcs fully contained in the surface then the surface is a subset of a sphere or a plane . perform inverse stereographic projection of the surface to @xmath7 . by remark [ rem - infinity ] the resulting surface is covered by at least 2-dimensional algebraic family of conics and hence circles . the circles of the family passing through a particular point must cover an open subset of the surface . project the surface back to @xmath0 from this point . we get a surface containing a line segment and infinitely many circular arcs through almost each point . by ( * theorem 1 ) the resulting surface is a subset of a quadric or a plane . since it is covered by a 2-dimensional family of circles , by the classification of quadrics the resulting surface , and hence initial one , is a subset of a sphere or a plane . we conclude the section by an open problem : does the lemma remain true in dimension @xmath448 ? in this section we prove the results concerning factorization of quaternionic polynomials : splitting lemma [ l - splitting - basic ] , corollary [ cor - splitting ] , and give some examples and final remarks . consider the polynomial @xmath453 obtained by substitution of the quaternion @xmath454 into the _ polynomial @xmath455 . on one hand , @xmath456 is divisible by @xmath457 of degree @xmath9 . on the other hand , @xmath458 has degree at most @xmath114 because @xmath459 is a constant . thus @xmath460 identically , i.e. , @xmath461 . [ ex - beauregard ] ( beauregard @xcite ) . the polynomial @xmath470 is irreducible in @xmath59 $ ] but @xmath471 is reducible in @xmath472 $ ] . in particular , @xmath473 are two decompositions in @xmath59 $ ] with different number of irreducible factors . first consider the case when one of these polynomials , say , @xmath37 , does not depend on one of variables , say , @xmath29 . then write @xmath476 . both @xmath477 and @xmath478 must be left - divisible by @xmath479 , a contradiction by taking @xmath232 . it remains to consider the case when @xmath480 . since @xmath481 and @xmath472 $ ] is a unique factorization domain it follows that @xmath482 is product of two quadratic factors . by splitting lemma [ l - splitting - basic ] the polynomial @xmath37 is reducible in @xmath59 $ ] . the left factor in a decomposition of @xmath483 is a left divisor of @xmath65 and does not depend on one of the variables , which leads to the first case already considered above . in both cases we get a contradiction which proves that @xmath65 is irreducible . it is interesting to obtain _ octonion _ counterparts of the results of this section . in particular , this could help to find all surfaces in @xmath16 containing two circles through each point for @xmath488 . let us give some final remarks . simple formula produces surfaces in @xmath17 containing two circles through each point . however , it does not give _ all _ such surfaces in @xmath17 and is not convenient to produce such surfaces in @xmath0 . thus we suggest the following alternative approach to surface construction . the _ ( top - left ) quasideterminant _ of a @xmath109 matrix @xmath489 with the entries from @xmath59 $ ] is the following expression @xcite : @xmath490 if @xmath491 is linear in @xmath27 , @xmath492 is linear in @xmath29 , and all the other entries @xmath493 are generic constants then by the homological relations ( * ? ? ? * theorem 1.4.2(i ) ) the quasideterminant is fraction - linear in each of the variables @xmath27 and @xmath29 . thus @xmath494 is a surface in @xmath17 containing @xmath9 circles through each point ( by lemma [ l - axial ] above ) . if the matrix @xmath495 is skew - hermitian then the surface actually lies in @xmath0 . similarly , if @xmath496 , @xmath497 is a constant , @xmath498 , @xmath499 , @xmath492 are linear in @xmath27 , @xmath491 , @xmath500 are linear in @xmath29 then @xmath494 is again a surface in @xmath17 containing @xmath9 circles through each point . it is interesting , if there is a natural class of matrices with the entries from @xmath59 $ ] such that _ all _ surfaces containing @xmath9 noncospheric circles through each point arise as the quasideterminants . the authors are grateful to n. lubbes for the movie in figure [ movie ] and many useful remarks , to l. shi for the photo in figure [ movie ] , to a. pakharev for pointing out that numerous surfaces we tried to invent have form , and to s. galkin , s. ivanov , w. khnel , n. lubbes , s. orevkov , r. pignatelli , f. polizzi , h. pottmann , g. robinson , j. schicho , k. shramov , v. timorin , m. verbitsky , s. zub for useful discussions . the first author is grateful to king abdullah university of science and technology for hosting him during the start of the work over the paper . w. degen , die zweifachen blutelschen kegelschnittflchen , manuscr . math . * 55:1 * ( 1986 ) , 938 . r. dietz , j. hoschek , b. juettler : an algebraic approach to curves and surfaces on the sphere and on other quadrics , computer aided geometric design 10 ( 1993 ) , 211 - 229 . graziano gentili , caterina stoppato , daniele c. struppa , regular functions of a quaternionic variable , springer monographs in math . 2013 b. gordon , t. s. motzkin , on the zeros of polynomials over division rings , trans . ( 1965 ) , 218 - 226 . k. kataoka , n. takeuchi , a system of fifth - order partial differential equations describing a surface which contains many circles , bulletin des sciences mathmatiques 137:3 ( 2013 ) , 325360 . j. kollr , rational curves on algebraic varieties , springer - verlag , berlin heidelberg , 1996 . r. krasauskas , s. zub , rational bzier formulas with quaternion and clifford algebra weights in : tor dokken , georg muntingh ( eds . ) , saga advances in shapes , geometry , and algebra , geometry and computing , vol . 10 , springer , 2014 , pp . lavicka , roman ; ofarrell , anthony g. and short , ian . reversible maps in the group of quaternionic moebius transformations . cambridge philos . soc . , 143:1 ( 2007 ) , 5769 . available online : http://oro.open.ac.uk/22456/1/quaternions.pdf mikhail skopenkov + faculty of mathematics , national research university higher school of economics , and + institute for information transmission problems , russian academy of sciences + ` skopenkov@rambler.ru ` http://skopenkov.ru

an abelian cover is a finite morphism @xmath0 of varieties which is the quotient map for a generically faithful action of a finite abelian group @xmath1 . this means that for every component @xmath7 of @xmath2 the @xmath1-action on the restricted cover @xmath8 is faithful . the paper @xcite contains a comprehensive theory of such covers in the case when @xmath2 is smooth and @xmath3 is normal . the covers are described in terms of the _ building data _ consisting of branch divisors @xmath9 ranging over cyclic subgroups @xmath10 , and line bundles @xmath11 with @xmath12 ranging over the character group of @xmath1 . this collection must satisfy the _ fundamental relations_. here , we extend this theory to the case of singular varieties . namely , we allow @xmath3 and @xmath2 to be varieties satisfying serre s condition @xmath4 and having double crossing singularities in codimension 1 , which we abbreviate to g.d.c . for `` generically double crossings '' ( see [ ssec : ysmooth ] for the precise definition ) . our interest in this case lies in applications to the moduli theory . such non - normal abelian covers appear in our work @xcite where we explicitly construct compactifications of moduli spaces of some campedelli and burniat surfaces by adding stable surfaces on the boundary . `` stable surfaces '' here are in the sense of @xcite : they have slc ( semi log canonical ) singularities and ample canonical class . in this paper , we give a comprehensive treatment of the situation . in section [ ssec : ysmooth ] we show that the theory of standard covers of @xcite has a very natural extension to the case when @xmath2 is still smooth but @xmath3 is possibly g.d.c .. in section [ ssec : ynormal ] we extend it to the case of normal base by an @xmath4-fication trick . in section [ ssec : ynonnormal ] we prove that a cover with non normal @xmath2 can be obtained by gluing a cover over the normalization @xmath13 , and we spell out which additional data must be specified . in section [ sec : geometry ] we study the singularities of covers . we determine the conditions for @xmath3 to have slc singularities , to be cohen - macaulay , and we determine the index of the canonical divisor in the situations appearing in common applications . in section [ sec : z2-covers ] we treat in detail the special case when the group @xmath1 is @xmath14 and @xmath15 , as in @xcite . we restrict ourselves to the situation where the base @xmath2 is smooth or has two smooth branches meeting transversally , and the components of branch divisors and the double locus are smooth and have distinct tangent directions at the points of intersection , i.e. locally they look like a collection of lines in the plane . in this situation , we give a complete classification of the covers and the singularities of @xmath3 . the answer is contained in nine tables . some of these covers appear on the boundary of moduli of campedelli and burniat surfaces , but the full list is longer . @xmath1 denotes a finite abelian group . we work with equidimensional varieties defined over an algebraically closed field @xmath16 whose characteristic does not divide the order of @xmath1 . we denote by @xmath17 the group @xmath18 of characters of @xmath1 , and we write it multiplicatively . the abbreviations _ lc _ and _ slc _ stand for _ log canonical _ and _ semi log canonical_. ( cf . [ sec : geometry ] for the definitions ) . @xmath19 , @xmath20 , etc . denote the normalization of @xmath3 , @xmath21 , etc . we use the additive and multiplicative notation for line bundles and divisors interchangeably . linear equivalence will be denoted by @xmath22 . the first author was partially supported by nsf under dms 0901309 . the second author wishes to thank miles reid and angelo vistoli for several useful communications . we also thank the referee for many useful comments and corrections . part of this work was done while both authors were visiting msri in the spring of 2009 . this project was partially supported by the italian prin 2008 project _ geometria delle variet algebriche e dei loro spazi di moduli_. the second author is a member of gnsaga of indam . we recall some basic facts about serre s condition @xmath4 and the @xmath4-fication of a coherent sheaf . for a comprehensive treatment , the reader may consult @xcite , where the latter appears under the name `` @xmath23-closure '' . all varieties below are assumed to be reduced , equidimensional , but possibly reducible . let @xmath24 be a coherent sheaf on @xmath3 all of whose associated components are irreducible components of @xmath3 . then there exists a unique _ @xmath4-fication _ , or _ saturation in codimension 2 _ , a coherent sheaf defined by @xmath25 the sheaf @xmath26 is @xmath4 , and @xmath24 is @xmath4 iff the map @xmath27 is an isomorphism . in particular , for @xmath28 one obtains the @xmath4-fication @xmath29 , which is dominated by the normalization of @xmath3 . on a normal variety @xmath3 , an @xmath4-sheaf is the same as a _ reflexive sheaf _ , satisfying @xmath30 , see @xcite . further , reflexive sheaves of rank 1 are the same as _ divisorial sheaves _ , isomorphic to @xmath31 for some weil divisor @xmath32 , see e.g. ( * ? ? 1 ) . on a smooth ( or factorial ) variety weil divisors are the same as cartier divisors , and rank 1 @xmath4 sheaves are the same as invertible sheaves . let @xmath1 be a finite abelian group . an _ abelian cover _ with galois group @xmath1 , or _ @xmath1-cover _ , is a finite morphism @xmath0 of varieties which is the quotient map for a generically faithful action of a finite abelian group @xmath1 . this means that for every component @xmath7 of @xmath2 the @xmath1-action on the restricted cover @xmath33 is faithful . an _ isomorphism _ of @xmath1-covers @xmath34 , @xmath35 is an isomorphism @xmath36 such that @xmath37 . the @xmath1-action on @xmath3 with @xmath38 is equivalent to a decomposition : @xmath39 where @xmath1 acts on @xmath40 via the character @xmath12 . if @xmath41 is galois then each @xmath40 has rank 1 : if @xmath42 is a general closed point , then @xmath1 acts freely on @xmath43 , so it acts on @xmath44 as the regular representation . thus , @xmath45 is @xmath46-dimensional for every @xmath12 . when the sheaves @xmath40 are locally free , it is customary to write @xmath47 , with @xmath11 a line bundle . [ lem : flatness - and - cm ] 1 . the sheaf @xmath48 is @xmath49 for some @xmath50 iff every @xmath40 is @xmath49 . 2 . if @xmath51 is flat then @xmath3 is cm iff @xmath2 is cm . if @xmath2 is smooth and @xmath3 is @xmath4 then @xmath41 is flat and @xmath3 is cm . \(1 ) is clear by definition of depth . \(2 ) @xmath41 is flat iff every @xmath52-module @xmath40 is invertible . then each @xmath40 is cm iff @xmath52 is . \(3 ) on a smooth variety every divisorial sheaf is invertible , and so flat . now ( 2 ) applies . a @xmath1-cover @xmath51 , where @xmath3 and @xmath2 are @xmath4 varieties , is determined by its restriction to the complement of a closed subset of codimension @xmath5 : [ lem : remove - codim2 ] let @xmath2 be an @xmath4 variety , @xmath53 an open subset with @xmath54 , and @xmath55 a @xmath1-cover with @xmath56 an @xmath4 variety . then there exist a unique @xmath4 variety @xmath3 and a @xmath1-cover @xmath51 whose restriction to @xmath57 is @xmath58 . for the existence , we take @xmath59 , where @xmath60 is the inclusion . then @xmath61 , where each @xmath40 is a rank 1 @xmath4-sheaf . the algebra structure on @xmath48 is defined as follows . for an open set @xmath62 and sections @xmath63 , @xmath64 , their product is @xmath65 since @xmath66 and @xmath40 is saturated in codimension 2 . thus , @xmath67 is an @xmath4 variety with a finite morphism to @xmath2 . the @xmath17-grading on @xmath48 defines the @xmath1-action on @xmath3 . by construction , the eigenspace @xmath68 for the trivial character is @xmath69 . therefore , @xmath38 . uniqueness follows from the uniqueness of the @xmath4-fication . given a @xmath1-cover @xmath51 and an irreducible subset @xmath70 , we define the _ inertia subgroup _ @xmath71 of @xmath72 to be the subgroup of @xmath1 consisting of the elements that fix @xmath73 pointwise , or , equivalently since @xmath1 is abelian , that fix an irreducible component of @xmath73 pointwise . the _ branch locus _ @xmath74 of @xmath41 is the set of points of @xmath2 whose inertia subgroup is not trivial ( notice that we regard @xmath74 simply as a set , without giving it a scheme structure ) . if @xmath41 is flat , then @xmath74 is a divisor by ( * ? ? ? if @xmath75 is an irreducible divisor of @xmath2 such that @xmath3 is generically smooth along @xmath76 , then the natural representation @xmath77 of @xmath78 on the tangent space @xmath79 at the generic point of an irreducible component @xmath80 of @xmath76 is faithful , hence @xmath81 is cyclic ( cf . @xcite ) . notice that @xmath77 does not depend on the choice of the component @xmath80 of @xmath76 since @xmath1 is abelian . in this section we recall , in a form which is convenient for our later applications , the definition of standard abelian covers , a class of flat abelian covers that can be constructed from a collection of line bundles and effective divisors on the target variety ( cf . @xcite , @xcite ) . the prototypical example is the classical construction of a double cover of a variety @xmath2 from the data of an effective divisor @xmath32 on @xmath2 and a line bundle @xmath82 such that @xmath83 . let @xmath2 be a variety . a set of _ building data for a standard @xmath1-cover _ @xmath51 consists of the following : * irreducible effective cartier divisors @xmath84 ( possibly not distinct ) , * for each @xmath85 a pair @xmath86 , where @xmath87 is a cyclic subgroup of @xmath1 of order @xmath88 and @xmath89 is a generator of the group of characters @xmath90 , * line bundles @xmath11 , for @xmath91 . moreover we assume that these data satisfy the so called _ fundamental relations _ : @xmath92 where for a character @xmath12 we write @xmath93 , with @xmath94 , and we define @xmath95 $ ] . observe that @xmath96 is equal either to @xmath97 or to @xmath46 . we call the divisors @xmath85 , together with the pairs @xmath98 , the _ branch data _ of the cover . an equivalent way of describing the branch data , and therefore the building data , is to give for each pair @xmath99 , with @xmath100 a cyclic subgroup and @xmath101 a generator , the divisor @xmath102 . this is the notation used in @xcite . [ rem : branchdata ] if the group @xmath103 has no @xmath104-torsion , where @xmath105 , then the branch data determine the building data by ( * ? ? ? 2.1 ) . in general , the branch data are enough to determine the local geometry of the cover ( cf . proposition [ prop : bdata ] , ( 2 ) ) . when @xmath106 , it is enough to associate with every divisor @xmath85 a nonzero element @xmath107 , the generator of @xmath87 . also , the definition of @xmath108 is simpler : @xmath109 is equal to 1 if @xmath110 and it is equal to 0 otherwise . we now explain how to construct a @xmath1-cover from a set of building data . choose @xmath111 such that @xmath17 is the direct sum of the cyclic subgroups generated by the @xmath112 . denote by @xmath113 the order of @xmath112 and write @xmath114 and @xmath115 . by ( * prop.2.1 ) for @xmath116 there exist isomorphisms : @xmath117 notice that the coefficients @xmath118 in the above formula are integers . using formulae ( 2.15 ) of @xcite and the isomorphisms @xmath119 above , one constructs for each pair @xmath120 of non trivial characters an isomorphism @xmath121 such that for every @xmath122 the following diagram commutes ( we set @xmath123 ) : @xmath124 where @xmath125 and the maps are induced by the @xmath126 in the obvious way . we denote by @xmath127 the maps induced by composing @xmath126 with the inclusion @xmath128 . by the commutativity of diagram , the collection of maps @xmath129 defines on @xmath130 a commutative and associative algebra structure compatible with the @xmath1-action defined by letting @xmath1 act trivially on @xmath131 and via the character @xmath12 on @xmath132 for @xmath133 . we define @xmath134 with the natural map @xmath135 to be a _ standard @xmath1-cover _ associated with the given set of building data . notice that , since the @xmath132 are locally free , @xmath41 is flat and @xmath3 is @xmath4 if @xmath2 is . @xmath3 can be described locally above a point @xmath42 as follows . up to shrinking @xmath2 , we may assume that all the @xmath11 are trivial and that the @xmath85 are defined by equations @xmath136 . if we denote by @xmath137 a coordinate on @xmath132 , @xmath138 , then @xmath3 is given inside the vector bundle @xmath139 by the following set of equations : @xmath140 where the @xmath141 are nowhere vanishing regular functions and for @xmath142 we set @xmath143 . for @xmath144 , denote by @xmath145 the order of @xmath12 and write @xmath146 , with @xmath147 . eliminating between the equations , one gets @xmath148 where @xmath149 is a nowhere vanishing function . it follows immediately that @xmath3 is a variety : indeed , using the decomposition of @xmath150 into @xmath1-eigenspaces , we may assume that a nilpotent element is locally of the form @xmath151 for some character @xmath12 and some regular function @xmath152 . then by , @xmath153 for some @xmath154 only if @xmath155 . using the local equations , one can also show the following : [ lem : inertia ] notation as above . let @xmath156 be a standard @xmath1-cover and @xmath42 be a point . the inertia subgroup @xmath157 of @xmath158 is equal to @xmath159 . since the question is local on @xmath2 , we may assume that @xmath3 is given by the equations . let @xmath160 be a point lying above @xmath158 . then by the coordinate @xmath161 does not vanish iff @xmath162 for every @xmath163 such that @xmath164 . since an element @xmath165 fixes @xmath166 if and only if for every @xmath167 such that @xmath168 the coordinate @xmath161 vanishes , this remark proves the claim . given a set of building data , the construction of the standard @xmath1-cover @xmath51 depends of course on the choice of the characters @xmath169 and of the isomorphims @xmath119 . assume that @xmath170 are another set of characters of @xmath1 such that @xmath17 is the direct sum of the cyclic subgroups generated by the @xmath171 . let @xmath172 be the order of @xmath171 , @xmath173 ; then by the multiplication maps for @xmath174 isomorphisms @xmath175 , where @xmath176 and @xmath177 . by the associativity and commutativity of the multiplication the algebra structure defined on @xmath178 by the @xmath179 is the same as that induced by the @xmath119 . hence it is enough to analyze to what extent the isomorphism class of @xmath41 depends on the @xmath119 : [ prop : bdata ] 1 . _ ( global case ) . _ if @xmath180 , then the building data determine @xmath51 up to isomorphism of @xmath1-covers . 2 . in general , given two standard covers @xmath181 , @xmath182 , with the same building data , there exists an tale cover @xmath183 such that , after base change with @xmath183 , @xmath184 and @xmath185 give isomorphic @xmath1-covers . \(2 ) we use the notation introduced above . let @xmath186 , @xmath187 be two @xmath188-algebra structures on @xmath189 given by isomorphisms @xmath119 , respectively @xmath190 . the isomorphisms @xmath119 , @xmath190 differ by an automorphism of @xmath191 , namely by multiplication by an element @xmath192 . this automorphism is induced by an automorphism of @xmath193 iff @xmath194 has a @xmath113-th root @xmath195 . so , up to taking an tale cover , one can assume that the roots @xmath196 exist . by formulae ( 2.15 ) of @xcite , the @xmath196 can be used to define for all @xmath91 automorphisms @xmath197 of @xmath198 that commute with the isomorphisms @xmath126 and @xmath199 . to prove statement ( 1 ) , just observe that if @xmath180 no base change is necessary to construct the isomorphism above . [ rem : local ] let @xmath51 be a @xmath1-cover with branch data @xmath200 , let @xmath42 and let @xmath136 be local equations for @xmath85 near @xmath158 . combining proposition [ prop : bdata ] with the local equations , we see that , up to passing to an tale cover of @xmath201 , @xmath3 is defined locally near @xmath158 by the equations : @xmath202 here we find conditions for a @xmath1-cover of a smooth variety to be standard . we keep the notation of the previous section . [ defn : hurwitz ] let @xmath2 be a smooth variety and let @xmath51 be a standard @xmath1-cover with building data @xmath11 , @xmath85 , @xmath98 . by lemma [ lem : inertia ] the branch locus @xmath74 of @xmath41 is the support of the divisor @xmath203 . we define the _ hurwitz divisor _ of @xmath41 as the @xmath204-divisor @xmath205 . notice that the support of @xmath32 is equal to @xmath74 . we say that a variety is _ _ ( has _ double crossings _ ) if every point is either smooth or analytically isomorphic to @xmath206 . we say that a variety is _ _ ( has _ generically double crossing_s ) if it is d.c . outside a closed subset of codimension @xmath5 the following result generalizes the main result of @xcite : [ thm : structure ] let @xmath51 be a @xmath1-cover such that @xmath2 is smooth and @xmath3 is @xmath4 . then : 1 . @xmath3 is normal iff @xmath41 is standard and every component of the hurwitz divisor @xmath32 has multiplicity @xmath207 . 2 . assume that @xmath41 is standard . then @xmath3 is g.d.c . iff every component of @xmath32 has multiplicity @xmath208 . 3 . assume that @xmath3 is g.d.c .. then @xmath41 is standard iff for every irreducible divisor @xmath75 of @xmath2 such that @xmath3 is singular above @xmath75 one has @xmath209 for some @xmath210 . in the case @xmath106 , which is of special interest to us because of the applications in @xcite , theorem [ thm : structure ] reads : [ cor : structurez2 ] let @xmath51 be a @xmath6-cover such that @xmath2 is smooth and @xmath3 is @xmath4 . then : 1 . @xmath3 is normal iff @xmath41 is standard and every component of @xmath32 has multiplicity @xmath207 . 2 . @xmath3 is g.d.c . iff @xmath41 is standard and every component of @xmath32 has multiplicity @xmath208 . let @xmath51 be a standard @xmath1-cover with @xmath2 smooth and @xmath3 g.d.c . and let @xmath75 be a component of the branch divisor @xmath74 . by lemma [ lem : inertia ] , we have @xmath211 . the pairs ( subgroup , character ) corresponding to @xmath75 can be determined as follows : * assume that @xmath75 has multiplicity @xmath207 in the hurwitz divisor @xmath32 . then there is precisely one index @xmath163 with @xmath212 . in this case , @xmath213 and the character @xmath89 is given by the action of @xmath87 on the tangent space to @xmath3 at the generic point of an irreducible component of @xmath76 ( cf . @xcite , 1 and 2 ) . * assume that @xmath75 has multiplicity @xmath214 in @xmath32 . then there are precisely two indices @xmath215 and @xmath216 such that @xmath217 and @xmath218 and @xmath219 have order 2 . so either @xmath220 or @xmath221 . in the latter case the proof of theorem [ thm : structure ] shows that @xmath218 and @xmath219 are generated by the elements of @xmath81 that interchange the two branches of @xmath3 at a general point of @xmath76 . statement ( 1 ) is @xcite , thm . 2.1 and cor.3.1 . so consider the non - normal case . the cover @xmath41 is flat since @xmath2 is smooth and @xmath3 is @xmath4 , hence we write as usual @xmath222 . the cover is standard if and only if there exist branch data @xmath85 , @xmath98 such that for every @xmath223 the zero divisor of the multiplication map @xmath224 is equal to @xmath225 , where the @xmath226 are defined in [ ssec : standard ] . notice that @xmath3 , being @xmath4 , is non - normal if and only if it is singular in codimension 1 . fix a component @xmath75 of @xmath32 such that @xmath3 is singular above @xmath75 . write @xmath227 . the cover @xmath41 factors as @xmath228 and @xmath75 is not contained in the branch locus of the map @xmath229 , hence @xmath230 is generically smooth over @xmath75 . it follows that there is an element of @xmath231 that exchanges the two branches of @xmath3 at a general point of @xmath76 . let @xmath232 be the normalization , let @xmath233 be the induced @xmath1-cover , let @xmath234 be the pair ( subgroup , character ) corresponding to @xmath75 for the cover @xmath235 and let @xmath236 be the order of @xmath237 ( if @xmath235 is not branched on @xmath75 , we take @xmath237 and @xmath238 to be trivial ) . since the normalization map @xmath232 is @xmath1-equivariant , we have a short exact sequence : @xmath239 we consider the @xmath231-covers @xmath240 and @xmath241 and we study the algebras @xmath242 and @xmath243 , where @xmath244 is an irreducible component of the inverse image of @xmath75 in @xmath245 . we denote by @xmath246 a local parameter . we distinguish three cases : in this case @xmath247 , and @xmath3 is given locally by @xmath248 , where @xmath249 . let @xmath101 be a generator that restricts to @xmath238 on @xmath237 . the algebra @xmath250 is generated by elements @xmath251 such that : @xmath252 where @xmath253 and @xmath231 acts on @xmath254 via the character @xmath77 and on @xmath255 via the character @xmath256 . the eigenspace corresponding to @xmath257 is generated by @xmath258 for @xmath259 , and by @xmath260 for @xmath261 . since the inclusion @xmath262 is @xmath1-equivariant , @xmath263 is generated by elements of the form @xmath264 for suitable @xmath265 . since @xmath231 fixes @xmath266 pointwise , by the argument in the proof of lemma [ lem : inertia ] @xmath267 is contained in the subalgebra @xmath268 of @xmath250 generated by @xmath269 @xmath270 is also generated by @xmath271 , with the only relation @xmath272 , hence @xmath273 is g.d.c . and the map @xmath274 is an isomorphism . so @xmath275 . in this case @xmath236 is even and @xmath276 . we denote by @xmath101 a character that restricts to @xmath238 on @xmath237 and by @xmath277 the character such that @xmath278 . @xmath250 is generated by @xmath251 such that : @xmath279 where @xmath253 and @xmath231 acts on @xmath254 via the character @xmath77 and on @xmath255 via the character @xmath277 . arguing as in the previous case , one checks that @xmath263 is generated by : @xmath280 @xmath267 can also be generated by @xmath281 with the only relation @xmath272 . for @xmath282 , denote by @xmath283 the order of vanishing on @xmath75 of the multiplication map @xmath284 . using the above analysis and arguing as in the proof of ( 2.1 ) , one obtains the following rules , up to exchanging @xmath285 and @xmath286 : * @xmath287 if @xmath288 , + @xmath289 otherwise . * for @xmath182 , write @xmath290 , where @xmath291 or @xmath46 and @xmath292 . then : + @xmath287 if @xmath293 , @xmath294 , + @xmath295 if @xmath296 , @xmath297 , @xmath298 , + @xmath299 $ ] in the remaining cases . * for @xmath182 , write @xmath300,where @xmath291 or @xmath46 and @xmath292 . then : + @xmath287 if @xmath293 , @xmath294 , + @xmath295 if @xmath296 , @xmath297 , @xmath298 + @xmath301 $ ] in the remaining cases . in the above analysis the group @xmath302 appears in case ( a ) and case ( c ) for @xmath303 . in case ( a ) , the cover @xmath41 is standard : @xmath75 appears twice among the branch data , both times with label @xmath231 . in case ( c ) , @xmath41 is standard for @xmath303 : @xmath75 appears twice among the branch data , with labels @xmath304 and @xmath305 corresponding to the subgroups of order 2 of @xmath231 distinct from @xmath237 . moreover , it is not difficult to check that in case ( b ) and in case ( c ) for @xmath306 the cover is not standard . so we have proven ( 3 ) and also that every component of the hurwitz divisor @xmath32 of a standard g.d.c . cover has multiplicity @xmath307 . vice versa , assume that @xmath41 is standard and @xmath75 appears in @xmath32 with multiplicity @xmath208 . if the multiplicity is @xmath207 then the cover is normal over @xmath75 . if the multiplicity is equal to @xmath46 , then @xmath75 appears twice among the branch data , and the corresponding subgroups @xmath308 and @xmath309 have order 2 . if @xmath310 , then the cover is given over the generic point of @xmath75 by the equation @xmath311 , with @xmath312 a unit ; so it is g.d.c . if @xmath313 , then the cover is given by the equations @xmath314 , @xmath315 , with @xmath316 and @xmath317 units . these equations are equivalent to @xmath318 , so the cover is g.d.c .. this completes the proof of ( 2 ) . let @xmath51 be a @xmath1-cover such that @xmath2 is normal and @xmath3 is @xmath4 . let @xmath57 be the nonsingular locus of @xmath2 . then the restriction @xmath55 is a @xmath1-cover , and by lemma [ lem : remove - codim2 ] @xmath41 is the unique @xmath4-extension of @xmath58 to @xmath2 . thus the theory in the normal case is the immediate extension of the nonsingular case . we record the changes : 1 . the sheaves @xmath40 are no longer invertible but they are @xmath4 , i.e. in this case reflexive , divisorial sheaves . the multiplication maps are @xmath319 2 . the branch divisors @xmath320 are weil divisors . otherwise , the same fundamental relations between @xmath40 and @xmath320 must hold . one has to be careful that the morphism @xmath41 may be not flat ; indeed , it is flat iff all @xmath40 are invertible . also , for a singular @xmath2 the branch locus may have non - divisorial components . let @xmath321 $ ] , @xmath322 acting by @xmath323 , @xmath324 , and let @xmath2 be the quotient @xmath325 $ ] , a quadratic cone . then @xmath41 is ramified only over the vertex @xmath326 of the cone . the divisors @xmath320 are zero . the eigensheaves are @xmath327 and @xmath328 , the divisorial sheaf corresponding to a line @xmath329 through the vertex . @xmath328 is also isomorphic to the @xmath52-submodule of @xmath48 generated by @xmath166 and @xmath158 . the fundamental relation in this case is @xmath330 . now we assume that @xmath2 is a non normal g.d.c . and @xmath4 variety . let @xmath21 be the divisorial part of the singular locus of @xmath2 , let @xmath331 be the normalization , let @xmath332 be the inverse image of @xmath21 in @xmath13 and let @xmath333 be the normalization . since @xmath2 is g.d.c . , there is a biregular involution @xmath334 on @xmath335 induced by the degree 2 map @xmath336 . ( if the components of @xmath2 are smooth , then @xmath335 is a union of several pairs of varieties , exchanged by the involution @xmath334 . in general , some components of @xmath337 map to themselves ) . consider a commutative diagram : where @xmath3 and @xmath338 are g.d.c . and @xmath4 varieties , the vertical arrows are @xmath1-covers , @xmath339 is a cover as in the previous section , and @xmath340 is a birational morphism . we denote by @xmath341 the preimages of @xmath342 in @xmath343 , and by @xmath344 the normalization of @xmath345 . @xmath346 \ar[dd ] \ar@(ul , dl)[]_{j } & & b ' \ar[rr ] \ar[dd ] \ar@{^(->}[dr ] & & b \ar@{^(->}[dr ] \ar[dd ] \\ & & & & & x ' \ar[rr ] \ar[dd ] & & x \ar[dd ] \\ & & \wt{c ' } \ar[rr ] \ar@(ul , dl)[]_{\iota } & & c ' \ar[rr ] \ar@{^(->}[dr ] & & c \ar@{^(->}[dr ] \\ & & & & & \wy \ar^{\nu}[rr ] & & y } \ ] ] we first give two constructions for the cover @xmath0 starting with @xmath347 and the appropriate data for the double locus . one construction proceeds by @xmath4-fication of the `` nice '' part . the second one is by a gluing procedure , and the result is very convenient for computing the invariants of @xmath3 . finally , we show that indeed every @xmath0 comes from these constructions . [ thm : glue ] suppose we are given 1 . @xmath2 , @xmath13 , @xmath332 , @xmath348 , 2 . a @xmath1-cover @xmath339 , with @xmath338 an @xmath4 and g.d.c . variety , let @xmath349 be the induced cover and let @xmath350 be its normalization . then @xmath338 can be glued to a cover @xmath0 with @xmath3 g.d.c . and @xmath4 if and only if it is generically smooth along @xmath345 , and there exists an involution @xmath351 that covers the involution @xmath352 and commutes with the action of @xmath1 on @xmath344 . assume that @xmath3 exists . then the map @xmath353 induces an involution @xmath354 as required . in addition , if @xmath338 were not generically smooth along a component @xmath75 of @xmath345 , then @xmath3 would have generically at least three branches along the image of @xmath75 . thus these two conditions on @xmath338 are necessary for the existence of @xmath3 . next we show that they are also sufficient . we start by identifying the `` bad locus '' . it includes the singular locus of @xmath13 , the intersection of branch divisors between themselves and with @xmath332 . the image of this bad locus in @xmath2 has codimension @xmath355 . let @xmath57 be its complement , and restrict all varieties and covers to @xmath57 . the condition that the involution @xmath354 commutes with the @xmath1-action implies that for any irreducible component @xmath75 of @xmath345 the subgroup @xmath231 of elements of @xmath1 that fix @xmath75 pointwise is the same as the supgroup of elements that fix @xmath356 pointwise . since @xmath338 is generically smooth along @xmath345 , one has ( cf . @xcite ) @xmath357 for some @xmath50 and , working tale - locally , @xmath231 acts locally by @xmath358 near @xmath75 and by @xmath359 near @xmath356 for some primitive root @xmath360 and @xmath361 . here @xmath362 , @xmath363 . we glue @xmath364 along @xmath365 to obtain a variety @xmath56 with a finite morphism to @xmath57 . the @xmath1-action extends to @xmath56 , because @xmath354 commutes with the @xmath1-action , and is of the type ( smooth ) @xmath366 ( compatible action of curves ) , where `` compatible '' means that , working tale - locally , @xmath367 acts on @xmath206 by @xmath368 , @xmath369 over the double locus we have @xmath370/(xy)$ ] and the ring of @xmath371-invariants is @xmath372/(uv)$ ] , where @xmath373 and @xmath374 . thus , @xmath56 has only normal crossings and @xmath375 is a @xmath1-cover . finally , we apply lemma [ lem : remove - codim2 ] to obtain an @xmath4 and g.d.c . cover @xmath0 by taking @xmath4-fication . we obtain @xmath3 by gluing @xmath338 along the involution @xmath376 , i.e. as the pushout of the following commutative diagram : since all varieties are affine over @xmath2 , @xmath48 is the fiber product of the corresponding diagram of @xmath52-algebras , in which we identify sheaves with their pushforwards on @xmath2 . we can rewrite this fiber product diagram by saying that @xmath48 is the kernel in the exact sequence @xmath377 further , we have @xmath378 where @xmath379 is the alternating part ( if @xmath380 then @xmath381 ) , and the image of @xmath382 contains @xmath383 . hence , we have induced exact sequences @xmath384 the thus defined variety @xmath3 is @xmath4 by the next lemma [ lem : sings - after - gluing ] , since @xmath385 is a subsheaf of @xmath379 and so obviously does not have embedded primes . it is g.d.c . again by looking in codimension 1 as in the previous proof . the @xmath1-action on @xmath338 descends to a @xmath1-action on @xmath3 since @xmath354 commutes with the @xmath1-action on @xmath344 and by construction the subalgebra of @xmath1-invariants is the algebra of @xmath386 glued along @xmath387 , i.e. @xmath52 . the varieties @xmath3 obtained in the two proofs coincide , since they both have finite morphisms to @xmath2 , are both @xmath4 and they coincide over an open subset @xmath388 with @xmath389 . [ warn : glue ] it may happen that there is no covering involution of @xmath345 but only of its normalization @xmath390 . then the double locus of @xmath3 is obtained from @xmath391 by some additional gluing in codimension 1 ( codimension 2 for @xmath3 ) . as a consequence , branches of @xmath3 may not be @xmath4 . but the variety @xmath3 is @xmath4 . @xcite contains multiple examples of this phenomenon . on the other hand , the involution @xmath354 need not be unique . for instance , if @xmath165 has order 2 , then @xmath392 is another involution satisfying the assumptions for gluing . the next example shows that gluing via different involutions can give rise to non isomorphic covers . [ ex : pinch ] let @xmath393 . the normalization of @xmath2 is the map @xmath394 defined by @xmath395 . here @xmath396 , @xmath397 and the involution @xmath334 of @xmath335 is given by @xmath398 . let @xmath399 and let @xmath400 be the trivial @xmath401 cover , given by the projection on the coordinates @xmath402 . the @xmath401-action is @xmath403 and @xmath404 . there are two involutions of @xmath344 that lift @xmath334 , namely @xmath405 and @xmath406 . the cover @xmath407 obtained by gluing via @xmath408 is obviously the trivial @xmath401-cover . we describe the cover @xmath409 obtained by gluing via @xmath410 following the second proof of theorem [ thm : glue ] . the map @xmath411 corresponds to the inclusion @xmath412\to \bk[s,\epsilon]/(\epsilon^2 - 1]$ ] and the map @xmath413 corresponds to the surjection @xmath414/(\epsilon^2 - 1 ) \to \bk[s,\epsilon]/(\epsilon^2 - 1]$ ] . the fiber product of these two ring maps can be identified with @xmath415/(\epsilon^2 - 1)\subset \bk[s , t,\epsilon]/(\epsilon^2 - 1)$ ] . the map @xmath416/(x^2-y^2)$ ] defined by @xmath417 , @xmath418 , @xmath419 is an isomorphism , hence @xmath420 is the union of two copies of @xmath421 glued along a line . the cover @xmath409 is given by @xmath422 and the @xmath401-action on @xmath3 is given by @xmath423 , thus @xmath424 is the only branch point . so the ramification locus of a standard @xmath1-cover has always pure codimension 1 but this not true for the @xmath1-covers obtained from a standard cover by gluing and the analogue of lemma [ lem : inertia ] does not hold . [ lem : sings - after - gluing ] with the notations as in the 2nd proof by gluing , assume that @xmath338 is @xmath49 for some @xmath425 . then @xmath3 is @xmath49 iff @xmath385 is @xmath426 . we use the cohomological interpretation of depth using local cohomology @xcite ( alternatively and equivalently one can use @xmath427 ) . a sheaf @xmath186 satisfies @xmath49 iff for every irreducible subvariety @xmath428 one has @xmath429 for all @xmath430 . looking at the long exact sequence of cohomologies corresponding to the short exact sequence , we get @xmath431 for all @xmath432 . the statement now follows . we spell out theorem [ thm : glue ] in a special case , which is of interest to us because of the applications in @xcite . [ ex:2surfaces ] take @xmath106 . for simplicity of exposition , we assume that @xmath433 is the g.d.c . union of two smooth projective surfaces that intersect along a smooth rational curve @xmath21 , but all our considerations generalize straightforwardly to the case of a g.d.c . surface with smooth components whose double locus is a union of smooth rational curves . we have @xmath434 , hence an @xmath4 and g.d.c . @xmath1-cover @xmath339 is the disjoint union of @xmath4 and g.d.c . covers @xmath435 , @xmath182 . by corollary [ cor : structurez2 ] , the covers @xmath436 are standard . we denote by @xmath437 , @xmath438 the branch data of @xmath436 , @xmath182 . we write @xmath439 , @xmath440 and @xmath441 . we denote by @xmath442 the generator of subgroup @xmath443 . an involution @xmath354 of @xmath344 as in theorem [ thm : glue ] exists if and only if there is an isomorphism @xmath444 compatible with the @xmath1-action . this is equivalent to the following conditions : 1 . @xmath445 , 2 . for @xmath446 , denote by @xmath447 the intersection multiplicity at @xmath158 of @xmath448 with @xmath449 , @xmath450 and by @xmath451 the intersection multiplicity at @xmath158 of @xmath452 with @xmath453 , @xmath454 . then : + @xmath455 indeed , ( 1 ) follows immediately by the fact that @xmath354 commutes with the action of @xmath1 . in addition , by the normalization algorithm of @xcite condition ( 2 ) is equivalent to requiring that the branch data of the normalizations @xmath456 and @xmath457 of the @xmath458-coverings of @xmath459 induced by @xmath184 and @xmath185 are the same . since @xmath21 is smooth rational , the branch data are enough to determine the building data ( cf . remark [ rem : branchdata ] ) . since @xmath21 is projective , the building data determine the cover up to isomorphism by proposition [ prop : bdata ] . assume that the gluing conditions are satisfied . giving an involution of @xmath344 that commutes with the @xmath1 action is the same as giving an isomorphism of @xmath1-covers @xmath460 . then any other such map @xmath461 is equal to @xmath462 for some @xmath165 and the automorphism of @xmath463 defined by @xmath464 if @xmath465 and @xmath466 if @xmath467 induces an isomorphism of the cover of @xmath2 obtained by gluing via @xmath468 with the one obtained by gluing via @xmath461 . so in this case all the possible involutions give isomorphic covers . [ thm : the - reverse ] vice versa , every @xmath1-cover @xmath0 with g.d.c . @xmath4 varieties @xmath469 is obtained via the gluing construction of theorem [ thm : glue ] . given @xmath0 and the normalization @xmath470 , let @xmath471 be the fiber product @xmath472 . we define @xmath338 as @xmath473 . thus , @xmath338 is @xmath4 by definition , and it maps to @xmath13 . by the universality of taking the reduced part and @xmath4-fication , there is an induced @xmath1-action on @xmath338 . by the universal property of @xmath1-quotients , we also have a morphism @xmath474 . we claim that it is an isomorphism . it is enough to check this in codimension one over the double locus . we claim that generically over the double locus of @xmath2 , the cover is ( smooth ) @xmath366 ( admissible action of curves ) , where `` admissible '' means that , working tale - locally , @xmath3 is given by @xmath206 , and the action is @xmath368 , @xmath369 for some primitive root @xmath360 and @xmath361 . indeed , let @xmath78 be the subgroup of elements that restrict to the identity on an irreducible component @xmath75 of the double locus of @xmath3 . then on the normalization on both branches we have the same subgroup for the preimages @xmath244 and @xmath475 . since generically @xmath476 are smooth , @xmath477 for some @xmath478 ( note that one possibly has @xmath479 ) . thus , tale locally the morphism @xmath0 can be written as @xmath480/(uv)\to \bk[x , y]/(xy ) , \qquad u\mapsto x^n,\ v\mapsto y^n,\ ] ] where @xmath1 acts as @xmath368 , @xmath369 , @xmath360 , @xmath361 . computing , we get that @xmath471 corresponds to ( smooth)@xmath481/(xy , y^n ) \oplus \bk[x , y]/(xy , x^n)$ ] , and @xmath338 to @xmath482\oplus \bk[y]$ ] . the quotient @xmath483 is then @xmath484\oplus \bk[v]$ ] , i.e. @xmath13 . this proves that @xmath485 is an isomorphism outside a closed subset of codimension @xmath355 . since both are finite over @xmath2 and @xmath4 , @xmath277 is an isomorphism . let @xmath245 be a variety , let @xmath486 , @xmath487 , be effective weil divisors on @xmath3 , possibly reducible , and let @xmath488 be rational numbers with @xmath489 . set @xmath490 . [ defn : lc ] assume that @xmath245 is a _ normal _ variety . then @xmath245 has a canonical weil divisor @xmath491 defined up to linear equivalence . the pair @xmath492 is called _ log canonical _ if 1 . @xmath493 is @xmath494-cartier , i.e. some positive multiple is a cartier divisor , and 2 . every prime divisor of @xmath245 has multiplicity @xmath208 in @xmath495 and for every proper birational morphism @xmath496 with normal @xmath497 , in the natural formula @xmath498 one has @xmath499 . here , @xmath500 are the irreducible exceptional divisors of @xmath501 , the pullback @xmath502 is defined by extending @xmath494-linearly the pullback on cartier divisors , @xmath503 is the strict preimage of @xmath495 . the coefficients @xmath504 are called _ discrepancies_. for the non - exceptional divisors , already appearing on @xmath245 , one defines @xmath505 . + if @xmath506 , then @xmath245 has a resolution of singularities @xmath496 such that @xmath507 is a normal crossing divisor ; then it is sufficient to check the condition @xmath508 for this morphism @xmath501 only . a pair @xmath492 is called _ semi log canonical _ if 1 . @xmath245 satisfies serre s condition @xmath4 , 2 . @xmath245 is g.d.c . , and no divisor @xmath486 contains any component of the double locus of @xmath245 , 3 . some multiple of the weil @xmath494-divisor @xmath493 , well defined thanks to the previous condition , is cartier , and 4 . denoting by @xmath509 the normalization , the pair @xmath510 is log canonical . [ lem : adjunction ] let @xmath511 be a finite morphism of degree @xmath145 between equidimensional @xmath4 varieties . assume that either @xmath512 , or @xmath152 is galois and @xmath513 does not divide @xmath145 . let @xmath57 be an open subset and denote by @xmath514 the induced cover . assume that : * @xmath515 and both @xmath56 and @xmath57 are d.c . , * there exist effective @xmath516-divisors @xmath517 of @xmath3 and @xmath518 of @xmath2 , not containing any component of the double locus , such that @xmath519 , where @xmath520 is the restriction of @xmath518 to @xmath57 and @xmath521 is the restriction of @xmath517 to @xmath56 . then : 1 . @xmath522 is @xmath516-cartier iff so is @xmath523 . 2 . the pair @xmath524 is slc iff so is the pair @xmath525 . \(1 ) let @xmath526 be the inclusion map . if the sheaf @xmath527 is invertible then we have a homomorphism @xmath528 which is an isomorphism outside of codimension 2 . so it must be an isomorphism by the @xmath4 condition . similarly , if the sheaf @xmath529 is invertible then the sheaf @xmath530 is isomorphic to the norm of @xmath531 , so is invertible . \(2 ) assume first that @xmath3 and @xmath2 are normal . in the case this statement , due to shokurov , is very well known . we recall the proof because usually it is only stated and proved in characteristic zero . let @xmath532 be some partial resolution with normal @xmath533 , @xmath338 be the normalization of @xmath534 , and let @xmath535 , @xmath536 be the induced maps . pick an irreducible divisor @xmath537 on @xmath533 , and let @xmath75 be an irreducible divisor on @xmath338 over it . by our condition on @xmath513 , the field extension @xmath538 is separable , and if @xmath539 , @xmath540 are uniformizing parameters in the dvrs @xmath541 and @xmath542 , then one has @xmath543 for a unit @xmath312 and some integer @xmath544 dividing @xmath145 and hence coprime to @xmath513 . then riemann - hurwitz formula applies and says that generically along @xmath537 and @xmath75 one has @xmath545 . comparing this to the identity @xmath546 and the definition of the log discrepancy , one obtains that @xmath547 . thus , @xmath548 @xmath549 @xmath550 . this proves that @xmath551 is lc iff @xmath524 is lc . now consider the general g.d.c . let @xmath552 be the normalization . we have @xmath553 and similarly for @xmath2 . thus , the double loci appear in the divisors @xmath554 , @xmath555 with coefficient 1 . by the riemann - hurwitz formula again , for the normalizations we still have @xmath556 . we conclude by applying the normal case . we now extend definition [ defn : hurwitz ] of hurwitz divisor to the case of a g.d.c . base @xmath2 : [ defn : hurwitz2 ] let @xmath51 be a @xmath1-cover of @xmath4 and g.d.c . varieties . for a prime weil divisor @xmath557 , we define @xmath558 as follows : * if @xmath75 is contained in the double locus of @xmath2 , then @xmath559 ; * if @xmath75 is not contained in the double locus of @xmath2 , but @xmath76 is contained in the double locus of @xmath3 , then @xmath560 , * if @xmath75 is not contained in the double locus of @xmath2 , @xmath76 is not contained in the double locus of @xmath3 and @xmath104 is the ramification order of @xmath41 at @xmath75 , then @xmath561 . we define the hurwitz divisor @xmath32 of @xmath41 to be the @xmath204-divisor @xmath562 . notice that if @xmath0 is a standard @xmath1-cover with @xmath3 g.d.c . this definition coincides with definition [ defn : hurwitz ] by theorem [ thm : structure ] . note that @xmath32 does not contain any components of the double locus of @xmath2 . [ prop : interesting - case ] let @xmath51 be a @xmath1-cover as in definition [ defn : hurwitz2 ] and let @xmath32 be the hurwitz divisor of @xmath41 , let @xmath339 be the corresponding @xmath4 and g.d.c . @xmath1-cover ( cf. [ ssec : ynonnormal ] ) then 1 . @xmath563 is @xmath516-cartier iff so is @xmath564 , and then @xmath565 . 2 . @xmath3 is slc iff so is the pair @xmath566 . recall that @xmath567 and @xmath568 are coprime by assumption . so lemma [ lem : adjunction ] applies and we may assume that @xmath2 is d.c .. we need to show that @xmath569 this is equivalent to the following equality for the cover @xmath570 , where @xmath19 is the normalization of @xmath338 ( and of @xmath3 ) : @xmath571 in view of definition [ defn : hurwitz2 ] the formula follows easily by the usual hurwitz formula . by lemma [ lem : flatness - and - cm ] , a @xmath1-cover over a smooth base is cm . here , we give a partial generalization of this case to the case of a non - normal base . we use the notations of theorem [ thm : glue ] and the exact sequence . [ prop : cm ] assume that @xmath338 is cm ( for example , @xmath13 is smooth ) . then @xmath3 is cm iff the sheaf @xmath385 is cm . immediate by lemma [ lem : sings - after - gluing ] . using proposition [ prop : cm ] it is not hard to give examples of abelian covers @xmath0 such that @xmath2 is cm and g.d.c . , and @xmath3 is g.d.c . and @xmath4 but not cm : [ ex : noncm ] we take @xmath572 and assume @xmath573 ; for any prime @xmath574 one can construct similar examples with @xmath575 and @xmath576 . let @xmath433 be the union of 2 copies of @xmath577 glued transversally along a plane @xmath21 . let @xmath578 and @xmath579 be distinct lines on @xmath21 and for @xmath182 let @xmath580 be a quadric that restricts to @xmath581 on @xmath21 . for a generic choice , @xmath85 is a quadric cone with vertex @xmath582 and the points @xmath583 , @xmath584 and @xmath585 are distinct . let @xmath586 be the double cover of @xmath7 branched on @xmath85 and let @xmath587 . then @xmath338 is gorenstein , has an ordinary double point over @xmath583 and @xmath584 and no other singularity . write @xmath588 and @xmath440 ; then @xmath589 is the union of two copies of @xmath590 glued transversally along @xmath591 and @xmath592 is the trivial @xmath401-cover . hence there exists an involution @xmath354 of @xmath344 that commutes with the @xmath401-action , and by theorem [ thm : glue ] @xmath338 can be glued to an @xmath4 and g.d.c . cover @xmath0 . locus of @xmath3 is the complement of the preimage of @xmath593 . in the exact sequence each term splits under the @xmath1-action and the maps are compatible with the splitting , so we get two exact sequences , one for each character of @xmath1 . since @xmath594 and @xmath401 acts on @xmath379 by switching the two summands , the sequence for the nontrivial character is : @xmath595 where @xmath596 ( resp . @xmath597 ) is the antiinvariant summand of @xmath598 ( resp . of @xmath379 ) . by definition , the map @xmath599 factorizes as @xmath600 . hence , @xmath601 coincides with @xmath602 , the maximal ideal of @xmath603 in @xmath21 , and therefore it is not @xmath4 . it follows by proposition [ prop : cm ] that @xmath3 is not cm over @xmath603 . let @xmath604 be a point distinct from @xmath603 ; in a neighbourhood of @xmath605 we have @xmath606 , thus @xmath607 is not @xmath516-cartier . since @xmath2 is gorenstein , it follows that @xmath608 is not @xmath516-cartier either , hence @xmath563 is not @xmath516-cartier by proposition [ prop : interesting - case ] . all the statements in this section are tale local , so we often pass to a smaller neighbourhood of a point without explicit mention of the fact . for convenience , we write `` @xmath563 '' to denote the divisorial sheaf @xmath609 ( recall that @xmath3 is gorenstein in codimension 1 and @xmath4 ) . we also use the additive notation @xmath607 for the sheaf @xmath610 . we consider the following situation : * @xmath2 is a normal variety and @xmath21 is a reduced effective divisor on @xmath2 such that @xmath611 is cartier ; * @xmath51 is a standard g.d.c . @xmath1-cover ( so @xmath3 is automatically @xmath4 by lemma [ lem : flatness - and - cm ] ) . we assume that @xmath3 is generically smooth over @xmath21 and we denote by @xmath495 the preimage of @xmath21 in @xmath3 . so @xmath495 is also a reduced effective divisor . let @xmath32 be the hurwitz divisor of @xmath41 ; then we have : @xmath612 thus , if @xmath145 is the exponent of @xmath1 , then the divisor @xmath613 is cartier ( recall that the divisors @xmath85 are cartier by the definition of a standard cover in section [ ssec : standard ] ) and thus @xmath614 is also cartier . fix a point @xmath42 ; the purpose of this section is to compute the cartier index of @xmath615 at a point @xmath160 such that @xmath616 . here we are interested mainly in the case @xmath617 , but the case of a pair is needed in the next section to treat the case @xmath2 non - normal . in order to state our result we need some notation . we label the branch data @xmath618 , @xmath619 , in such a way that @xmath620 iff @xmath621 . since the question is local on @xmath2 we may assume that @xmath164 for every @xmath163 . consider the map @xmath622 . by lemma [ lem : inertia ] the image of this map is the inertia subgroup @xmath157 ; we denote by @xmath623 the kernel . we let @xmath624 be the character @xmath625 . since the group @xmath1 is finite abelian , the map @xmath626 is surjective . so the character @xmath627 is the pullback of a character of @xmath157 iff it is the pullback of a character of @xmath1 . [ prop : index - normal ] notation and assumptions as above . + the cartier index of @xmath615 at @xmath166 is equal to the order of @xmath628 . in particular , @xmath615 is cartier iff @xmath627 is the pull back of a character @xmath167 . since the question is local , we may assume that the line bundles @xmath11 , @xmath629 and @xmath630 are trivial . the map @xmath631 is tale , hence up to replacing @xmath2 by @xmath632 we may assume that @xmath633 , or , equivalently , that @xmath634 . we denote by @xmath635 local equations of @xmath84 near @xmath158 . by remark [ rem : local ] , up to passing to an tale cover of @xmath2 we may assume that @xmath3 is given by : @xmath636 the equations : @xmath637 define inside @xmath638 a @xmath639-cover @xmath640 ( @xmath639 acts on @xmath641 via the character @xmath89 ) , the _ maximal totally ramified cover _ of @xmath2 with branch data @xmath642 ( here we regard @xmath87 as a subgroup of @xmath639 ) . since @xmath2 is g.d.c . by assumption and @xmath0 and @xmath640 have the same hurwitz divisor , @xmath643 is also g.d.c . by theorem [ thm : structure ] . for every @xmath167 , write @xmath644 , with @xmath645 for @xmath619 ; then setting @xmath646 defines a map @xmath647 which is the quotient map for the action of the kernel @xmath623 of @xmath648 . the map @xmath574 is unramified in codimension 1 and @xmath649 consists of just one point @xmath650 . denote by @xmath651 the preimage of @xmath21 ( and of @xmath495 ) in @xmath643 ; observe that @xmath652 pulls back to @xmath615 on @xmath3 and to @xmath653 on @xmath643 . if @xmath654 is a generator of @xmath630 then @xmath655 is generated by the residue @xmath656 on @xmath643 of the rational differential form : @xmath657 thus @xmath655 is invertible and @xmath1 acts on the local generator @xmath658 via the character @xmath659 . set @xmath660 . the map @xmath661 is unramified in codimension 1 and @xmath658 descends on @xmath245 to a generator of @xmath662 , where @xmath663 is the image of @xmath651 . the map @xmath664 is a cyclic cover with galois group @xmath665 with the following properties : * it is unramified in codimension 1 and the preimage of @xmath166 consists only of one point , * the pull back of @xmath666 is a line bundle on which the galois group acts via a primitive character . it follows that @xmath664 is a canonical cover and that the cartier index of @xmath615 at @xmath166 is equal to @xmath667 $ ] . [ cor : gor - smooth ] let @xmath668 be a standard abelian with @xmath3 and @xmath2 g.d.c . and @xmath2 gorenstein , let @xmath42 and let @xmath160 be a point such that @xmath616 . then @xmath3 is gorenstein at @xmath166 iff the character @xmath669 descends to a character @xmath12 of @xmath157 . @xmath3 is cohen - macaulay by lemma [ lem : flatness - and - cm ] and @xmath563 is cartier by proposition [ prop : index - normal ] . corollary [ cor : gor - smooth ] is proven in @xcite under the assumption that @xmath3 is normal and @xmath2 is smooth . here we consider the problem of determining the cartier index of @xmath563 at a point @xmath160 of a @xmath1-cover @xmath0 with @xmath2 non - normal of cartier index 1 . the situation is much more complicated than in the case @xmath2 normal and we are able to give only a partial answer , that is however sufficient for the applications in @xcite . the main difficulty is that one does not know how to write down an analogue of the maximal totally ramified cover used in the proof of proposition [ prop : index - normal ] . we consider the following setup : * @xmath670 , where @xmath7 is irreducible for @xmath173 , is a g.d.c . and @xmath4 variety ; @xmath671 is the normalization , * @xmath51 is an @xmath4 and g.d.c . @xmath1-cover obtained by gluing a cover @xmath672 such that @xmath673 is standard for every @xmath163 , * @xmath42 and @xmath160 are points such that @xmath616 ; we assume that @xmath158 lies on every component of the branch locus of @xmath41 . we denote by @xmath618 , @xmath619 the branch data of the standard cover @xmath674 and we assume that @xmath85 is contained in the preimage @xmath332 of the double locus of @xmath2 if and only if @xmath621 . consider the map @xmath622 . as in the case @xmath2 normal , we denote by @xmath675 the character @xmath625 . then : [ prop : index - non - normal ] in the above setup , if @xmath563 is cartier , then : 1 . @xmath564 is @xmath204-cartier , 2 . @xmath669 is the pullback of a character @xmath167 . \(1 ) follows immediately by proposition [ prop : interesting - case ] . \(2 ) for every @xmath173 denote by @xmath676 ( resp . @xmath677 ) the preimage of the double locus of @xmath2 in @xmath678 ( resp . in @xmath679 ) . let @xmath167 be the character via which @xmath1 acts on @xmath680 at @xmath166 . let @xmath681 be a point that maps to @xmath166 and let @xmath682 be the image of @xmath683 in @xmath678 . since @xmath563 pulls back to @xmath684 on @xmath679 , the inertia subgroup @xmath685 acts on @xmath686 via the restriction of @xmath12 . set @xmath687 and let @xmath688 be the restriction of @xmath669 to @xmath689 ; the map @xmath690 is a surjection by lemma [ lem : inertia ] . by the proof of proposition [ prop : index - normal ] @xmath12 pulls back on @xmath689 to @xmath688 . since @xmath691 , it follows that @xmath12 pulls back to @xmath669 on @xmath639 . we now prove a partial converse of proposition [ prop : index - non - normal ] . assume that for every component @xmath678 of @xmath13 the map @xmath692 induces a homeomorphism @xmath693 onto its image ( this is always true up to an tale cover ) . then we associate to @xmath201 an incidence graph @xmath694 as follows : the vertices of @xmath694 are indexed by the branches of @xmath201 , the edges are indexed by the components of the double locus @xmath21 of @xmath2 , the edge corresponding to a component @xmath75 of @xmath21 connects the vertices corresponding to the two branches of @xmath2 through @xmath75 . [ prop : index - graph ] in the above setup , assume that : 1 . the graph @xmath694 is a tree , 2 . @xmath695 is cartier and there exists @xmath104 such that @xmath696 is cartier and @xmath697 , 3 . @xmath669 is the pullback of a character @xmath167 . then @xmath563 is cartier . let @xmath698 the restriction of the double locus @xmath332 of @xmath386 and let @xmath677 be the preimage of @xmath590 . let @xmath699 be the only point that maps to @xmath42 ; let @xmath700 and @xmath701 be defined as in the proof of proposition [ prop : index - non - normal ] . by assumption ( 3 ) , the divisor @xmath684 is cartier by proposition [ prop : index - normal ] . by the following lemma [ lem : character ] , up to replacing @xmath201 by an tale neighbourhood we may assume that for @xmath173 the sheaf @xmath702 is trivial and has a generator @xmath136 on which @xmath1 acts via @xmath12 . by proposition [ prop : interesting - case ] , there exists a local generator @xmath654 of @xmath703 near @xmath166 . for every @xmath163 , by lemma [ lem : character ] , @xmath654 pulls back on @xmath679 to @xmath704 where @xmath705 is a nowhere vanishing regular function on @xmath678 . up to passing to an tale cover of @xmath2 we may assume that @xmath705 has an @xmath104-th root @xmath706 for every @xmath163 . so we may replace @xmath136 by @xmath707 and assume that @xmath654 pulls back to @xmath708 for every @xmath163 . now let @xmath62 be an open set such that @xmath709 is d.c . and the complement of @xmath709 has codimension @xmath710 . let @xmath75 be an irreducible component of the double locus @xmath21 of @xmath2 and let @xmath711 , @xmath712 be the components of @xmath2 that contain @xmath75 . choose an irreducible component @xmath537 of the inverse image of @xmath75 in @xmath709 . it makes sense to compare @xmath713 and @xmath714 along @xmath537 , since they both restrict to local generators of @xmath715 . since @xmath716 , there exists @xmath717 such that @xmath718 along @xmath537 . since @xmath1 acts on @xmath713 and @xmath714 via the same character @xmath12 and @xmath1 acts transitively on the components of the preimage of @xmath75 , @xmath719 depends only on @xmath75 . so @xmath720 represents a class in @xmath721 . since @xmath694 is a tree , we can find @xmath722 such that the local generators @xmath723 glue to give a local generator @xmath658 of @xmath724 on which @xmath1 acts via @xmath12 . we complete the proof of proposition [ prop : index - graph ] by proving the following : [ lem : character ] let @xmath725 be a standard @xmath1-cover with building data @xmath726 . let @xmath727 be a point and let @xmath231 be the inertia subgroup of @xmath255 . let @xmath82 be a @xmath1-linearized line bundle of @xmath245 , let @xmath728 be a point that maps to @xmath255 and let @xmath729 be the character via which @xmath231 acts on @xmath730 . then : 1 . let @xmath167 be such that @xmath731 ; then , up to replacing @xmath732 by an tale neighbourhood of @xmath255 , there exists a generator @xmath656 of @xmath82 such that @xmath1 acts on @xmath656 via the character @xmath12 ; 2 . @xmath656 is uniquely determined by @xmath12 up to multiplication by a nowhere vanishing regular function of @xmath732 . \(2 ) assume that @xmath733 are generators of @xmath82 on which @xmath1 acts via the character @xmath12 . then @xmath734 is a regular @xmath231-invariant function on @xmath245 , so it is a function on @xmath732 . \(1 ) we break the proof into three steps . let @xmath210 be a generator of @xmath82 near @xmath254 . the group @xmath231 acts on the vector space @xmath735 of local sections of @xmath82 spanned by the elements @xmath736 , @xmath737 . @xmath735 is finite - dimensional , and decomposes under the @xmath1-action as a direct sum of eigenspaces . since @xmath738 and @xmath739 , there exists an eigenvector @xmath740 such that @xmath741 . since @xmath1 acts on @xmath730 via @xmath12 , @xmath658 belongs to the eigenspace corresponding to @xmath12 . consider the factorization @xmath742 . the map @xmath743 is an @xmath231-cover such that the preimage of @xmath255 consists of one point @xmath744 . the subgroup @xmath623 acts freely on @xmath245 , hence @xmath82 descends to an @xmath231-linearized line bundle @xmath531 on @xmath497 . then by step 1 there exists a local generator @xmath745 of @xmath531 near @xmath746 such that @xmath231 acts on @xmath745 via @xmath277 . pulling back to @xmath245 we get a generator @xmath654 of @xmath82 on which @xmath231 acts via @xmath277 and @xmath623 acts trivially . denote by @xmath747 the restriction of @xmath12 to @xmath623 , so that @xmath748 . consider the factorization @xmath749 . the map @xmath750 is a tale @xmath623-cover . so there exists a nowhere vanishing function @xmath152 on @xmath751 such that @xmath623 acts on @xmath152 via the character @xmath277 . thus @xmath1 acts on @xmath752 via the character @xmath12 . choose a finite abelian group @xmath623 with a surjective map @xmath753 that extends the inclusion @xmath754 and let @xmath755 be the kernel of @xmath756 . by proposition [ prop : bdata ] , up to replacing @xmath732 by an tale neighbourhood of @xmath255 , we may also assume ( cf . ) that @xmath757 is given inside @xmath758 by the equations : @xmath759 where @xmath760 is a local equation for @xmath85 , @xmath619 . the branch data for @xmath245 can be interpreted in an obvious way as branch data for a @xmath761-cover . letting @xmath762 be the @xmath761-cover given by the equations analogous to , we have @xmath763 by construction . let @xmath764 be the pull back of @xmath82 to @xmath765 . @xmath764 has a natural @xmath761-linearization and @xmath231 is a direct summand of @xmath761 , hence by step 2 there exists a generator @xmath766 of @xmath764 on which @xmath761 acts via the character @xmath767 of @xmath761 induced by @xmath12 . since @xmath755 acts freely on @xmath765 and @xmath768 by construction , @xmath766 descends to a generator @xmath658 of @xmath82 on @xmath245 on which @xmath1 acts via @xmath12 . in this section we make a detailed study of @xmath6-covers of surfaces . we use freely the notation introduced in [ ssec : ynormal ] . in particular , we refer the reader to the commutative diagram ( [ big - diagram ] ) and theorem [ thm : glue ] . * @xmath2 is a g.d.c . surface with smooth irreducible components @xmath769 . the irreducible components @xmath770 of the double curve @xmath21 of @xmath2 are smooth , @xmath2 is d.c . at the smooth points of @xmath21 and it is analytically isomorphic to the cone over a cycle of rational curves at the singular points of @xmath21 . in particular , @xmath2 is gorenstein . * @xmath106 and @xmath51 is a @xmath1-cover with @xmath3 g.d.c . and @xmath4 , obtained as in theorem [ thm : glue ] by gluing a cover @xmath771 such that for every @xmath772 the restricted cover @xmath435 is standard with building data @xmath773 , @xmath774 . * the @xmath775 and the components of the double curve @xmath332 are `` lines '' of @xmath2 , namely they are smooth and meet pairwise transversally . * the intersection points of the support of the hurwitz divisor @xmath32 of @xmath41 with the double curve @xmath21 of @xmath2 are smooth points of @xmath21 . * @xmath564 ( or , equivalently , @xmath32 , since @xmath2 is gorenstein ) is @xmath776-cartier and the pair @xmath566 is slc , so that by proposition [ prop : interesting - case ] @xmath3 is slc and @xmath563 is @xmath776-cartier . recall that , since we assume that the components of @xmath777 and of @xmath332 are lines , the pair @xmath566 is slc iff on @xmath13 the divisor @xmath778 has components of multiplicity @xmath208 and has multiplicity @xmath779 at every point . * for every @xmath42 that is singular for @xmath21 , label the components @xmath780 of @xmath2 containing @xmath158 in such a way that for every @xmath781 the surfaces @xmath7 and @xmath782 meet along an irreducible curve @xmath783 containing @xmath158 ( the indices are taken modulo @xmath784 ) and let @xmath107 be the generator of the inertia subgroup of @xmath783 . by theorem [ thm : glue ] , for every @xmath163 we have @xmath785 . we assume that the natural map @xmath786 is an isomorphism for every @xmath781 . + these conditions imply that the fibre of @xmath0 over @xmath158 consists of @xmath787 points . at each of these points @xmath3 is analytically isomorphic to the cone over a cycle of @xmath784 smooth rational curves . to compute the cohomology of @xmath790 , we are going to write down explicitly in the above situation the sequences in the second proof of theorem [ thm : glue ] ( as usual we push forward to @xmath2 all the sheaves ) . since all the maps are @xmath1-equivariant , the sequences split as sums of exact sequences : @xmath791 where @xmath12 varies in @xmath17 and @xmath1 acts in @xmath40 , @xmath792 and @xmath793 via @xmath12 . to describe the sheaves @xmath792 and @xmath793 , we need to introduce some more notation . given a component @xmath794 of @xmath21 we denote by @xmath795 the generator of the inertia subgroup of @xmath794 and by @xmath796 and @xmath797 the two components of @xmath2 that contain @xmath794 . we denote by @xmath798 ( resp . @xmath799 , @xmath800 ) the preimages of @xmath794 in @xmath3 ( resp . @xmath801 ) and by @xmath802 the common normalization of @xmath798 , @xmath799 , @xmath800 ( cf . example [ ex:2surfaces ] ) . in the following commutative diagram : * we identify @xmath806 with @xmath807 and for every @xmath808 we restrict @xmath809 to @xmath794 , * for every @xmath810 with @xmath811 , we label each point of @xmath812 with the image of @xmath813 in @xmath803 . the same can be done of course for @xmath814 . let @xmath815 be a point such that @xmath777 has multiplicity 1 at the points of @xmath386 that map to @xmath158 ( since we assume that @xmath816 is cartier , the multiplicity of @xmath777 is the same at all points lying over @xmath158 ) . recall that by assumption @xmath2 is d.c . at @xmath158 ; denote by @xmath817 @xmath818 the elements of @xmath1 associated to the two branch lines of @xmath819 containing @xmath158 and by @xmath820 , @xmath821 the elements of @xmath1 associated to the two branch lines of @xmath822 containing @xmath158 . we have @xmath823 modulo @xmath824 ( cf . example [ ex:2surfaces ] ) . then @xmath825 is singular over @xmath158 iff @xmath817 and @xmath818 are both different from @xmath824 , namely iff there exists a character @xmath12 with @xmath826 and @xmath827 . for each @xmath167 and @xmath828 such that @xmath826 we denote by @xmath829 the set of points @xmath830 such that @xmath827 , and we take @xmath829 to be the empty set if @xmath831 . we define in a similar way @xmath832 by considering the cover @xmath833 . we have the following : in our setup , the map @xmath845 is the disjoint union of two copies of @xmath846 that are switched by the involution @xmath354 . so by lemma [ lem : data - c ] the first sequence in can be rewritten as : @xmath847 in addition , if @xmath794 is a component of @xmath21 contained in @xmath796 and @xmath797 , then again by lemma [ lem : data - c ] the image of the map @xmath848 is equal to @xmath849 , so we have an exact sequence : @xmath850 where the cokernel @xmath851 is concentrated on the set @xmath840 . using the description of the singularities of @xmath3 at these points given in [ ssec : setup - z2 ] , one checks that @xmath851 has length 1 at points @xmath158 such that @xmath841 is trivial and it is 0 elsewhere , so @xmath852 . [ rem : inertia ] let @xmath446 be a smooth point , let @xmath75 be the irreducible component of @xmath21 that contains @xmath158 and let @xmath853 , @xmath854 be the two components of @xmath2 that contain @xmath75 . let @xmath231 the subgroup of @xmath1 generated by the inertia subgroups of @xmath75 and of the components of @xmath32 that contain @xmath158 . of course one has @xmath855 , but in the present setup equality actually holds . indeed , if @xmath856 is a non trivial character , then by proposition [ prop : fchi ] the second sequence in can be written near @xmath158 as @xmath857 , where @xmath858 is given by @xmath859 . by lemma [ lem : inertia ] , there exist @xmath860 , @xmath182 , that correspond to functions on @xmath679 that do not vanish at any point of @xmath861 . up to multiplying , say , @xmath862 by a nowhere vanishing regular function on @xmath853 we can arrange that @xmath863 . so @xmath137 corresponds to a function on @xmath3 that is nonzero near @xmath861 and on which @xmath1 acts via the character @xmath12 . it follows that @xmath864 acts freely on @xmath861 , i.e. that @xmath865 . we say that a point @xmath446 is _ relevant _ iff either it is singular for @xmath21 or there exists @xmath828 , @xmath12 with @xmath826 such that @xmath866 . observe that , in view of the assumptions of [ ssec : setup - z2 ] , by proposition [ prop : index - graph ] and by the description of singularities of [ subsec : sings - red - base ] the set of relevant points can be described intrinsically as the set of points of @xmath21 over which @xmath3 is gorenstein but not d.c .. [ ex1 ] let @xmath871 , let @xmath872 be the nonzero elements of @xmath1 and for @xmath873 let @xmath874 be the nonzero character such that @xmath875 . let @xmath876 , @xmath877 and let @xmath2 be the surface obtained by gluing @xmath853 and @xmath854 along a smooth rational curve @xmath21 which is of type @xmath878 on @xmath853 and is a line on @xmath854 . fix three distinct points @xmath879 . for @xmath873 , let @xmath880 be the union of a fibre and a section through @xmath881 and let @xmath882 be a pair of lines through @xmath883 ( the index @xmath354 varies in @xmath884 ) . in the picture below @xmath853 is represented on the left and @xmath854 on the right , the curve @xmath21 is shown in green , red lines correspond to @xmath885 , black lines to @xmath886 and blue lines to @xmath887 . for @xmath182 , we let @xmath435 be the standard @xmath1-cover with branch data @xmath888 , @xmath813 , @xmath889 . solving , we get @xmath890 and @xmath891 , @xmath889 , where @xmath892 denotes the subsheaf of @xmath893 corresponding to the character @xmath112 . notice that the line bundles @xmath892 have no cohomology , hence in particular @xmath894 . by @xcite , for @xmath182 the normalization of the cover of @xmath21 induced by @xmath436 is the trivial @xmath1-cover . so by theorem [ thm : glue ] , we can glue @xmath895 to a cover @xmath51 . by we have : @xmath896 the curve @xmath21 is smooth and the points @xmath897 and @xmath603 are relevant points with @xmath898 , so corollary [ cor : chi ] gives : @xmath899+[g : h_{y_2}]+[g : h_{y_3}]=1 + 1 - 4 + 1 + 1 + 1=1.\ ] ] for @xmath142 , we have an isomorphism @xmath900 , hence @xmath901 has no cohomology in degree @xmath902 and the exact sequence : @xmath903 implies that @xmath904 for @xmath902 . next we compute the cohomology of the sheaves @xmath40 . by proposition [ prop : fchi ] , for @xmath889 we have @xmath905 . so gives an exact sequence : @xmath906 therefore @xmath907 for @xmath889 and thus @xmath908 . as in the previous example , let @xmath871 and for @xmath911 let @xmath435 be the @xmath1-cover branched on the colored lines in the picture . for every @xmath163 , two of the sheaves @xmath912 are @xmath913 and the remaining one is @xmath914 . so the @xmath915 have no cohomology and @xmath916 . it s easy to check using theorem [ thm : glue ] that the cover @xmath917 can be glued to a @xmath1-cover @xmath51 . the normalization @xmath918 of the induced cover of the double curve @xmath21 is the disjoint union of 6 smooth rational curves , each mapping 2-to-1 onto a component of @xmath21 . the only relevant point is the singular point @xmath158 of @xmath21 . so applying and corollary [ cor : chi ] , we get : @xmath919 let @xmath920 be the irreducible components of @xmath21 . for @xmath142 , as in the proof of corollary [ cor : chi ] we have an exact sequence : @xmath921 which gives @xmath922 for @xmath902 . by proposition [ prop : fchi ] , for @xmath923 the sheaf @xmath924 is isomorphic to the direct sum of two copies of @xmath925 , hence it has no higher cohomology . so by we have @xmath926 for @xmath902 and therefore @xmath908 . we wish describe the singularities of a @xmath6-cover @xmath927 as in [ ssec : setup - z2 ] . since the question is local , we fix @xmath42 and we study @xmath3 locally above @xmath2 in the tale topology . by the assumptions in [ ssec : setup - z2 ] , the singularities of @xmath3 over a point @xmath42 lying on @xmath928 components of @xmath2 are degenerate cusps such that the exceptional divisor of its minimal semiresolution is a cycle of @xmath784 rational curves ( cf . * def . 4.20 ) ) . so it is enough to analyze two cases : all the singularities listed in tables [ tab : reduced - lines][tab : r - d12non ] , actually occur on some stable surface of general type . to give examples of the singularities that appear when the base @xmath2 of the cover is smooth , one can take @xmath106 , @xmath929 , a set of generators @xmath930 of @xmath1 , @xmath931 , and lines @xmath932 through a point @xmath933 such that the pair @xmath934 is lc . if @xmath935 , define @xmath936 , where @xmath937 is a general curve of even degree and for @xmath938 let @xmath320 be a general curve of odd degree . the divisors @xmath320 so defined are the branch data for a @xmath1-cover @xmath939 ( equations are easily seen have a solution in this case ) . by proposition [ prop : interesting - case ] the surface @xmath3 is slc and it is of general type as soon as the the degree of the hurwitz divisor @xmath32 is @xmath940 . there is only one point @xmath160 that maps to @xmath158 and all the singularities listed in tables 1 , 2 and 3 with can be realized as @xmath941 in this way and @xmath942 ( for the definition of @xmath231 , see below ) . the singularities with @xmath943 can be obtained by taking a double cover @xmath944 , branched on the sum of @xmath154 lines through @xmath158 and a general curve of degree @xmath145 such that @xmath945 is even and @xmath946 . we let @xmath949 be the branch data of @xmath41 . we may assume that @xmath164 for every @xmath163 . so by the condition that @xmath32 is slc we have @xmath931 and no three of the @xmath85 coincide . whenever the @xmath85 are not all distinct , we assume @xmath950 . all the possible cases are listed in tables [ tab : reduced - lines ] , [ tab : one - double - line ] , [ tab : two - double - lines ] below . the first digit in the label given to each case is equal to the number @xmath154 of components through @xmath158 , followed by @xmath951 if @xmath950 and by @xmath952 if @xmath950 and @xmath953 ( obviously this case occurs only for @xmath954 ) . so , for instance , a label of the form @xmath955 , where @xmath104 is any positive integer , means that @xmath158 belongs to three components of @xmath32 , two of which coincide . * @xmath956 : the order of the subgroup @xmath231 the subgroup generated by @xmath930 . * _ relations : _ describes the relations between @xmath957 . for instance , @xmath958 means @xmath959 . * _ singularity : _ the notations are mostly standard . @xmath960 denotes a cyclic singularity @xmath961 with weights 1,1 . @xmath962 denotes an arrangement consisting of four disjoint @xmath963-curves @xmath964 and of a smooth rational curve @xmath75 intersecting each of the @xmath965 transversely at one point . the self intersection @xmath966 is given in the table . in the non - normal case ( tables 2 and 3 ) we use the notations of @xcite , where kollr and shepherd - barron classified all slc surface singularities over @xmath967 . we work in any characteristic @xmath968 but only the singularities from the list in @xcite appear . `` deg.cusp@xmath969 '' means a degenerate cusp ( cf . * def . 4.20 ) ) such that the exceptional divisor in the minimal semiresolution has @xmath154 components . * @xmath334 : the index of @xmath160 . it is equal to 1 if all the relations have even length and it is equal to 2 otherwise ( cf . proposition [ prop : index - normal ] ) . * @xmath19 : denotes the normalization of @xmath3 ( the entries refer to the cases in table 1 ) . * @xmath970 : @xmath971 is the inverse image in @xmath972 of the double curve @xmath973 of @xmath3 and @xmath974 is the image of @xmath973 in @xmath2 . the symbol @xmath975 denotes the germ of a smooth curve , and @xmath976 is the seminormal curve obtained by gluing @xmath154 copies of @xmath975 at one point . the notation @xmath977 means that the map restricts to a degree @xmath504 map on the @xmath163-th component of @xmath976 ( we do not specify the @xmath504 when they are all equal to 1 ) . * @xmath978 : is the minimal semiresolution of @xmath3 . we write `` d.c . '' when @xmath978 has only normal crossings and `` pinch '' if it has also pinch points . * we always assume @xmath979 . indeed , the cover @xmath41 factors as @xmath980 . by lemma [ lem : inertia ] the map @xmath184 is tale near @xmath158 , while for every @xmath981 the fiber @xmath982 consists only of one point . since @xmath1 acts transitively on each fiber of @xmath41 , it is enough to describe the singularity of @xmath3 above any point @xmath983 . * the cover @xmath3 is normal at @xmath166 iff @xmath984=0 $ ] . it is nonsingular at @xmath166 iff either @xmath985 or @xmath986 , @xmath987 , @xmath988 . assume that @xmath3 is not normal , and let @xmath75 be an irreducible divisor that appears in @xmath32 with multiplicity 1 . this means that , say , @xmath989 and @xmath990 . the normalization of @xmath3 along @xmath75 is a @xmath1-cover of @xmath2 with branch data @xmath991 , for @xmath992 , and , if @xmath993 , @xmath994 ( cf . @xcite ) . * the cover @xmath3 is said to be _ simple _ if the @xmath995 is a basis of @xmath956 ( for instance , @xmath3 is simple if the @xmath996 are all equal ) . in this case @xmath3 is a complete intersection and it is very easy to write down equations for it ( see case @xmath997 below ) . * the double curve @xmath973 maps onto the divisors that appear in @xmath32 with multiplicity @xmath214 . since for a semismooth surface the double curve is locally irreducible , @xmath3 is never semismooth in the cases @xmath998 . in addition , if @xmath3 is semismooth then the pull back @xmath971 of @xmath973 to the normalization is smooth . using this remark , it is easy to check that @xmath3 is never semismooth in the cases @xmath999 , either . * in order to compute the minimal semiresolution @xmath978 , we consider the blow up @xmath1000 of @xmath2 at @xmath158 , pull back @xmath3 and normalize along the exceptional curve @xmath537 to get a cover @xmath1001 . the branch locus of @xmath1002 is supported on a d.c . divisor and , by construction , the singularities of @xmath1003 are only of type @xmath46 , @xmath776 or @xmath1004 . looking at the tables , one sees that either @xmath1003 is semismooth or it has points of type @xmath1005 or @xmath1006 ( cf . table [ tab : reduced - lines ] ) . in the former case @xmath1003 is the minimal semiresolution . in the latter case , blowing up @xmath1007 at the non semismooth points and taking base change and normalization along the exceptional divisor , one gets a semismooth cover @xmath1008 . the semiresolution @xmath1009 is minimal , except in cases @xmath1010 , @xmath1011 . in these cases the minimal semiresolution @xmath978 is obtained by contracting the inverse image in @xmath1012 of the exceptional curve of the blow up @xmath1000 . by remark ( 2 ) above , the normalization @xmath972 is an @xmath231-cover with branch data @xmath1013 , @xmath1014 and @xmath1015 . so @xmath1016 acts on @xmath3 without fixed points and @xmath3 is the disjoint union of two copies of the cover @xmath1017 . we choose local parameters @xmath1018 on @xmath2 such that @xmath950 is given by @xmath1019 , @xmath1020 is defined by @xmath1021 and @xmath1022 by @xmath1023 . the cover @xmath3 is defined tale locally above @xmath158 by the following equations : @xmath1024 in particular @xmath3 is a complete intersection ( see remark ( 3 ) above ) . the element @xmath996 acts on @xmath1025 as multiplication by @xmath1026 . the double curve @xmath973 is the inverse image of @xmath1019 , hence it is defined by @xmath1027 and the map @xmath1028 is given by @xmath1029 , so @xmath973 is isomorphic to @xmath1030 , with each component mapping @xmath776-to-@xmath46 to @xmath1031 . the curve @xmath971 is the inverse image of @xmath1032 in @xmath972 , so it has two connected components , each isomorphic to @xmath1030 , that are glued together in the map @xmath1033 . to compute the minimal semiresolution , consider the blow up @xmath1000 of @xmath2 at @xmath158 and the cover @xmath1002 obtained by pulling back @xmath1034 and normalizing along the exceptional curve @xmath537 . the branch data for @xmath1003 are @xmath1035 and , for @xmath1036 , @xmath1037 , where @xmath1038 indicates the strict transform . the cover is singular precisely above @xmath1039 and it is easy , using the local equations , to check that it is d.c . there . so @xmath1003 is the minimal semiresolution of @xmath3 . the exceptional divisor is the inverse image @xmath75 of @xmath537 in @xmath3 . applying the normalization algorithm to the restricted cover @xmath1040 , one sees that the normalization @xmath1041 of @xmath75 is the union of two smooth rational curves @xmath1042 and @xmath1043 . the map @xmath1044 identifies the two points of @xmath1042 that lie over the point @xmath1045 with the corresponding two points of @xmath1043 . hence @xmath1003 is the minimal semiresolution of @xmath3 and the singularity is a degenerate cusp solved by a cycle of two rational curves . as in the previous case , @xmath972 and @xmath971 can be computed by the normalization algorithm . one obtains that @xmath972 is the disjoint union of two copies of @xmath1046 and @xmath971 is the disjoint union of two copies of @xmath975 . this singularity is the quotient of a cover @xmath56 of type @xmath1047 by the element @xmath1048 . since this element has odd length , the index @xmath334 of @xmath3 at @xmath166 is equal to 2 . since the only fixed point of @xmath1049 on @xmath3 is @xmath1050 , the double curve @xmath973 is the quotient of the double curve @xmath1051 of @xmath56 . the two components of @xmath1051 are identified by @xmath1049 , thus @xmath973 is irreducible and maps @xmath776-to-@xmath46 onto @xmath1032 . to compute the minimal semiresolution , again we blow up @xmath1000 at @xmath158 and consider the cover @xmath1002 obtained by pull back and normalization along the exceptional curve @xmath537 . as usual , we denote by @xmath1052 the strict transform on @xmath1007 of a curve @xmath75 of @xmath2 . the branch data for @xmath1003 are @xmath1053 , @xmath1054 , @xmath1055 , @xmath1056 , and @xmath1057 . so @xmath1003 has normal crossings over @xmath1058 , it has four @xmath948 points over the point @xmath1059 and it is smooth elsewhere ( cf . tables 1 and 2 ) . we blow up at @xmath1060 and take again pull back and normalization along the exceptional curve @xmath1061 . we obtain a cover @xmath1062 which is d.c . over the strict transform @xmath1063 of @xmath1064 and has no other singularity , so @xmath1065 is a semismooth resolution . let @xmath1066 denote the strict transform on @xmath1067 of the exceptional curve @xmath537 of the first blow up . arguing as in case @xmath997 , one sees that inverse image of @xmath1066 is the union of two smooth rational curves @xmath1068 and @xmath1069 that intersect transversely precisely at one point of the double curve , and the inverse image of @xmath1061 consists of 4 disjoint curves @xmath1070 . all these curves pull back to -2 curves on the normalization of @xmath1012 and , up to relabeling , @xmath1071 and @xmath1072 form two disjoint @xmath1073 configurations . so @xmath1074 is the minimal semiresolution of @xmath3 . in the notation of @xcite[def . 4.26 ] , @xmath1012 is obtained by gluing two copies of @xmath1075 along @xmath975 . here we repeat the local analysis of the previous section for the case in which @xmath433 is d.c . , keeping as far as possible the same notations . so we fix @xmath446 , where @xmath21 is the double curve of @xmath2 , and describe @xmath3 locally over @xmath158 . we assume that @xmath0 is obtained by gluing standard covers @xmath435 , @xmath182 , such that @xmath158 lies on all the components of the hurwitz divisor @xmath32 . we let @xmath1076 be the union of the branch data of @xmath184 and @xmath185 such that @xmath85 is distinct from the double curve @xmath21 of @xmath2 ( hence @xmath1077 ) . we denote by @xmath1049 the generator of the inertia subgroup of @xmath21 for @xmath184 and @xmath185 . by remark [ rem : inertia ] the inertia subgroup @xmath157 is equal to @xmath1078 , so up to an tale cover we may assume that @xmath979 and that @xmath634 . since @xmath32 is @xmath204-cartier , there are the same number of @xmath85 on @xmath853 and on @xmath854 . we order them so that all components on @xmath853 come first . recall that @xmath931 by the assumption that @xmath566 is slc . the cases in the tables are labeled @xmath537 ( `` tale '' ) if @xmath1079 and @xmath80 ( `` ramified '' ) if @xmath1080 . the first digit of the label is the number @xmath154 of branch lines through @xmath158 . it is followed by @xmath951 if @xmath950 and by @xmath952 if @xmath950 and @xmath953 . for instance , in the cases e@xmath1081 the map @xmath41 is generically tale over @xmath21 and there are four branch lines @xmath1082 with @xmath950 , and @xmath1083 . the singularities that we get here are non - normal , and as in ( * ? ? ? 4.21 , 4.23 ) they turn out to be either semismooth or degenerate cusps in the gorenstein case and @xmath401-quotients of these otherwise . the tables here contain the same columns as those of [ subsec : sings - smooth - base ] plus an extra one , denoted @xmath12 : this is the contribution of @xmath158 in the formula for @xmath1084 of corollary [ cor : chi ] ( recall @xmath1085 ) . by propositions [ prop : index - non - normal ] and [ prop : index - graph ] the index @xmath334 is equal to 1 if all relations have even length when reduced modulo @xmath1049 and it is equal to 2 otherwise . the analysis of the singularities in the reducible case is similar to the case @xmath2 smooth . one blows up @xmath2 at the point @xmath158 and takes pull back and normalization of @xmath3 along the exceptional divisor . repeating this process , if necessary , one obtains a semiresolution @xmath1086 . if @xmath56 is not minimal , then the minimal semiresolution @xmath1087 is obtained by blowing down the @xmath1088-curves of @xmath56 . the normalization @xmath1089 is equal to @xmath1090 , where @xmath1091 is the normalization of @xmath679 . the branch data of @xmath1092 are @xmath1093 , @xmath1094 , so @xmath1095 is tale locally the disjoint union of four copies of the cover @xmath1096 . @xmath1097 is tale locally the disjoint union of two copies @xmath1017 . the image @xmath974 of the double curve @xmath973 is equal to @xmath1098 . the preimage in @xmath1095 of @xmath974 is the disjoint union of four copies of @xmath1030 . the preimage of @xmath974 in @xmath1099 is equal to two copies of @xmath1030 . hence @xmath1100 . each component of @xmath1101 maps 2-to-1 onto its image . the map @xmath1102 identifies in pairs the four components of the preimage of @xmath1032 and the eight components of the preimage of @xmath21 . so @xmath973 is @xmath1103 , with two components mapping 2-to-1 onto @xmath1032 and four components mapping 2-to-1 onto @xmath21 . to compute the semiresolution , blow up @xmath42 to get @xmath1000 let @xmath1104 and @xmath1105 be the irreducible components of the exceptional divisor . let @xmath1106 be the @xmath1-cover obtained from @xmath0 by taking pull back and normalizing along @xmath1066 and @xmath1061 . denoting by @xmath1038 the strict transform on @xmath1007 , the branch data of @xmath1107 are : @xmath1108 , @xmath1109 , @xmath1053 , @xmath1110 , @xmath1111 , @xmath1112 , @xmath1113 . so @xmath1003 is d.c . by the tables of [ subsec : sings - smooth - base ] and it is therefore the semiresolution @xmath978 of @xmath3 . the preimage of @xmath1066 is the union of four smooth rational curves meeting in pairs over the point @xmath1114 . the preimage of @xmath1061 is the disjoint union of two rational curves , which together with the components of the preimage of @xmath1066 form a cycle of six rational curves . the singularity @xmath160 is gorenstein by proposition [ prop : index - graph ] hence it is `` deg.cusp@xmath1115 '' . this is a @xmath401-quotient of r@xmath1116 and it is not gorenstein by proposition [ prop : index - non - normal ] . the normalization @xmath1089 is equal to @xmath1090 , where @xmath1117 is the normalization of @xmath1118 . the branch data of @xmath1119 are @xmath1120 , @xmath1094 , so @xmath1095 is tale locally the disjoint union of two copies of the cover @xmath1096 . the image @xmath974 of the double curve @xmath973 is equal to @xmath1098 . the preimage in @xmath1121 of @xmath974 is the disjoint union of two copies of @xmath1030 . the preimage of @xmath974 in @xmath1099 is @xmath1030 . hence @xmath1122 . each component of @xmath1101 maps 2-to-1 onto its image in @xmath974 . the map @xmath1102 identifies glues two itself each of the two components of the preimage of @xmath1032 and it identifies in pairs the four components of the preimage of @xmath21 . so @xmath973 is @xmath1123 , with two components mapping 1-to-1 onto @xmath1032 and two components mapping 2-to-1 onto @xmath21 . to compute the semiresolution , blow up @xmath42 to get @xmath1000 let @xmath1104 and @xmath1105 be the irreducible components of the exceptional divisor . let @xmath1106 be the @xmath1-cover obtained from @xmath0 by taking pull back and normalizing along @xmath1066 and @xmath1061 . denoting by @xmath1038 the strict transform on @xmath1007 , the branch data of @xmath1107 are : @xmath1124 , @xmath1125 , @xmath1126 , @xmath1110 , @xmath1111 , @xmath1112 , @xmath1127 . by the tables of [ subsec : sings - smooth - base ] , @xmath1003 has two pinch points over the point @xmath1128 and is at most d.c . elsewhere , hence it is equal to the minimal semiresolution @xmath1129 . the preimage of @xmath1066 is a pair of smooth rational curves meeting over the point @xmath1130 . the preimage of @xmath1061 is a smooth rational curve , meeting each component of the preimage of @xmath1066 at a point lying over @xmath1131 . in the notation of @xcite[def . 4.26 ] , @xmath978 is a chain consisting of copy of @xmath1132 ( namely the second component of @xmath978 ) in the middle and two copies of @xmath1133 with @xmath975 pinched at the ends . barbara fantechi and rita pardini , _ automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers _ , comm . algebra * 25 * ( 1997 ) , no . 5 , 14131441 . mr 1444010 ( 98c:14028 ) miles reid , _ canonical @xmath1134-folds _ , journes de gometrie algbrique dangers , juillet 1979/algebraic geometry , angers , 1979 , sijthoff & noordhoff , alphen aan den rijn , 1980 , pp . 273310 . mr 605348 ( 82i:14025 )

it is many years since the first description of how radio sources fueled by agn interact with and inject energy into the surrounding medium @xcite . now , largely due to _ chandra _ , we have many observational examples . x - ray signatures of the mechanisms involved are varied . gas cavities crafted by current or past radio lobes are common ( see * ? ? ? * for a review ) . agn - driven radio lobes are seen to shock the gas strongly , as in cena and pks b2152 - 699 @xcite or weakly , as in ngc4636 @xcite . while these phenomena mold the gas surface - brightness distribution , in other interactions it is the gas structures that shape the distribution of radio emission . this may be through the radio structures becoming buoyant , as in ngc326 @xcite or m87 @xcite , or because radio plasma is riding on a pressure wave of gas , as in 3c442a @xcite . moreover , as galaxies within groups and clusters interact with one another , their interstellar and intracluster media ( ism and icm ) get churned up through ram - pressure stripping , and wakes are observed as x - ray - gas density enhancements which are sometimes cooler and sometimes hotter than the surrounding gas ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? most of the best x - ray - studied cases are of radio galaxies of relatively low radio power hosted by the brightest central galaxy in the cluster or group environment , and less attention has been paid to cases where there are multiple radio galaxies in relatively close proximity . the radio galaxies hosted by ngc7016 and ngc7018 are a remarkable pair @xcite which reside in the central regions of the cluster abell3744 at @xmath3 @xcite and have escaped attention in recent years . they were mapped at high resolution in the radio with the vla by @xcite , and an image showing the salient features of figure [ fig : radio ] appears in @xcite in the context of rotation - measure modeling of ngc7018 . the radio source hosted by ngc7016 has asymmetric bent jets . lower - resolution radio data show a very long bent extension on the jet side one of the ` tendrils ' to which we refer later . on the counterjet side there is extreme looping , making a feature we refer to as the ` swirl ' . the radio source hosted by the eastern nucleus of the dumbbell galaxy ngc7018 is a classical double , of @xcite type ii ( fr ii ) morphology but unusual extended structure . we refer to the bright western extension from the southern lobe as the ` filament ' . at lower radio resolution , two long extensions ( tendrils ) are seen to the w and nw . ) and marking the ` filament ' and ` swirl ' referred to in this work . an unassociated radio source lies to the nw . ] while the complicated radio structures suggest interaction with the cluster atmosphere , little has been published on abell3744 s x - ray properties . it is detected in the rosat all - sky survey ( rass ) and is one of the 447 members of the reflex cluster catalog @xcite . however , it lies amongst the 12 per cent least luminous reflex clusters , with a cataloged 0.12.4-kev luminosity of only @xmath4 ergs s@xmath2 , and has not appeared in samples for deeper study . the rosat high resolution imager pointed at the field for 16.8 ks , and although investigation of the archival data shows a clear detection of the centers of both galaxies , the data are insufficiently sensitive for investigation of cluster substructure . in this paper we present new sensitive _ chandra _ observations of the system , together with new radio data obtained with the jansky very large array ( jvla ) . [ sec : obs ] describes the new observations and data processing . [ sec : cluster ] and [ sec : galaxies ] describe the cluster and galaxy / group x - ray features , respectively . in [ sec : rmorphs ] we highlight the fact that the two radio galaxies lie within the range of power that dominates jet - mediated feedback in the universe as a whole . after examining available galaxy velocity data in [ sec : velocities ] , with reference to the distribution of cluster gas and possible evidence for a merger , we discuss in [ sec : discussion ] relationships between the radio and x - ray structures and likely underlying causes . [ sec : summary ] summarizes our results . we adopt a luminosity distance for abell3744 of @xmath5 mpc ( appropriate for a hubble constant of 70 km s@xmath2 mpc@xmath2 ) , and 1 arcmin is equivalent to 45.3 kpc at the source . flux densities and spectral indices are related in the sense @xmath6 . we made a 75 ks observation of the system in full - window and vfaint data mode with the advanced ccd imaging spectrometer ( acis ) on board _ chandra _ on 2010 september 11 ( obsid 12241 ) . ngc7016 was positioned close to the nominal aimpoint of the front - illuminated i3 chip . the other three ccds of acis - i and the s2 chip of acis - s were also on during the observations , giving a frame time of 3.14 s. details of the instrument and its modes of operation can be found in the _ chandra _ proposers observatory guide . results presented here use ciao v4.5 and the caldb v4.5.6 calibration database . we re - calibrated the data to take advantage of the sub - pixel event reposition routine ( edser ) , following the software threads from the _ chandra _ x - ray center ( cxc ) , to make new level 2 events files . only events with grades 0,2,3,4,6 were retained . after screening to exclude intervals of high background at a threshold appropriate for use of the blank - sky background files , the calibrated dataset has an exposure time of 71.553 ks . since the cluster fills a large part of the detector array , background was measured from blank - sky fields following procedures described in the cxc software threads . after cleaning the background data using the same criteria as for the source data , and reprojecting to the same coordinate system , a small normalization correction was applied ( 2% ) so that the count rates matched in the @xmath7-kev energy band where particle background dominates . the ciao wavdetect task was used to find point sources with a threshold set to give 1 spurious source per field . their regions were subsequently masked from the data for the analysis of extended structure . all spectral fits are performed in xspec on binned spectra using @xmath8 statistics over the energy range @xmath9 kev . the models include galactic absorption of @xmath10 @xmath11 @xcite . parameter uncertainties are 90% confidence for 1 interesting parameter unless otherwise stated . we made sensitive , high - resolution , observations of the field containing ngc7016 and ngc7018 using the jvla in its a configuration at l ( 1.4 ghz ) and c ( 5 ghz ) bands ( table [ tab : vlatab ] ) . at the time of the observations , only part of the full bandwidth of the jvla correlator was available . the data were calibrated and flagged for extensive interference before being passed through the normal clean and gain self - calibration cycles in casa . for more diffuse structures we also downloaded archival l - band data taken in c configuration with the vla , and mapped them using standard procedures in aips . details of the heritage of the data sets and properties of the resulting maps are given in table [ tab : vlatab ] . for l to c - band spectral - index measurements we made a version of the c - band map with the same restoring beam as the l - band jvla map . lllllll ab1389 & 2011 jun 27 & 1.39 & jvla - a & @xmath12 & 0.066 & 1 + ab1389 & 2011 jun 27 & 4.96 & jvla - a & @xmath13 & 0.056 & 1 + ac105 & 1984 mar 31 & 1.525 & vla - c & @xmath14 & 0.16 & 2 + [ tab : vlatab ] from a map of our 1.4-ghz jvla data made with a restoring beam of @xmath12 arcsec . the outer lines mark the perimeter of the acis - i chip array . the white cross is at @xmath15 , the position adopted as the center of the cluster gas . ] figure [ fig : large ] is a smoothed , exposure - corrected 0.3 - 5 kev image of the _ chandra _ data , after removal of point sources but not the atmospheres of the three labeled ngc galaxies . 1.4-ghz radio contours are overlayed . the gaseous atmosphere of ngc7017 is seen in projection on the s jet of ngc7016 . the extended gas distribution very obviously deviates from spherical symmetry . interestingly , the filament and swirl ( fig . [ fig : radio ] ) both correspond to regions where the x - ray emission is less prominent . despite the lack of spherical symmetry , we have characterized the overall extent of the cluster atmosphere by fitting a @xmath16-model profile to a background - subtracted exposure - corrected radial profile centered on the position @xmath17 ( the approximate centroid of the diffuse emission contained within a circle of radius 227 arcsec ) shown as a cross in figure [ fig : large ] . the result is shown in figure [ fig : profile ] . we note the core radius of 220 arcsec inferred by @xcite based on earlier , less sensitive , rosat data falls within the @xmath18 error bound for @xmath19 . -model profile with the residuals ( shown as a contribution to @xmath20 ) in the lower panel . the best fit is for a core radius of @xmath21 arcsec and @xmath22 ( @xmath23dof = 84/69 ) . * right : * uncertainty contours ( @xmath18 , 90% and 99% for 2 interesting parameters ) of @xmath24 and @xmath16 for the radial - profile fit.,title="fig : " ] -model profile with the residuals ( shown as a contribution to @xmath20 ) in the lower panel . the best fit is for a core radius of @xmath21 arcsec and @xmath22 ( @xmath23dof = 84/69 ) . * right : * uncertainty contours ( @xmath18 , 90% and 99% for 2 interesting parameters ) of @xmath24 and @xmath16 for the radial - profile fit.,title="fig : " ] we have fitted the spectrum of the brightest extended emission in figure [ fig : large ] to a single - temperature thermal ( apec ) model absorbed by gas in the line of sight . emission from the three galaxy / group atmospheres has been excluded . we find very similar results for an on - source circle of radius 227 arcsec centered on @xmath25 as for a polygon of similar area tracing better a contour of constant surface brightness . the spectral contours for the circular extraction region and two outer annuli ( radii 227 and 300 arcsec , and 300 and 960 arcsec ) are shown in figure [ fig : clustertempabun ] . there is a weak trend for the best - fit temperature to decrease with increasing radius . the 90% confidence contour for the 227-arcsec - radius circle and @xmath18 contour for the outer annulus do not touch , giving less than 3 per cent probability of the spectral parameters being the same in these regions . the measurable difference is only in temperature , with the abundances consistent with a constant value of roughly 0.3 times solar ( i.e. , @xmath26 = 0.3 ) . and 90% for 2 interesting parameters ) of a fit of background - subtracted data ( with point - sources and galaxy / group atmospheres removed ) to a single - temperature thermal model . from right to left : ( a ) solid contours : from a circle of radius 227 arcsec ( best - fit @xmath8/dof = 352.3/348 ) , ( b ) dotted contours : from an annulus of radii 227 and 330 arcsec ( best - fit @xmath23dof = 315.6/319 ) , and ( c ) dashed contours : from an annulus of radii 330 and 960 arcsec ( best - fit @xmath23dof = 485.9/448 ) . ] beyond a radius of 330 arcsec some of the cluster lies off the acis ccds . we have therefore estimated the total cluster luminosity using a spectral fit to the emission within this circle , with point sources and galaxy emission excluded , and have extrapolated to account for missing flux based on the radial profile . the spectrum gives an acceptable fit to a model with @xmath27 kev ( @xmath18 error , see also fig . [ fig : clustertempabun ] ) . the radial profile ( fig . [ fig : profile ] ) finds that the counts within 330 arcsec should be multiplied by a factor of 2.39@xmath28 0.09 to account for emission out to @xmath29 , corresponding to @xmath30 arcsec for the measured temperature based on @xcite . the result is a bolometric luminosity for the cluster gas of ( 3.2 @xmath28 0.2 ) @xmath1 erg s@xmath2 ( @xmath18 error ) . the luminosity is too low to be a good match to the luminosity - temperature relations of clusters see e.g. , figure 5 of @xcite or , in other words , the temperature is too hot by @xmath31 kev for the luminosity . from the sample of @xcite at @xmath32 , the cluster with x - ray properties most closely resembling those of abell3744 is rxj2247 + 0337 at @xmath33 , with @xmath34 kev and a luminosity of @xmath35 erg s@xmath2 ( @xmath18 errors ) . that cluster is described as being neither relaxed nor having a cool core an appropriate description also of abell3744 . abell3744 appears as rxc j2107.2 - 2526 in the reflex cluster catalog @xcite . the 0.1 - 2.4 kev cluster luminosity of @xmath36 erg s@xmath2 is based on 100 detected counts and used an iterative method with an embedded temperature - luminosity relationship which will have settled on a lower temperature than now measured . a better - constrained value for the 0.1 - 2.4 kev luminosity to @xmath29 based on the _ chandra _ data ( calculated as above ) is ( @xmath37 ) @xmath1 erg s@xmath2 . we find that the total gas mass out to @xmath38 arcsec is ( @xmath39 ) @xmath40 m@xmath41 , and under the assumption of isothermal gas in hydrostatic equilibrium we can use equations 30 and 32 of @xcite to estimate a total mass within this radius of ( @xmath42 ) @xmath43 m@xmath41 . the relatively low gas - mass fraction is driven by the unexpectedly high temperature that is measured ( total mass is proportional to temperature while the gas mass is largely independent of it ) . the central proton number density in the x - ray - emitting gas is relatively low , at about 940 m@xmath44 , which from figure 5 of @xcite we see corresponds to a long cooling time of about 48 billion years . are shown with outer contours at 0.5 and 5 mjy beam@xmath2 from a map with restoring beam of @xmath14 arcsec made from the 1.5-ghz vla data of program ac105 ( solid line ) . we refer to the three extended radio arms as ` tendrils ' . the contours are overlayed on an adaptively smoothed grey - scale image showing the brightest region of cluster gas . the black diagonal cross marks the center of the distribution of dominant cluster galaxies while the white cross is that for the small group whose center is closer to that of the gas distribution ( [ sec : velocities ] ) . the dashed lines outline x - ray extraction regions : within the 227-arcsec - radius circle the arm region avoids extended radio emission while the cavity region does not . ] ( including same axis scales ) for * left * : arm ( best fit gives @xmath45 for 73 dof ) and * right * : cavity ( best fit gives @xmath46 for 103 dof ) shown in fig . [ fig : armhole ] , title="fig : " ] ( including same axis scales ) for * left * : arm ( best fit gives @xmath45 for 73 dof ) and * right * : cavity ( best fit gives @xmath46 for 103 dof ) shown in fig . [ fig : armhole ] , title="fig : " ] the gas morphology indicates a cavity encompassing the filament of ngc7018 and the swirl of ngc7016 , and so temperature structure in the inner regions might be expected . however , here we find no obvious statistically - significant temperature structure and no cool core . in figure [ fig : armhole ] we mark regions selected based on x - ray and radio morphology that lie within the 227-arcsec - radius circle and that we refer to as the ` arm ' and ` cavity ' , and in figure [ fig : armholetempabun ] we show their spectral contours for temperature and abundance . the axes of figure [ fig : armholetempabun ] are the same as in figure [ fig : clustertempabun ] . model for @xmath47 arcsec . the dashed curve is the inner extension of the @xmath16 model . the solid curve is the @xmath16 model from which has been subtracted the counts in the cavity which , since lying well within the core radius , is modeled as a zero - density sphere of radius 100 arcsec . ] the lower average x - ray surface brightness towards the cavity with respect to the arm ( a factor of roughly 0.6 ) is difficult to understand for gas of constant temperature . the apparent opening to the sw seen in figure [ fig : large ] is very well aligned with the run of the chip gap , and so , despite exposure corrections having been applied , it is not possible to conclude that the region is unbounded in that direction . we have therefore investigated what scale of spherical cavity is consistent with the data . to do this we have extracted a radial profile centered roughly on the cavity , at @xmath48 . as shown in figure [ fig : cavityplotit ] , the central data are clearly depressed relative to the extrapolation of the best - fit @xmath16 model fitted at angular radii larger than 120 arcsec ( dashed line ) . since the angular scale of the counts deficit lies within the core radius , we have tested the simple model of a central evacuated spherical region by subtracting the profile of counts that would otherwise lie there ( normalized to the central volume density of the @xmath16 model ) from the profile of the @xmath16 model . figure [ fig : cavityplotit ] shows that such a central spherical evacuated region is consistent with observations if of angular radius roughly 100 arcsec . it is not obvious how to construct a static cavity of this type in a gas of constant pressure . it would either need to be a dynamical effect , or there needs to be a source of additional pressure in the cavity . it is likely that extra pressure is provided by radio - emitting relativistic particles ( note from fig . [ fig : armhole ] that radio emission covers the cavity but not the arm ) , in which case a power - law x - ray component from inverse - compton scattering ( discussed in [ sec : discuss - hole ] ) might be detectable . to test how well a power law can be accommodated within the region where the radio emission is brightest , we have defined a further region , extending that covered by the cavity and guided by the radio contours . a spectral fit to either a single - component thermal or power - law model is acceptable , at @xmath23dof = 105.1/94 and 106.7/95 , respectively . when the two models are combined and the power - law slope is fixed to @xmath49 , we find @xmath23dof = 97.8/93 and the flux contribution of the power law is @xmath50 per cent ( 90% confidence ) . while we would not claim this as strong evidence for a power - law component , it is consistent with a contribution of inverse compton emission , as discussed further in [ sec : discuss - hole ] . the residuals in the radial - profile fit ( fig . [ fig : profile ] ) show structure at radii between 100 and 400 arcsec . profiles of different pie slices were made , from which it became apparent that the main feature was an x - ray excess that lies between ngc7018 s tendrils of 1.5-ghz radio emission ( fig . [ fig : armhole ] ) outside the gas cavity . we refer to this gas feature as the ` plateau ' and it is marked in figure [ fig : armhole ] . radial profiles for specific pie slices ( figure [ fig : profilesew ] ) show excess counts in the profile corresponding to the position angles of the plateau . , plotted separately for the subset of counts in pie slices of position angles 275325 degrees ( filled triangles , solid curve ; plateau region ) and 20180 degrees ( open circles , dashed curve ; e segment ) . the curves have the shapes of @xmath16 models , but are only present to guide the eye . the dotted curve is for the combined n and sw segments ( points suppressed ) . * bottom : * adapted from fig . [ fig : armhole ] to show the locations of the pie slices . the e segment is truncated by the edge of the detector ( solid diagonal line ) . , title="fig : " ] , plotted separately for the subset of counts in pie slices of position angles 275325 degrees ( filled triangles , solid curve ; plateau region ) and 20180 degrees ( open circles , dashed curve ; e segment ) . the curves have the shapes of @xmath16 models , but are only present to guide the eye . the dotted curve is for the combined n and sw segments ( points suppressed ) . * bottom : * adapted from fig . [ fig : armhole ] to show the locations of the pie slices . the e segment is truncated by the edge of the detector ( solid diagonal line ) . , title="fig : " ] after identifying the plateau as a distinct morphological structure , we investigated the spectral properties of the gas on these larger scales in more detail . while the gas in the plateau fits a temperature similar to the average for the cluster , gas to the east at radii beyond 227 arcsec was found to be cooler , and that to both sides between the plateau and e segment was found to be hotter . in particular , the temperature of gas in the e segment ( fig . [ fig : profilesew ] ) is cooler than that from the combined n and sw segments at high significance ( fig . [ fig : segtempabun ] ) . for regions in lower panel of fig . [ fig : armholetempabun ] . left to right : ( a ) e segment , dashed contours ( 3894 net counts , best - fit @xmath8/dof = 182.0/168 ) , ( b ) plateau , solid contours ( 3225 net counts , best - fit @xmath23dof = 181.3/204 ) , and ( c ) combined n and sw segments , dotted contours ( 5152 net counts , best - fit @xmath23dof = 228.2/204 ) . ] to probe further the locations of temperature changes , we have used the contbin software of @xcite to define spectral regions from our adaptively - smoothed image , and map xspec model - fitting results onto the image . we used a constant signal to noise ( s / n ) based on our adaptively smoothed exposure - corrected ( but not background subtracted ) map , which provided regions for spectral fitting with a s / n ranging between 21 and 40 . the s / n was on purpose chosen to be slightly lower than that in the regions of figure [ fig : segtempabun ] , in order to investigate to what extent the sector boundaries of figures [ fig : profilesew ] and [ fig : segtempabun ] were supported by a less subjective examination of the data . the fitted spectral model is a single - temperature thermal , and we show results for @xmath51 and metallicity in figure [ fig : contbin ] . spectral results are listed by region in table 2 . the fits to the individual regions are acceptable . when the outer regions of most extreme temperature ( 3,6 , 9 , 11 ) are combined , we find an unacceptable fit to a single - temperature model , as expected from figure [ fig : segtempabun ] which shows that gas in the e separates in temperature from that in the n and sw . our main conclusions are that gas to the e is cool ( @xmath51 about 2.1 kev ) and of metallicity less than about 0.36 . gas within a radius of 227 arcsec and in the plateau is of intermediate temperature ( @xmath51 about 3.6 kev ) and average cluster metallicity , except in a region corresponding to the brightest radio emission from ngc7017 and ngc7018 , where a possibly depressed abundance might be due to dilution with non - thermal power - law emission ( [ sec : armhole ] and [ sec : discuss - hole ] ) . gas wedged between the plateau and the cool gas to the e is significantly hotter ( @xmath51 about 5 kev ) and generally of normal metallicity . lllcl 1 & @xmath52 & @xmath53&103.5/92 & 0.19 + 2 & @xmath54 & @xmath55&103.8/108 & 0.60 + 3 & @xmath56 & @xmath57&73.9/79 & 0.64 + 4 & @xmath58 & @xmath59&155.9/138 & 0.14 + 5 & @xmath60 & @xmath61&86.5/113 & 0.97 + 6 & @xmath62 & @xmath63&56.3/56 & 0.46 + 7 & @xmath64 & @xmath65&120.2/102 & 0.11 + 8 & @xmath66 & @xmath67&85.6/88 & 0.55 + 9 & @xmath68 & @xmath69&114.8/95 & 0.08 + 10 & @xmath70 & @xmath71&101.4/105 & 0.58 + 11 & @xmath72 & @xmath73&73.2/67 & 0.28 + 12 & @xmath74 & @xmath75&57.8/66 & 0.75 + 3 + 6 + 9 + 11 & @xmath76 & @xmath77 & 342.4/292 & 0.02 + [ tab : regspec ] we have extracted spectra of the x - ray atmospheres surrounding the three galaxies marked on figure [ fig : large ] using local background from a surrounding annulus rather than the blank - sky background , as the galaxies and their group atmospheres are embedded within cluster gas . ngc7016 and ngc7018 both lie on the i3 ccd of acis , and ngc7017 is on i2 . we have used 2mass j - band data accessed with skyview to associate x - ray and radio features with their host galaxies . ngc7016 is a relatively isolated galaxy ( fig . [ fig : n7016x-2mass ] ) . the 886 net counts from a 12-arcsec - radius source - centered circle ( fig . [ fig : n7016 ] ) give a poor fit to a thermal model alone ( @xmath23dof = 79.9/30 ) , but the fit is acceptable when a power law ( with no excess absorption ) is added to account for emission from the active nucleus . the abundance is poorly constrained and was fixed at 0.3 times solar in the two - component fit . results are given in table [ tab : gspectab ] . the temperature is cooler than that of the surrounding cluster , as expected for a galaxy atmosphere . from a smaller circle of radius 1.5 arcsec , again centered on the radio - core position of @xmath78 , the counts are more highly dominated by the power - law component and spatially sharply peaked , but the fit was still improved by a small contribution from thermal gas ( from @xmath23dof = 26.9/17 to 10.7/15 ) . the power - law parameter values were consistent with those from the larger extraction region , demonstrating an active - nucleus origin of the non - thermal emission . the power - law x - ray flux density ( table [ tab : gspectab ] ) and 4.96-ghz core flux density of 58.3 mjy lie within the scatter of the correlation of unabsorbed nuclear emission components of @xcite , arguing in favor of a common non - thermal origin . an extension of the x - ray emission in the direction of the radio jet ( fig . [ fig : n7016 ] ) strongly suggests x - ray synchrotron emission from an inner radio jet , found by _ chandra _ to be a common feature of fri radio galaxies @xcite . for ngc7016 on * left * unsmoothed 0.3 - 5 kev x - ray image in native pixels and * right * 2mass j - band image . the host galaxy does not have a companion of similar brightness . the swirl has not obviously been shaped by a feature seen in the x - ray or near ir . ] , increasing by factors of 2 to 38.4 mjy beam@xmath2 , from a map of our 5-ghz jvla data made with a restoring beam of @xmath13 arcsec . the circle of radius 12 arcsec indicates the on - source spectral extraction region , with background taken from the partially shown annulus of radii 12 and 30 arcsec . ] there is a bright pair of galaxies associated with ngc7017 ( fig . [ fig : n7017x-2mass ] ) , although their velocity difference of @xmath79 km s@xmath2 @xcite is large . the centers of each were detected as x - ray point sources by wavdetect . we excluded the counts in circles of radii 3 and 1.2 arcsec around the centers of the western and eastern galaxy , respectively , and 176 net counts were then detected from an ellipse of semi - axis lengths of 17.4 and 14.2 arcsec shown in figure [ fig : n7017x-2mass ] . a good fit to a thermal model ( table [ tab : gspectab ] ) was found . shown ( for reference only ) projected on ngc7017 in * left * unsmoothed 0.3 - 5 kev x - ray image in native pixels , with spectral extraction region shown and * right * 2mass j - band image . x - ray emission is seen predominantly from the envelope of the galaxy pair , with excess counts associated with the core of each . ] ngc7018 is also a galaxy pair . the eastern member hosts the radio - galaxy nucleus , and here point - like nuclear x - ray emission that strongly outshines diffuse gas emission is seen ( fig . [ fig : n7018 ] ) . 709 net counts from a circle of radius 5.6 arcsec centered on @xmath80 can be fitted well to a power - law model with no excess absorption ( table [ tab : gspectab ] ) . as for ngc7016 the power - law x - ray flux density ( table [ tab : gspectab ] ) and 4.96-ghz core flux density of 36.0 mjy lie within the scatter of the correlation of unabsorbed nuclear emission components of @xcite , arguing in favor of a common non - thermal origin . @xcite find a velocity difference of only @xmath81 km s@xmath2 between the radio host galaxy and its western companion . diffuse x - ray emission , associated largely with the companion galaxy , is seen to the west of the radio nucleus . 385 net counts were extracted from an ellipse of semi - axis lengths 20.6 and 11.7 arcsec , excluding the circle described above and shown in figure [ fig : n7018 ] . the data give a good fit to a thermal model ( table [ tab : gspectab ] ) . we note that there appears to be a weak excess of radio emission associated with the western galaxy . for ngc7018 on * left * unsmoothed 0.3 - 5 kev x - ray image in native pixels , with spectral extraction region shown and * right * 2mass j - band image . the eastern galaxy of the pair hosts the core of the radio source together with predominantly point - like x - ray emission . the western galaxy is associated with a weak radio excess and predominantly diffuse x - ray emission . ] the radio core of ngc7018 is within its northern lobe . while a broad trunk of radio emission connects it to the southern lobe , the jet to the north is narrow and more distinct , terminating in a double hotspot ( fig . [ fig : radio ] ) . there is excess x - ray emission to the north of the nucleus associated with the tip of the inner jet ( fig . [ fig : n7018 ] ; 14 counts detected in a circle of radius 2 arcsec where an average of 2.5 is expected based on local background ) . the northern hotspot region is also detected in x - rays , with 18 counts detected in a circle of radius 3.5 arcsec where 5 counts are expected based on local background . hotspot counts appear to be associated with each component of the double system , with a small misalignment between x - ray and radio that is probably within the tolerance of off - axis mapping uncertainties in the radio ( fig . [ fig : n7018nhots ] ) . the southern hotspot region is undetected in x - rays . outlining the northern edge of the northern lobe of ngc7018 ( bold line ) together with contours from our higher - resolution 5-ghz map at 0.3 , 0.9 , 1.5 , 3 , and 4.5 mjy beam@xmath2 , showing the double hotspot . x - rays are seen from both parts of the double hotspot : the image is 0.3 - 5-kev _ chandra _ data in native pixels smoothed in ds9 with a 2d gaussian of radius 2 pixels . the apparent misalignment of x - ray and radio emission may be due to inaccuracies in the 5-ghz radio mapping 3.5 arcmin off axis . ] llcccccc ngc7016 & 0.03685 & @xmath82 & @xmath83 ( fixed ) & @xmath84 & @xmath85 & @xmath86 & 29.9/29 + ngc7017 & 0.03465 & @xmath87 & @xmath88 & @xmath89 & - & - & 7.2/8 + ngc7018-nucleus & 0.03842 & - & - & - & @xmath90 & @xmath91 & 26.9/25 + ngc7018-gas & - & @xmath92 & @xmath93 & @xmath94 & - & - & 17.5/14 + [ tab : gspectab ] the small radio source in the nw of the field of figure [ fig : radio ] has a curious structure , as shown in figure [ fig : nwradio ] . wavdetect finds an x - ray source at the nw tip of the se lobe with about 7 excess counts ( 0.3 - 5 kev ) . this seems to be chance coincidence , since it does not match the location of the radio core or any other radio feature . there is possible excess x - ray emission associated with bright parts of the lobes , but it is not possible to establish a level of significance since x - ray counts are generally sparse in this part of the image . there is no strong 2mass association of the radio source . . the contours are a crude representation of groupings of x - ray counts , made from a lightly smoothed image . the x - rays at the nw tip of the se lobe represent a real source containing roughly 7 excess counts ( 0.3 - 5 kev ) , but it is most likely unrelated to the radio source . ] the flux density of ngc7018 at 1.525 ghz is roughly 6.2 jy , of which about 80 per cent lies in the classical - double structure and the rest in its tendrils : a nw tail of the n lobe and the w extension of the radio filament of the s lobe . while data at lower frequencies have been published , the large beam sizes confuse the structures , and total flux densities combine emission from ngc7016 and ngc7018 . adopting a spectral index of @xmath95 we extrapolate to find a 178-mhz power for the double source ngc7018 of @xmath96 w hz@xmath2 sr@xmath2 , placing it in the range where fanaroff and riley types i and ii ( fris and friis ) overlap in luminosity @xcite . these are the sources with the correct range of power to dominate jet - mediated feedback in the universe as a whole , and so are of particular importance @xcite . the best - studied example is pks b2152 - 699 , where not only have the lobes evacuated cavities in the x - ray - emitting gas , but also relatively strong ( mach number between 2 and 3 ) shocks are seen , and the kinetic and thermal energy of shocked gas dominates over the cavity power @xcite . in ngc7018 the s lobe may have helped to bore out the large x - ray cavity in which it resides ( fig . [ fig : armhole ] , and see below ) , but there is no evidence for shocked gas . there are differences in radio morphology that may explain why strong shocks are not detected around ngc7018 . in particular , ngc7018 has bright terminal hotspots rather than the lobe - embedded hotspots seen in pks b2152 - 699 , perhaps suggesting that strong shocks are detected more readily either during a phase of the source s evolution or at preferred source orientations . x - ray study of more sources of these radio powers are needed to resolve the issues . ngc7016 , with a total flux density of 4.9 jy at 1.525 ghz , is almost as bright as ngc7018 , although it exhibits no frii morphology . just over half of the flux density is from the region beyond the sharp southern bend of the se jet that is seen well in the high - resolution radio data , and about 64 per cent of that is in the region of the tendril that lies beyond the extent of the jet in our high - resolution map . the gas of abell3744 is highly disturbed . merging can produce irregularities , and so we have examined the available galaxy velocities . in order to search for sub - clumps we used velocities listed in ned for galaxies within 30 arcmin of ngc7016 , noting that the mean redshift of @xmath97 and velocity dispersion of @xmath98 km s@xmath2 of 72 galaxies in the field , and likely associated with the cluster , are in good agreement with the redshift , and dispersion of 559 km s@xmath2 , found by @xcite . we applied an algorithm that looks for objects close to one another in three dimensions , ignoring velocity differences less than 800 km s@xmath2 . the search scale for these associations begins at 1 kpc and increases slowly to 1 mpc , and pairs are combined into lumps on each scale at their mean position and velocity . the input parameters were then adjusted to check for stability in the results . as shown in figure [ fig : tree ] , the dominant structure is a cluster of 56 galaxies centered at @xmath99 , about 4 arcmin west of the nucleus of ngc7018 , with mean velocity 11290 km s@xmath2 and velocity dispersion 350 km s@xmath2 . the cluster includes ngc7016 ( 11046 km s@xmath2 ) and both nuclei of ngc7018 ( 11517 and 11694 km s@xmath2 ) , which are among the bright galaxies in the cluster , but none of which is clearly identifiable as a central brightest cluster galaxy . several groups can also be identified , including one of velocity 13030 km s@xmath2 associated with the eastern component of ngc7017 ( 12892 km s@xmath2 ) and two other galaxies . this group , at roughly @xmath100 , is closer to the center of the gas distribution than that of the cluster galaxies . both centers are marked in figure [ fig : armhole ] . the second largest grouping is of eleven galaxies centered about 8 arcmin to the wsw , at @xmath101 with velocity 12140 km s@xmath2 . the sparseness of velocity information for galaxies that might be in these distinct groups makes it difficult to interpret the neighborhoods of ngc7016 and ngc7018 with certainty , but interactions of the group associated with the western component of ngc7017 and the main cluster also seems plausible . this component of ngc7017 is the brightest galaxy in the field but , at a velocity of 10387 km s@xmath2 , appears to be dynamically distinct from the main cluster , and may have an associated infalling group of galaxies without measured velocities . the position of the main galaxy centroid explains the excess plateau gas , although not why it is so well positioned between the radio tendrils ( fig . [ fig : contbin ] ) . the velocity data are suggestive that a merger interaction between a group or groups with significant velocity offsets from the main cluster is responsible for the non - uniformities in gas distribution and temperature . the x - ray cavity is a region of distinctly reduced x - ray surface brightness in the direction where the radio emissions from ngc7018 and ngc7016 overlap . due to coincidence in projection , we assume the cavity is co - located within the core radius of the cluster , and we have demonstrated that the contrast in x - ray surface brightness is consistent with such an interpretation . many clusters and groups are now known to harbor gas cavities containing the radio plasma of active galaxies ( e.g. , * ? ? ? since the pressure in relativistic plasma should not be less than that of surrounding x - ray gas , insight can be gained into the extent to which the relativistic plasma departs from a state of minimum energy or requires extra pressure to be provided by non - radiating particles or a reduced filling factor ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . where a detection of inverse - compton radiation from the cavities can be used to measure the magnetic field , the minimum - energy assumption is tested , and results concerning the other pressure components are more secure ( e.g. , * ? ? ? * ) . in abell3744 , while there is not a clean spectral separation of inverse - compton and thermal emission in the cavity , the suggestion of lower metallicity implies that inverse compton emission is not completely negligible , providing a modeling constraint . the average gas pressure surrounding the x - ray cavity can be calculated from the x - ray radial profile , giving ] @xmath102 pa ( @xmath18 error ) . a sphere of radius 100 arcsec evacuated of x - ray - emitting gas provides a model for the cavity that is consistent with the surface - brightness distribution ( fig . [ fig : cavityplotit ] ) . we model the cavity s radio emission from our map using data from program ac105 as synchrotron radiation from electrons with a power - law distribution of @xmath95 extending to a minimum lorentz factor of @xmath103 , to find a minimum - energy field strengthg ] in the cavity of @xmath104 nt ( no protons ) or 1.0 nt ( equal energy in protons and electrons ) . we do nt have radio data to constrain @xmath105 , but the adopted value is supported by the x - ray data if modeled as power - law emission from inverse compton scattering by the electron population ( [ sec : armhole ] ) . the resulting pressure is a factor of 5.0 ( 3.5 ) too low to match that in surrounding x - ray gas , where here and in what follows the first value is for no protons and the second for equal energy in protons and electrons . while the cavity pressure can be increased by adding further non - radiating particles or reducing the filling factor , the prediction for inverse compton scattering on the cosmic microwave background radiation is still then less than 6 per cent of the x - ray counts from the cavity , not explaining the reduced metallicity when the spectrum is fitted entirely to thermal emission . in contrast , a small departure from minimum energy in the sense of a reduced magnetic field can both provide the required pressure and explain the metallicites . specifically , reducing the magnetic field strength to 30 ( 37 ) per cent of the minimum - energy value , i.e. , to 0.26 ( 0.37 ) nt , matches the pressure and provides an increase in electron number density ( to match the radio synchrotron emission ) such that 52 ( 27 ) per cent of the 0.35-kev x - ray counts from this region would be from inverse compton emission . a modest departure from minimum energy is consistent with findings elsewhere , and in particular with the lobes of radio galaxies for which x - ray inverse compton is detected and modeled using the same value of @xmath106 as adopted here @xcite . we then note that the cavity shape would need to be somewhat spheroidal and elongated along the line of sight , to explain the surface brightness contrast of figure [ fig : cavityplotit ] in the presence of extra ( inverse compton ) x - ray counts in the cavity . the sound speed in x - ray gas of temperature @xmath51 ( kev ) is roughly @xmath107 km s@xmath2 , meaning that radio plasma should have been developing and filling the cavity for at least @xmath108 yrs for the cavity not to have collapsed . this is within the lifetime of typical radio galaxies based on spectral aging measurements ( e.g. * ? ? ? ngc7018 , whose southern lobe is currently at the edge of the cavity , would have taken a comparable time ( @xmath109 yr ) to cross the cavity at a speed characteristic of the cluster velocity dispersion , and faster speeds are expected here , near the cluster center . thus it seems likely that the radio plasma of ngc7018 is primarily responsible for carving out the cavity . in contrast , the radio emission of ngc7016 is broken into the swirl inside the cavity . while this might be due to collision with radio plasma of ngc7018 , it may also result from interaction with a wake left by the motion of ngc7018 and its companion galaxy ( see e.g. , * ? ? ? if ngc7018 is responsible for the cavity , its lobes should be at least mildly overpressured with respect to the surrounding gas . the s lobe lies at the entrance to the cavity where the average gas pressure ( calculated using the x - ray radial profile for an off - center angular distance of 65 arcsec ) is @xmath110 pa ( @xmath18 error ) . we model the lobe using our 1.39-ghz jvla data with parameters as for the cavity . ] , but with a sphere of radius 18 arcsec , to find a minimum - energy field strength of @xmath111 ( 2.6 ) nt and a pressure of @xmath112 pa , slightly above that in the external gas . if the magnetic field is below the minimum - energy value by the amount we argue is likely for the cavity , then the s lobe is overpressured by a factor of 4.9 with respect to its surrounding gas . such an overpressure would lead to a shock of mach number 2 being driven into the external medium ( e.g. , equation 49 of * ? ? ? * ) and direct heating in the immediate vicinity of the lobe , not ruled out by the data due to masking by much foreground and background gas along the line of sight . direct evidence of shock heating is difficult to verify around cluster - embedded radio lobes , but there is growing evidence for moderately strong shocks around intermediate - power radio galaxies like ngc7018 @xcite . it is striking that the radio filament of the s lobe of ngc7018 ( fig [ fig : radio ] ) runs along the northern inside edge of the cavity ( fig [ fig : large ] ) . the structure contains strands , and the brightening is suggestive of preferential particle acceleration along the interface between the cavity and the external x - ray gas something which in principle could be tested by radio spectral index mapping . the tendrils of ngc7018 run along the outside of the plateau and look like they are buoyant , so that their deprojected pressures should match those in the plateau , as measured using the spectral fit and profile in figure [ fig : profilesew ] . to test this we take rectangular sections at the end of each tendril modeled as cylinders of radius 21 arcsec and lengths 126 and 162 arcsec centered at distances from the center of the gas distribution of 380 and 530 arcsec for the sw and nw tendrils , respectively . if the tendrils are in the plane of the sky , the external pressure in the sw and nw is @xmath113 and @xmath114 pa , respectively . for minimum energy the two tendrils then agree in being under - pressured by a factor of 5.5 ( 3.9 ) , where again the two values correspond to a lepton - only plasma or one where the lepton energy density is matched by that in protons . if in the plane of the sky with no significant entrainment , pressure balance can be restored if the magnetic field is 29 ( 35 ) percent of the minimum - energy value . these departures from minimum energy are remarkably similar to those required in the cavity , and by the region in the nw tendril the magnetic field strength so estimated would have dropped to about half that in the cavity . however , while there may be reasons for dynamic structures to depart from minimum energy , it is more appealing for these buoyant flows to have reached a state of minimum energy . the tendrils extend into the region where the x - ray gas pressure is falling with radius , @xmath115 , as roughly @xmath116 , and they would be in pressure balance with this gas if lying at about @xmath117 degrees to the line of sight . the actual value of @xmath118 is likely to be set by the unknown direction of the velocity vector of ngc7018 , such that the buoyant tendrils trail , and such values of @xmath118 are plausible . some increase in internal pressure in the tendrils through gas entrainment is also likely , allowing @xmath118 to increase from these estimates . not being in the plane of the sky has the advantage of explaining the rather dramatic fading of the tendrils which occurs at larger @xmath115 for smaller @xmath118 . strong fading is expected sufficiently far out in the cluster outer atmosphere that the tendrils are no longer supported , and the relativistic plasma expands adiabatically . this is easily accommodated by a further steepening in the radial profiles beyond the values of @xmath115 for which we are able to make measurements in the current data . the broad tendril to the se is the extension of the brighter ( approaching ) jet of ngc7016 . the two jet bends , first to the south and then to the east ( fig . [ fig : radio ] ) and presumably the result of shocks , are sufficiently large in projection to suggest that this jet is at a relatively small angle to the line of sight . we test a cylinder of radius 33 arcsec and length 98 arcsec ( position angle 70 degrees ) at 340 arcsec from the center of the gas distribution where the external pressure ( in the cooler gas here ) is @xmath119 pa . as for ngc7018 s tendrils , there is an underpressure by a factor of a few that can be restored if the radio emission is lying at 24 ( 31 ) degrees to the line of sight . x - ray cavities are sufficiently common that they are regarded as an important heat source , with enough power to balance radiative cooling in dense cluster cores ( e.g. , * ? ? ? * ; * ? ? ? * ) . in the case of the abell3744 s cavity , we can estimate the change in enthalpy as @xmath120 , where @xmath121 is the pressure of the x - ray gas , @xmath122 is the volume of the cavity , and @xmath123 is the ratio of specific heats for relativistic gas . the resulting value is roughly @xmath124 j. in [ sec : integrated ] we pointed out that in its integrated properties the cluster is too hot by @xmath31 kev for its luminosity , based on scaling relations . this is equivalent to an excess enthalpy of @xmath125 , where @xmath126 is the total gas mass , @xmath127 is the particle mass , and @xmath128 is the ratio of specific heats for the x - ray - emitting gas . the resulting value is roughly @xmath129 j. a staggering 85 cavities like those attributed to ngc7018 would be needed to explain the excess heat in abell3744 that causes it to deviate from scaling relations . the most likely source of the excess heat is therefore a merger . the temperature structure seen in figure [ fig : contbin ] points to such a merger being roughly along a nw - se direction , such that gas heated in collisions escapes to the sides . it is also consistent with the velocity data discussed in [ sec : velocities ] and the two centers marked as crosses in figure [ fig : armhole ] . if we take the merging group to have about 5 per cent of the mass of abell3744 , based on the number of galaxies within known velocities , and use the radial - velocity offset of roughly 1700 km s@xmath2 as the total velocity difference , then the conversion of 20 per cent of the kinetic energy of the group would be sufficient to produce the required excess heating . that is , the heating requires only a minor merger event . the absence of a shock feature in the x - ray image is not unexpected if this merger is indeed responsible for the heating , since such shocks are only seen when the merger is in the plane of the sky , and the radial velocity difference would be excessive if the merger axis is far from the line of sight in the present case . numerical simulations have shown that mergers lead to persistent ( gigayears ) relative motion of gas streams which are in pressure equilibrium ( sloshing ) , and this can explain x - ray features seen rather commonly in relaxed cool - core galaxy clusters and known as cold fronts , where a large density discontinuity is in pressure balance due to the dense gas being cooler @xcite . abell3744 is far from relaxed , and shows no obvious evidence of sloshing . we clearly see density substructure , but the temperature and density features lie in different parts of the gas , the former having a closer relationship to the location of the radio tendrils . the most likely explanation is that we are witnessing a relatively recent merger encounter , one which may also have helped trigger the radio galaxies into their current phase of activity . the excess x - ray emission in the region we name the ` plateau ' can be understood since the centroid of the distribution of galaxies with reported velocities ( biased towards the more massive ones ) lies in this neighborhood . its temperature is unremarkable compared with that in the x - ray brightest region , including the temperatures in the arm and surrounding the cavity . what is remarkable is that the radio tendrils of ngc7018 appear to border the hottest gas ( fig . [ fig : contbin ] ) , and envelop the roughly cylindrical gas region seen in figure [ fig : armhole ] that forms the bright x - ray bridge between the arm and the plateau . it is tempting to invoke cause and effect , such that the radio plasma of ngc7018 helps to reduce thermal transport between the warmer and cooler gas , and to reduce momentum transport and hence viscous drag , so that x - ray gases of different temperatures can slide by one another with little heat exchange or mixing . the regions in the temperature map were defined automatically from the smoothed x - ray data , with no reference to the radio structures . it is also notable that the southern tendril of ngc7016 lies along a temperature boundary . the morphology of the radio and x - ray extensions suggest that the radio extensions of ngc7018 s lobes were caused by the relative motion of the host galaxy and the gas extending into the plateau . the attachment of radio plasma to this infalling gas would cause the extension of the radio lobes into the tendrils and cause the gas bridge to become at least partially enveloped by the radio plasma . because the radio plasma is moderately strongly magnetized , it will act as an effective barrier to transport between the inner arm and the general cluster environment . not only are the particle gyroradii smaller ( by a factor of about 10 ) in the radio plasma than in the general cluster environment , but the field will also tend to be ordered by the stretching along the radio tendrils . this will reduce the ( perpendicular ) thermal conductivity between the two gas regions . the degree of thermal protection offered by the radio features depends both on the reduction in transport through the radio plasma and the fraction of the interface between regions of different temperature covered by the radio plasma . taking spitzer conductivity @xmath130 @xcite appropriate for intracluster gas , and modeling the gas as a cylinder of radius @xmath131 kpc , the timescale for a temperature difference @xmath132 to be erased is of the order @xmath133 or about 100 myr , taking the proton density , @xmath134 , to be roughly 940 m@xmath44 ( [ sec : integrated ] ) . this is significantly shorter than the time ( about 300 myr ) taken for the gas to travel from ngc7018 to the plateau at the sound speed . however , it appears that at least half the surface area is covered by radio plasma . if this radio plasma cuts @xmath135 by a factor of about 10 , then effective heat transport occurs over only about half the surface of the cylinder and its thermal lifetime is raised by a factor of order 3 . the associated drop in viscosity may also help the gas to slide in towards the plateau . there is good evidence of x - ray emission from both the east and west components of the northern double hotspot of ngc7018 , each with similar ratios of x - ray to radio flux density . the x - ray emission is about two orders of magnitude higher than expected due to inverse - compton scattering if the electrons and magnetic field are at minimum energy . this result is in common with many other _ chandra_-observed hotspots , particularly those in fr ii radio galaxies at the lower end of the spectrum of total radio power @xcite , the large reduction in magnetic field ( about a factor of 24 for ngc7018 ) and the subsequent high increase in source energy needed to explain the x - ray emission by the inverse - compton mechanism led @xcite to suggest that the electron spectrum extends to high enough energies in such hotspots for synchrotron radiation to dominate the x - ray output . in ngc7018 s hotspots , if the electron spectrum breaks from a power law slope of @xmath136 ( giving the observed @xmath137 ) by @xmath138=1 at a lorentz factor of @xmath139 , the synchrotron radiation from 100-tev electrons could be responsible for the x - rays in both components in minimum - energy magnetic fields of about 10 nt . while this is an attractively simple explanation , we note that @xcite have used _ spitzer _ data to claim that a single - component broken - power - law synchrotron spectrum can be excluded for 80 per cent of 24 hotspots they study . they point to the work of @xcite as possible support for magnetic fields in hotspots not having increased to minimum - energy levels . an alternative reason that hotspots on the jet side of a source ( as in ngc7018 ) may be unusually x - ray bright is if relativistic beaming is a factor . an applicable mechanism suggested by @xcite is that for a jet approaching a hotspot with high bulk lorentz factor the available photon fields for compton scattering will be boosted by the strongly directional ( particularly in the jet frame ) radio synchrotron emission from the terminal hotspot , resulting in inverse compton x - rays beamed in the forward direction of the jet and offset ( upstream ) from the peak of the radio emission . at high redshift the spatial separation of components necessary to test this mechanism is not possible , although a claim at low redshift has been made for pks b2152 - 699 @xcite . in ngc7018 there is some evidence for an offset between radio and x - ray emission , although we can not rule out residual uncertainties in our 5-ghz radio mapping 3.5 arcmin off axis as responsible . a non - detection of the southern hotspot region , at the termination of the jet pointed away from the observer , is consistent with beaming playing a part in the detection of the x - ray hotspots , although statistics are such that the formal upper limit on the x - ray flux is at a similar level to the detections in the north , and so without more sensitive x - ray data firm conclusions can not be drawn . we have presented results using new _ chandra _ and jvla observations of abell3744 , together with archival lower - resolution vla data . we find the cluster to be far from relaxed , and devoid of a cool core . the gas is too hot on average by @xmath31 kev for the cluster to agree with temperature - luminosity correlations , and while there is a 100 kpc - scale cavity carved out by radio - emitting plasma , the excess enthalpy is insufficient to explain the heating . much of what is observed seems likely to have been caused by the recent merger of a small subgroup of galaxies with fast relative motion at relatively small angle to the line of sight in a roughly se - nw projected direction . this can easily produce enough heating , and seems to explain the temperature distribution , where hotter gas lies in directions perpendicular to the inferred line of encounter . existing galaxy - velocity data provide some support for this minor - merger hypothesis , but a more detailed survey of galaxy velocities in the field is needed to provide a more complete test . while the x - ray data provide no evidence for shocks ( understandable since velocity data suggest the encounter is not in the plane of the sky ) , or isobaric density drops ( cold fronts ) that would be explainable by gas sloshing , there is much interplay between the x - ray gas and the relativistic plasma hosted by ngc7016 and ngc7018 . this radio - emitting plasma terminates in buoyant tendrils reaching the cluster s extremities . in the case of ngc7018 , a tendril trails from each radio lobe , almost certainly as a result of the galaxy s motion and consequent drag . in contrast , the counter jet of ngc7016 runs into the x - ray cavity , where it produces a dramatic swirl perhaps due to collision with the radio plasma from ngc7018 or in the wake of its motion , but a tendril extends from the highly bent southeastern jet . an important and noticeable feature of the tendrils is that they run along boundaries between gas of different temperatures . because the radio plasma is moderately strongly magnetized , it will act as an effective barrier to transport between gas layers and reduce the effective viscosity , helping to preserve post - merger gas flows . the radio galaxies hosted by ngc7016 and ngc7018 both have powers in the range that dominates output in the universe as a whole , and so are most important for understanding radio - mode feedback . the most apparent ways in which this feedback is mediated by these sources is through the large x - ray cavity and the capacity for the tendrils to allow the sliding of gas flows and the containment of temperature structures . in other respects the x - ray properties of the radio galaxies are normal for their radio powers . the power - law x - ray emission from both nuclei is unabsorbed and lies within the scatter of radio / x - ray correlations , arguing in favor of a common non - thermal origin for the radio and x - ray core emission . resolved x - ray emission is detected from the brighter jet in each source . x - rays are detected from both components of the northern double hotspot of ngc7018 , which we have discussed in some detail given the on - going debate concerning the flow - speed into hotspots and the origin of their x - ray emission . we acknowledge support from nasa grant go1 - 12010x . we thank the anonymous referee for constructive comments , and paul giles for discussion of the temperature luminosity relation . results are largely based on observations with _ chandra _ , supported by the cxc . the national radio astronomy observatory is a facility of the national science foundation operated under cooperative agreement by associated universities , inc . this research has made use of the nasa / ipac extragalactic database ( ned ) which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this publication makes use of data products from the two micron all sky survey , which is a joint project of the university of massachusetts and the infrared processing and analysis center , funded by the national aeronautics and space administration and the national science foundation .

pseudodiffusive transmission refers to the @xmath1 scaling of the electrical current transmitted over a distance @xmath0 through a clean sheet of undoped graphene @xcite . the same @xmath1 scaling characterizes diffusion in a random potential , but now it applies in the absence of any disorder . there is a large number of theoretical @xcite and experimental @xcite studies of this phenomenon , which is understood as a general property of massless dirac fermions in the limit of vanishing excitation energy . the optical analogue in a photonic crystal with a dirac spectrum has been studied as well @xcite . layered superconductors with a _ d_-wave symmetry of the order parameter ( notably the high-@xmath2 cuprates @xcite ) form an altogether different system in which massless dirac fermions are known to exist @xcite . these are so - called nodal fermions , located in the two - dimensional brillouin zone near the intersections ( nodal points ) of the fermi surface with lines ( nodal lines ) of vanishing excitation gap . elastic mean free paths @xmath3 as large as @xmath4 have been reached in @xmath5 single - crystals @xcite , much larger than the superconducting coherence length @xmath6 . it is the purpose of this work to demonstrate theoretically the pseudodiffusive @xmath1 scaling of the transmission through a _ d_-wave superconductor over the range of lengths between @xmath7 and @xmath3 . this anomalous scaling was not noticed in earlier studies of similar systems @xcite . the problem is interesting from a conceptual point of view , because it highlights both the differences and similarities between dirac fermions produced by a bandstructure ( as in graphene or photonic crystals ) or produced by a _ d_-wave order parameter . in undoped graphene , the transmitted electrical current @xmath8 in response to a voltage difference @xmath9 scales as @xcite @xmath10 the length @xmath0 over which the current is transmitted should be large compared to the fermi wave length @xmath11 in the metal contacts , but small compared to the mean free path @xmath3 . the length @xmath0 should also be small compared to the transverse width @xmath12 of the graphene sheet ( to avoid edge effects ) . potential barriers ( smooth on the scale of the lattice constant ) at the interfaces between the metal contacts and the graphene sheet have no effect on the current , because of the phenomenon of klein tunneling @xcite . for the _ d_-wave superconductor , we find a transmitted electrical current per layer equal to @xmath13 for @xmath14 . here @xmath15 are the tunnel probabilities through the potential barriers at the two normal - metal superconductor ( ns ) interfaces . the dirac equation for nodal fermions is anisotropic @xcite , with different velocities @xmath16 and @xmath17 parallel and perpendicular to the nodal lines . this anisotropy ( with @xmath18 in @xmath5 ) increases the slope of the @xmath1 scaling . remarkably enough , the anisotropy does not introduce a dependence of the transmitted current on the angle @xmath19 between the direction of the current and the nodal lines . the result holds generically for any orientation , except for a narrow range of angles of order @xmath20 around @xmath21 . the tunnel barriers reduce the slope of the @xmath1 scaling of the transmitted electrical current , by a factor @xmath22 for small tunnel probabilities . this does not imply that the nodal fermions are only weakly transmitted , but rather that the transmission probabilities for transmission as an an electron or as a hole are almost the same for @xmath23 . indeed , we find that the electrical shot noise power @xmath24 as well as the transmitted thermal current @xmath25 ( both of which do not depend on the sign of the carriers charge ) remain finite in the limit @xmath26 . we interpret this result in terms of a resonant coupling via the nodal lines of the mid - gap states @xcite extended along the two ns interfaces . we also find , quite surprisingly , that the thermal conductivity is _ independent _ of the tunnel probabilities @xmath27 . the outline of this paper is as follows . in sec . [ nodalt ] we formulate the scattering problem and calculate the transfer matrix of the nodal dirac fermions through the _ d_-wave superconductor . the matching of wave functions at the interface with the metal electrodes is done in sec . [ wavematching ] , both for ideal ns interfaces and for interfaces containing a tunnel barrier . the transmission matrix of electrons and holes follows in sec . [ transmission ] . we then apply this result to the calculation of transport properties : the electrical current ( sec . [ electric ] ) , the thermal current ( sec . [ thermal ] ) , and the electrical shot noise ( sec . [ shotnoise ] ) . we conclude in sec . [ discuss ] with a discussion of our results and an outlook . geometry to measure the transmission of nodal fermions through a @xmath28-wave superconductor . a current @xmath29 is injected into the superconductor from metal contact @xmath30 ( at a voltage @xmath9 ) and drained to ground via the superconductor ( current @xmath31 ) or via a second metal contact @xmath32 ( current @xmath33 ) . if the separation @xmath0 of the metal contacts is large compared to the superconducting coherence length @xmath7 , the current @xmath33 is predominantly due to transmission parallel to the nodal lines @xmath34 or @xmath35 of vanishing excitation gap . ] we consider a two - dimensional spin - singlet superconductor ( s ) , connecting two normal metal contacts with parallel ns interfaces , separated by a distance @xmath0 . the transverse dimension @xmath12 of the superconducting strip ( in the @xmath36-@xmath37 plane ) is assumed to be large compared to @xmath0 , in order to avoid edge effects . the order parameter @xmath38 is assumed to have @xmath39 symmetry : it vanishes for wave vectors along two nodal lines , which are taken to be the @xmath36 and the @xmath37 axis . all our results also apply to @xmath40-superconductors , for our purposes , a simple @xmath41 rotation relates the two systems . to be specific , the @xmath36-@xmath37 plane can represent a single @xmath42 layer of a cuprate superconductor @xcite , with the @xmath43 $ ] direction at an angle @xmath41 . ellipsoidal equal - energy contours of low - energy excitations in the brillouin zone of a superconductor with @xmath39 symmetry . long and short axes have ratio @xmath44 . the contours are centered at the four nodal points ( solid dots ) , where the order parameter vanishes on the fermi surface . the normal @xmath45 to the ns interfaces is indicated . the dashed line , displaced from the nearest nodal point by @xmath46 , indicates points of constant wave vector component parallel to the interface . ] low - energy excitations in the superconductor are found in the brillouin zone near the four intersections @xmath47 , @xmath48 ) of the fermi surface with the nodal lines of the order parameter . ( these nodal points are labeled @xmath49 in fig . [ fig_brillouin ] . ) around these points , both the pair potential @xmath38 and the kinetic energy can be linearized : the dynamics of the nodal fermions is governed by an anisotropic dirac equation @xcite . for example , near node @xmath50 at @xmath51 this can be written in the form @xmath52 or more compactly with the help of pauli matrices , @xmath53\psi=\varepsilon\psi . \label{dirac2}\ ] ] we have set @xmath54 to unity , restoring units in the final expressions . the spinor @xmath55 contains the envelope wave functions of electron and hole excitations ( slowly varying on the scale of the fermi wavelength @xmath56 ) . the fermi velocity @xmath16 is larger than the velocity @xmath57 by a factor of order @xmath58 ( with @xmath59 the superconducting coherence length ) , which is in the range 1020 for cuprate superconductors . the equal - energy contours in the brillouin zone of the nodal fermions thus have an elongated ellipsoidal shape , @xmath60 as a function of the displacement @xmath61 of the wave vector from the nodal point . since the system is translation invariant along the ns interfaces , the component of the wave vector along these interfaces , @xmath62 , is a conserved quantity . here @xmath19 is the angle between the normal to the ns interface and the nodal line pointing to node @xmath50 , which we restrict to @xmath63 without loss of generality . moreover , since mirror reflection along the ns interface , followed by the transformation @xmath64 , while leaving all the other parameters unchanged , maps @xmath19 on @xmath65 , we can further restrict @xmath19 to @xmath66 . in all our formulas , to obtain the corresponding formulas for @xmath65 , replace @xmath46 by @xmath67 and @xmath68 by @xmath69 . we write @xmath70 , with @xmath71 the coordinate perpendicular to the ns interfaces and @xmath72 the coordinate parallel to them . we substitute @xmath73 into eq . and find that the spinor @xmath74 satisfies the wave equation @xmath75 \psi(s)=\varepsilon\psi(s ) , \label{dirac3}\ ] ] where @xmath76 is differentiation perpendicular to the ns interface , and @xmath77 and @xmath78 are the operators of particle current perpendicular and parallel to the ns interface , @xmath79 we note that the operator @xmath77 squares to a scalar , its magnitude giving the particle velocity @xmath80 perpendicular to the ns interface : @xmath81 to solve eq . , we multiply it by @xmath82 and rearrange to obtain @xmath83 with @xmath84 the solution to eq . can then be written as @xmath85 , \label{msdef}\end{aligned}\ ] ] where the second equation defines the _ transfer matrix _ @xmath86 . as expected , the particle current @xmath77 is conserved by eq . : @xmath87 = \psi^\dagger(s ) [ -i{\mathcal{a}}_0^\dagger { j}+i { j}{\mathcal{a}}_0 ] \psi(s)=0 $ ] . at the two ns interfaces the coupled electron - hole excitations in the superconductor are converted into uncoupled electrons and holes in the normal metal . we thus need to match , at @xmath88 and @xmath89 , the envelope wave functions @xmath55 of the nodal fermions in s to the bloch wave functions @xmath90 of free fermions in n. this is similar to the matching of dirac equation to helmholtz equation considered in the context of transmission through a photonic crystal @xcite . translational invariance parallel to the ns interfaces requires that the coupling conserve the wave vector component @xmath46 parallel to the interfaces . particle flux conservation imposes further constraints , as we determine here . at the surface of the superconductor , the order parameter @xmath91 attains its bulk value over a short length scale , the healing length @xmath92 . the two - component wave function on the s side of the interface ( at @xmath93 ) can be linked to that on the n side ( @xmath88 ) by an _ interface matrix _ @xmath94 , defined by @xmath95 in the normal metal , the operator of particle flux perpendicular to the ns interface can be written as @xmath96 with @xmath97 possibly different from @xmath98 because of a fermi energy mismatch . the requirement of particle flux conservation reads @xmath99 to derive the most general form of the interface matrix fulfilling this requirement , notice that a unitary rotation through angle @xmath100 , where @xmath101 , \label{theta_def}\end{aligned}\ ] ] transforms @xmath77 into @xmath102 up to a scalar factor : @xmath103 { j}\exp[-i\theta \sigma_y /2 ] . \label{jndef}\ ] ] this allows us to write the interface matrix as @xmath104 { \mathcal{m}}_0 , \label{mnsdef}\ ] ] where @xmath105 is a @xmath106 matrix fulfilling a generalized unitarity condition , @xmath107 eq . restricts @xmath108 to a three - parameter form @xmath109 ( ignoring an irrelevant scalar phase factor ) , with arbitrary real parameters @xmath110 . to understand better where the nontrivial interface matrix arises from , and to show that we may set @xmath111 , we have to extend the dirac equation to the interface layer , where @xmath68 varies in space . this is done in appendix [ mns_derivation ] . so far we have considered only _ intranode _ scattering at the ns interface . we refer to such an interface as an `` ideal interface '' . a nonideal interface contains a tunnel barrier , which introduces _ internode _ scattering . we will consider the transfer matrices through the _ d_-wave superconductor for both cases in the next two subsections . the complete transfer matrix for a strip of _ d_-wave superconductor with ideal ns interfaces reads @xmath112 where @xmath113 is the @xmath114 from eq . with @xmath89 , describing propagation inside the superconductor , and @xmath94 from eq . , with @xmath115 , describes an ns interface . upon substitution , we obtain @xmath116\nonumber\\ & \quad\mbox{}+\frac{\sinh[\kappa_{\alpha}(q)l]}{v_{\alpha}^{2}\kappa_{\alpha}(q)}\left(i\varepsilon v_{\alpha}\sigma_{z } + qv_{f } v_{\delta}\sigma_{y}\right)\biggr ] , \label{mideal}\end{aligned}\ ] ] with the definitions @xmath117 notice , how as a result of accounting for the two ns interfaces the transfer matrix has simplified from that of eq . . the change is that the particle flux operator @xmath77 in eq . is replaced by @xmath118 in eq . . also note that the determinant of the transfer matrix has norm one , @xmath119 , as required by the generalized unitarity relation @xmath120 which holds for any transfer matrix as a consequence of particle current conservation . to appreciate the effects of the dirac cone anisotropy , we can perform a linear transformation on our system to obtain one with an isotropic dirac cone : contraction along the nodal line by a factor @xmath121 , and expansion perpendicular to it by a factor @xmath122 . the dispersion of the new , isotropic dirac cone has a single velocity parameter @xmath123 . the superconducting strip is deformed by the transformation : its width @xmath12 is unchanged , but its length @xmath0 becomes @xmath124 an effective propagation length we define here for later use . the matrix derived above is the transfer matrix for nodal fermions near point @xmath125 on the fermi surface , with @xmath126 the transverse wave vector component relative to @xmath127 . similarly , the transfer matrices near each of the four nodal points can be written as [ mabcd ] @xmath128 the basis at each nodal point is the same spinor @xmath129 , but the electron states @xmath130 are `` right - movers '' ( propagating from @xmath30 to @xmath32 ) at nodal points @xmath50 or @xmath131 and `` left - movers '' ( from @xmath32 to @xmath30 ) at nodal points @xmath132 and @xmath133 . the complete fermi surface of the normal metal ( n ) might differ in many ways from that of the superconductor . however , when we study transport near a specific nodal point , due to transverse momentum conservation , we can effectively reduce the fermi surface to the two @xmath134 points where transverse momentum has the same value as at the nodal point . these two @xmath134 points in n each couple to different nodal points in s , for example to nodal points @xmath50 and @xmath132 in fig . 2 . a nonideal ns interface couples different nodal points , by reversing the component of the momentum perpendicular to the interface . such _ internode scattering _ may be caused by an insulating layer at the ns interface , or it may result from the fermi velocity mismatch between n and s. note that only internode scattering is possible in the absence of superconducting order any _ intranode _ scattering has to happen inside the superconductor . we will generically describe a nonideal ns interface by a tunnel barrier , with tunnel probability @xmath135 ( which we take mode independent for simplicity ) . for @xmath136 , the tunnel barrier couples electrons near nodal points @xmath50 and @xmath132 . the transfer matrix @xmath137 for a tunnel barrier at position @xmath138 , defined by @xmath139 has the form @xmath140 with @xmath141 . the tunnel barrier at @xmath138 also couples holes near nodal points @xmath50 and @xmath132 , with transfer matrix @xmath142 ( the basis states are chosen such that the upper component is a right - mover and the lower component a left - mover . ) finally , we can write down the full transfer matrix of the superconducting strip , in the basis @xmath143 , including nonideal contacts with tunneling probabilities @xmath144 at @xmath88 and @xmath145 at @xmath89 . it is obtained by matrix multiplication , @xmath146 with @xmath9 a unitary matrix that switches bases from @xmath147 to @xmath143 : @xmath148 if @xmath149 , the nodal point @xmath50 is coupled to the nodal point @xmath133 , so the above formulas still hold , with @xmath132 replaced by @xmath133 . if both @xmath150 and @xmath151 , the tunnel barriers at the interfaces do not couple nodal point @xmath50 to any other nodal points . since we assume @xmath152 , this case of misaligned nodes is the generic case . in that case , @xmath153 is to be replaced by the singular transfer matrix @xmath154 corresponding to andreev reflection with reflection amplitude @xmath155 , @xmath156 since @xmath154 is also the @xmath157 limit of @xmath158 in eq . ( up to an irrelevant phase factor ) , eq . is valid as it stands for misaligned nodes as well . referring to the geometry of fig . [ fig_layout ] , a scattering state ( for a given value of @xmath46 ) has the form @xmath159 at the normal side of the left ns interface and @xmath160 at the normal side of the right ns interface . the complex number @xmath161 is the amplitude for andreev reflection ( from electron to hole ) and the complex number @xmath162 is the amplitude for electron transmission . we calculate this transmission amplitude using the relation @xmath163_{11}\right)^{-1 } = \left([{\mathcal{m}}_a^\dagger]_{11 } \,\right)^{-1 } , \label{tmrelation}\ ] ] where the first equality follows from @xmath164 , and the second equality from particle current conservation , eq . . substitution of eq . gives the expression @xmath165-\frac{i\varepsilon\sinh[\kappa_{\alpha}(q_a ) l]}{v_{\alpha}\kappa_{\alpha}(q_a)}\right]^{-1}.\label{teeresult}\ ] ] for nonideal interfaces we have to consider both the transmission amplitude @xmath162 from electron to electron and the transmission amplitude @xmath166 from electron to hole . it is convenient to define the @xmath106 transmission matrix @xmath167 which contains also the transmission amplitudes @xmath168 and @xmath169 from hole to electron and from hole to hole . this matrix @xmath170 is a @xmath106 subblock of the @xmath171 unitary scattering matrix @xmath172 , which we derive in appendix [ fulls ] . to obtain @xmath170 from the @xmath171 transfer matrix @xmath173 , we make a change of basis from the basis @xmath143 used in eq . to a basis @xmath174 in which the upper two components are right - movers and the lower two components are left - movers . the change of basis is carried out by the unitary matrix @xmath175 we can then follow the same reasoning as in the previous subsection , to conclude that @xmath170 is determined by the @xmath106 upper - left block @xmath176 of @xmath177 , @xmath178 cf . . substitution of @xmath173 from eq . gives , after some algebra , @xmath179 turning now to observable quantities , we will work in the linear response regime @xmath180 , when the transmission amplitudes may be evaluated at the fermi level ( @xmath181 ) . the current @xmath182 ( per layer ) transmitted into metal contact @xmath32 through nodal point @xmath50 is obtained by integrating the transmission probability @xmath183 over @xmath184 , @xmath185 ( the conductance quantum @xmath186 includes a twofold spin degeneracy . ) the integrand decays exponentially for @xmath187 . for @xmath188 the effective integration range is much smaller than @xmath189 and may be extended to @xmath190 . substituting eq . ( for @xmath181 ) we arrive at @xmath191 as expected , the conductance of a single nodal point has the same form as that of a single valley in a graphene strip , with @xmath0 replaced by the effective propagation length @xmath192 of eq . . the current @xmath193 transmitted through nodal point @xmath131 is given by the same formula with @xmath80 replaced by @xmath194 . because of the identity @xmath195 the total current @xmath196 becomes _ independent _ of @xmath19 . the conductivity @xmath197 per layer for the case of ideal ns interfaces is then equal to @xmath198 as discussed in sec . [ bulk ] , eq . differs [ by a factor @xmath199 from the bulk electrical conductivity of refs . @xcite . for nonideal ns interfaces , tunnel barriers couple the nodal points , and the calculation of the current @xmath200 becomes more involved . in this section we treat the generic case of misaligned nodal points . the case of ( perfectly ) aligned nodal points is considered in appendix [ galigned ] . we first calculate the current through nodal point @xmath50 . as discussed in sect . [ nonideal ] , we can substitute @xmath153 with @xmath154 of eq . , and using eq . we obtain the transmission matrix ( at @xmath181 ) @xmath201 where the denominator @xmath202 has the form @xmath203.\label{zdefa2}\end{aligned}\ ] ] here @xmath192 is the effective propagation length , while @xmath204 is the transverse wave number defined by @xmath205.\end{aligned}\ ] ] both @xmath162 and @xmath166 are peaked at @xmath206 . this peak momentum lies at the nodal point ( @xmath207 ) only for ideal interfaces . in the presence of tunnel barriers the sign of @xmath204 is such that the order parameter has opposite sign at the two intersections of the line @xmath206 with the fermi surface . integration over @xmath184 of electron current minus hole current gives the net ( electrical ) current , @xmath208\nonumber\\ & = g_{0}v\frac{w}{l}\frac{v_{\alpha}^{2}}{\pi v_{f}v_{\delta}}\frac{1}{2-\gamma_1 } \frac{\gamma_2}{2-\gamma_2}.\label{i2anonideal}\end{aligned}\ ] ] similarly , for the current through nodal point @xmath132 we take the limit @xmath209 of eq . and obtain the transmission matrix @xmath210 , \label{zdefc}\end{aligned}\ ] ] and then the current @xmath211 note the minus sign in the formula for @xmath212 . the current has opposite sign to that at nodal point @xmath50 , since here holes rather than electrons tunnel across the system to contact 2 . the total current ( per layer ) through nodal points @xmath50 and @xmath132 becomes @xmath213 comparison with eq . reveals that each tunnel barrier changes the sum of the current transmitted through a nodal point and the one opposite to it in momentum space , its time - reversed partner , by a factor of @xmath214 . as in the case of ideal ns contacts , the pair of nodal points @xmath131 and @xmath133 contribute a same amount , but with @xmath215 replaced by @xmath216 . the @xmath19-dependence again drops out of the total current @xmath217 . for the conductivity @xmath218 per layer we finally obtain @xmath219 the conductivity vanishes in the weak tunneling limit @xmath220 , because the electron and hole contributions to the electrical current @xmath33 then become equal but of _ opposite sign_. electrons and holes contribute with the _ same sign _ to the thermal current , @xmath221,\label{ithermal}\ ] ] with @xmath222 the lorenz number . the thermal current flows from contact @xmath30 at temperature @xmath223 into contact @xmath32 at temperature @xmath224 . ( eq . requires @xmath225 and @xmath224 sufficiently small that the transmission amplitudes may be evaluated at the fermi energy @xmath181 . ) we consider the ( generic ) case of misaligned nodes . substitution of the expressions for @xmath170 from sec . [ gmisaligned ] , and summing over the pair of nodal points @xmath50 and @xmath132 , we find that @xmath226 quite surprisingly , this turns out to be _ independent of the tunnel probabilities _ @xmath144 and @xmath145 . the total thermal current ( per layer ) also includes contributions from the nodal points @xmath131 and @xmath133 , and is just as the electrical conductivity independent of the angle @xmath19 : @xmath227 as discussed in sec . [ bulk ] , the thermal conductivity @xmath228 extracted from eq . coincides with the bulk thermal conductivity of ref . @xcite . the zero - frequency noise power of time dependent electrical current fluctuations @xmath229 measured in contact number 2 , @xmath230 is given in terms of the transmission matrix elements by the general expression @xcite @xmath231.\label{p22}\end{aligned}\ ] ] as with the conductance , we work in the linear response regime , so the transmission matrix is to be evaluated at @xmath181 . we restrict ourselves to the case of misaligned nodes and substitute the expressions for @xmath170 from sec . [ gmisaligned ] . the integral over @xmath46 contains four separate contributions , from @xmath46 near nodes @xmath50 , @xmath131 , @xmath132 , and @xmath133 . the total result ( per layer ) is @xmath232 the fano factor @xmath233 is given by @xmath234 in the ideal limit @xmath235 we find a fano factor @xmath236 , three times smaller than the value @xmath237 associated with a poisson process . as discussed in the context of graphene @xcite , this is the same one - third reduction as in a diffusive metallic conductor and is a hallmark of pseudodiffusive transmission . in the weak tunneling limit @xmath220 the noise power remains finite , @xmath238 while the electrical current vanishes , @xmath239 . the electrical current fluctuations therefore become large relative to the time - averaged current in the presence of tunnel barriers . this is discussed in the context of resonant tunneling through midgap states in sec . [ midgap ] . the electrical current @xmath33 and thermal current @xmath25 that we have calculated describe transmission of electrons and holes over a finite length @xmath0 of a clean _ d_-wave superconductor . earlier work @xcite calculated the electrical and thermal conductivities @xmath240 and @xmath241 of a disordered infinite system . these are in principle different systems , but we can still compare them by formally converting the currents through the finite system into bulk conductivities by means of @xmath242 and @xmath243 . the thermal conductivity obtained in this way from the finite - system thermal current , @xmath244 is the same as the bulk thermal conductivity of durst and lee @xcite . the results for the electrical conductivity differ , however . the bulk result @xcite @xmath245 differs from the finite - system result even if we assume ideal ns interfaces . the difference between the factors @xmath44 in eq . and @xmath246 in eq . is small in practice ( because @xmath247 ) , but the difference does illustrate that these are different systems . we have found that tunnel barriers at the ns interfaces reduce the transmitted electrical current , but not the thermal current nor the electrical noise . this result has a natural interpretation in terms of the midgap states at the ns interfaces . midgap states are zero - energy edge states of the _ d_-wave superconductor , which exist at momentum @xmath46 along the edge if the order parameter has opposite sign at the two intersections of the line of constant @xmath46 with the fermi surface @xcite . the midgap states at the two ns interfaces have a small overlap , and therefore acquire a nonzero energy @xmath248 ( tunnel splitting ) . moreover , the coupling to the metal electrodes at @xmath249 introduces partial widths @xmath250 , @xmath251 of the midgap states ( tunnel broadening ) . tunneling through a pair of midgap states was studied in ref . @xcite , in the context of majorana bound states ( which are a special type of nondegenerate midgap states ) . we can compare the transmission probabilities resulting from that work , @xmath252 with the results from sec . [ gmisaligned ] in the tunneling limit @xmath253 , @xmath254 we have defined @xmath255 this comparison leads to the identification @xmath256 resonant tunneling , with all transmission probabilities equal to @xmath257 , occurs when @xmath258 , hence when @xmath259 ( tunnel splitting of the midgap states equal to tunnel broadening ) . because transmission from electron to electron and from electron to hole happens with the same probability ( to leading order in @xmath260 ) , the transmitted electrical current vanishes in the limit of small @xmath135 . the thermal current @xmath25 and electrical noise @xmath261 remain finite , because @xmath183 and @xmath262 contribute with the same sign to these quantities . this interpretation explains the finite small-@xmath135 limit for @xmath261 and @xmath25 , but it does not explain why the thermal current turns out to be completely independent on the values of @xmath144 and @xmath145 . that remains a surprising result of our calculation , for which we have no qualitative explanation . we have shown how ballistic transport through a clean _ d_-wave superconductor ( such as single - crystal @xmath5 ) has features in common with graphene @xcite : a pseudodiffusive @xmath1 scaling of the electrical current transmitted over a distance @xmath0 , and a @xmath263 suppresssion of the electrical shot noise with respect to the poisson value of uncorrelated current pulses . these effects have been observed in graphene @xcite and it would be interesting to search for them in the high-@xmath2 cuprates . the @xmath1 scaling should persist , with a modified slope , in the presence of tunnel barriers at the ns interfaces , and in the case of the thermal current we find that even the slope is independent of the tunnel barrier height . there are more areas of correspondence between massless dirac fermions in _ d_-wave superconductors and in graphene , in addition to the pseudodiffusive transport studied in this work . we mention two such effects , as directions for future research . * in graphene an electrostatic potential can displace the fermi level away from the dirac point of vanishing density of states . d_-wave superconductor the supercurrent velocity @xmath264 enters into the dirac equation as a scalar term @xmath265 @xcite , and therefore has the same effect of displacing the dirac point relative to the fermi level . there is one curious difference with respect to graphene : the _ d_-wave superconductor has two pairs of valleys and the dirac point can be displaced independently in each pair ( relative to the same fermi level ) . with reference to fig . [ fig_brillouin ] , the component of @xmath264 in the @xmath36-direction acts on valleys at the nodal points @xmath50 and @xmath132 , while the component in the @xmath37-direction acts on those at @xmath131 and @xmath133 . * while the role of an electrostatic potential in graphene is played by the supercurrent , an electric field in the _ d_-wave superconductor plays the role of a magnetic field in graphene . if a sufficiently strong electric field could be induced in a thin - film cuprate superconductor , it might be possible to see effects analogous to the effects of landau level quantization in graphene @xcite . this research was supported by the dutch science foundation nwo / fom and by an erc advanced investigator grant . in sec . [ tmideal ] we derived the most general form of the transfer matrix of an ns interface , consistent with the requirement of particle flux conservation . the result in eq . has three undetermined parameters @xmath266 , @xmath267 , and @xmath268 . here we calculate the interface matrix by solving the dirac equation in the interface layer and determine these unknown parameters . the interface layer is the region where the order parameter increases from @xmath269 to its bulk value , over a healing length @xmath92 ( which is typically of the same order of magnitude as the coherence length @xmath7 ) . as discussed in ref . @xcite , in order to preserve hermiticity , the dirac equation needs to be supplemented by terms containing the spatial derivatives of @xmath17 : @xmath270 \psi = \varepsilon \psi . \label{dirac4}\ ] ] we assumed that the phase of @xmath91 is constant , and set it to 0 ( without loss of generality ) , thus @xmath68 is real throughout , and is only a function of the distance @xmath271 from the ns interface . an eigenstate @xmath74 of momentum @xmath46 parallel to the ns interface satisfies @xmath272 \psi(s ) = \varepsilon \psi(s),\ ] ] with the derivative of @xmath68 denoted by the shorthand @xmath273 . accordingly , the matrix @xmath274 in eq . becomes @xmath271-dependent and gets a new term : @xmath275 since @xmath276 ( in the relevant range of @xmath46 s near the nodal point ) , the integral of @xmath277 over the interface layer is @xmath278 and may be neglected . then @xmath279 commutes with @xmath280 for @xmath281 , and therefore we can simply integrate eq . over the interface layer : \nonumber\\ \quad & = \exp\left[\frac{-i}{2}\int_0^{\tan \theta } \frac{1}{1+u^2 } \left ( \sigma_y - i u \right ) du\right ] \nonumber\\ \quad & = \sqrt{\frac{v_f \cos \alpha } { v_\alpha } } \exp[-i\theta \sigma_y /2],\end{aligned}\ ] ] with @xmath283 $ ] as defined in eq . . the result agrees with eq . with @xmath111 , and @xmath284 . the fermi velocity mismatch contributes an additional factor @xmath285 to the interface matrix , and in addition may cause internode scattering ( as detailed in sec . [ nonideal ] ) . in secs . [ transmission ] and [ electric ] we have calculated the @xmath106 transmission matrix @xmath170 , which is the quantity we need for the transport properties considered . for reference , we give here the full @xmath171 scattering matrix , @xmath286 containing the @xmath106 transmission matrices @xmath170 ( from left to right ) and @xmath287 ( from right to left ) , as well as the reflection matrices @xmath288 ( from left to left ) and @xmath289 ( from right to right ) . these matrices can be obtained from transfer matrix @xmath173 by constructing the four @xmath106 sub - blocks @xmath290 , @xmath291 and then evaluating @xmath292 cf . eqs . and . we restrict ourselves to @xmath181 and misaligned nodes . near node @xmath50 we find the reflection matrices @xmath293 the transmission matrix @xmath294 is given by eq . and the resulting scattering matrix is unitary , @xmath296 , as it should be . similarly , near node @xmath132 we find @xmath297 given by eq . , @xmath298 , and the reflection matrices @xmath299 for @xmath301 the two nodal points @xmath50 and @xmath132 line up with the normal to the ns interface , while nodes @xmath131 and @xmath133 remain misaligned . restricting ourselves again to @xmath181 , we may put @xmath302 , @xmath303 , @xmath304 , and @xmath305 in eq . . the result is @xmath306\cosh l_0 q\quad & i[\sqrt{1-\gamma_1}+e^{-2ik_{f}l}\sqrt{1-\gamma_2}]\sinh l_0 q\\ -i[\sqrt{1-\gamma_1}+e^{2ik_{f}l}\sqrt{1-\gamma_2}]\sinh l_0 q\quad & [ 1+e^{-2ik_{f}l}\sqrt{1-\gamma_1}\sqrt{1-\gamma_2}]\cosh l_0 q \end{pmatrix},\label{talignedresult}\\ z_{ac}={}&2\sqrt{1-\gamma_1}\sqrt{1-\gamma_2}\cos(2k_{f}l)+2-\gamma_1-\gamma_2+\gamma_1\gamma_2\cosh^{2 } l_0 q.\label{zaligneddef}\end{aligned}\ ] ] dependence on the separation @xmath0 of the ns interfaces of the current @xmath33 into contact @xmath32 , for the interface orientation @xmath310 of aligned nodes @xmath50 and @xmath132 . calculated from eqs . for parameters @xmath311 , @xmath312 . ] as shown in fig . [ fig_aligned_a ] , the current @xmath33 oscillates as a function of @xmath313 , between minima @xmath314 at @xmath315 and maxima @xmath316 at @xmath317 . ( similar oscillations were found in ref . @xcite . ) simple expressions for these two values follow for the case @xmath318 of equal tunnel barriers , @xmath319 with abbreviation @xmath320 . for @xmath321 we recover the ideal limit @xmath322 . for @xmath323 we have instead @xmath324 , @xmath325 . same as fig . [ fig_brillouin ] , but now for an angle @xmath328 between the normal @xmath329 to the ns interface and the lines @xmath34 , @xmath35 of vanishing order parameter . for this orientation the nodal points @xmath330 and @xmath331 are pairwise aligned with @xmath329 ( dashed lines ) , so that they are pairwise coupled by a tunnel barrier at the interfaces . ] for @xmath332 , nodal points @xmath330 and @xmath331 are pairwise aligned with the normal to the ns interface ( see fig . 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in the current picture of structure formation , inflation ends in reheating , which produces gaussian random field density fluctuations in the radiation , matter , and dark matter . as a gaussian random field , the density is described completely by its mean and 2-point correlation function ( 2pcf ) , which measures the probability of finding a certain value of the density at one point given the density at another . however , the subsequent evolution of the density field introduces additional correlations as gravity drives the convergence of overdense regions towards each other . in particular , a 3-point correlation function ( 3pcf ) is produced by this evolution ( bernardeau et al . 2002 or szapudi 2005 for reviews ) . since the evolution is itself sensitive to the cosmological parameters , measuring the 3pcf of galaxies offers an independent probe of these parameters . it is typically used to break the degeneracy between galaxy bias ( encoding the fact that galaxies do not trace the matter density field with perfect fidelity ) and the clustering on a given scale ( e.g. @xmath1 ) ( gaztaaga & frieman 1994 ; jing & borner 2004 ; guo et al . the 3pcf measurements also can probe primordial non - gaussianity ( desjacques & seljak 2010 ) ; while the constraints on this are currently dominated by cmb experiments such as planck , it is expected that the increasing quality and number of galaxy redshift surveys will provide interesting independent information . since the first measurement by peebles & groth ( 1977 ) , numerous studies have presented 3pcf measurements , summarized in kayo et al . ( 2004 ) , mcbride et al . ( 2011a , b ) , guo et al . ( 2014 ) and references therein . in this work , we present a new algorithm for measuring the 3pcf of galaxies through its multipole moments . this decomposition of the 3pcf was first advanced in szapudi ( 2004 ) and to a limited extent ( measurement of the monopole moment ) used in pan & szapudi ( 2005 ) on two - degree - field galaxy redshift survey ( 2dfgrs ) data . slepian & eisenstein ( 2015 ) ( hereafter se15 ) found this decomposition to be particularly useful for distinguishing linear and non - linear bias as well as isolating a possible relative velocity bias . current algorithms , such as that used for the mcbride et al . ( 2011 ) measurement ( presented in moore et al . 2001 , gray et al . 2004 , nichol et al . 2006 , and gardner et al . 2007 ) fundamentally scale as the number of possible triangles in a survey . if one wishes to measure the 3pcf out to some scale @xmath2 , there are @xmath3 relevant triangles , where @xmath4 is the number of objects in the survey , @xmath5 is the survey number density and @xmath6 . the algorithm presented in the series of references above , whose most recent incarnation is developed in march ( 2013 ) , uses multiple mrkd - trees . here `` mr '' means the kd - tree caches additional information , in this case the number of galaxies within each node of the tree as well as the bounding box of the node . this algorithm is faster than simply counting all triangles . it is particularly effective if the galaxies are close to each other , so that there are many triangles whose side lengths fall within a given combination of radial bins . however , typical galaxy surveys are sparse , particularly those mapping the largest volumes . for example , the baryon oscillation spectroscopic survey ( boss ) has an average separation of 13 @xmath7 , too large to permit many galaxies to be in the same bin . this means the algorithm will not be as fast for such large - scale measurements . the use case tested in march ( 2013 ) is triangles with three sides of @xmath8 mpc each , much smaller than the scales that are well - described by linear perturbation theory and hence most useful for cosmology . furthermore , even with the speed - ups coming from the multi - tree structure of the algorithm , it is still fundamentally scaling as the number of galaxies in the survey times the square of the number within @xmath2 ( march 2013 , figure 21 ) . in this paper , we present an algorithm that does better : it scales as the number of galaxies in the survey times the number within @xmath2 , and so by construction is significantly faster than any previous algorithm that is exact in angle . in brief , we write the opening angle dependence of the triangles about a given vertex in terms of legendre polynomials of @xmath9 , where these are two unit vectors describing two triangle sides . the dot product seems to require explicitly considering all pairs of galaxies about a given vertex ( i.e. third galaxy ) , but using the spherical harmonic addition theorem , this representation can be factored into a product of spherical harmonics each depending on only one unit vector . therefore from the spherical harmonic expansion of the radially binned density field one can obtain the multipole moments without ever needing to consider pairs about a given vertex . this is the central insight of this paper . in section [ sec : algorithm ] , we present the algorithm in more detail , and show in section [ sec : proj3pcf ] how this framework goes through to the projected 3pcf . section [ sec : edgecorrxn ] discusses edge correction , while section [ sec : implementation ] describes our implementation . section [ sec : covariance ] computes the covariance of this multipole decomposition in the gaussian random field limit , and section [ sec : mock_results ] presents the results of using the algorithm on the lasdamas sdss - ii data release 7 ( sdss - dr7 ) luminous red galaxy mock catalogs . we conclude in section [ sec : conclusions ] . in this paper , we parametrize triangle configurations by two side lengths , @xmath10 and @xmath11 , and the angle between them with cosine @xmath12 . we will decompose the 3pcf as a function of these three variables into a sum over legendre polynomials for the angular dependence times radial coefficients encoding the side length dependence , as @xmath13 szapudi ( 2004 ) first advanced this decomposition , and he puts a factor of @xmath14 in front of his analogous expansion coefficients ; we absorb this into @xmath15 . there are three major advantages to this decomposition . first , the shape of the 3pcf for fixed side lengths as a function of angle is smooth and slowly varying ( see e.g. bernardeau 2002 , figure 11 ) , without much fine structure . thus we expect that only a few multipoles will be required to capture the angle dependence . second , this decomposition provides a natural way to visualize the 3pcf for all triangle configurations ; one can make several panels for different @xmath16 , each with all @xmath10 and @xmath11 and amplitudes indicated by a colorbar , as in se15 . in contrast to many previous works , this allows immediate appraisal of the information in all triangles and not just a particular set of configurations ( e.g. isosceles , two - to - one , etc . ) third , as we will see , the multipole moments of the 3pcf can be obtained with much greater speed than other decompositions of the 3pcf . however , in contrast to other fast methods , such as tree methods that fix a critical angular scale below which they are approximate ( e.g. zhang & pen 2005 ) or fourier methods that choose a grid with some minimum spacing , we do not sacrifice accuracy to obtain this speed . our method is exact in angle . we will bin in side length , but even were speed of no concern this would be necessary to keep the covariance matrix to a reasonable size . the 3pcf describes the number of triangles of a given configuration whose vertices are the galaxies in a survey . while nine coordinates are required to completely describe any individual triangle connecting three galaxies , the 3pcf averages over both translations and rotations of the triangle configuration . the presumed losslessness of this averaging corresponds to the two usual cosmological assumptions of isotropy ( rotation - invariance about a given point ) and homogeneity ( translation - invariance ) . this ultimately reduces the 3pcf to a function of only three variables ; as indicated already , we will use two triangle sides and the angle between them . we will now show explicitly how to go from nine coordinates to three . we begin with averaging over rotations . we will show explicitly that legendre polynomials are an angular basis for the 3pcf after this averaging . to do so , we first step back and write an estimate ( denoted by a hat ) of the 3pcf for a triangle with sides @xmath17 extending from a vertex whose absolute position within the survey is @xmath18 . we have @xmath19 we now wish to average over all rotations of the triangle about @xmath18 . writing a rotation as @xmath20 ( simply a matrix involving the three euler angles ) , we have @xmath21 where subscript `` iso '' abbreviates `` isotropy . '' noting that @xmath22 , where @xmath23 is a wigner matrix ( e.g. arfken , weber & harris 2013 ( hereafter awh13 ) , equation ( 16.52 ) ) , we find @xmath24 the integral over wigner matrices is simply evaluated by orthogonality ( e.g. brink & satchler 1993 , appendix v ) as @xmath25 , @xmath26 the kronecker delta . using the spherical harmonic addition theorem ( awh13 , equation ( 16.57 ) ) @xmath27 and defining @xmath28 we find @xmath29 in what follows we drop the subscript `` iso '' as we will always be considering the isotropic 3pcf . we now move to averaging over translations . recalling that @xmath18 is the vertex of the triangle from which the two sides given by @xmath17 extend , the densities on a particular triangle of points will be @xmath30 . averaging over translations means allowing every point in the survey to serve as the vertex @xmath18 , so we must integrate over @xmath31 . we thus find that the @xmath32 radial coefficient of the 3pcf is @xmath33 where @xmath34 is the survey volume . our algorithm will bin radially ( denoted with a bar ) , so we seek @xmath35 with @xmath36 a binning function demanding that we are in the bin given by its second argument . binning averages the radial coefficient over some interval in each side length , and in that sense is not lossless . it is also necessary for the speed advantage of our algorithm , as will become clear shortly . we will not compute using equation ( [ eqn : binned_zeta ] ) . rather , we will bin radially around each possible origin @xmath18 _ before _ averaging over rotations and translations , so it will be useful also to define the binned estimator before translation - averaging as @xmath37 where @xmath38 is the radially binned density field about an origin @xmath39 . hence in practice we never compute @xmath40 using equation ( [ eqn : binned_zeta ] ) , but rather measure @xmath41 via equation ( [ eqn : binned_zetahat_l ] ) and then compute @xmath42 as the radially binned multipole coefficients of the 3pcf . a direct way to measure @xmath43 would be to sit on every possible origin and compute the angle between pairs of vectors pointing to all possible sets of two galaxies out to the radius @xmath2 to which one wishes to measure the 3pcf . this scales as @xmath3 . as discussed in the introduction , this scaling applies to other algorithms as well ( e.g. the gardner ( 2007 ) and march ( 2013 ) kd - tree approach ) , fundamentally because the number of possible triangles within @xmath2 with one vertex fixed scales as @xmath44 . however , as is the case for angular power spectra , we can exploit a property of multipole decompositions to enormously accelerate the measurement . we can use the spherical harmonic addition theorem ( [ eqn : sph_addition_theorem ] ) to decompose the legendre polynomial into factors that depend only on one angular variable each . inserting this into equation ( [ eqn : binned_zetahat_l ] ) , we find @xmath45 this equation immediately shows how to reduce the quadratic scaling in the number density to a linear scaling . the two angular integrals have now been separated , and each simply asks for a particular expansion coefficient of the density field ( as a function of angle alone ) in spherical harmonics , in a fixed radial bin . in other words , if we compute for each radial bin @xmath46 @xmath47 we can construct all combinations dictated by @xmath10 and @xmath11 , without ever needing to do an @xmath48 operation . explicitly , inserting equation ( [ eqn : almsdef ] ) into equation ( [ eqn : zetal_int ] ) , we find @xmath49 this is why radial binning is essential for the speed - up of our algorithm ; we can precompute the @xmath50s in each radial bin . for @xmath51 , we then only need to construct @xmath52 combinations of these coefficients . a schematic about a single possible origin is shown in figure [ fig : kernel ] . for a 3pcf measurement , one might use a bin width @xmath53 mpc , and so if one measures out to @xmath54 there will be only @xmath55 distinct bin combinations . meanwhile , computing the @xmath50s themselves takes only as long as performing the integral ( [ eqn : almsdef ] ) , which should scale as @xmath56 . we still must integrate over all possible choices of origin as dictated by equation ( [ eqn : translation_avg ] ) . because the galaxies are discrete , this will reduce to a sum with @xmath4 terms . thus our algorithm will scale as @xmath57 : linear in both the total number of galaxies _ and _ the number within a sphere of radius @xmath2 , and a factor of order @xmath58 faster than the naive counting approach . our algorithm thus provides a route to the 3pcf that on large scales is no more computationally intensive than calculating the multipole moments ( standardly calculated are monopole and quadrupole ) of the 2-point correlation function ( 2pcf ) . ) . the @xmath59 can be combined to yield the multipole moments around this galaxy ( sum over @xmath60 , equation ( [ eqn : zetal_ito_alms ] ) ) and then translation - averaged to yield @xmath15 for the survey . ] finally , to obtain the spherical harmonic coefficients of the galaxy density as in equation ( [ eqn : almsdef ] ) , one might think a spherical harmonic transform is required . this scales as @xmath61 , @xmath62 the number of spatial grid cells on the surface of a sphere . a large number of grid cells is necessary for accuracy even if there are very few galaxies , much as a small @xmath63 is needed when taking a numerical fourier transform to avoid ringing . however , because only low - order multipoles are needed here ( @xmath64 ) , we can avoid this transform and instead directly evaluate the @xmath65s , which are simply spherical harmonics evaluated at angles given by a galaxy s location with respect to a given choice of origin . the required @xmath66s can be easily computed using the cartesian expressions for the spherical harmonics ( e.g. awh13 , equation ( 15.139 ) and table 15.4 ) . indeed , about a given origin , the cartesian components @xmath67 and @xmath68 and their powers for each galaxy can be pre - calculated just once and subsequently combined to form all of the required multipoles . redshift - space distortions ( rsd ) are differences between the true position of a galaxy along the line of sight and its position as inferred from assuming its redshift is purely cosmological . they arise from peculiar velocities , ultimately generated by the growth of large - scale structure ( hamilton 1998 , for a review ) . the projected 3pcf is insensitive to these distortions because it is integrated along the line of sight . below we show how our approach extends to measuring it . we work in the flat - sky approximation , where there is a single line of sight to all galaxies in the survey . sitting around a given central galaxy and projecting corresponds to drawing cylindrical shells around that central with bases that are concentric annuli . all of the galaxies in a given cylinder project down into the cylinder s base annulus . we thus have a planar problem with circular symmetry . this permits simplification of our spherical harmonic basis . recall that @xmath69 where here @xmath70 are the angular coordinates of a galaxy in the system where the central is at the origin . since all the ( projected ) positions are coplanar with the central , the separation along the @xmath71-axis is zero , so @xmath72 . defining @xmath73 we see from equation ( [ eqn : zetal_int ] ) that the multipole moments of the projected , radially binned 3pcf will simply involve fourier coefficients of the projected , radially binned density field weighted by @xmath74 : @xmath75 above , the integrals over @xmath76 and @xmath77 of equation ( [ eqn : zetal_int ] ) have already been performed using that the projected density field is only non - zero at @xmath78 . with this in mind , we observe that if one is solely interested in the projected 3pcf , it is probably optimal simply to use the fourier basis directly . one parametrizes the projected 3pcf estimator about a given central as @xmath79 and writes the exponential as @xmath80 where @xmath76 and @xmath77 are now angles in the plane in polar coordinates , with @xmath81 . using orthogonality of the plane waves , one may then extract the expansion coefficients @xmath82 in equation ( [ eqn : fourier_param ] ) as @xmath83 just as in the non - projected case , these integrals can be explicitly evaluated using the cartesian expressions for the exponentials , and precomputing @xmath84 and @xmath85 . again , one never explicitly considers pairs of galaxies about a given central ; one simply constructs the coefficients @xmath86 for all radial bins , then computes @xmath87 for all desired bin combinations , and finally averages over translations by integrating out @xmath39 . we should note that chen & szapudi ( 2005 ) advanced a similar scheme to measure the 3pcf of cosmic microwave background ( cmb ) maps , analogous to the projected 3pcf since both are on 2-d manifolds . however their method evaluates the fourier transform of the ( continuous ) temperature anisotropy map by gridding , whereas here we suggest the ( discrete ) galaxy density field be fourier - transformed using direct evaluation of the cartesian expressions for @xmath84 and @xmath85 . surveys have jagged and complicated boundaries , and these can produce a spurious contribution to the 3pcf that is the signature of the survey geometry rather than physics the survey hopes to probe . this spurious contribution must be removed . in fourier space , boundaries lead to gibbs phenomenon ringing in the bispectrum , and are challenging to remove . however , in configuration space , edge correction is fairly straightforward for popular estimators ( see kayo et al . 2004 , appendix , for comparison of several ) . we focus here on the szapudi & szalay ( 1998 ) estimator , which kayo et al . ( 2004 ) find preferable to the others they consider ; it has now become the standard in the field . it is @xmath88 with @xmath89 , @xmath90 the data and @xmath91 the random counts . note that , if one inserted @xmath92 for @xmath93 in section [ sec : algorithm ] , one would need to compute integrals of this fraction against the spherical harmonics , requiring definition of @xmath92 at every point in space . however , the estimator ( [ eqn : hat_zeta_est ] ) really represents the function @xmath94 averaged over rotations and translations with weights @xmath95 , which in the shot noise limit is just inverse variance weighting ( we include radial binning represented by @xmath96 ) . in short , @xmath97 thus the estimator ( [ eqn : hat_zeta_est ] ) should be interpreted as demanding the triple count @xmath98 divided by the triple count @xmath99 . therefore we can insert @xmath4 and @xmath91 separately in turn for @xmath93 in section [ sec : algorithm ] , processing random and data catalogs serially . the division required can be done as a post - processing step . we now turn to how this division translates to the legendre basis . working now in our legendre basis , we have @xmath100 @xmath101 and @xmath102 inserting the multipole expansions ( [ eqn : mp_nr])-([eqn : mp_r ] ) into the estimator ( [ eqn : hat_zeta_est ] ) and multiplying through by @xmath99 we find @xmath103 using a linearization formula for the product of two legendre polynomials ( ferrers ( 1877 ) , adams ( 1878 ) , neumann ( 1878 ) , park & kim ( 2006 ) ; se15 equation ( a11 ) ) we find , with angular arguments suppressed , @xmath104 the wigner 3j - symbol above describes angular momentum coupling ; see e.g. brink & satchler ( 1993 ) or awh13 . the vector addition of angular momenta means that the upper row must satisfy triangle inequalities , so @xmath105 and at fixed @xmath16 and @xmath106 the sum is finite . using orthogonality , separating out the @xmath107 term , dividing through by @xmath108 , and defining @xmath109 , we obtain @xmath110 for a boundary - free survey the random field would generate only a monopole ( @xmath108 ) , leaving only @xmath111 on the righthand side ; this is the limit where there is no need for edge - correction , but just division by the randoms . the form of equation ( [ eqn : edgecorrxn_fund ] ) suggests that this problem can be cast as a matrix multiplication , so we define the multipole coupling matrix @xmath112 with elements @xmath113 note that while these matrix elements describe the off - diagonal couplings of different multipoles to each other , they need not be zero along the diagonal . a given multipole in the data may couple to that same multipole in @xmath114 because the 3j - symbol allows @xmath115 for @xmath116 . but the dominant coupling of a given multipole in the data to the same multipole in @xmath114 is described by @xmath111 in equation ( [ eqn : edgecorrxn_fund ] ) , since the @xmath117 are expected to be much less than unity . this term translates to the identity matrix @xmath118 . the edge - correction equation ( [ eqn : edgecorrxn_fund ] ) thus becomes @xmath119 where @xmath120 and analogously for @xmath121 . the system of equations this represents can then be solved for @xmath121 by matrix inversion . to explore this matrix for a realistic use case , we use the lasdamas sdss dr7 real space mock catalogs , using 15 radial bins and a maximum scale of @xmath0 ( further details are given in section [ sec : mock_results ] ) . we show @xmath122 for a particular bin in @xmath123 in figure [ fig : mkl ] , and the leading order edge correction factor @xmath124 in figure [ fig : f_one ] . @xmath112 is not symmetric , but @xmath125 is ; this is why the upper off - diagonal , where @xmath126 , exceeds the lower in figure [ fig : mkl ] . there are two approximations implicit in our approach to solving equation ( [ eqn : matrix_edgecorrxn ] ) . first , to obtain a given matrix element @xmath122 , formally one requires @xmath117 for all values of @xmath106 . however , for @xmath127 these factors fall rapidly . for the lasdamas real space mock catalogs for which we present results here , they are @xmath128 by @xmath129 even for the largest - scale radial bin combination ( the values are listed in the caption to figure [ fig : mkl ] ) , so we simply truncate the series there . if one wished one could easily expand our code to measure higher multipoles of the randoms at the cost of slightly more computation time . however we expect that going to @xmath129 will already render the edge correction error negligible compared to the total error budget . importantly , the smallness of the @xmath117 for @xmath127 means that the coupling between multipoles @xmath130 and @xmath16 is nearly diagonal . coupling between multipoles separated by more than one angular momentum step is suppressed as @xmath131 or higher because the 3j - symbol in the coupling matrix elements ( [ eqn : mkl ] ) requires that @xmath132 . the second approximation relates to the matrix inversion when we solve equation ( [ eqn : matrix_edgecorrxn ] ) . formally one has an infinite dimensional matrix where at fixed @xmath16 , all @xmath130 enter the correction . thus this matrix will not be square ( and hence invertible ) unless we go to an infinite number of @xmath16 as well . however , in practice the matrix is so diagonally - dominant that we believe it is accurate enough simply to invert the sub - matrix given by truncating @xmath16 and @xmath130 at some maximum multipole . we verify this approximation by constructing @xmath122 using solely the dominant @xmath124 edge - correction factor , letting @xmath130 and @xmath16 go to @xmath133 , inverting , and comparing to the result where both go to @xmath134 . were the matrix purely diagonal , truncation would not affect the inverse at all . in the limit where only @xmath124 is non - zero ( in reality , it does dominate the other edge correction factors ) , the matrix is tridiagonal , and so truncation at @xmath134 affects @xmath135 at order @xmath124 , @xmath136 at order @xmath137 , and @xmath138 at order @xmath139 . using a simple toy model , we can estimate the edge correction factors @xmath140 to confirm that they really should be small . consider a spherical ball of random galaxies with radius @xmath91 about a given central , and assume this sphere is cut by a planar survey boundary . orient the @xmath71-axis perpendicular to this boundary , with the central galaxy a distance @xmath71 from it . the problem now has symmetry about this axis , so we need only compute the @xmath141 spherical harmonic coefficients @xmath65 ; @xmath142 , with @xmath143 . for a galaxy at distance @xmath91 from the central , there will be some critical angle with cosine @xmath144 such that , for smaller @xmath145 , the galaxy is outside the survey . we have @xmath146 . \end{aligned}\ ] ] we used the recursion formula @xmath147 $ ] to evaluate the integral and noted that the terms at the lower bound cancel off because they have the same parity . we now compute the @xmath148 required by equations ( [ eqn : almsdef ] ) and ( [ eqn : zetal_int ] ) and average over @xmath149 ( denoted by angle brackets ) . we have @xmath150.\end{aligned}\ ] ] since each term above has even parity , we can integrate from @xmath151 to @xmath152 , divide by @xmath153 , and then invoke orthogonality , to find that @xmath154 finally , we compute @xmath155 explicitly , to find that @xmath156 @xmath157 , @xmath158 , @xmath159 , @xmath160 , falling to @xmath161 . it should be kept in mind that in a large survey volume such as sdss , many centrals will have spheres around them that do not impinge on a large - scale survey boundary at all , further reducing these factors ; for instance , for the sdss boss dr10 footprint only of order @xmath162 of spheres impinge on a boundary , so our rough estimates should be scaled down by a factor of 5 . on the other hand , the true survey mask is far more complicated than the simple planar boundary model above , so this model should not be taken too literally . ) for the lasdamas sdss dr7 real space mock catalogs with 15 radial bins out to @xmath0 . higher @xmath163 fall off very rapidly . even the leading order coefficient is small . this means that one does not need to measure many multipoles of the randoms to obtain a highly accurate edge correction : since the higher @xmath117 fall off so rapidly they contribute very little to the matrix @xmath112 that must be inverted ( equation ( [ eqn : matrix_edgecorrxn ] ) ) . as we expect , @xmath124 becomes larger at larger scales , as larger scale triangles are more likely to impinge on a survey boundary . ] ) at each @xmath130 and @xmath16 for the largest combination of radial bins we test here . this illustrates that all of the couplings are @xmath164 , even for the largest scales we test , which should have the largest correction factors as they are most likely to impinge on a survey boundary ( see figure [ fig : f_one ] ) . while the diagonal appears zero in this plot , it is actually just small , as we discuss in the main text . the @xmath140 entering the matrix elements @xmath122 for this radial bin combination are @xmath165 , and @xmath166 we next describe our c++ implementation of the ideas in sections [ sec : algorithm ] and [ sec : edgecorrxn ] . the basic program flow is to loop over each central galaxy . for each , we find all neighbors within @xmath2 and accumulate the @xmath65 for each of the radial bins . once finished with the neighbor finding , we compute all of the bin cross - powers and add them to our accumulators as a function of bins @xmath10 and @xmath11 and multipole @xmath16 . all of the accumulations include a user - supplied weight per galaxy . we accelerate the finding of neighbors by sorting the particles into a grid , so that the search for neighbors need only consider grid cells that include some point closer than @xmath167 . ideally one wants the grid spacing to be a few times smaller than @xmath167 , so that the inefficiency of doing a cubic search for a spherical region is mild . one also wants the grid spacing to be large enough to contain at least several particles , so that the overhead of storing and accessing the grid is modest . these criteria are not hard to satisfy : for the lasdamas mocks , we use a grid spacing of @xmath168 when searching out to @xmath169 ; this typically contains a dozen galaxies ( and somewhat more random points ) . for the sdss - iii baryon oscillation spectroscopic survey , the density is three times higher . once a neighbor is found , we need to add its contribution to the spherical harmonics . we do not use an angular binning to compute the spherical harmonics . rather , as mentioned in section [ sec : algorithm ] , we use the fact that the spherical harmonics can be written as powers of the cartesian coordinates of unit vectors . in particular , for a unit vector @xmath170 , we can write @xmath171 as a polynomial of terms of the form @xmath172 where @xmath173 . to compute all @xmath65 up to multipole order @xmath174 , we therefore accumulate sums over all neighbors of the cartesian powers @xmath172 with @xmath175 , using the unit vectors of the separation of the neighbor from the central galaxy . there are @xmath176 such power combinations for each radial bin . having finished with all neighbors , we convert these powers into the @xmath65 using the appropriate coefficients from the spherical harmonics , then form all of the bin - to - bin cross powers . for values of @xmath174 of order 10 , the computation of the cartesian powers is much faster than doing the spherical harmonic transform of a fine angular grid . this is particularly true because we use custom assembly code , supplied by marc metchnik as part of the abacus project ( metchnik & pinto , in prep . ) , to accumulate these powers using advanced vector extension ( avx ) instructions . in double precision , 8 neighbors are computed at once , using two sets of avx registers . though we do not present the 3pcf measurement here , we also run our algorithm on the sdss - iii boss dr10 data . in the north galactic cap footprint for the cmass sample , we consider the rrr count of 642,619 random particles . we count 6.7 billion pairs with @xmath177 , an average of 10,400 neighbors per central , divided into 10 linearly spaced radial bins . using @xmath178 , the code runs in 170 seconds on a 6-core 4.2 ghz i7 - 3930k . if we use @xmath179 , thereby reducing the problem to the pair finding and a simple accumulation per radial bin , then the code runs in 53 seconds . loading the particles and sorting them into the grid is a small fraction of that total , so we infer that each pair found and processed at @xmath179 takes about 200 clock cycles . given @xmath178 , we have 286 powers to track per neighbor , each requiring a separate multiply and add . hence , we are computing about 3.8 trillion double - precision operations in 120 extra seconds , a rate of 32 double - precision gflops . this is about 30% of the maximum performance of the cpu ( assuming 4 double precision operations in avx per clock cycle per core ) , a high mark for a practical calculation . the code is sustaining 22 gflops for the full problem , including the pair finding . a two - point correlation function code would only need to count half as many pairs , since the particles are indistinguishable in that application , and so at these speeds would take of order @xmath180 seconds . we therefore find that our computation of the three - point correlation function up to @xmath181 is only about @xmath182 times slower than the equivalent two - point correlation function calculation . we would expect an explicit counting of triples to be about 3,000 times slower than the two - point pair counting , given the 10,000 neighbors ( divide by a factor of 3 for the number of indistinguishable triples compared to indistinguishable pairs ) . as our method is only six times slower than a two - point measurement , it is a factor of five hundred faster than an explicit triple count for this large - scale example . in any method that compares the data points to a random set , we have to consider the effect of poisson noise in the randoms . for example , in the landy - szalay ( 1993 ) estimator for the two - point function , @xmath183 , we will have noise in the data - random @xmath184 and random - random @xmath185 counts that would go to zero in the limit of infinite numbers of random points . one therefore usually wants to use many more randoms than data ( but note the important optimization presented in padmanabhan et al . ( 2007 ) in which one fits these counts to smooth functions of scale so as to reduce the poisson noise ) . a common inefficiency , however , is to use the same number of randoms for each of the terms . this results in spending far too much computational resource on @xmath186 , whose poisson noise will be dwarfed by the @xmath187 noise . for example , if the number of randoms is @xmath60 times the number of data points , then ( assuming uniform galaxy weights ) the variance on @xmath186 will be @xmath188 of that of @xmath189 since the number of @xmath186 pairs is @xmath190 the number of @xmath189 pairs . in contrast , the variance of @xmath187 will only be reduced to @xmath191 of that of @xmath189 ; the factor of 4 comes from the factor of 2 in the landy - szalay estimator , and the @xmath192 enters because the @xmath189 and @xmath186 pairs have double the variance since each pair is counted twice . meanwhile the work in the two terms is scaling as @xmath193 and @xmath60 , respectively . a simple way to avoid this is to compute the @xmath187 and @xmath186 counts with a smaller set of randoms and then repeat this numerous times , averaging over the answers . by choosing the number of randoms in each set , one can optimize the work . for example , in the above two - point case , at fixed total work , the number of random catalogs @xmath194 one can use scales as @xmath195 . the total variance scales as the variance per random catalog divided by @xmath194 , so as @xmath196 . this is minimized for @xmath197 , i.e. , it is optimal to use random catalogs equal in size to the data set . a further advantage of this method is that in addition to averaging all of the sets to get the best answer , one can compute the variance to explicitly measure the contribution of the random catalog density relative to one s estimate of the irreducible on - sky variance . for our three - point algorithm , the work scales as @xmath198 , while the poisson variance of @xmath199 for each random catalog scales as @xmath200 for @xmath99 and @xmath201 for @xmath202 and @xmath203 ; the @xmath204 enters due to the @xmath205 in the szapudi - szalay estimator , while the @xmath206 comes from a 6-fold counting symmetry in @xmath207 and @xmath99 compared to a 2-fold one in @xmath203 and @xmath202 . the total variance is thus @xmath208 . at fixed total work @xmath194 scales as @xmath209 , and so the total variance scales as @xmath210 . this is minimized for @xmath211 but with only @xmath212 variation between @xmath213 and @xmath153 . we implement this strategy in our three - point method by supplying a single list of particles , with the randoms concatenated to the data but with negative weights . notationally , this is @xmath214 , as in section [ sec : edgecorrxn ] . we then compute the three - point correlations of this @xmath4 list . we then re - run repeatedly with new random points @xmath91 . we avoid the small amount of repeated counting of the @xmath189 pairs and @xmath207 triples by the following trick . we first run the code with only the data particle list and save a file that contains the cartesian multipoles for each radial bin and each primary particle , in the enumerated order of the particles . when next running with @xmath215 lists , whenever a data particle is the primary ( as marked by its having a non - negative weight ) , we initialize the multipole accumulators with the saved values and then skip any secondary particles that are also from the data list . the resulting sums pass transparently to the rest of the analysis code . we also run a separate case with only the randoms , so that we can compute the denominator and edge - correction terms in equation ( [ eqn : edgecorrxn_fund ] ) . this requires much less precision , as the denominator of the estimator is much larger than the numerator for large - scale correlations . we therefore do this with only a single set of random points . finally , we have also written a python implementation of the algorithm presented here and tested it on a periodic box with sides of @xmath216 containing @xmath217 galaxies ( roughly the sdss boss number density ) . rather than using gridding , this code exploits kd - trees for galaxy finding , using a fast c implementation ( wrapped to python ) in the @xmath218 library within @xmath219 . we verified the accuracy of this code on a sample of 500 galaxies by comparing with a simple direct - counting algorithm that just counts triplets and then projects onto multipoles . this provides an important cross check on our spherical harmonics since the simple triple counting never uses spherical harmonics . we then ran both the multipole python code and the multipole c++ code on a larger , 20,000 galaxy sample to verify the c++ code . runtime for the python version on a dual core ( 2014 ) macbook air was about @xmath220 minutes ; since the box is periodic , scaling to larger numbers of galaxies is linear . parameter fitting requires weighting the data points according to how independent they are , with two highly independent points contributing more than two less independent points all else equal . the covariance matrix describes how independent the measured multipoles at each @xmath123 are . for our algorithm to be useful , we must show that the covariance matrix can be controlled ; here we compute it with this end in mind . the general 3pcf covariance has been computed before ( szapudi 2001 ) as a 6-d integral , but it is not straightforward to obtain the covariance of our multipole decomposition from this result . here we derive the covariance for our multipole decomposition and show that it can be reduced to a sum of 2-d integrals . this reduction offers a significant improvement in the computation speed possible at a given accuracy . we begin with some definitions and conventions . while we wish to compute the covariance matrix of the configuration space 3pcf , we will end up working in fourier space to do the computation because simplifications are available there by appeal to the power spectrum . we define the fourier transform as @xmath221 with inverse @xmath222 for the earlier stages of our computation we will in fact need to use the discrete fourier transform and its inverse , defined as @xmath223 and @xmath224 where these discrete transforms are over a volume @xmath225 with quantized wavenumbers such that @xmath226 , @xmath227 . we define the power spectrum as @xmath228 @xmath26 is the kronecker delta , unity when its argument is zero and zero otherwise . one can check easily that this definition allows one to recover the familiar relation that the correlation function is the fourier transform of the power spectrum . we will also use the fact that @xmath229 finally , note that one can convert from the discrete to the continous case by replacing @xmath230 with @xmath231 . we now obtain the covariance of our multipole decomposition of the 3pcf . here we begin with an estimator for the translation - averaged but not rotation - averaged full 3pcf ; we will project onto multipoles ( which also averages over rotations ) and bin radially later . @xmath232 for a gaussian random field , @xmath233 . the covariance is thus @xmath234\nonumber\\ & \times \left<\tilde{\delta}(\vec{k})\tilde{\delta}(\vec{q})\tilde{\delta}(\vec{p})\tilde{\delta}(\vec{k}')\tilde{\delta}(\vec{q}')\tilde{\delta}(\vec{p}')\right>.\end{aligned}\ ] ] peforming the integrals over @xmath31 and @xmath235 we have @xmath236\nonumber\\ & \times\left<\tilde{\delta}(\vec{k})\tilde{\delta}(\vec{q})\tilde{\delta}(\vec{p})\tilde{\delta}(\vec{k}')\tilde{\delta}(\vec{q}')\tilde{\delta}(\vec{p}')\right>\end{aligned}\ ] ] where @xmath26 is a kronecker delta whose argument is the sum of the subscripted vectors . we now use wick s theorem to reduce the 6-point expectation value to triple products of 2-point functions ; this is where gaussianity enters . we need to consider all possible contractions . @xmath237}\nonumber\\ & \times\bigg\{(qq')(pp')(kk')+(pq')(qp')(kk')+(kq')(qk')(pp')\nonumber\\ & + ( kp')(qk')(pq')+(kq')(pk')(qp')+(kp')(pk')(qq')\bigg\ } \label{eqn : contracted_covar}\end{aligned}\ ] ] where parentheses represent contractions of @xmath238 evaluated at the arguments in the parentheses . using equation ( [ eqn : powerspec_def ] ) , the term in curly brackets above becomes @xmath239.\end{aligned}\ ] ] doing the sums over @xmath240 , and @xmath241 in equation ( [ eqn : contracted_covar ] ) we find @xmath242}\nonumber\\ & \times\bigg\ { e^{-i\left[\vec{q}\cdot\vec{r}_{1}'+\vec{p}\cdot\vec{r}_{2}'\right]}+e^{-i\left[\vec{p}\cdot\vec{r}_{1}'+\vec{q}\cdot\vec{r}_{2}'\right]}+e^{-i\left[\vec{k}\cdot\vec{r}_{1}'+\vec{p}\cdot\vec{r}_{2}'\right]}\nonumber\\ & + e^{-i\left[\vec{p}\cdot\vec{r}_{1}'+\vec{k}\cdot\vec{r}_{2}'\right]}+e^{-i\left[\vec{k}\cdot\vec{r}_{1}'+\vec{q}\cdot\vec{r}_{2}'\right]}+e^{-i\left[\vec{q}\cdot\vec{r}_{1}'+\vec{k}\cdot\vec{r}_{2}'\right]}\bigg\}. \label{eqn : sumdone}\end{aligned}\ ] ] notice each pair of exponentials in the curly brackets is obviously symmetric under switching @xmath243 . also notice from the first line that equation ( [ eqn : sumdone ] ) is symmetric under @xmath244 if we also flip @xmath245 and @xmath246 . applying this to all of the terms in curly brackets too , we find @xmath247 if we had originally labeled each exponential in the curly brackets as @xmath248 hence the equation has the desired symmetries . converting this into an integral we have @xmath249}\left(\vec{q}+\vec{p}+\vec{k}\right)e^{-i\left[\vec{q}\cdot\vec{r}_{1}+\vec{p}\cdot\vec{r}_{2}\right]}\bigg\ { \cdots\bigg\ } , \label{eqn : fullcovar}\end{aligned}\ ] ] where above we have not rewritten the terms in curly brackets from equation ( [ eqn : sumdone ] ) . we now consider the covariance projected onto multipoles , defining @xmath250 noticing that in equation ( [ eqn : fullcovar ] ) the exponentials contain the only @xmath251 dependence , we first define the projection of one exponential onto one legendre polynomial as @xmath252}p_{l}(\hat{r}_{1}\cdot\hat{r}_{2})d\omega_{r1}d\omega_{r2}\nonumber\\ & = ( 2l+1)(-1)^{l}\mathcal{j}_{l}(k_{1},k_{2})p_{l}(\hat{k}_{1}\cdot\hat{k}_{2 } ) . \label{eqn : iproj_def}\end{aligned}\ ] ] we will have this factor from projecting the exponential outside the curly brackets in equation ( [ eqn : fullcovar ] ) , and then six analogous factors within the curly brackets from projecting each exponential of @xmath253 onto @xmath254 . we have defined @xmath255 and will also use @xmath256 . we performed the projection integral by expanding the exponential in spherical harmonics using awh13 equation ( 16.61 ) and expanding the legendre polynomial in spherical harmonics using the spherical harmonic addition theorem ( [ eqn : sph_addition_theorem ] ) ; the integral can then be evaluated by orthogonality . writing out the projection integrals explicitly using equation ( [ eqn : iproj_def ] ) , we thus have the projected covariance as @xmath257}(\vec{p}+\vec{q}+\vec{k})\nonumber\\ & \times(2l+1)(2l'+1)(-1)^{l+l'}\mathcal{j}_{l}(q , p)p_{l}(\hat{q}\cdot\hat{p})\nonumber\\ & \times\bigg\{\mathcal{j}'_{l'}(q , p)p_{l'}(\hat{q}\cdot\hat{p})+\mathcal{j}'_{l'}(p , q)p_{l'}(\hat{p}\cdot\hat{q})+\mathcal{j}'_{l'}(k , p)p_{l'}(\hat{k}\cdot\hat{p})\nonumber\\ & + \mathcal{j}'_{l'}(p , k)p_{l'}(\hat{p}\cdot\hat{k})+\mathcal{j}'_{l'}(k , q)p_{l'}(\hat{k}\cdot\hat{q})+\mathcal{j}'_{l'}(q , k)p_{l'}(\hat{q}\cdot\hat{k})\bigg\}. \label{eqn : proj_covar_qp}\end{aligned}\ ] ] note that in equation ( [ eqn : proj_covar_qp ] ) the legendre polynomial dependence is the same for each of the first pair in the curly brackets , the second pair , and the third pair because the dot product is symmetric . thus we have three possible angular integrals to do , corresponding to these three pairs : @xmath258}(\vec{k}+\vec{p}+\vec{q})\nonumber\\ & i_{{\rm ang},ll'}^{{\rm asymm}}=\int d\omega_{p}d\omega_{q}d\omega_{k}p_{l}(\hat{q}\cdot\hat{p})p_{l'}(\hat{k}\cdot\hat{p})(2\pi)^{3}\delta_{d}^{[3]}(\vec{k}+\vec{p}+\vec{q})\nonumber\\ & i_{{\rm ang},ll'}^{{\rm asymm}}=\int d\omega_{p}d\omega_{q}d\omega_{k}p_{l}(\hat{q}\cdot\hat{p})p_{l'}(\hat{k}\cdot\hat{q})(2\pi)^{3}\delta_{d}^{[3]}(\vec{k}+\vec{p}+\vec{q } ) . \label{eqn : projxn_integrals}\end{aligned}\ ] ] note that the second and third integrals above are really the same under @xmath259 . we term the first integral above the symmetric integral and the second and third asymmetric . figure [ fig : covar_cycles_diagram ] explains these equations and their symmetries diagrammatically to illustrate the underlying structure of the covariance calculation up to this point . ) , which in turn derive from the structure of equation ( [ eqn : sumdone ] ) ; one can directly compare the arguments of the exponentials in this latter with the diagram . the leftmost triangle represents the term @xmath260 outside the curly brackets in equation ( [ eqn : proj_covar_qp ] ) , showing also the radial arguments implicit in the @xmath261 , and the six triangles inside the curly brackets above represent the six terms in the curly brackets . the legendre polynomials are always evaluated about a particular vertex , as shown in the diagram , and the real - space variables match to fourier - space variables differently in each triangle ( see equation ( [ eqn : sumdone ] ) ) . one can see from above that each pair of triangles , or pair of terms in curly brackets in equation ( [ eqn : proj_covar_qp ] ) , has switch symmetry @xmath262 . these are rotation symmetries about the vertex between @xmath263 and @xmath264 in each pair . one can also see that if we switch @xmath265 and @xmath266 , the leftmost triangle is symmetric . the topmost pair will also be symmetric under this switch as well , which is why it gives rise to two symmetric projection integrals , but the middle and bottom pairs will not be , which is why they give rise to four asymmetric projection integrals ( see equation ( [ eqn : covar_ito_ints ] ) ) . ] to evaluate these angular integrals , we write the dirac delta as the fourier transform of unity , @xmath267}(\vec{k}+\vec{p}+\vec{q})=\int d^3\vec{r}\;e^{i\left[\vec{k}\cdot\vec{r}+\vec{p}\cdot\vec{r}+\vec{q}\cdot\vec{r}\right]}\ ] ] expand each exponential in spherical harmonics using awh13 equation ( 16.61 ) , and perform the angular integral over @xmath268 . defining @xmath269 we obtain @xmath270}(\vec{k}+\vec{p}+\vec{q})=\nonumber\\ & \left(4\pi\right)^{3}\sum_{l_{1}l_{2}l_{3},m_{1}m_{2}m_{3}}\mathcal{d}_{l_{1}l_{2}l_{3}}\mathcal{c}_{l_{1}l_{2}l_{3}}\mathcal{r}_{l_{1}l_{2}l_{3}}(k , p , q)\nonumber\\ & \times\left(\begin{array}{ccc } l_{1 } & l_{2 } & l_{3}\\ 0 & 0 & 0 \end{array}\right)\left(\begin{array}{ccc } l_{1 } & l_{2 } & l_{3}\\ m_{1 } & m_{2 } & m_{3 } \end{array}\right)\nonumber\\ & \times y_{l_{1}m_{1}}^{*}(\hat{k})y_{l_{2}m_{2}}^{*}(\hat{p})y_{l_{3}m_{3}}^{*}(\hat{q } ) . \label{eqn : delta_as_ylms}\end{aligned}\ ] ] this is equivalent to mehrem ( 2002 ) equation ( 5.1 ) if the 3j - symbols above are translated to clebsch - gordan symbols . inserting equation ( [ eqn : delta_as_ylms ] ) into equation ( [ eqn : projxn_integrals ] ) and then expanding the legendre polynomials in equation ( [ eqn : projxn_integrals ] ) into spherical harmonics using the spherical harmonic addition theorem ( [ eqn : sph_addition_theorem ] ) , we now simply have integrals over products of three spherical harmonics , which can be done analytically with 3j - symbols . the result can then be simplified by explicitly evaluating some of the 3j - symbols ( using nist digital library of mathematical functions ( dlmf ) 34.3.1 ) and summing over all of the spin angular momenta ( using nist dlmf 34.3.10 and 34.3.18 ) . for the symmetric integral we find @xmath271 where we have separated @xmath130 with a semicolon because it is the only argument that does not appear in the legendre polynomials in the integral . a simple case to check is setting @xmath107 and @xmath272 in equation ( [ eqn : projxn_integrals ] ) . then @xmath273 so by direct computation @xmath274}(p - q)}{q^2 } \label{eqn : symm_direct}\end{aligned}\ ] ] where we used @xmath275 . in equation ( [ eqn : symm_int ] ) , @xmath107 sets @xmath276 and the 3j - symbol s square is @xmath277 . using the orthogonality relation for spherical bessel functions , @xmath278}(p - q)/(2q^2)$ ] ; inserting this in equation ( [ eqn : symm_int ] ) and simplifying yields agreement with the direct computation . for the asymmetric integral we find @xmath279 where now @xmath174 is separated by a semicolon because it appeared in two legendre polynomials in the integrand . note that for @xmath107 , the symmetric and asymmetric integrals of equation ( [ eqn : projxn_integrals ] ) are equal , so equation ( [ eqn : asymm_int ] ) should reduce to equation ( [ eqn : symm_int ] ) in this limit , as can be verified by noting @xmath107 implies @xmath280 . thus @xmath281 we now interchange the order of integration so that the integrals over @xmath282 and @xmath130 are done first , since they are separable , and the linking integral over @xmath46 implied by @xmath283 is done last . we also make the sum over @xmath284 explicit and do it after evaluating the @xmath282 and @xmath130 integrals . finally we define @xmath285 and @xmath286 in terms of these functions , @xmath287+(-1)^{(l+l'+l_{2})/2}\nonumber\\ & \times\bigg[f_{ll}(r;r_{1})f_{l'l'}(r;r_{1}')f_{l_{2}ll'}(r;r_{2},r_{2}')\nonumber\\ & + f_{ll}(r;r_{1})f_{l'l'}(r;r_{2}')f_{l_{2}ll'}(r;r_{2},r_{1}')\nonumber\\ & + f_{ll}(r;r_{2})f_{l'l'}(r;r_{1}')f_{l_{2}ll'}(r;r_{1},r_{2}')\nonumber\\ & + f_{ll}(r;r_{2})f_{l'l'}(r;r_{2}')f_{l_{2}ll'}(r;r_{1},r_{1}')\bigg]\bigg\}. \label{eqn : fullcovar_final}\end{aligned}\ ] ] one can see that this is symmetric under switching @xmath288 and @xmath289 , as expected . we have thus shown how to reduce the covariance of our multipole decomposition to a sum of 2-d integrals . this is a significant computational benefit : the @xmath290 and @xmath291 can be pre - computed once to give all the terms in the sum above , and then integrated over @xmath292 . further , since the problem is now 2-d one can simply evaluate the integrals using a grid and avoid appealing to more complicated higher - dimensional integration techniques . in closing , we note that for @xmath293 ( i.e. @xmath294 ) , @xmath290 and @xmath295 can be computed analytically . we find @xmath296 using @xmath297 and gradshteyn & ryzhik ( 2007 ) equation ( 6.512.1 ) . @xmath298 is the hypergeometric function and we assume @xmath299 ; the result for @xmath300 is given by switching @xmath301 . @xmath295 can be computed using techniques outlined in fabrikant ( 2013 ) and is given by his equation ( 9 ) ; since the expression is rather long we do not reproduce it here . we mention this since one could imagine scenarios in which high speed was desirable for computing the covariance , such that these approximate forms might suffice . finally , to incorporate shot noise in the covariance , one takes @xmath302 , @xmath5 the survey number density . this will introduce a cross term where one of the @xmath290 or @xmath295 in each pair in equation ( [ eqn : fullcovar_final ] ) no longer involves the power spectrum , and also a term in @xmath303 where both functions in each pair do not . the required integrals are also analytic : @xmath304}(r - r_1),\end{aligned}\ ] ] while @xmath295 is rather longer and given by mehrem ( 2002 ) equation ( 5.14 ) , assuming a much simpler form ( his equation ( 5.15 ) ) if @xmath305 , or @xmath306 . the above calculation used exact values for @xmath307 and @xmath308 , but we can easily integrate over bins in radius . we simply integrate the @xmath295 and @xmath290 functions defined above over bins , equivalent to replacing @xmath309 and @xmath310 with their bin - averaged values . we define @xmath311 and @xmath312 with @xmath313 where @xmath314 is a dummy variable and we recall that @xmath315 is the binning function ensuring @xmath314 is in the bin @xmath316 ( see section [ subsec : binning ] ) . we display the binned reduced covariance , @xmath317 for fixed @xmath318 and a number of @xmath319 and @xmath320 combinations in figure [ fig : covar_grid ] . one can see clear features when @xmath321 , especially when @xmath322 as well ( e.g. the 11 , 22 , 33 , and 44 panels ) . the computation was done in @xmath323 bins but we display with an interpolated color scheme because the underlying radial variables are continuous , in contrast to the multipoles @xmath16 and @xmath106 . we used the linear - theory matter power spectrum from camb ( lewis 2000 ) and checked convergence of the integrals by varying the endpoints and spacing of the grids in @xmath46 and @xmath130 we used . cosmology with @xmath324 , and @xmath325 . ] for the spherical bessel functions we used high - order taylor series for small values of the arguments , with the change - over point to the series depending on the order @xmath16 , and cross - checked with direct computation using _ scipy s _ built - in functions . for the @xmath326 ( equation ( [ eqn : jlbar ] ) ) we used analytical results , cross - checked with numerical integrations of the @xmath327 . in figure [ fig : covar_diag_grid ] , we show the binned covariance when @xmath328 and @xmath329 versus all @xmath16 and @xmath106 , and for a number of choices of @xmath330 and @xmath308 , indicated in the upper left of each panel . notice that the strongest covariance is , as one might expect , when @xmath322 as well , along the diagonal . we display with no color interpolation because the multipoles are discrete . we show larger radial bins than the lasdamas mock results contain because these will be relevant for the baryon acoustic oscillation ( bao ) scale analysis planned for future work . we present the results of running our algorithm on the publicly available lasdamas mock catalogs for the sdss - ii dr7 in both real and redshift space . we used 15 radial bins with @xmath331 . we show first the results at each multipole versus the two triangle side lengths @xmath10 and @xmath11 in figure [ fig:3pcf_colorgrid ] . this shows that the largest amplitude contribution to the 3pcf , especially for triangles well away from the diagonal , is @xmath332 . this is what we expect from se15 , figure 9 , third row , leftmost panel , showing the perturbation theory results and focusing on the linear bias @xmath333 , which dominates the non - linear bias @xmath334 . in @xmath335 , there is a hint of a large - scale decrement , to be compared with the slight feature close to the diagonal around @xmath336 in se15 figure 9 , second row , leftmost panel . as in se15 , @xmath332 and @xmath337 look similar but with @xmath332 having higher amplitude away from the diagonal . for @xmath338 , the panels all begin to look the same , agreeing with our expectation from se15 figure 9 . this is because these higher multipoles , in particular near the diagonal , are dominated by a small population of squeezed triangles where two sides are equal ( e.g. @xmath10 and @xmath11 ) and the third side nears zero . in the hierarchical ansatz for the 3pcf , one has @xmath339 , so we expect the amplitude to become very large as any side approaches zero . also discussed in se15 is another reason for the similarity of the @xmath338 panels : before cyclic summing over vertices of the triangle , the leading order pre - cyclic perturbation theory 3pcf only has structure for @xmath340 . in particular , at leading ( fourth ) order , the 3pcf receives one contribution from the second - order density field , @xmath341 , which is in turn calculated by integrating a kernel @xmath342 against the linear density field ( goroff et al . 1986 ; jain & bertschinger 1994 ; bernardeau et al . this kernel has only @xmath343 , and @xmath153 terms . if one chooses the second - order density point to be at the origin , the 3pcf therefore has multipole structure only to @xmath332 . in reality we do not know which point contributes @xmath341 , so we must cyclically sum around the triangle and can not choose @xmath341 at the origin . this cyclic summing generates additional angular structure , but it just stems from the geometric effect of writing a simple @xmath344 and @xmath153-only multipole expansion with argument e.g. @xmath345 in terms of @xmath346 . we note that the @xmath343 , and @xmath153 terms that enter pre - cyclically have a physical meaning . @xmath342 is formed by summing two mode - coupling kernels @xmath347 and @xmath348 ( bernardeau et al . 2002 equations ( 39 ) and ( 156 ) ) . these in turn come from solving respectively the full continuity equation and the euler equation ( compare bernardeau et al . 2002 equations ( 16 ) and ( 17 ) with their equations ( 37 ) and ( 38 ) ) . @xmath347 produces all of the @xmath349 and @xmath350 of the @xmath335 terms in @xmath342 . the @xmath349 contribution is from the product of the velocity divergence and the density , while the @xmath335 contribution is from gradients of the density field parallel to the velocity . meanwhile , @xmath348 generates the remaining @xmath351 of the @xmath335 term and all of the @xmath332 term in @xmath342 ; these stem from gradients of the velocity divergence parallel to the velocity . figures [ fig : recon ] and [ fig : ratios ] show that the full 3pcf of the data can be reconstructed well from only a few multipoles . figure [ fig : recon ] reconstructs the 3pcf from coefficients @xmath15 , up to and including the @xmath16 indicated in the legend , for a particular triangle configuration with @xmath352 , @xmath353 . one recovers an accurate shape versus @xmath354 even using only multipoles up to @xmath355 , and that adding in @xmath356 and finally @xmath357 changes the shape very little . in figure [ fig : ratios ] , we illustrate the same idea for three different triangle configurations : the higher multipoles fall off relative to @xmath358 , meaning they contribute less to reconstructing the full 3pcf . this plot likely is conservative in that it makes the effect of the higher multipoles appear larger than it is ; the plot shows the ratio of each higher multipole to @xmath358 , but the change in the 3pcf produced by adding in a higher multipole is actually roughly the ratio of the multipole to the sum of _ all _ the lower multipoles . since , in detail , legendre polynomial weights also enter , one might consider an angle - averaged version of this ratio . however since figure [ fig : recon ] effectively already shows the unimportance of the highest multipoles , we have in figure [ fig : ratios ] just chosen to show @xmath359 because it offers more granular information . indicated in the legend , as described in section [ sec : mock_results ] . the reconstruction converges even for modest @xmath16 . the @xmath360 points lie essentially directly under those for @xmath181 . ] coefficients to @xmath358 for several triangle configurations ( again using the lasdamas real space mocks ) . the decline of the higher multipoles with @xmath16 indicates that not many multipoles are needed for accurately reconstructing the full 3pcf . this is especially true for the largest scale triangle we show , which is also the least likely to be altered by non - linear effects . the relative magnitudes of the higher multipoles here may seem large when recalling from figure [ fig : recon ] that the reconstruction appears well converged by @xmath355 ; but note that a given multipole s contribution to the reconstruction is roughly its ratio to the sum of all the lower multipoles , not just to @xmath358 ; this reduces the importance of the higher multipoles . finally , the strength of @xmath332 shows the quadratic or u - shaped behavior of the 3pcf traditionally associated with gravitational growth of structure ( see also figure [ fig : recon ] ) . gravity generates gradients of the density and velocity divergence mostly parallel to the velocity , in turn enhancing roughly collinear structures with @xmath96 near @xmath361 or @xmath362 ( bernardeau et al . 2002 ) . ] se15 presented a compression scheme for the multipole moments of the 3pcf . this was designed to avoid the squeezed limit where two galaxies are so nearby that perturbation theory is invalid and also to reduce the dimension of the covariance matrix required for parameter fitting . this approach integrated each multipole moment over @xmath11 from @xmath363 at each value of @xmath10 . in the current work the data is binned coarsely enough in both @xmath10 and @xmath11 that this approach must be adapted slightly . we simply choose to , for a given bin @xmath10 , sum over all bins with @xmath364 . @xmath365 is the set of all bins in @xmath11 where @xmath11 is greater than @xmath366 and less than @xmath367 . one might wish to select a different multiple of @xmath368 in defining @xmath369 . ] this assures that the minimum value of @xmath11 is @xmath370 and that the minimum difference between @xmath10 and @xmath11 is also @xmath370 , meaning by the triangle inequality that @xmath371 . this avoids the squeezed limit while reducing the dimension of the problem . mathematically , the compression is defined here as @xmath372 where bar denotes `` binned '' , superscript `` c '' denotes `` compression '' , and @xmath373 is the @xmath374 binned 3pcf multipole ( see section [ subsec : binning ] ) . @xmath375 is the volume of bin @xmath11 . the denominator is for normalization . in figures [ fig : low_ell_comps ] and [ fig : hi_ell_comps ] we show the results of this compression . we also compressed the leading ( fourth ) order perturbation theory predictions , calculated as outlined in se15 , and show them for comparison . the theory requires linear ( @xmath333 ) and non - linear ( @xmath334 ) bias parameters as an input ; we use a least - squares fit with points weighted by the inverse compressed variance . this latter is computed from the scatter between mocks and ignores noise in the random catalog used for edge correction , which due to the large number of randoms is negligible . our mocks constitute a volume of order 7 times that used for the theoretical covariance calculation here , so , as explained in figure [ fig : covar_diag_grid ] , we might expect error bars on the compressions of order @xmath376 . this is indeed what we find . we offer the caveat that a full , rigorously correct fit of theory to observation would require inversion of the full covariance matrix . we leave this for future work ; here our goal is simply to indicate that the results of our algorithm roughly agree with perturbation theory predictions . using the simple procedure above , the results are well - fit with @xmath377 and @xmath378 ; note that to compute the theory predictions we matched the lasdamas cosmology . there is quoted at @xmath379 ; @xmath380 . the lasdamas mocks are at @xmath381 , so when normalizing the power spectrum we should use @xmath382 $ ] , with @xmath90 the linear growth factor ( e.g. mo van den bosch & white ( 2010 ) equations ( 4.75 ) , ( 4.76 ) , and ( 3.77 ) ; carroll et al . ( 1992 ) ) . ] there is some deviation noticeable at large scales in @xmath349 ( about @xmath383 ) and @xmath335 ( about @xmath384 ) , with nearly all the other multipoles deviating only within the error bars or at most in a few cases just slightly outside them . the larger deviations in @xmath349 and @xmath335 are likely because non - linear corrections to the perturbation theory results can not be neglected . in particular , in @xmath349 we expect non - linear evolution might smooth structure on smaller scales , making the slope of the perturbation theory compression shallower and allowing a better global fit to the @xmath349 mock results . importantly , the error bars become much larger for @xmath385 as compared to those for @xmath386 . this suggests when doing a full parameter fit using the covariance matrix , one might not gain much by including these higher multipoles . one might choose simply to drop these modes to reduce the dimension of the covariance matrix to be inverted . figures [ fig : low_ell_comps ] and [ fig : hi_ell_comps ] also show that the redshift space results at each multipole appear to be roughly a constant rescaling of the real space results , with a constant that only weakly depends on the multipole . to illustrate this we show the ratio of redshift space to real space results in each radial bin at each multipole ( figure [ fig : redshift_space_rescale ] , left panel ) and the radially - averaged ratio versus multipole ( figure [ fig : redshift_space_rescale ] , right panel ) . more detailed discussion is in the caption to this figure ; the key point is that for @xmath387 , there is little radial dependence to the rescaling factor and also little multipole dependence . both dependences are more pronounced for @xmath388 ; we suspect this is because these higher multipoles are dominated , even in the compression , by a small subset of relatively squeezed triangles that are more strongly affected by rsd . this issue might merit further attention in subsequent work . ) . the right panel shows the radial average at each multipole . while there is some radial scale dependence in the left panel , it is modest for all but @xmath388 . thus averaging over the radial dependence does not lose much information for the lower multipoles . the averages ( right panel ) are similar for all but @xmath388 , and even these differ by less than a factor of 2 from the averages of the lower multipoles . ] given that we now have compressed data , we must also apply our compression scheme as in the previous section to the binned covariance of section [ subsec : binning ] . we denote the compressed , binned covariance as @xmath389 , noting that the two superscript `` c ' 's denote that we compress over @xmath11 and @xmath308 , leaving the quantity a function only of @xmath10 and @xmath330 . it would be computationally intensive to compute the binned covariance using equation ( [ eqn : fullcovar_final ] ) with the @xmath290 and @xmath295 replaced by equations ( [ eqn : flbar_2])-([eqn : jlbar ] ) and then compress , and a faster approach is available . this is to compress @xmath390 and @xmath391 as necessary first and from them obtain the compressed binned covariance . @xmath390 need only be compressed at most once ( if its argument is @xmath11 or @xmath308 ) , but @xmath391 may be compressed once or twice depending on if one or both of its arguments have subscript @xmath153 . we thus define three functions , where `` cc '' again denotes a double compression : @xmath392 note that in the second line above , the @xmath391 being compressed is a function @xmath10 and @xmath393 , and hence need only be compressed once it is not compressed over @xmath10 . however in the third line above , the @xmath391 being compressed depends on @xmath11 and @xmath308 and so must be compressed twice . making these replacements as appropriate in equation ( [ eqn : fullcovar_final ] ) yields the compressed , binned covariance . this shows that the framework of compression can be easily generalized to the covariance . we have presented a novel algorithm to compute the multipole moments of the 3pcf . it is especially apt for large cosmological datasets such as sdss and upcoming surveys like euclid , large synoptic survey telescope ( lsst ) , and dark energy spectroscopic instrument ( desi ) , which will have tens of millions to billions of objects ( jain et al . 2015 ) . for these datasets , an approach that scales with @xmath394 would be wholly infeasible . we have shown that our algorithm scales as @xmath395 , handles edge correction easily , and permits computation of the 3pcf of a large dataset quickly even with modest computing resources . we have also computed the covariance matrix of this decomposition in the gaussian random field limit . finally , we have developed the compression scheme first presented in se15 and shown its application both to the data and to the covariance matrix . this compression scheme offers a compelling way to visualize the results of the algorithm that loses little information , in contrast to the plots of the 3pcf or reduced 3pcf versus opening angle @xmath96 for particular triangle configurations that previous literature supplies . the algorithm presented here is unique in that it fundamentally reduces the scaling of the 3pcf measurement to that of the two - point function , while remaining exact in angle . this did not have to be the case . formally , for a complete representation of the 3pcf , one needs an infinite number of multipoles @xmath16 . however , because the physics generating the 3pcf does not have a great deal of angular structure , in practice a finite , modest number of multipoles suffices . furthermore , we have shown that in our lasdamas test case , the 3pcf is already well - reconstructed by @xmath355 ( figure [ fig : recon ] ) . since our algorithm fundamentally requires pair - counting , using a fast fourier transform ( fft ) for this step may in some cases offer an additional acceleration ; we present this in slepian & eisenstein 2015c . finally , one might worry that jagged survey boundaries could easily introduce high multipoles into the measured 3pcf . but we have shown that the coefficients required for the edge correction , at least for our lasdamas test case , fall off quickly enough that one need only measure a few multipoles of the randoms for an accurate solution ( figures [ fig : f_one ] and [ fig : mkl ] ) . the 3pcf contains important information on the non - gaussianity of large - scale structure ( lss ) due to growth under gravity and also perhaps that remaining from primordial non - gaussianity . with measurements of only the 2-point function , the amplitude of clustering ( e.g. @xmath1 ) and the linear bias are degenerate however , the 3pcf is sensitive to a different power of the linear bias than the 2-point function ( cube versus square ) , and so measuring it exposes a raw factor of the bias and helps break this degeneracy . as for primordial non - gaussianity , while the cmb has been the dominant constraint up to now ( see ade et al . ( 2015 ) ) , it is expected that even maximally improved cmb measurements can only enhance the cmb constraint by a factor of a few . thus lss will become a vital complementary probe . generically , inflation must couple to ordinary matter so as to produce it during reheating , and this coupling produces some level of non - gaussianity ( desjacques & seljak 2010 , for a recent review ) . thus the 3pcf can be used to probe the dynamics of inflation in principle and perhaps , soon , in practice . the 3pcf also contains information on redshift space distortions . it should be emphasized that in the current work , the multipole moments are averaged over rotations of the triangle configurations , and so we lose any information about anisotropy . however , our algorithm can easily be adapted to retain the full , unaveraged information ( @xmath65s ) around each possible origin . this would allow tracing the anisotropy rsd induce . preliminary calculations indicate that rsd introduce couplings between multipoles @xmath396 which are absent without rsd . these couplings have selection rules due to the underlying symmetries under rotation about the line of sight and parity flips . there will also be @xmath60 dependence induced by the preferred direction defined by the line - of - sight . we therefore expect that the off - diagonal terms in a tensor of spherical harmonic coefficients will have structure that can be used to probe rsd - induced anisotropies . it is already important that the spherical harmonics , with the introduction of @xmath397 and @xmath60 , offer a natural 5-d basis for redshift - space measurements . work on these questions from both analytic and numerical perspectives is underway . we plan to apply our algorithm and analysis approach to sdss dr12 , with the goals of assessing the presence of bao features , measuring the linear and non - linear bias , and constraining the baryon - dark matter relative velocity ( tseliakhovich & hirata 2010 ; yoo , dalal & seljak 2011 ; yoo & seljak 2013 ; se15 ) . previous literature has found a bao feature in the reduced 3pcf @xmath398 , with @xmath399 the 2pcf ( gaztaaga et al . 2009 ) . that work used only one triangle configuration , @xmath400 and @xmath401 . with the additional signal - to - noise the large number of galaxies in sdss dr12 offers , as well as our algorithm s ability to consider all triangle configurations quickly , this problem is ripe for revisiting . furthermore , se15 suggests that the multipole decomposition clearly isolates a strong bao feature , especially in the @xmath335 multipole . this is also a particularly informative multipole for the relative velocity , further discussed in se15 . that work additionally shows that , in principle , the multipole decomposition can clearly separate the effects of linear and non - linear bias significant because , as noted above , the 3pcf has traditionally been an important tool for constraining these parameters . finally , the significant speed advantage of our algorithm will permit much finer and much faster calibration of any 3pcf measurements against large cosmological simulations . such improved calibration should greatly enhance the leverage of the 3pcf as a fundamental probe of large - scale structure . zs thanks simeon bird , doug finkbeiner , lehman garrison , jr gott iii , robert marsland , philip mocz , cameron mcbride , stephen portillo , david spergel , and yucong zhu for useful discussions . we thank the anonymous referee for several helpful suggestions as well . this material is based upon work supported by the national science foundation graduate research fellowship under grant no . dge-1144152 . = 1.5em = 1 adams j. c. , 1878 , proc . r. soc . , 27 , 63 = 1.5em = 1 ade p.a.r . et al . , 2015 , preprint ( arxiv:1502.01592 ) = 1.5em = 1 frieman j.a . & gaztaaga e. , 1999 , apj , 521 , l83 - 86 . = 1.5em = 1 gardner j. p. , connolly a. & mcbride c. , 2007 , in asp conf ser . 376 , astronomical data analysis software and systems xvi , ed . r. a. shaw , f. hill , & d. j. bell ( san francisco , ca : asp ) , 69 = 1.5em = 1 gaztaaga e. & frieman j.a . , 1994 , apj , 437 , l13 . = 1.5em = 1 gaztaaga e. , cabr a. , castander f. , crocce m. & fosalba p. , 2009 , mnras 399 , 2 , 801 - 811 = 1.5em = 1 gradshteyn i.s . & ryzhik i.m . , 2007 , table of integrals , series , and products , ed . a. jeffrey & d. zwillinger ( amsterdam : academic press ) = 1.5em = 1 mehrem r. , 2011 , journal of applied mathematics and computation 217 5360 - 5365 = 1.5em = 1 moore a. w. et al . , 2001 , in mining the sky , ed . a. j. banday , s. zaroubi , & m. bartelmann ( berlin : springer ) , 71 = 1.5em = 1 nist digital library of mathematical functions ( dlmf ) . + http://dlmf.nist.gov/ , release 1.0.10 of 2015 - 08 - 07 . + online companion to w. j. olver , d. w. lozier , r. f. boisvert , and c. w. clark , eds . nist handbook of mathematical functions . cambridge university press , new york , ny , 2010 . = 1.5em = 1 nichol r. c. et al . , 2006 , mnras , 368 , 1507 = 1.5em = 1 padmanabhan , n. , white , m. , & eisenstein , d.j . , 2007 , mnras , 376 , 1702 = 1.5em = 1 szapudi i. , 2001 , proc . of `` the onset of nonlinearity in cosmology '' , ed . fry , j.r . buchler & h. kandrug , in annals of the new york academy of sciences , vol . 927 . = 1.5em = 1 szapudi i. , 2004 , apj , 605 , l89 = 1.5em = 1 szapudi i. , 2005 , chapter in `` data analysis in cosmology '' , ed . v.j . martinez , e. martinez - gonzalez , m.j . pons - borderia & e. saar , springer - verlag lecture notes in physics . = 1.5em = 1 tseliakhovich d. & hirata c.m . , 2010 , prd 82 , 083520

when matter is strongly coupled to light , the interaction can not simply be thought of in terms of absorption and emission processes . instead we must consider the eigenstates of the fully coupled matter - light system . the paradigmatic example of this is the existence of exciton polaritons , hybrid matter - light particles formed by the strong interaction between excitons and photons @xcite . matter - light coupling can be engineered by confining light in optical cavities , so as to modify the density of states and the coupling to matter . for weak coupling , or a bad cavity , cavity losses are fast so one can eliminate virtual processes where photons are in the cavity . this gives fermi s golden rule , but with the cavity density of states modifying the emission rate , as first discussed by purcell @xcite . when coupling is strong , first order perturbation theory ( i.e. fermi s golden rule ) fails , as there can instead be coherent emission and reabsorption of photons before light leaks out of the cavity @xcite . a natural context in which strong matter - light coupling arises is between organic molecules and light in semiconductor microcavities . because of the existence of conjugated @xmath0 bonds in organic molecules , electronic transitions can acquire large dipole moments @xcite , leading to very strong coupling to light . when such molecules are placed in optical microcavities this leads to huge polariton splittings @xcite . these scales allow such experiments to be performed at room temperature , whereas for many inorganic materials , cryogenic temperatures are required . the polariton splitting is due to a collective phenomenon : the electronic transitions of many molecules couple to radiation , and as such the polariton splitting grows as the square root of the molecule density . in contrast , in weak coupling , the rate at which one molecule emits is independent of whether or not any other molecules are present @xcite . much of the recent work on organic microcavity polaritons ( see , e.g.ref . @xcite for a recent review ) has been focused on condensation and lasing @xcite , involving a strongly pumped system , and the appearance of macroscopic quantum coherence . there has however also been significant recent work on the effects of matter - light coupling in the vacuum state , i.e. without strong pumping . such work aims to understand how the physical and chemical properties of organic molecules are affected by strong coupling to electromagnetic modes . examples of this include modifying the rates of photochemical reactions @xcite , or modifying the transport properties of organic semiconductors @xcite . more recently , there has also been experimental @xcite and theoretical @xcite work on coupling the vibrational state of organic molecules to infra - red radiation , leading to molecular optomechanics . theoretical work @xcite has also studied how strong matter - light coupling to electronic states can suppress the effects of disorder and vibronic features in the polariton spectrum . of particular interest for the present paper is a recent work from the ebbesen group @xcite , in which the optical spectra of strongly - coupled organic microcavities were studied by varying molecular concentration and temperature , and paying particular attention to the relative weights of the resonant features in the absorption spectra : the two polariton peaks , and a third peak at the bare excitonic energy @xcite . our aim in this manuscript is to examine the behavior of such strongly - coupled organic microcavities starting from various microscopic models , allowing quantitative predictions of the extent to which a self - consistent adaptation of the molecular state , driven by coupling with light , may occur . to understand the variation of the optical spectra with both concentration and temperature the models which we consider all contain a variable degree of coupling to light . this is because a molecule that has strictly zero coupling to light is not visible in the absorption spectrum , while molecules with a small but non - zero coupling will lead to absorption at the bare molecular energy . in order that the coupling to light can vary self consistently ( in response to the rabi splitting ) , it must depend on some adaptable feature of the state or environment of the molecule . i.e. there must be some physical property that can vary , which determines the strength of matter - light coupling . we refer to this concept hereafter as `` self - consistent molecular adaptation '' . we refer to this process as `` self - consistent '' because the effective matter - light coupling depends on ( some aspect of ) the molecular state , and the molecular state is modified because of how its energy depends on the matter - light coupling . in the first part of this manuscript we investigate in detail two candidates that could lead to self - consistent adaptation : rotational and vibrational degrees of freedom , we also consider an extension of these models to generic ( classical ) aspects of the molecules physical or chemical state . while we do find a temperature dependence of the optical spectra , the involved energy scales turn out to be incompatible with the observation of ref . @xcite . in the final part of the present work , we examine how our model of vibrationally dressed polaritons naturally predicts an effect whose energy scale is of the right magnitude to explain the data . this effect does not involve the renormalization of the coupling strength , but instead involves the effect of vibrational replicas and their coupling to the excitonic transition on the optical spectra . our results thus show a first example of the rich , and presently poorly understood behavior that can stem from the interplay of strong matter - light coupling with strong coupling to vibrational / conformational modes of the molecules . we start by noting that the existence of a peak at the bare energy of the exciton , brought forward as evidence of novel physics in ref . @xcite , is not unexpected ; such a `` residual excitonic peak '' has been seen in many cases , for example ref . @xcite discussed theoretically , and demonstrated experimentally , the appearance of such a feature in a gaas / algaas heterostructure , containing quantum wells inside a dbr microcavity . while such a peak comes from the spectral weight of the exciton line , it is important to note that this peak _ can not _ be viewed simply as excitons which do not couple to light : if they did not couple , they would not be visible in the absorption or transmission spectrum . the origin of the peak can be understood physically as coming from the subradiant excitonic states due to inhomogeneous broadening . in a disordered system , the coupling to the photon mode picks out a specific superradiant state , which forms the polaritons , while the other states orthogonal to this superradiant state remain at the bare exciton energies . however , because of energetic disorder , the superradiant state is not an energy eigenstate , and a residual coupling between the superradiant and subradiant states exists , so that the spectral weight of the subradiant states is visible in the optical spectrum @xcite . a simple analytical treatment shows that the weight of this residual excitonic peak does decrease as the matter - light coupling increases . the existence of this peak is thus consistent with the behavior seen in @xcite . however , on its own this explanation can not account for the temperature dependence observed , as the residual excitonic peak should be unaffected by temperature , unless @xmath1 approaches optical energies , of the order of @xmath2 ( for comparison 300k@xmath3mev ) . one of the main goals of this paper is to address how this temperature dependence may occur . as will become clear in the following , to discuss the microscopic theory of such effects , it will be crucial to consider the physics of ultrastrong matter - light coupling @xcite , and the breakdown of the rotating wave approximation ( rwa ) . this requires retaining `` counter - rotating '' terms in the matter - light coupling hamiltonian . these terms , which involve simultaneous creation of pairs of excitations , are typically considered to be non - resonant and so are often neglected . however , if the matter - light coupling is a significant fraction of the bare exciton and photon energies , then these terms have a non - negligible impact . such behavior has been seen in both inorganic @xcite and organic @xcite systems , with a current record of a coupling strength @xmath4 of the bare oscillator frequency @xcite . our focus in this paper is on the more typical regime where such counter - rotating terms can not be neglected , but remain sufficiently small to be treated perturbatively . the rest of the paper is structured in two main sections . in section [ sec : rotational - freedom ] we consider how temperature dependence can arise due to self - consistent adaptation of the rotational and vibrational degrees of freedom of the molecules , via a mechanism very similar to that proposed in ref . @xcite . for the orientational degree of freedom , we consider both free molecules , and molecules with randomly pinned orientations as appropriate in a polymer matrix . we will see that in such systems we do predict a temperature dependence of the residual excitonic peak . however , while this effect could potentially be observed in other experimental realizations , the energy scales ( temperatures ) required and the scaling with molecular concentration are not compatible with the experimental observations reported in ref . @xcite . in section [ sec : vibrational - freedom ] we instead consider a different effect , arising from the interplay of vibrational modes with the matter - light coupling , which is able to reproduce similar behavior to that observed in experiments . specifically we find that vibrational excitations dress the residual excitonic peak in a strongly temperature dependent manner . moreover , the form of the vibrational dressed spectrum shows that the spectral feature at the exciton energy can have a more complex interpretation than that previously considered @xcite . a brief but self - contained account of the main theoretical methods used throughout this paper is given in the appendices . in this section we consider whether self - consistent molecular adaptation can enhance matter - light coupling by renormalizing the bare matter - light coupling strength . we consider models in which the effective matter - light coupling strength of a given molecule depends on the configuration of that molecule , such as its orientation , or its vibrational state . we then ask how this same matter - light coupling modifies the energy landscape for the auxiliary parameters describing the configuration . this leads to the idea of self consistency if strong coupling leads to a reduction of the ground state energy , the energy landscape is deformed so as to favor auxiliary parameters for which the effective matter - light coupling is as large as possible . our aim is to derive this from a microscopic model , and so quantify this effect . in the following we consider two potential scenarios involving adaptation of either orientational or vibrational degrees of freedom . if such a self - consistent enhancement of matter - light coupling occurs , then this can lead to a temperature dependent effective coupling , and thus to a temperature dependence of the residual excitonic peak . we show that such an effect exists , but that its strength is relatively weak , and that the relevant energy scale shows no collective enhancement , i.e. the presence of @xmath5 molecules does not lead to a @xmath5 enhancement of this energy scale , because it must compete with the extensive entropy gain from orientational or vibrational disorder . as such , while increasing the molecular concentration will increase the polariton splitting , it has little effect on the self - consistent orientation . changing the bare oscillator strength of the molecules does however affect both the polariton splitting and the self - consistent molecular adaptation energy scale . before introducing any auxiliary variables , the basic hamiltonian which we consider is an extended dicke model , including diamagnetic terms : @xmath6,\end{gathered}\ ] ] where the field @xmath7 is written in terms of the bosonic creation and annihilation operators @xmath8 describing photon modes labeled by their in - plane momentum @xmath9 , and energy @xmath10 . the pauli matrices @xmath11 describe the electronic state of the molecules . for completeness , we therefore also included the diamagnetic terms , arising from the @xmath12 term of the minimal coupling hamiltonian @xcite . furthermore , in order to consider varying the cavity mode volume while respecting the thomas - reiche - kuhn sum rule @xcite we use @xmath13 where @xmath14 is the oscillator strength of the given molecule . the coefficient @xmath15 depends on the electric field strength of a single photon , and the properties of the effective charges that respond to the field . if we assume an uniform distribution of molecules , momentum conservation can be used to write the diamagnetic term in eq . ( [ eq:1 ] ) as @xmath16 . this term can then be removed by a bogoliubov transformation @xmath17 , yielding the effective hamiltonian : @xmath18 where the on - site hamiltonian is given by @xmath19 and the renormalized parameters @xmath20 are : @xmath21 with @xmath5 the number of molecules in the mode volume . in the dipole approximation , for a molecule with a single mobile electron @xmath22 with @xmath23 the mode volume and @xmath24 quantifying the electronic response of a single electron in terms of the vacuum permittivity @xmath25 and its reduced mass @xmath26 . for molecules involving many conjugated bonds , the coefficient @xmath27 is replaced by a sum over all mobile charges . it is important to note that changing the cavity length changes both the mode volume @xmath23 ( and hence both @xmath15 and @xmath28 ) and the spectrum of photon modes @xmath29 . in the following numerical results we will fix the values of the polariton splitting @xmath30 , exciton energy @xmath31 , and coupling strength @xmath32 , and use these to determine @xmath33 . before considering the role of vibrational and orientational degrees of freedom , we consider how the existence of a temperature dependent matter - light coupling strength , @xmath34 would be seen in the absorption spectrum . for this purpose , it is sufficient to consider the absorption spectrum of eq . ( [ eq:2 ] ) , and its dependence on @xmath35 . appendix [ sec : absorpt - transm - refl ] summarizes how the absorption spectrum can be calculated , including the counter - rotating terms in the matter - light coupling @xcite . in performing these calculations , as noted above , it is necessary to include disorder in order to see the residual excitonic feature . we will consider disorder in the exciton energies , denoted by the energy distribution @xmath36 for simplicity we consider only disorder in the energies and we ignore the subleading effect of the exciton energy distribution on the coupling strength @xmath28 , by using the averaged value @xmath37 . as discussed in the appendix , we consider the quantity @xmath38 $ ] , in terms of the retarded green s @xmath39 , which is proportional to the absorption spectrum for a good cavity . the green s function has the form : @xmath40 where the excitonic self energy for eq . ( [ eq:2 ] ) , can be written as @xmath41 with : @xmath42 here @xmath43 is the inverse temperature and @xmath44 is the homogeneous linewidth of the excitons . in the following we will present results both for @xmath45 and for small but non - zero @xmath44 as indicated in the figure captions . the spectrum is shown in fig . [ fig : abs_simple ] , focusing on the residual excitonic peak , to show its dependence on the polariton splitting . as noted above , such a feature has been observed and commented on several times before , e.g. @xcite . when the polariton splitting @xmath46 is increased , the splitting between the lower and upper polaritons increases , and the exciton spectral weight of the feature at the exciton energy decreases . the asymmetry between the shift of the lower and upper polaritons arises due to the diamagnetic term @xmath47 renormalizing the photon energy . if the coupling to light is weak , so that the rwa is valid , this asymmetry vanishes . in fig . [ fig : abs_simple ] ( b ) we show how the spectrum is modified by including the effects of cavity losses and non - radiative excitonic decay as described in appendix [ sec : absorpt - transm - refl ] . the main effect these processes have is to broaden and thus reduce the height of the polariton peaks , there is also additional broadening to the central peak . because the only temperature dependence of eq . ( [ eq:3],[eq:4 ] ) is via the combination @xmath48 , the spectrum is temperature independent while @xmath49 ( with @xmath50 of the order of the exciton bare energy ) . therefore , as noted in the introduction , the experimentally observed temperature dependence can not occur from this mechanism alone , unless the effective value of @xmath28 is made temperature dependent via its dependence on configuration . we now go on to consider if this effect is plausible . absorption spectrum , @xmath51 vs polariton splitting @xmath52 for the system with no auxiliary degrees of freedom . plotted for @xmath53ev ( i.e. @xmath54k ) , @xmath55ev , and a truncated gaussian distribution @xmath56 with @xmath57ev , @xmath58ev . panel ( a ) shows the results with a perfect cavity while ( b ) includes the effects of cavity losses at rate @xmath59ev and excitonic non - radiative decay at rate @xmath60ev also the width of the energy distribution is larger , @xmath61ev as discussed in appendix [ sec : absorpt - transm - refl].,title="fig:",width=307 ] absorption spectrum , @xmath51 vs polariton splitting @xmath52 for the system with no auxiliary degrees of freedom . plotted for @xmath53ev ( i.e. @xmath54k ) , @xmath55ev , and a truncated gaussian distribution @xmath56 with @xmath57ev , @xmath58ev . panel ( a ) shows the results with a perfect cavity while ( b ) includes the effects of cavity losses at rate @xmath59ev and excitonic non - radiative decay at rate @xmath60ev also the width of the energy distribution is larger , @xmath61ev as discussed in appendix [ sec : absorpt - transm - refl].,title="fig:",width=307 ] to consider molecular adaptation , the hamiltonian in eq . ( [ eq:2 ] ) will be modified to include either orientational or vibrational degrees of freedom . these are illustrated in fig . [ fig : cartoon ] . we next introduce these modifications and then discuss how they may be treated . the hamiltonian including an orientational degree of freedom takes the same form as eq . with the on - site part @xmath62 where the angle @xmath63 parametrizes the orientation of the dipole moment of the @xmath64th molecule with respect to the polarization of the cavity electric field , thus reducing the oscillator strength . the term @xmath65 represents the bare dependence of the hamiltonian on orientation . this term allows one to model pinning of the orientation @xmath63 . for simplicity we consider only a classical orientational degree of freedom ; the corresponding quantum theory would require us to also include a rotational kinetic energy term , and diagonalize the resulting hamiltonian . the effective matter - light coupling strength , which depends on the distribution of angles @xmath63 adopted by the molecule , can be written as @xmath66 , where the double angle brackets represent both an ensemble and thermal average . to consider vibrational degrees of freedom , we again start from the transformed hamiltonian , eq . ( [ eq:2 ] ) , and now consider the following modification : @xmath67 here @xmath68 describe the @xmath69th harmonic vibrational mode of molecule @xmath64 , with the mode having frequency @xmath70 and its coupling to the electronic state being parametrized by the huang - rhys parameter @xmath71 . in this case , defining the effective oscillator strength is more involved : the effective oscillator strength depends on the matrix element describing the overlap between the vibrational states in the ground and excited state manifold . we return to this point in later sections . for both the orientational and vibrational degrees of freedom , our aim is to find how the matter - light coupling is self - consistently modified by these auxiliary degrees of freedom : i.e. how the presence of matter - light coupling modifies the distribution of orientational or vibrational states , and how this in turn affects the effective matter - light coupling strength . we consider a case without any strong pumping , and with a temperature such that @xmath72 , which is typically satisfied even at room temperature for organic polaritons . as such , the origin of the `` self - consistent '' dependence of configuration on the matter - light coupling coupling arises due to the existence of the counter - rotating terms in the original hamiltonian : if these terms were neglected then the energy in the ground state sector can be trivially found as the ground state would correspond to the empty state , and its energy would therefore not involve the matter - light coupling strength at all @xcite . the presence of counter - rotating terms means that the ground state sector also involves an admixture of all even parity sectors , and the degree of admixture depends on the effective matter - light coupling . as discussed below , while exact solutions are possible in some limiting cases of the orientational problem , these are not generally possible at finite temperature , nor for the vibrational problem . this is because thermal or quantum fluctuations of the auxiliary degrees of freedom break translational symmetry , preventing simple exact diagonalization . as such , we proceed using the schrieffer - wolff formalism @xcite , which allows us to consider perturbatively the effects of these counter - rotating terms , and how they modify the energy landscape seen by the auxiliary orientational or vibrational degrees of freedom . for completeness , appendix [ sec : schr - wolff - transf ] provides a brief summary of the schrieffer - wolff formalism . the essential point is to separate @xmath73 , where @xmath74 are the terms treated perturbatively . at leading order this gives an effective hamiltonian : @xmath75 , \quad { \hat{g } } : \ [ { \hat{g } } , \hat{h}_0 ] \equiv i \hat{h}_1.\ ] ] taking the counter - rotating terms as @xmath74 , the perturbation theory is controlled by the small parameter @xmath76 , which is indeed small for the physical parameters we consider can still be comparable to @xmath77 , as is indeed the case for the parameters we consider . ] . we are thus in a regime where the counter - rotating terms can not be ignored , but where they can be included perturbatively . in the following sections we apply this approach in turn to the orientational and vibrational degrees of freedom , and see how the effective matter - light coupling can be derived self - consistently . as discussed above , we consider first the classical orientational degrees of freedom @xmath63 , subject to a pinning potential @xmath65 . if all molecules are identical , @xmath78 , then one can find the zero temperature ground state by choosing @xmath79 . for the ground state , all that is required is to find the quantum ground state of eq . ( [ eq:6 ] ) as a function of @xmath80 and then minimize over @xmath80 . since eq . ( [ eq:6 ] ) is translationally invariant in the case @xmath79 , it is possible to find the exact ground state in the bosonic approximation by fourier transforming . the bosonic approximation assumes the occupation of each excited molecule is small , a result that is valid unless @xmath81 @xcite . however , at finite temperature , even for identical molecules , it is crucial to allow independent fluctuations of each @xmath63 ; assuming @xmath79 massively underestimates the entropy at finite temperature . at zero temperature , one may compare the exact solution to the schrieffer - wolff perturbative expansion used below , and one finds that these indeed match to leading order . for this rotational case , the form of @xmath82 required in eq . ( [ eq:8 ] ) can be found trivially , and the resulting hamiltonian can most conveniently be written as @xmath83 , with the molecular hamiltonian in the form : @xmath84 where @xmath85 is the bare molecular hamiltonian , including the rwa coupling to light and including the pinning term @xmath65 . in order to consider the thermal distribution of @xmath63 , we must specify the orientational potential @xmath65 . we consider a form @xmath86 which tries to pin the molecules at angle @xmath87 , relative to the cavity electric field , with strength @xmath88 . we may thus consider both the free orientation case , @xmath89 , and the pinned case simultaneously . in what follows we will assume that the pinning angles have a uniform distribution such as would be found in a polymer matrix . this treatment , however , ignores effects which would be important in systems such as organic crystals in which the pinning angle has a fixed direction . the energy landscape for each angle @xmath63 , given the pinning angle @xmath87 is then : @xmath90 where for simplicity we have assumed that each molecule has the same values of @xmath91 and the only disorder is in the pinning angles . in the following we will define the quantity : @xmath92 the quantity @xmath93 characterizes the self - consistent energy favoring alignment of molecules . replacing the summation by an integral , and inserting the explicit forms of @xmath94 and @xmath15 written above we have that @xmath95 where @xmath96 is again the cavity length , @xmath27 the combination defined following eq . ( [ eq:2 ] ) , and @xmath97 is a cutoff reflecting the breakdown of the dipole approximation . to evaluate such integrals it is useful to write the dispersion in the form @xmath98 which allows us to find the exact result @xmath99 a notable feature of eq . ( [ eq : k0exact ] ) , as anticipated above , is that this quantity _ does not simply increase _ as one increases the polariton splitting by varying the density of emitters , @xmath100 . there is a sub - leading dependence on the molecule density , via the renormalization @xmath101 given in eq . ( [ eq:14 ] ) . however this effect is only significant in the deep strong coupling limit @xcite . physically this lack of scaling with @xmath5 is because this `` molecular adaptation energy '' depends on the shift seen for _ each _ molecule . inserting typical experimental values for an organic system @xmath102 , @xmath103ev , @xmath104 nm , @xmath105 , @xmath106 nm , @xmath107ev@xmath108nm@xmath109 which correspond to the values extracted from ref . @xcite , one finds that @xmath110 which is much smaller than @xmath111 at room temperature . as noted earlier , the effective matter - light coupling strength is given by @xmath112 at zero temperature , this corresponds to minimizing the energy , leading to @xmath113 . however , since the value of @xmath93 given above is such that @xmath114 , it is crucial to consider finite temperatures . the smallness of the ratio @xmath115 will also allow us to make further perturbative expansions in the following . defining @xmath116 , this ratio can be calculated as @xmath117,\ ] ] where the partition function is @xmath118.\ ] ] as noted above @xmath119 , and so we may taylor expand in this small parameter to get a closed form for @xmath120 . assuming that the pinning angle distribution @xmath121 is uniform we obtain the simple result for the coupling @xmath122\ ] ] with @xmath123 the imaginary bessel function @xmath124 at this point we have made no assumption about the pinning strength @xmath88 , and the value of the effective coupling is controlled by the combination @xmath125 . if the pinning is strong @xmath126 then we find the asymptotic form @xmath127 this no longer depends on temperature as the strong pinning limit means entropy become unimportant . thus , in the @xmath128 limit the coupling takes on the isotropic value of @xmath129 , corresponding to the uniform distribution of angles @xmath130 . the effective coupling increases as the pinning @xmath88 decreases . in the limit of vanishing pinning @xmath131 , the imaginary bessel function @xmath132 vanishes and so we have : @xmath133 both eq . ( [ eq:9 ] ) and eq . ( [ eq:10 ] ) indicate that as long as @xmath134 , the modification and temperature dependence of @xmath135 is very small . we note that while in the organic systems which we focus on here @xmath93 is relatively small , in systems which have very small mode volumes this parameter could be engineered to be much larger and hence the coupling strength renormalization would be much more pronounced . i.e. , since there is no scaling with @xmath5 , the crucial feature to see a strong renormalization is to minimize the mode volume in absolute terms , and not the mode volume _ per molecule_. this suggests evanescently confined radiation modes in plasmonic @xcite or phonon polariton @xcite systems may be a promising venue to explore this physics . in the vibrational case , even without disorder , an exact solution of eq . ( [ eq:7 ] ) via fourier transformation is no longer possible , because the vibrational degrees of freedom are modeled as quantum degrees of freedom with their own quantum dynamics . as such , they break translational invariance i.e. localized vibrational excitations can scatter between different polariton momentum states . thus , once again we must use the schrieffer - wolff formalism . if we start from eq . ( [ eq:7 ] ) , solving the equation @xmath136=i\hat{h}_1 $ ] is now more challenging than for the rotational case , as @xmath137 involves terms that couple the electronic state to the vibrational quantum state . the equation can however be solved in the form of a power series , @xmath138 where the operators @xmath139 are defined by the recursion relation @xmath140 \\- \hat{o}_{j-1 } \sqrt{s_m}({\hat{b}^{}}_m+{\hat{b}^\dagger}_m ) \biggr\},\end{gathered}\ ] ] with the base case @xmath141 . this expansion then allows one to write out the effective hamiltonian in the same form as above , but with the molecular hamiltonian , @xmath142 \sum_{{\mathbf{k}},j } \frac{\tilde{g}_{{\mathbf{k}}}^2 } { ( \tilde{\omega}_{{\mathbf{k } } } + \epsilon_n)^{j+1 } } ( \hat{o}_{j } + \hat{o}_{j}^{\dagger}),\end{aligned}\ ] ] with @xmath85 the bare molecular hamiltonian , including the rwa coupling to light , and vibrational terms of eq . ( [ eq:7 ] ) . this expression can be considered as a multinomial power series in the quantities @xmath143 for each vibrational mode @xmath69 . for typical parameters , such quantities are small , and so we may truncate at first order , i.e. keep terms up to @xmath144 . beyond @xmath144 , the expression becomes considerably more complicated , as cross terms between different vibrational modes appear . up to @xmath144 , we find an effective molecular hamiltonian which we write out in full ( neglecting constant terms ) : @xmath145 \sqrt{s_m } ( b_m+{\hat{b}^\dagger}_m ) \right).\end{gathered}\ ] ] the @xmath146 term gave an energy shift of two - level systems with the same form @xmath93 found previously . the @xmath144 term is on the last line , and describes a shift to the vibrational modes . the coefficients for these terms are defined by a generalization of that used in eq . for the rotational case above , @xmath147 where we may find an analytic form for the resulting integral @xmath148 we may once again note that none of the terms @xmath149 are proportional to the number of molecules @xmath5 : vacuum - state molecular adaptation is not collectively enhanced . from the form of eq . ( [ eq:12 ] ) it is clear that the term in @xmath150 describes a reduction in the offset between the vibrational ground state in the two electronic states , as the virtual excitations admix the excited electronic state configuration into the ground state . this can be viewed as a reduction of the huang - rhys parameter , @xmath151 $ ] . this point is illustrated in fig . [ fig : cartoon ] . as noted earlier , the dependence of @xmath120 on the vibrational degrees of freedom is more complicated : one must calculate the overlap between the vibrational states of the ground and excited electronic state manifolds . the reason that the vibrational states differ in these manifolds is the existence of the terms @xmath152 , which correspond to displacement of the vibrational coordinate dependent on the electronic state of the molecule . as such , a reduction of @xmath153 would lead to an enhanced matter - light coupling . however , using the same typical experimental values as before we find that this dimensionless shift has a value of approximately @xmath154 . thus , once again one may conclude that the characteristic scale of any vibrational molecular adaptation ( determined by @xmath150 ) is negligibly small . the discussion so far has focused on two specific microscopic mechanisms which might have led to self - consistent adaptation of the molecules so as to enhance their coupling to light . here we note those aspects of the above results which can be easily generalized to other microscopic mechanisms . examples of such other potential mechanisms include solvation of the molecule , molecular configuration , and charge transfer state . in some cases , these will require detailed modeling of the specific process , particularly for degrees of freedom with energy scales larger than temperature , where quantum effects become important , as in the example of vibrational modes above . however , the basic idea can be illustrated in the simplest case , where some aspect of configuration can be parametrized by a classical variable . in the case where classical parametrization applies , the description is very similar to the discussion of orientational degrees of freedom : one considers a classical variable @xmath155 which parametrizes some aspect the state for molecule @xmath64 . associated with this will be an energy function @xmath156 , and , for self - consistent adaptation to occur , the matter - light coupling must depend on this variable as @xmath157 . the same perturbative analysis as discussed above will then lead to the effective energy function : @xmath158 . this in turn means that the self - consistent effective matter - light coupling will take the form : @xmath159.\ ] ] since this involves the same sum @xmath93 as defined in eq . ( [ eq : defk0 ] ) , the same absence of scaling with number of molecules occurs i.e. it is the single - molecular , rather than collective , coupling which is important . moreover , since @xmath134 , then : @xmath160 where @xmath161 indicates thermal averaging with the bare energy function @xmath156 , i.e. any such vacuum - state self - consistent adaptation of molecules is suppressed by the small parameter @xmath162 . in the previous section we have seen that while self - consistent molecular adaptation due to matter - light coupling is possible , neither the strength nor the dependence upon molecular concentration are consistent with the results reported in ref . @xcite . in this section we show how directly calculating the absorption spectrum from the model of vibrationally dressed polaritons introduced in the previous section can lead to a rather different mechanism which could however be responsible for the features observed . this mechanism arises from the combination of vibrational dressing of the spectrum and the effects of disorder . we focus in particular on the peak in the spectrum near the bare excitonic resonance . we will see that the shape of this feature depends strongly on the vibrational state of the molecules . this section is divided into two subsections , in section [ sec : self - energy - incl ] we first discuss how vibrational excitations should be included in the calculation of the disordered polariton spectrum . this follows the method outlined in appendix [ sec : schr - wolff - transf ] , and so all that is required is to calculate the excitonic self energy . we then discuss the resulting form of the spectrum in section [ sec : evol - absorpt - spectr ] . as discussed in appendix [ sec : absorpt - transm - refl ] , the absorption , emission and transmission spectra can all be found in terms of the photon retarded green s function . in this approach , all the properties of the molecules ( i.e. inhomogeneous broadening , vibrational dressing etc . ) are incorporated via the excitonic self energy @xmath163 . we must therefore calculate this quantity , defined in eq . ( [ eq : self - energy],[eq:13 ] ) , for the vibrationally dressed hamiltonian , eq . ( [ eq:7 ] ) . to do this , it is useful to label the eigenstates as @xmath164 , corresponding to the vibrational eigenstates in the excited electronic state manifold and the ground electronic state manifold . we denote the energies of these states as @xmath165 , and @xmath166 and we introduce the overlap matrix element : @xmath167 . this matrix element describes the extent to which the state with @xmath168 vibrational excitations in the electronic ground manifold overlaps with the state in the excited electronic manifold with @xmath169 excitations . in order to find these eigenvalues and eigenfunctions we must diagonalize the vibrational problem , @xmath170 in the electronic ground and excited states . as discussed in sec . [ sec : vibr - degr - freed ] , the renormalization of these parameters due to virtual pair creation is very small , and so we neglect it in the following . in terms of these quantities we may write the self energy , including inhomogeneous broadening of excitonic energies in the form : @xmath171 } { \nu+i\gamma + ( e_q^{\downarrow}-e_p^{\uparrow } ) } , \label{eq : rwa - self - energy}\end{aligned}\ ] ] where @xmath36 is again the distribution of exciton energies , as in sec . [ sec : rotational - freedom ] , and we have again ignored the subleading effects of the exciton energy distribution on the coupling strength , by using @xmath172 . it should be noted that the energies @xmath173 depend ( linearly ) on the energy @xmath50 , but that the matrix elements @xmath174 are not dependent on this energy scale . in the following , we model the inhomogeneous broadening of excitons by using the distribution @xmath175 , i.e. a truncated gaussian distribution of excitonic energies . the truncation has little effect , but is formally required , as negative energy states can not physically exist , and cause problems in the ultrastrong coupling limit @xcite . the exciton spectrum in the absence of vibrational dressing was shown previously in fig . [ fig : abs_simple ] . before exploring the effect of vibrational modes , it is helpful to summarize how the residual exciton peak arises . mathematically , the excitonic feature in the polaritonic spectrum can be understood directly from the form of eq . ( [ eq:3 ] ) and the definition of the absorption spectrum . the peak at the exciton frequency occurs because the imaginary part of the retarded green s function has a numerator involving the imaginary part of the self energy , @xmath176 $ ] , and this self energy has a peak at the excitonic energy . however , the weight of the residual excitonic peak reduces as the matter - light coupling increases , because the excitonic self energy also appears , squared , in the denominator . it is also important to note that this residual excitonic peak does not appear in the transmission spectrum : since @xmath177 , there is no term in the numerator of @xmath178 arising from the exciton self energy . thus , the residual exictonic peak is a feature of the absorption ( and reflection ) , but not the transmission spectrum . for a system with a single vibrational mode with @xmath179ev . the insets show the equivalent absorption spectra for bare molecules . the other parameters are as in fig . [ fig : abs_simple ] . panel ( a ) shows the spectrum for a perfect cavity while panel ( b ) includes the effects of cavity losses and non - radiative excitonic decay exactly as in fig . [ fig : abs_simple](b).,title="fig:",width=307 ] for a system with a single vibrational mode with @xmath179ev . the insets show the equivalent absorption spectra for bare molecules . the other parameters are as in fig . [ fig : abs_simple ] . panel ( a ) shows the spectrum for a perfect cavity while panel ( b ) includes the effects of cavity losses and non - radiative excitonic decay exactly as in fig . [ fig : abs_simple](b).,title="fig:",width=307 ] figure [ fig : abs_vs_g ] shows an equivalent set of spectra to fig . [ fig : abs_simple ] , but with vibrational dressing . for comparison in the insets we show the absorption spectra of a bare molecule @xmath176 $ ] , i.e. the spectra without a cavity . panel ( a ) shows the simple case where cavity losses are ignored , while panel ( b ) shows the more experimentally relevant case where these effects are included ( discussed in appendix [ sec : absorpt - transm - refl ] ) . in the following we use the notation @xmath180 which denotes transitions in the molecule from the state with @xmath169 vibrational excitations in the electronic ground state to the state with @xmath168 vibrational excitations in the electronic excited state . the presence of the vibrational modes causes dramatic changes to the residual excitonic peak in the polariton spectrum not seen in the bare excitonic absorption . for the bare molecule the spectral weight associated with transition with the `` zero phonon line '' , i.e. the transition denoted @xmath181 in the notation introduced above is completely dominant , only a small amount of weight visible in the sideband which corresponds to @xmath182 transition . when the molecule is placed inside a cavity , the vibrational sidebands corresponding to the @xmath183 and @xmath182 transitions become much more prominent . physically this is because the spectral weight that had been associated with the @xmath181 transition in the bare molecular spectrum has been moved into the polariton peaks of the spectrum . the mathematical form of the green s function makes clear that formation of the polariton spectral feature is predominantly at the expense of whatever feature dominates the excitonic emission spectrum ; here this is the @xmath181 feature . the excitonic feature corresponds to the `` left - over '' spectral weight associated with the subradiant states . thus , the vibrational sidebands have been `` excavated '' by removing the dominant feature from the zero - phonon line . including the the effects of cavity losses and non - radiative excitonic decays , as can be seen in fig . [ fig : abs_vs_g ] ( b ) , washes out this complex sideband structure of the central peaks and the spectrum as a function of coupling strength looks very similar to that obtained without coupling to vibrational modes , as in fig . [ fig : abs_simple ] . however as we discuss below , there are still effects which can be observed which are a direct consequence of the vibrational structure . , ( b ) coupling to a single vibrational mode with @xmath184ev . the inset shows the bare molecular spectrum . ( c ) including the effects of cavity losses and non - radiative excitonic decay . all panels are for @xmath185ev , and other parameters as in previous figures.,title="fig:",width=307 ] , ( b ) coupling to a single vibrational mode with @xmath184ev . the inset shows the bare molecular spectrum . ( c ) including the effects of cavity losses and non - radiative excitonic decay . all panels are for @xmath185ev , and other parameters as in previous figures.,title="fig:",width=307 ] , ( b ) coupling to a single vibrational mode with @xmath184ev . the inset shows the bare molecular spectrum . ( c ) including the effects of cavity losses and non - radiative excitonic decay . all panels are for @xmath185ev , and other parameters as in previous figures.,title="fig:",width=307 ] fig . [ fig : abs_vs_t ] illustrates the evolution of spectra with temperature . panel ( a ) shows that without coupling to vibrational modes there is no notable temperature dependence . in the presence of the vibrational dressing , a strong temperature dependence appears . at higher temperatures , there is a greater thermal occupation of the vibrational modes hence the spectral weight under the vibrational peaks rises . while this has a small effect in the bare molecular spectrum ( where the @xmath181 transition dwarfs all other features ) , see inset in ( b ) , it is very pronounced in the polariton spectrum and is even visible in the presence of large cavity losses as in fig . [ fig : abs_vs_t](c ) . . the main panel shows the optical spectrum of the strongly coupled system , the inset shows the spectrum of the bare excitons . in this case , we include two vibrational modes : @xmath186ev , and @xmath1874ev . other parameters as in fig . [ fig : abs_vs_t].,width=307 ] the figures so far have shown results where disorder is relatively small , and so vibronic replicas can be clearly observed for the good cavity , but merge for the bad cavity limit . to show that small disorder is not required for the strong temperature dependence to occur fig . [ fig : abs_large_disorder ] shows the effects of large disorder ( i.e. inhomogeneous broadening ) . just as seen for the homogeneous broadening in fig . [ fig : abs_vs_t](c ) , a temperature dependence of the residual excitonic peak is still visible . in this figure , we have also included a more complicated vibrational spectrum , involving two vibrational modes . such a system will exhibit behavior similar to that of a single mode but with a large vibrational coupling , then the effective huang - rhys parameter will be @xmath188 . in this paper we have presented two microscopic models which could in principle describe self - consistent molecular adaptation so as to maximize the vacuum - state coupling to light . in both cases , the crucial feature of the model is the counter - rotating terms in the matter - light coupling . these allow virtual fluctuations in the ground state , that lower the ground state energy depending on the configuration of the molecules . this energy gain is the only energy gain that can be relevant in the linear response regime i.e. the question of whether the excited states would have lower energy is not of relevance while the system is only weakly pumped . we found that while such a mechanism for molecular adaptation does exist , it does not show any collective enhancement , in contrast to the polariton splitting , and does not therefore lead to significant molecular adaptation , even when the polariton splitting @xmath189 . the appearance of a residual exciton peak in the polariton spectrum would be affected by any such self - consistent molecular adaptation if its scale were sufficient . however , for relevant parameters , such effects are dwarfed by a far more dramatic effect , of vibrational dressing of the residual exciton peak . this leads to a pronounced temperature dependence of the feature near the exciton energy in the absorption spectrum . while the molecular adaptation energy scale is not collectively enhanced , the basic underlying physics could potentially be relevant in single molecule strong coupling , e.g. with plasmonic resonances @xcite . in such cases , rather than having many molecules in a large mode volume , the mode volume is reduced so that strong , or even ultra - strong coupling occurs at the single molecule level , so that both the polariton energy and the molecular adaptation energy become large . in such a case , one may hope to see either reorientation , or renormalization of the huang - rhys parameter due to strong coupling . another intriguing direction for future research is to consider how the physics discussed in this manuscript interacts with the physics of polariton condensation and lasing @xcite . polariton condensation has been seen in both inorganic @xcite and organic @xcite systems . in addition , condensation of photons has been seen for weakly coupled systems of organic molecules @xcite . theoretical work @xcite has begun to address some of the peculiarities of the organic polariton system , including effects of disorder and of vibrational modes . however , features as seen in this paper , resulting from the interplay of these may lead to further exotic behavior in the high density condensed phase . we are grateful for comments from t. ebbesen on an earlier version of this paper . jk acknowledges helpful discussions with david lidzey and brendon lovett . jac acknowledges support from epsrc . jk and pgk acknowledge financial support from epsrc program `` topnes '' ( ep / i031014/1 ) . jk acknowledges support from the leverhulme trust ( iaf-2014 - 025 ) . sdl is royal society research fellow . sdl acknowledges financial support from epsrc grant ep / m003183/1 . pgk acknowledges support from epsrc grant ep / m010910/1 . jk , pgk and sdl acknowledge support from the british council for the meeting which initiated this work . [ [ note - added ] ] note added : + + + + + + + + + + + during the final preparation of this manuscript , another paper @xcite appeared , also reporting the fact that ground - state bond length depends on the single - molecule coupling @xmath190 , not the collective coupling @xmath46 . in this appendix we summarize the calculation of the absorption , transmission and reflection spectra . some subtleties arise because we wish to calculate the spectrum of a model with ultrastrong coupling , i.e. without making the rotating wave approximation . such results were first calculated by @xcite , here we present a synopsis of these results , as well as a `` dictionary '' to translate the results of that paper into the language of green s functions . we begin by defining the retarded green s function @xcite . because we consider both co- and counter - rotating terms , we must consider both normal and anomalous green s functions . i.e. we must include number non - conserving terms which appear for ultrastrong coupling , and thus we consider a matrix green s function : @xmath191 \rangle & \langle [ \psi^{\dagger}_{-{\mathbf{k}}}(t ) , \psi^\dagger_{{\mathbf{k}}}(t^\prime ) ] \rangle \\ \langle [ \psi^{}_{{\mathbf{k}}}(t ) , \psi^{}_{-{\mathbf{k}}}(t^\prime ) ] \rangle & \langle [ \psi^{\dagger}_{-{\mathbf{k}}}(t ) , \psi^{}_{-{\mathbf{k}}}(t^\prime ) ] \rangle \end{pmatrix}\ ] ] in terms of the bogoliubov transformed operators , i.e. the operators appearing in eq . ( [ eq:2 ] ) the inverse green s function takes the form @xmath192^{-1 } = \begin{pmatrix } \nu + i \tilde{\kappa}(\nu ) - \tilde{\omega}_{{\mathbf{k } } } + \tilde{\sigma}^{}_{{\mathbf{k}},xx}(\nu ) & + i \tilde{\kappa}(\nu ) + \tilde{\sigma}_{{\mathbf{k}},xx}(\nu ) \\ - i \tilde{\kappa}^\ast(-\nu ) + \tilde{\sigma}^\ast_{{\mathbf{k}},xx}(-\nu ) & -\nu - i \tilde{\kappa}^\ast(-\nu ) - \tilde{\omega}_{{\mathbf{k } } } + \tilde{\sigma}^\ast_{{\mathbf{k}},xx}(-\nu ) \end{pmatrix},\ ] ] where @xmath193 is the self energy for a photon of in - plane momentum @xmath9 , arising from the excitonic response ( discussed further below ) , and @xmath194 is the loss rate . here , following @xcite we have used a frequency dependent complex loss rate @xmath194 . frequency dependence is required for physical consistency in the case of ultrastrong coupling markovian loss and ultrastrong coupling would predict a perpetual light source . frequency dependent loss requires , via the kramers - kronig relation , a corresponding lamb shift , which is incorporated into the imaginary part of @xmath194 . both @xmath195 and @xmath196 are written for the bogoliubov transformed operators , and so both these terms incorporate a pre - factor @xmath197 to account for the bogoliubov transformation of the combination @xmath198 . the specific self energy required , @xmath193 , corresponds to correlation functions of the @xmath199 excitonic operators . in the absence of strong ( i.e. beyond rwa ) excitonic damping ( see @xcite for the more general case ) , this self energy can however be related to results in the rotating wave approximation by : @xmath200^{\star } \right ) , \label{eq : self - energy}\ ] ] where the expression @xmath201 is the `` standard '' self energy that would appear in the rotating wave approximation , depending on the correlation of @xmath202 operators . these can most easily be found by analytic continuation from imaginary time to real time , starting from the matsubara self energy , @xmath203 and replacing the matsubara frequency by @xmath204 . the sum over @xmath169 appearing here is over all states of the excitonic system , and @xmath205 is the partition function . for the `` vacuum '' state we consider i.e. in the absence of strong pumping these are the bare exciton states , including the quantum states of any auxiliary degrees of freedom . using the input - output formalism @xcite adapted to the ultrastrong coupling regime @xcite , one can write a frequency dependent scattering matrix relating input and output fields at the left and right sides of the cavity , @xmath206 where @xmath207 with @xmath208 are the real parts of the loss rates arising from the left and right mirrors and the quantity @xmath39 relates to the matrix retarded green s function as @xmath209 where @xmath210 . this structure means the bogoliubov transformation corresponds to @xmath211 and the bogoliubov transformed green s function takes the form : @xmath212 one can then find the transmission @xmath178 , reflection @xmath213 and absorption @xmath214 coefficients by considering the modulus square of various coefficients . clearly @xmath215 is independent of which direction light is incident from , while the absorption coefficient @xmath216 takes the form @xmath217 + \kappa(\nu ) \left\vert g^r_{{\mathbf{k}},xx}(\nu)\right\vert ^2 \right],\ ] ] with @xmath218 . the prefactor in this expression shows the obvious dependence on the transmissivity of the input mirrors . in order to separate mirror transmissivity dependent features from the `` intrinsic '' properties of the ultrastrong coupling we will consider below the two quantities @xmath219 as being proportional to the transmission , and @xmath38 $ ] as controlling the absorption in the limit of a good cavity , i.e. @xmath220 . the quantity @xmath221 differs from the full absorption as it neglects interference effects at the input mirror . ) , plotted for @xmath222 , @xmath223 , @xmath224ev , @xmath225ev , @xmath226ev and various cavity loss rates , @xmath227 . other parameters are as in fig . [ fig : abs_large_disorder ] . as discussed in the text , in order that the polaritonic peaks have a finite width , one must include the effects of excitonic absorption or non - mirror cavity losses . for this figure , we include an excitonic linewidth @xmath60ev.,width=307 ] for comparison to the results in ref . @xcite where silver mirrors were used , we present in figures [ fig : abs_simple ] , [ fig : abs_vs_g ] , and [ fig : abs_vs_t ] the effects of large cavity linewidth . figure [ fig : true - absorption ] also shows how the full absorption spectrum given by eq . ( [ eq:15 ] ) evolves with varying linewidth . in calculating these spectra for large linewidth , an issue arises regarding the form of the absorption spectrum : the equations written above assume that the only photon loss is due to escape through the mirrors . this means that the @xmath228 appearing explicitly in eq . ( [ eq:15 ] ) ( describing interference effects from the mirror ) is the same as the @xmath228 appearing in the denominator of the photon green s function , eq . ( [ eq:11 ] ) , and one may check that for any frequency where @xmath229 $ ] is small , this causes a near cancellation between the two contributions . for the gaussian exciton density of states @xmath36 used in this paper , this cancellation almost completely suppresses the polariton peaks . such a cancellation in the absorption spectrum can be expected on physical grounds : if the only loss channel for photons is the mirrors , then there is no absorption . all photons that enter eventually leave . in a real device there are other photon loss sources ( absorbers , scattering by surface roughness ) . similarly , in a real device , the excitons have a non - zero rate of non - radiative decay . the effect of this is included by retaining a non - zero value of @xmath44 in the denominator of the self - energy , eq . . this leads to lorentzian tails of @xmath229 $ ] , giving a finite weight to the polariton peak in the absorption spectrum ; such an effect was included by @xcite . we follow this approach in plotting fig . [ fig : true - absorption ] . in section [ sec : rotational - freedom ] we make use of the schrieffer - wolff approximation ; for completeness we provide here a brief explanation of this formalism . the approach is based on dividing the hamiltonian into two parts , @xmath230 , where the term @xmath231 takes one between different `` sectors '' . in our case , these sectors correspond to different numbers of polaritons i.e. @xmath232 is the `` counter - rotating '' part of the hamiltonian which simultaneous creates a photon and excites a molecule . the aim of the schrieffer - wolff formulation is to make a unitary transformation @xmath233 such that the transformed hamiltonian no longer has any coupling between sectors . physically , this corresponds to eliminating the effect of virtual pair creation and destruction , and deriving how such virtual processes renormalize the hamiltonian within a given sector . if the hamiltonian @xmath74 can be treated perturbatively by replacing @xmath234 with @xmath235 a small parameter , then one can consider a series solution @xmath236 . in order to make the first - order terms in @xmath235 vanish , one must choose @xmath237=i\hat{h}_1 $ ] . this then leads ( setting @xmath238 ) to the expression : @xmath239 + \text{h.o.t}\end{aligned}\ ] ] where the higher order terms involve all @xmath240 . stopping at leading order gives the expression in eq . ( [ eq:8 ] ) , corresponding to the leading order effects of virtual pair creation and annihilation . to solve @xmath241=i\hat{h}_1 $ ] in practice is straightforward if one knows the eigenspectrum of @xmath242 , which then allows one to write @xmath243 58ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty http://prola.aps.org/abstract/pr/v112/i5/p1555_1 [ * * , ( ) ] http://adsabs.harvard.edu/abs/1958jetp....6..785p [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physreva.44.669 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.69.3314 [ * * , ( ) ] @noop _ _ ( , , ) @noop _ _ ( , , ) @noop _ _ ( , , ) link:\doibase 10.1038/25692 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.82.3316 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.106.196405 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.95.036401 [ * * , ( ) ] link:\doibase 10.1103/physrev.93.99 [ * * , ( ) ] link:\doibase 10.1007/978 - 3 - 662 - 45082 - 6 [ _ _ ] , edited by , nano - optics and nanophotonics ( , , ) link:\doibase 10.1103/physrevlett.101.116401 [ * * , ( ) ] link:\doibase 10.1038/nphoton.2010.86 [ * * , ( ) ] link:\doibase 10.1038/nmat3825 [ * * , ( ) ] link:\doibase 10.1038/nmat3874 [ * * , ( ) ] link:\doibase 10.1002/ange.201107033 [ * * , ( ) ] link:\doibase 10.1038/nmat4392 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.114.196403 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.114.196402 [ * * , ( ) ] link:\doibase 10.1038/ncomms6981 [ * * , ( ) ] link:\doibase 10.1038/nnano.2015.264 [ * * , ( ) ] link:\doibase 10.1088/1367 - 2630/17/5/053040 [ * * , ( ) ] @noop `` , '' ( ) , link:\doibase 10.1063/1.4919348 [ * * , ( ) ] link:\doibase 10.1002/anie.201301861 [ * * , ( ) ] link:\doibase 10.1002/cphc.201200734 [ * * , ( ) ] link:\doibase 10.1103/physreva.53.2711 [ * * , ( ) ] link:\doibase 10.1103/physrevb.64.235101 [ * * , ( ) ] link:\doibase 10.1103/physrevb.72.115320 [ * * , ( ) ] link:\doibase 10.1103/physrevb.72.115303 [ * * , ( ) ] link:\doibase 10.1103/physrevb.92.125433 [ * * , ( ) ] link:\doibase 10.1103/physrevb.79.201303 [ * * , ( ) ] link:\doibase 10.1021/ph500266d [ * * , ( ) ] @noop * * ( ) link:\doibase 10.1103/physrevb.90.205309 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.35.432 [ * * , ( ) ] link:\doibase 10.1103/physreva.74.033811 [ * * , ( ) ] link:\doibase 10.1103/physrev.149.491 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.105.263603 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.016401 [ * * , ( ) ] link:\doibase 10.1021/acs.nanolett.5b01204 [ * * , ( ) ] link:\doibase 10.1021/nl401590 g [ * * , ( ) ] link:\doibase 10.1103/physreva.80.053810 [ * * , ( ) ] http://stacks.iop.org/0034-4885/78/i=1/a=013901 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.85.299 [ * * , ( ) ] link:\doibase 10.1038/nature05131 [ * * , ( ) ] link:\doibase 10.1126/science.1140990 [ * * , ( ) ] link:\doibase 10.1038/nature09567 [ * * , ( ) ] link:\doibase 10.1002/pssa.200304067 [ * * , ( ) ] link:\doibase 10.1103/physrevb.74.165320 [ * * , ( ) ] link:\doibase 10.1103/physrevb.77.155325 [ * * , ( ) ] link:\doibase 10.1103/physrevb.79.035325 [ * * , ( ) ] link:\doibase 10.1103/physrevb.80.235313 [ * * , ( ) ] link:\doibase 10.1103/physrevb.80.235314 [ * * , ( ) ] link:\doibase 10.1103/physrevb.88.075321 [ * * , ( ) ] link:\doibase 10.1209/0295 - 5075/105/47009 [ * * , ( ) ] link:\doibase 10.1103/physrevx.5.041022 [ * * , ( ) ] @noop _ _ ( , , ) link:\doibase 10.1103/physreva.30.1386 [ * * , ( ) ]

the interaction between neutral objects is dominated by fluctuation forces due to the coordinated behavior of fluctuating charges or collective modes inside the objects . at zero temperature or at sufficiently small distances , the interactions result from quantum fluctuations . they are important on atomic scales as well as between macroscopic bodies . a prominent example for the latter is the casimir force between parallel metallic plates due to current or electromagnetic field fluctuations @xcite . during the last decade experimental verifications of this effect have been performed with increasing precision . these high precision measurements were enabled by the use of curved surfaces instead of parallel planar mirrors in order to avoid the problem of parallelism . the most commonly employed geometry is a sphere - plate setup that was used in the first high - precision tests of the casimir effect @xcite . this geometry has been successfully used ever since in most of the recent experimental studies of casimir forces between metallic surfaces @xcite . in order to keep the deviations from two parallel plates sufficiently small , spheres with a radius much larger than the surface distance have been used . the effect of curvature has been accounted for by the `` proximity force approximation '' ( pfa ) @xcite . this scheme is assumed to describe the interaction for sufficiently small ratios of radius of curvature to distance . however , this an uncontrolled assumption since pfa becomes exact only for infinitesimal separations , and corrections to pfa are generally unknown . at the other extreme , the interaction between a planar surface and objects that are either very small or at asymptotically large distance is governed by the casimir - polder potential that was derived for the case of an atom and a perfectly conducting plane @xcite . this limit has been probed experimentally with high precision for a bose - einstein condensate that was trapped close to a planar surface @xcite . recently , there have been first attempts to go beyond the two extreme limits of asymptotically large and small separations by measuring the casimir force between a sphere and a plane over a larger range of ratios of sphere radius to distance @xcite . so far , no theoretical prediction is available that can describe the electromagnetic casimir interaction between a compact object and a planar surface at all distances , including the important sphere - plate geometry . until recently , progress in understanding the geometry dependence of fluctuation forces was hampered by the lack of practical methods that are applicable at all separations . conceptually , the effect of geometry and shape is difficult to study due to the non - additivity of fluctuation forces . explicit consequences of this non - additivity and also non - monotonic changes in the force have been recently predicted for a pair of cylinders next to planar walls @xcite . this behavior has been interpreted in terms of collective charge fluctuations inside the bodies @xcite . for some decades , there has been considerable interest in the theory of casimir forces between objects with curved surfaces . two types of approaches have been pursued . attempts to compute the force explicitly in particular geometries and efforts to develop a general framework which yields the interaction in terms of characteristics of the objects like polarizability or curvature . within the second type of approach , balian and duplantier studied the electromagnetic casimir interaction between compact perfect metals in terms of a multiple reflection expansion and derived also explicit results to leading order at asymptotically large separations @xcite . for parallel and partially transmitting plates a connection to scattering theory has been established which yields the casimir interaction of the plates as a determinant of a diagonal matrix of reflection amplitudes @xcite . for non - planar , deformed plates , a general representation of the casimir energy as a functional determinant of a matrix that describes reflections at the surfaces and free propagation between them has been developed in ref . later on , an equivalent representation has been applied to perturbative computations in the case of rough and corrugated plates with finite conductivity @xcite . functional determinant formulas have been used also for open geometries that do not fall into the class of parallel plates with deformations . for the electromagnetic casimir interaction between a planar plate and infinitely long cylinders , a partial wave expansion of the functional determinant has been developed @xcite . the same results have been reobtained and used to compute corrections to the pfa for the cylinder - plate geometry in ref . kenneth and klich identified the inverted green s function in the functional determinant as a t - matrix and derived a formal result for the casimir interaction in terms of this matrix in the case of scalar fields in a medium with a space and frequency dependent speed of light @xcite . recently we described a new method based on a multipole expansion of fluctuating charges that makes possible accurate and efficient calculations of casimir forces and torques between any number of compact objects @xcite . the method applies to electromagnetic fields and dielectrics as well as perfect conductors . it also applies to other fields , such as scalar and dirac , and to any boundary conditions . in this approach , the casimir energy is given in terms of the fluctuating field s scattering amplitudes from the individual objects , which encode the effects of the shape and boundary conditions . an equivalent partial wave expansion has been applied to a scalar model for dielectrics @xcite . casimir interactions due to scalar field fluctuations serve as a simplified model for the full electromagnetic interaction . this model is usually easier to analyze and provides an important tool in developing conceptually new approaches and in estimating geometry dependencies . for the sphere - plate geometry with dirichlet boundary conditions bulgac _ _ obtained the casimir interaction over a wide range of separations from a modified krein trace formula @xcite . most notably in the context of scalar fields , a versatile numerical world - line algorithm based on monte carlo methods has been developed and applied to a number of interesting geometries , including the here studied sphere - plate interaction @xcite . in this work , we extend our approach developed in refs . @xcite to describe the interaction of compact objects in the presence of a plane mirror . our general result holds for scalar and electromagnetic fields . in the electromagnetic case the mirror is assumed to be perfectly conducting but the compact objects can be dielectrics or perfect conductors . the derivation of the general result for the casimir interaction , see eqs . and , is given in section [ sec : derivatiobn ] by combining a functional integral approach and the method of images . in section [ sec : plate - sphere ] we apply our approach to compute the interaction of a sphere with a plane mirror over a wide range of separations for the scalar dirichlet and neumann problem and for the electromagnetic field that is most relevant to applications . we provide a large distance expansion of the interaction , generalizing the casimir - polder potential to include higher order multipole polarizabilities . for small separations , we compute the two leading correction terms to the pfa . in this section we first review the functional integral formulation of casimir interactions between compact objects for a scalar field and the electromagnetic field . the fluctuating field is integrated out in order to obtain an effective action for the fluctuating sources on the objects . then we show that the interaction of compact objects with an infinite plane mirror can be described by the equivalent problem of the interaction of compact objects with their mirror images in otherwise empty space , i.e. , without the plane mirror . this equivalence holds for dirichlet or neumann boundary conditions ( at the mirror and the objects ) for a scalar field and for a perfectly conducting mirror plane and arbitrary dielectric objects in the case of electromagnetic fluctuations . we first consider a real quantum field @xmath0 in order to introduce the method of images in the path integral formulation of casimir interactions . we assume that the space is divided into two half spaces by a mirror plane at @xmath1 . @xmath2 fixed closed surfaces @xmath3 , @xmath4 , are located in the right half space ( @xmath5 ) . these surfaces are regarded as the boundaries of objects on which either dirichlet or neumann boundary conditions are imposed . the action of the unconstrained field in minkowski space is @xmath6 = \frac{1}{2}\int dt \int _ > d{{\bf x}}\left\ { \frac{1}{c^2 } ( \partial_t \phi)^2 - ( \nabla\phi)^2\right\ } \ , .\ ] ] here the @xmath7-integration runs over the right half space , indicated by @xmath8 . the free energy @xmath9 of the constraint field @xmath0 at inverse temperature @xmath10 is represented by the euclidean functional integral @xmath11_{{\mathcal c}}\exp\left ( -s_e[\phi]/\hbar \right ) \ , , \ ] ] where the euclidean action @xmath12 follows from eq . after the wick rotation @xmath13 , @xmath14 with @xmath15 . the functional integral is over all fields that are periodic in the @xmath16-interval from @xmath17 to @xmath18 and that obey the boundary conditions at the surfaces , indicated by the subscript @xmath19 . the surfaces are fixed and the boundary conditions are time independent . hence each fourier component of the field with respect to @xmath20 obeys the constraints at the surfaces separately . to make use of this property , we expand @xmath21 as @xmath22 with matsubara frequencies @xmath23 and @xmath24 . the functional integral splits into independent functional integrals over the @xmath25 and the logarithm of @xmath26 can be written as the sum @xmath27_{{\mathcal c}}\exp \left[-\frac{\lambda}{2\hbar c } \int _ x}}\left\ { \kappa_n^2 |\phi_n|^2 + |\nabla \phi_n|^2 \right\}\right ] \nonumber\ , .\end{aligned}\ ] ] in the following we are interested in the limit of zero temperature . then @xmath28 , and the sum over @xmath29 can be replaced by the integral @xmath30 and @xmath25 is replaced by @xmath31 . combining positive and negative @xmath32 , we get @xmath33 with @xmath34_{{\mathcal c}}[{{\mathcal d}}\phi^*({{\bf x}},\kappa)]_{{\mathcal c}}\exp \left[-\frac{\lambda}{\hbar c } \int _ x}}\left\ { \kappa^2 |\phi|^2 + |\nabla \phi|^2 \right\}\right ] \ , .\ ] ] the ground state energy is obtained from @xmath26 as @xmath35 the casimir energy is the difference between the ground state energy of the surfaces at their actual distance and the sum of the ground state energies of the separate surfaces which is obtained by removing the surfaces to infinite separation , @xmath36 where @xmath37 is the partition function for the surfaces with infinite separation . in the following we suppress the label @xmath32 . next , the constraints at the objects are implemented by functional @xmath38-functions @xcite . for dirichlet boundary conditions , @xmath39 , on the surfaces @xmath3 , the constraint functional integral can be expressed in terms of an unconstrained integral by using @xmath40_{{\mathcal c}}[{{\mathcal d}}\phi^*]_{{\mathcal c}}= \\ & & \!\!\!\!\!\int \!\!{{\mathcal d}}\phi { { \mathcal d}}\phi^ * \prod_{\alpha=1}^n \int \!\!{{\mathcal d}}\varrho_\alpha { { \mathcal d}}\varrho^*_\alpha \exp\left [ i\!\!\int_{\sigma_\alpha}\!\!\!\!\ ! d{{\bf x}}\,\left\{\varrho_\alpha^*({{\bf x } } ) \phi({{\bf x } } ) + { { \mbox{c.c.\,}}}\right\ } \right ] , \nonumber\end{aligned}\ ] ] where the @xmath38-functions at each position of the surfaces have been written as an integral over a source field @xmath41 that is non - zero on the surfaces @xmath3 only . when we use this representation of the constraints in eq . , the now unconstrained functional integral over @xmath42 is gaussian and yields @xmath43 \nonumber\\ & \equiv & { \mathcal z}_0 \prod_{\alpha=1}^n\int { { \mathcal d}}\varrho_\alpha{{\mathcal d}}\varrho^*_\alpha e^{-(\hbar c/\lambda ) \tilde s[\varrho]}\ , , \end{aligned}\ ] ] where @xmath44 is the partition function of the field in the right half space without the surfaces @xmath3 . for neumann boundary conditions at the surfaces @xmath3 the field @xmath42 in the exponential of eq . is replaced by @xmath45 and @xmath46 in eq . is replaced by @xmath47 , where @xmath48 is the normal derivative pointing out of the objects . here @xmath46 is the free green s function in the right half space which is given by @xmath49 where @xmath50 is the mirror image of @xmath51 and the @xmath52(@xmath53 ) sign applies to dirichlet ( neumann ) boundary conditions at the mirror plane at @xmath1 . the green s function of free unbounded space is given by @xmath54 using eq . the action @xmath55 $ ] defined by eq . can be expressed in terms of the original sources @xmath41 and their mirror sources @xmath56 , @xmath57 = \frac{1}{2 } \sum_{\alpha\beta } \left\ { \int_{\sigma_\alpha } d{{\bf x}}\int_{\sigma_\beta } d{{\bf x } } ' \varrho^*_\alpha({{\bf x } } ) g_0({{\bf x}},{{\bf x}}',\kappa ) \varrho_\beta({{\bf x } } ' ) \right . \nonumber\\ & + & \!\!\ ! \left . \int_{\sigma_\alpha } d{{\bf x}}\int_{\sigma^r_\beta } d{{\bf x } } ' \varrho^*_\alpha({{\bf x } } ) g_0({{\bf x}},{{\bf x}}',\kappa ) \varrho^r_\beta({{\bf x } } ' ) + { { \mbox{c.c.\,}}}\right\ } \end{aligned}\ ] ] for dirichlet boundary conditions at the surfaces @xmath3 and with @xmath58 replaced by @xmath59 for neumann boundary conditions at the surfaces @xmath3 . here we have introduced the mirror sources @xmath60 where the @xmath52 ( @xmath53 ) sign applies to a dirichlet ( neumann ) mirror . the mirror sources are located on the mirror surfaces @xmath61 that are obtained from the @xmath3 by @xmath62 for all surface positions , see fig . [ fig : objects ] . the first term of the action of eq . describes the interaction of the surface sources in the absence of the mirror plane . the second term couples each surface source to all mirror sources . since the mirror problem is now described by an action in _ free _ space with sources and mirror sources , we can apply the concepts of the previously developed approach for casimir interactions between compact objects in unbounded space @xcite . below , we provide an explicit derivation for the case of dirichlet boundary conditions at the surfaces @xmath3 but we shall also indicate how the derivation has to be modified for neumann boundary conditions . ( objects @xmath63 ) in the right half space ( @xmath5 ) and mirror surfaces @xmath61 ( objects @xmath64 ) . @xmath65 , @xmath66 are local coordinate vectors measured from the object s origins , @xmath67 is the center - to - mirror distance of surface @xmath68 ( object @xmath69 ) . ] the action of eq . is composed of two qualitatively different terms that we will now consider separately . firstly , there are terms that couple sources on different surfaces ( @xmath70 ) where we use the term `` surface '' in the following for the original and the mirror surfaces . we shall call these terms _ off - diagonal_. as _ diagonal _ terms we shall denote those which couple sources on the same surface ( @xmath71 possible ) . both type of terms can be expressed in terms of the multipole moments of the sources . _ off - diagonal terms _ these terms couple sources on different objects , @xmath72 for @xmath73 and the original sources to the mirror sources , @xmath74 for _ all _ @xmath75 , @xmath10 . here we have introduced local coordinates @xmath65 that are measured relative to an arbitrarily chosen origin that is located inside the surface @xmath75 , see fig . [ fig : objects ] . we have also defined the fields @xmath76 which are the classical fields generated by the sources . for neumann boundary conditions on the surfaces @xmath3 , @xmath77 and @xmath78 in eqs . and have to be replaced by @xmath79 and @xmath80 , respectively . also , @xmath58 in eq . has to be replaced by @xmath81 . since we can assume that every position on @xmath3 is outside a sphere enclosing @xmath68 or @xmath82 , we can use the partial wave expansion of @xmath58 for @xmath83 when we consider eq . with coordinates @xmath84 relative to the origin inside surface @xmath68 , the field that is generated by the source @xmath85 can be written as @xmath86 where we have defined the multipole moments of the source @xmath85 as @xmath87 the field of the mirror sources can be expressed in the same form , @xmath88 where @xmath89 denotes the local coordinates that are measured relative to the origin inside surface @xmath90 . the multipole moments of the mirror source are given by @xmath91 due to eq . and for neumann boundary conditions at the surfaces @xmath3 the above expressions remain valid if the multipole moments @xmath93 are replaced by neumann multipoles which have the form of eq . but with @xmath94 replaced by @xmath95 $ ] . in order to express the action of eq . in terms of multipole moments , we have to write the field generated by the surface source @xmath85 as a function of the local coordinate @xmath65 that is regular at the origin inside @xmath3 . this can be done using translation matrices @xmath96 which relate outgoing ( @xmath97 ) and regular ( @xmath98 ) spherical bessel functions , @xmath99 the matrix elements @xmath100 of @xmath96 depend on the vector @xmath101 from the origin inside @xmath3 to the origin inside @xmath68 , see fig . [ fig : objects ] , and are given by @xcite @xmath102 where we have assumed that the cartesian coordinate frames associated with the two origins have identical orientation , i.e. , they are related by a translation . the summation over @xmath103 involves only a finite number of terms since the 3-@xmath104 symbols vanish for @xmath105 and @xmath106 . using the translation formula of eq . , the field generated by the source on surface @xmath68 , given by eq . , can be written as function of the coordinate @xmath65 as @xmath107 where we have introduced @xmath108 . similarly we obtain for the field generated by the mirror sources of eq . , now expressed as function of @xmath65 , @xmath109 where we defined @xmath110 and used eq . . notice that the latter formula applies also to the case @xmath111 which describes the translation between the surface @xmath3 and its mirror image so that the argument of the translation matrix becomes @xmath112 where @xmath113 is the normal distance between the origin of surface @xmath3 and the mirror plane . for this case of translations along the @xmath114-direction the translation matrix simplifies to @xcite @xmath115 when we substitute the result for the fields of eqs . , into eqs . , , we obtain the action in terms of the original multipole moments , @xmath116 which applies to dirichlet as well as neumann boundary conditions at the surfaces @xmath3 . to simplify notation , we have defined the modified translation matrix elements @xmath117,\ , ( \alpha\neq\beta ) \nonumber\\\end{aligned}\ ] ] here we have used the definition of eq . and @xmath118^*=(-1)^lj_l(i\kappa r)$ ] . in addition , we applied the symmetry relations @xmath119 and @xmath120 which follow from eqs . , , symmetry properties of 3-j symbols and spherical harmonics , and @xmath121 where @xmath122 is the ( real valued ) modified bessel function of second kind . notice that the actions of eqs . , couple sources on different surfaces and hide the dependence on the particular boundary conditions and shape of the surfaces . _ diagonal terms _ these are the self - action terms @xmath123 in eq . , which we have expressed here in terms of the classical field defined in eq . for neumann boundary conditions on the surface @xmath3 we have to replace again @xmath124 by @xmath125 in eq . . here and in the following we only use the coordinate system associated with the origin inside @xmath3 , and hence drop the label @xmath75 on the coordinates . the classical field generated by the source @xmath41 on surface @xmath3 obeys the helmholtz equation @xmath126 for positions @xmath7 that are located on the surface @xmath3 the _ total _ field @xmath42 generated by all sources must obey the same helmholtz equation . the part of the total field that is generated by sources other than @xmath41 can be regarded as incident field @xmath127 at the surface @xmath3 , which obeys in region around @xmath3 , that is free of sources other than @xmath41 , the homogeneous helmholtz equation . hence the total field can be written as @xmath128 for all @xmath7 located on @xmath3 . for neumann boundary conditions at @xmath3 the green s function in eq . is replaced again by @xmath129 . we would like to evaluate the action of eq . in terms of multipole moments . hence , we must consider field configurations with a fixed source on surface @xmath3 that is characterized by its multipole moments . this implies that we have to find the incident field @xmath127 that induces a prescribed set of multipoles @xmath130 on @xmath3 . the multipole moments can be identified as the amplitudes of the scattered field which is given by @xmath124 with @xmath7 located outside of the surface @xmath3 . using the partial wave expansion of eq . , we get @xmath131 from scattering theory we know that the amplitudes of the scattered field are related to the amplitudes of the regular incident field by the transition matrix @xmath132 which is related to the scattering matrix @xmath133 by @xmath134 . if we expand the incident field as @xmath135 the amplitudes of the scattered field are given by @xmath136 where the @xmath137 denote the matrix elements of the @xmath132-matrix of the surface @xmath3 . hence , the amplitudes of the incident field have to be given by @xmath138^{-1}_{lml'm ' } q_{\alpha , l'm ' } \ , .\ ] ] for dirichlet boundary conditions , the total field @xmath42 of eq . has to vanish on @xmath3 so that @xmath139 on the surface . hence , using eqs . and , the action of eq . can be expressed in terms of the multipole moments of @xmath41 , @xmath140^{-1}_{lml'm ' } \,q_{\alpha , l'm'}\ ] ] with the modified @xmath141-matrix defined by @xmath142 here we have used the definition of the multipoles in eq . to integrate over the surface and applied the relation @xmath143 . for neumann boundary conditions on @xmath3 , the normal derivative of the total field of eq . has to vanish on @xmath3 so that @xmath144 on the surface . when we use the definition of the multipole moments for neumann boundary conditions , we obtain again eq . but with the matrix @xmath145 for neumann boundary conditions . now we can combine all results for off - diagonal and diagonal terms and express the total action of eq . in terms of the original multipole moments . since we have @xmath146 we obtain from eqs . , and the total action of the multipole moments @xmath147 = \sum_{\alpha\beta } q^*_\alpha { \mathbb m}^{\alpha\beta } q_\beta \ , , \ ] ] where we have suppressed the sum over the indices @xmath148 , @xmath149 , @xmath150 , @xmath151 and defined the matrix @xmath152^{-1 } \delta_{\alpha\beta } - \tilde { \mathbb u}^{\alpha\beta } ( 1-\delta_{\alpha\beta } ) \pm \tilde { \mathbb u}^{r,\alpha\beta } \right\}\ ] ] for dirichlet ( @xmath53 ) or neumann ( @xmath52 ) boundary conditions at the mirror plane . the partition function of eq . is then obtained by integrating over all multipole moments , @xmath153 \ , .\ ] ] the gaussian integral over the multipoles is proportional to the inverse determinant of @xmath154 . finally , we substitute into eq . to obtain the casimir energy , @xmath155 where the determinant is taken with respect to the partial wave indices @xmath148 , @xmath149 and the surface indices @xmath75 , @xmath10 . the matrix @xmath156 is the result of moving the surfaces to infinite separation , where the translation matrices @xmath157 , @xmath158 vanish so that @xmath159^{-1 } \delta_{\alpha\beta } \ , .\ ] ] in the special case of one compact surface in front of the mirror plane eq . simplifies to @xmath160 for dirichlet ( + ) or neumann ( - ) boundary conditions at the mirror plane . this expression applies to dirichlet , neumann and even more general boundary conditions at the compact surface which enter only via the matrix @xmath161 . notice that this result depends on the original matrix elements ( without tilde ) since the phase factors of eqs . and drop out when taking the matrix product of @xmath162 and @xmath163 . this general result shows that the casimir interaction between a mirror and an object with arbitrary shape and boundary condition can be obtained from the transition matrix @xmath161 of the object and the translation matrix @xmath164 that describes the ( classical ) interaction between the induced source and its mirror image . the derivation of the casimir energy for a scalar field can be extended to electromagnetic field fluctuations in the presence of dielectric objects @xcite . the result will have the form of an effective action for electric and magnetic multipoles of the current densities @xmath165 inside the objects . we consider again @xmath2 objects that are located in the right half space that is bounded by a perfectly conducting plane at @xmath1 . at his plane the tangential electric field and the normal magnetic field vanish , @xmath166 , @xmath167 . the material objects are assumed to be dielectrics that are characterized by a frequency dependent dielectric function @xmath168 and permeability function @xmath169 . the partition function can be factorized again into a product of factors @xmath170 at a fixed wick rotated frequency @xmath32 . hence , in the following we consider all expressions at fixed @xmath32 and suppress the label @xmath32 . the euclidean action for the electromagnetic field in the presence of macroscopic media without external sources can be expressed as @xmath171\ ] ] in terms of the macroscopic fields @xmath172 , @xmath173 . here the _ energy _ density of the field is integrated since under a wick rotation to imaginary time the lagrangian in real time is generally transformed to the hamiltonian in imaginary time . in this description the induced ( bound ) currents inside the material objects have been absorbed into the definition of the macroscopic fields . the partition function for this action is given by a functional integral over the vector potential @xmath174 and the scalar potential @xmath175 ( after introducing a faddeev - popov gauge fixing term ) where the fields are expressed in terms of the potentials as @xmath176 an alternative description in terms of the fields @xmath177 , @xmath178 only is obtained if the bound charges ( @xmath179 ) and currents ( @xmath180 ) density inside the objects are not substituted by the macroscopic fields but considered explicitly . then the wick rotated action can be written in terms of the potentials as @xmath181 & = & \int _ d{{\bf x}}\bigg [ \kappa^2 ( |{\mathbf a}|^2+|\phi|^2 ) + \sum_{j=1}^3 |\nabla a_j|^2 + |\nabla\phi|^2 \nonumber\\ & + & \left({\mathbf a}{\mathbf j}^ * + \phi \rho^ * + { { \mbox{c.c.\,}}}\right)\bigg ] \ , , \end{aligned}\ ] ] where we have chosen the feynman gauge . the partition function is obtained by integrating over both potentials @xmath174 , @xmath175 and sources @xmath180 , @xmath179 . however , in the latter integration the currents and charges must be weighted according to the energy cost for inducing them on the objects . this energy cost must depend on shape and material of the objects . we will see below that this can be achieved by rewriting the self - energies of the separate objects in terms of the incident field that generates the polarizations and magnetizations which give rise to the induced current . hence the action of eq . is independent of material and shape of the objects and these properties enter the partition function through proper weights on the currents that measure the susceptibility of the objects to current fluctuations . we proceed by integrating out the unconstrained fluctuations of the potentials @xmath174 , @xmath175 in the action of eq . . this integration yields the partition function as a _ weighted _ functional integral over sources which we indicate at this stage by a subscript @xmath182 on the integration variable . when we denote the current density in the interior @xmath63 of object @xmath75 by @xmath165 , we get @xmath183_w [ { { \mathcal d}}{\mathbf j}^*_\alpha]_w \exp\left[-\frac{1}{2}\sum_{\alpha\beta } \int_{d_\alpha } \!\!d{{\bf x}}\int_{d_\beta } \!\!d{{\bf x } } ' \left\ { { \mathbf j}_\alpha^*({{\bf x } } ) { { \mathcal g}}_>({{\bf x}},{{\bf x}}',\kappa ) { \mathbf j}_\beta({{\bf x } } ' ) + { { \mbox{c.c.\,}}}\right\}\right ] \nonumber\\ & = & { \mathcal z}_0 \prod_{\alpha=1}^n \int [ { { \mathcal d}}{\mathbf j}_\alpha]_w [ { { \mathcal d}}{\mathbf j}^*_\alpha]_w\ , e^{-\tilde s[{\mathbf j } ] } \ , , \end{aligned}\ ] ] where we have used the continuity equation @xmath184 to eliminate the charge density by introducing the tensor green s function for the half space @xmath185 \ , , \ ] ] where @xmath58 is the free scalar green s function of eq . , @xmath186 is the identity matrix and @xmath187 is defined below eq . . the action of eq . can be expressed in terms of the original current densities @xmath165 and the mirror current densities . their components parallel and perpendicular to the mirror plane are @xmath188 the action then reads @xmath189 & = & \frac{1}{2 } \sum_{\alpha\beta } \left\ { \int_{d_\alpha } d{{\bf x}}\int_{d_\beta } d{{\bf x } } ' { \mathbf j}^*_\alpha({{\bf x } } ) { { \mathcal g}}_0({{\bf x}},{{\bf x}}',\kappa ) { \mathbf j}_\beta({{\bf x } } ' ) \right . \nonumber\\ & + & \left . \int_{d_\alpha } \!\!d{{\bf x}}\int_{d^r_\beta } \!\!d{{\bf x } } ' { \mathbf j}^*_\alpha({{\bf x } } ) { { \mathcal g}}_0({{\bf x}},{{\bf x}}',\kappa ) { \mathbf j}^r_\beta({{\bf x } } ' ) + { { \mbox{c.c.\,}}}\right\ } \ , .\nonumber\\\end{aligned}\ ] ] now the sources are coupled by the free , infinite space green s function @xmath190 the mirror currents are located on the mirror objects @xmath64 that are obtained from the original objects by reflection at the mirror plane . this action has the same structure as in the case of scalar fields . it can be expressed in terms of multipole moments of the current densities very similarly to the scalar case . again , we consider diagonal and off - diagonal terms separately . _ off - diagonal terms _ we introduce again local coordinates @xmath65 that are measured relative to an origin inside @xmath63 . the terms that couple the original sources on different objects can be written as @xmath191 for @xmath73 and the terms involving mirror sources become @xmath192 for all @xmath75 , @xmath10 . here we have introduced the electric fields @xmath193 which are generated by the current densities . for positions @xmath84 that are located outside a sphere that encloses the object @xmath69 , the electric field can be expressed in terms of the electric and magnetic multipole moments of the current density @xmath194 for @xmath195 , @xmath196 , @xmath197 where @xmath198 , @xmath199 are the regular , divergence - less solutions @xmath200 \\ { \mathbf n}_{lm}^{j}({{\bf x } } ) & = & \frac{1}{\lambda}\frac{1}{i\kappa } \nabla\times \nabla\times [ { { \bf x}}j_l(i\kappa r)y_{lm}(\hat{{\bf x}})]\end{aligned}\ ] ] of the vector helmholtz equation and @xmath201 . the electric fields of the original sources can then be written as @xmath202 \ , , \ ] ] which is an expansion in the outgoing solutions @xmath203 \\ { \mathbf n}_{lm}^{h}({{\bf x } } ) \!\!&= & \!\!\frac{1}{\lambda}\frac{1}{i\kappa } \nabla\times \nabla\times [ { { \bf x}}h_l^{(1)}(i\kappa r)y_{lm}(\hat{{\bf x}})]\end{aligned}\ ] ] of the vector helmholtz equation . the electric fields of the mirror sources can be also expressed in terms of the multipole moments of the original sources , @xmath204 \ , , \ ] ] where @xmath205 denotes the local coordinates of the mirror object @xmath206 . using eq . , the definition of the vector solutions of eq . and @xmath207 , the moments of the mirror currents are given by @xmath208 the expansion of the electric fields into outgoing vector waves with respect to the origin of the object that generates the field does not allow us to perform the integrations in eqs . , . we would like to expand the electric field generated by object @xmath69 in terms of vector waves that are regular at the origin of object @xmath63 . when the coordinate systems associated with the two objects have identical orientation , this can be done by relating the outgoing and regular vector waves by a translation matrix @xmath209 . generalizing the result for scalar fields , the translation matrix couples both types of vector solutions , @xmath210 with @xmath211 . the matrix elements are given by @xcite , @xmath212 \sqrt{\frac{(2l+1)(2l'+1)(2l''+1)}{l(l+1)l'(l'+1 ) } } \nonumber\\ & & \times\begin{pmatrix}l&l'&l''\\0&0&0\end{pmatrix } \begin{pmatrix}l&l'&l''\\m&-m'&m'-m\end{pmatrix } h^{(1)}_{l''}(i\kappa|{{\bf x}}_{\alpha\beta}| ) y_{l''m - m'}(\hat{{\bf x}}_{\alpha\beta } ) \ , , \\ \label{eq : em - trans - mat - elements - gen - c } c_{l'm'lm}({{\bf x}}_{\alpha\beta } ) & = & \frac{\kappa}{\sqrt{l(l+1)l'(l'+1 ) } } { { \bf x}}_{\alpha\beta } \cdot \bigg [ \hat{{\bf x}}\frac{1}{2 } \left(\lambda^+_{lm } a_{l'm'lm+1}({{\bf x}}_{\alpha\beta } ) + \lambda^-_{lm } a_{l'm'lm-1}({{\bf x}}_{\alpha\beta } ) \right)\nonumber\\ & & \hspace*{3.6cm}+\ , \hat{{\bf y}}\frac{1}{2i } \left(\lambda^+_{lm } a_{l'm'lm+1}({{\bf x}}_{\alpha\beta } ) - \lambda^-_{lm } a_{l'm'lm-1}({{\bf x}}_{\alpha\beta } ) \right)\nonumber\\ & & \hspace*{3.6cm}+\ , \hat{{\bf z}}\ , m \,a_{l'm'lm}({{\bf x}}_{\alpha\beta } ) \bigg]\end{aligned}\ ] ] with @xmath213 and @xmath214 . it is useful to combine the translation matrix elements to the matrix @xmath215 that acts on the vector @xmath216 of multipole moments . using the translation formula of eqs . , , we can express the field @xmath217 in eq . in terms of regular vector solutions and perform the integration . this integration yields the multipole moments defined by eqs . , . the action of eq . can then be written as @xmath218 where we have defined the modified translation matrix @xmath219 \ , , \ ] ] where @xmath220 denotes the conjugate transpose of the matrix of eq . . for translations along the z - axis , @xmath221 , the latter expression is diagonal in @xmath149 and simplifies to @xmath222 due to the symmetries @xmath223 and @xmath224 . the action of eq . that couples the original and mirror sources can be cast into a form similar to eq . with a translation matrix @xmath225 that is defined by @xmath226 where the phase factors relating the original and mirror multipole moments in eqs . , have been absorbed in the translation matrix . with this definition eq . becomes @xmath227 with the modified translation matrix @xmath228 \ , .\ ] ] for @xmath111 , the action of eq . describes the interaction between a source and its mirror image . in this case the translation vector is @xmath229 where @xmath113 is the normal distance between the origin of object @xmath63 and the mirror plane . hence , the translation is along the z - axis and the translation matrix elements of eqs . , simplify to @xmath230 \sqrt{\frac{(2l+1)(2l'+1)}{l(l+1)l'(l'+1 ) } } ( 2l''+1)\nonumber\\ & & \times\begin{pmatrix}l&l'&l''\\0&0&0\end{pmatrix } \begin{pmatrix}l&l'&l''\\m&-m&0\end{pmatrix } h^{(1)}_{l''}(i\kappa 2l_\alpha ) \ , , \\ \label{eq : em - trans - mat - elements - z - c } c_{l'm'lm}(-2l_\alpha\hat{{\bf z } } ) & = & -\delta_{m'm } \kappa 2l_\alpha m ( -1)^m i^{l - l ' } \sum_{l '' } i^{-l '' } \sqrt{\frac{(2l+1)(2l'+1)}{l(l+1)l'(l'+1 ) } } ( 2l''+1)\nonumber\\ & & \times\begin{pmatrix}l&l'&l''\\0&0&0\end{pmatrix } \begin{pmatrix}l&l'&l''\\m&-m&0\end{pmatrix } h^{(1)}_{l''}(i\kappa 2l_\alpha ) \ , .\end{aligned}\ ] ] these matrix elements obey the symmetry relations @xmath231 , @xmath232 so that the matrix of eq . simplifies for @xmath111 to @xmath233 . _ diagonal terms _ so far we have described the interaction between _ arbitrary _ multipoles inside the material objects . but the functional integral over currents of eq . contains weights that measure the energy cost for inducing currents on an object with particular shape and material composition . in the following we will show that these weights can be implemented straightforwardly when we express the diagonal terms of the action of eq . , @xmath234 in terms of the incident field which generates a prescribed current that corresponds to the induced polarization and magnetization inside the object . then the relation between the induced current and the incident field ensures that the currents are weighted properly . to see this we employ the macroscopic formulation of the electromagnetic action , see eq . . the energy of eq . is associated with the process of building up the induced current @xmath165 inside the object . this energy must therefore be equal to the change in the total macroscopic field energy that results when the object is placed into an external field . thus we obtain @xmath235 \ , , \ ] ] where integration extends over all space . field vectors with subscript @xmath17 represent the incident field that is generated by some fixed external sources in otherwise empty space and field vectors without label stand for the total field from the external and induced sources after adding the object . @xmath236 can be also written as @xmath237 \nonumber\\ & + & \frac{1}{2 } \int \!d{{\bf x}}\left [ ( { \mathbf e}+{\mathbf e}_0 ) ( { \mathbf d}^*-{\mathbf d}^*_0 ) \right . \nonumber\\ & & \quad \quad\,\,\,\left . + ( { \mathbf b}+{\mathbf b}_0 ) ( { \mathbf h}^ * -{\mathbf h}_0^ * ) + { { \mbox{c.c.\,}}}\right ] \ , . \end{aligned}\ ] ] the second integral of this expression vanishes . this can be seen by setting @xmath238 , @xmath239 and using that @xmath240 , @xmath241 since the external sources @xmath242 , @xmath243 are fixed when the object is added . notice that the bound ( induced ) sources do not appear explicitly since they are included in the macroscopic fields . since @xmath244 , @xmath245 and @xmath246 , @xmath247 with @xmath248 , @xmath249 inside the objects and @xmath250 , @xmath251 outside the objects , we get @xmath252 \ , , \ ] ] where integration runs only over the interior of the object . the material dependent functions @xmath253 and @xmath254 can be expressed in terms of the polarization @xmath255 and magnetization @xmath256 of the object . the relations between macroscopic fields yield @xmath257 and @xmath258 . when we substitute @xmath259 in eq . ( [ eq : em - mac - action-2 ] ) we can integrate by parts to obtain @xmath260 \ , , \ ] ] where we have combined the polarization and magnetization to yield the induced current density @xmath261 notice that the incident field @xmath262 in eq . ( [ eq : em - mac - action - final ] ) depends on the current density @xmath165 since @xmath262 has to induce the prescribed current density . hence , the problem of expressing the diagonal part of the action in terms of multipole moments has been reduced to computing the incident field that has to act on the object to generate a given current density . in scattering theory one usually encounters the opposite problem . for an incident field one would like to compute the scattered field which can be expanded in outgoing partial vector waves , see eq . ( [ eq : e - of - mps - orig ] ) . here the situation is slightly different . we seek to determine the incident field that generates a given set of multipole moments inside the object . in other words , for a given scattered field , which according to eq . ( [ eq : e - of - mps - orig ] ) is given by the multipole moments , we would like to obtain the corresponding incident field . we expand the incident field as @xmath263 \ , .\ ] ] the relation between the multipole moments and the amplitudes of the incident field is determined by the t - matrix @xmath264 , where @xmath265 is the scattering matrix of the object . when we solve this relation for the incident field amplitudes @xmath266 , we get @xmath267^{-1}_{lml'm ' } { \mathbf q}^\alpha_{l'm ' } \ , , \ ] ] where @xmath268 is a @xmath269 matrix acting on magnetic and electric multipoles . this relation together with eq . ( [ eq : e - inci - expanded ] ) yields the incident field that generates the multipoles @xmath270 . with this result we can perform the integration in eq . ( [ eq : em - mac - action - final ] ) which yields @xmath236 in terms of the multipoles , @xmath271^{-1}_{lml'm ' } + \left [ [ \tilde { { \mathcal t}}^\alpha ] ^{-1}\right]^\dagger_{l'm'lm } \right\ } { \mathbf q}^{\alpha}_{l'm'}\ ] ] with the modified t - matrix @xmath272 -t^{\alpha,\textsc{em}}_{lml'm'}&t^{\alpha,\textsc{ee}}_{lml'm ' } \end{pmatrix } \ , , \ ] ] where the @xmath273 are the elements of the matrix @xmath268 with @xmath274 , @xmath275 , @xmath276 labeling magnetic and electric elements , respectively . due to the symmetry @xmath277 t^{\alpha,\textsc{em}}_{lml'm'}&t^{\alpha,\textsc{ee}}_{lml'm ' } \end{pmatrix}^ * \!\!=(-1)^{l+l ' } \begin{pmatrix } t^{\alpha,\textsc{mm}}_{l'm'lm}&-t^{\alpha,\textsc{em}}_{l'm'lm}\\[1em ] -t^{\alpha,\textsc{me}}_{l'm'lm}&t^{\alpha,\textsc{ee}}_{l'm'lm } \end{pmatrix}\ ] ] of the t - matrix eq . ( [ eq : em - diag - action - of - mps ] ) can be simplified to @xmath278^{-1}_{lml'm ' } { \mathbf q}^{\alpha}_{l'm ' } \ , .\ ] ] the total action @xmath279 $ ] for the multipole moments is given by the sum of the actions of eqs . ( [ eq : em - off - dia - mps ] ) , ( [ eq : em - off - diag - terms - mirror - mps ] ) and ( [ eq : em - diag - action - of - mps - simp ] ) . in compact matrix notation it can be written as @xmath280 = \sum_{\alpha\beta } { \mathbf q}^*_\alpha { \mathbb m}^{\alpha\beta } { \mathbf q}_\beta \ , , \ ] ] with the matrix @xmath281^{-1 } \delta_{\alpha\beta } - \tilde { \mathbb u}^{\alpha\beta } ( 1-\delta_{\alpha\beta } ) + \tilde { \mathbb u}^{r,\alpha\beta } \right\ } \ , , \ ] ] where @xmath282 , @xmath283 and @xmath284 stand for the matrices with matrix elements given by eqs . ( [ eq : em - mod - t ] ) , ( [ eq : em - u - mat - mod ] ) and ( [ eq : em - u - mat - mod - mirror ] ) , respectively . in analogy to the scalar case , eq . ( [ eq : z - og - kappa - final ] ) , the partition function @xmath170 is obtained by integrating over all multipoles . notice that by construction of the diagonal parts of the action of eq . ( [ eq : em - total_action - final ] ) the proper weights are assigned to the multipole configurations . the gaussian integral over multipoles and eq . ( [ eq : casimir - energy ] ) lead to the final result @xmath285 for the electromagnetic casimir energy where the determinant is taken with respect to the partial wave indices @xmath148 , @xmath149 , polarization indices @xmath286 , @xmath276 and the object indices @xmath75 , @xmath10 . the matrix @xmath156 is given by eq . ( [ eq : em - def - matrix - m ] ) with the translation matrices @xmath283 and @xmath284 set to zero . for one object in front of a perfectly reflecting mirror plane , the casimir energy can be written as @xmath287 the matrix @xmath288 that describes the interaction between fluctuating currents on the object and its mirror image is universal , and the shape and material composition of the object enters through its t - matrix @xmath162 . in this section we consider a geometry that is most relevant to a large number of experimental studies of casimir interactions carried out in the last decade . this geometry , consisting of a plane mirror and a sphere , is experimentally favorable since it avoids the problem of parallelism for plane surfaces facing each other . despite its experimental importance , this geometry lacks a theoretical description of the electromagnetic casimir interaction . for a scalar field with dirichlet boundary conditions at the sphere and the plane , the interaction over a wide range of distances has been obtained recently @xcite , including an analytic expression for the lowest order correction to the proximity force approximation @xcite . we consider a sphere of radius @xmath289 in front of a plane mirror with distance @xmath290 between the center of the sphere and the mirror , see fig . [ fig : plane - sphere ] . for the scalar field we study dirichlet or neumann boundary conditions at the mirror and the sphere . for the electromagnetic field , the sphere consists of a material with arbitrary dielectric function @xmath291 and permeability @xmath292 while the mirror is assumed to be perfectly reflecting ( implying vanishing of the parallel components of the electric field and the normal component of the magnetic field ) . we start by studying the casimir interaction at large separations . for separations @xmath290 that are large compared to the size @xmath289 of the object , the casimir energy can be expressed as an asymptotic series in @xmath293 . using that @xmath294 in eqs . ( [ eq : casimir - energy-1-surface ] ) and ( [ eq : em - casimir - energy-1-surface ] ) , we get by expanding the logarithm @xmath295 where we have defined the operator @xmath296 with @xmath52(@xmath53 ) for a scalar field in the presence of a dirichlet ( neumann ) mirror and @xmath297 for the electromagnetic field and a perfectly reflecting mirror . the operator @xmath298 describes a wave that travels from the mirror to the object and back , involving one scattering at the object . all information about the shape of the surface and the boundary conditions is provided by the @xmath132-matrix . for a spherically symmetric surface the matrix is diagonal and completely specified by scattering phase shifts @xmath299 that do not depend on @xmath149 , @xmath300 where @xmath301 is the real frequency . the scattering phase shifts for a sphere of radius @xmath289 with dirichlet and neumann boundary conditions are @xmath302 where @xmath303 and @xmath98 ( @xmath304 ) are spherical bessel functions of first ( second ) kind . to make use of the general result of eq . , we have to evaluate the matrix elements for imaginary frequencies @xmath305 . this can be done using @xmath306 , @xmath307 , where @xmath308 and @xmath309 are modified bessel functions of first and second kind , respectively . we obtain @xmath310 for dirichlet and neumann boundary conditions where @xmath311 . the distance dependence of the casimir energy of eq . is given by the translation matrix that describes the interaction between the original and mirror multipole moments . since the origins ( centers ) of the original and mirror sphere are related by a translation along the @xmath312-axis , the matrix elements of the translation matrix @xmath313 are given by eqs . and . after a rotation of frequency to the imaginary axis we obtain @xmath314 with the elements of the transition and translation matrices we can compute the casimir interaction from eq . . the scaling of the @xmath315-matrix for small @xmath32 shows that partial waves of order @xmath148 start to contribute to the energy at order @xmath316 if the @xmath132-matrix is diagonal in @xmath148 . hence , we can truncate the infinite matrix @xmath298 in eq . at finite multipole order to obtain the large distance expansion of the energy at a given order in @xmath293 . also , we can truncate the series over @xmath317 in eq . since the @xmath318 power of @xmath298 becomes important only at order @xmath319 . the first @xmath317 powers of @xmath298 describe @xmath317 scattering off the sphere . we have used mathematica to perform the matrix operations and to expand the integrand of eq . in @xmath293 . from this we find that the casimir energy can be written as @xmath320 where @xmath321 is the coefficient of the term @xmath322 . in the following we give the results for a mirror plane with dirichlet or neumann boundary conditions separately . _ dirichlet mirror _ in this case we have @xmath323 . if the field obeys dirichlet boundary conditions at the sphere , we obtain the for leading coefficients @xmath324 indicating an attractive force . this result agrees with recent findings of wirzba @xcite . if we impose neumann boundary conditions at the sphere , we find @xmath325 this result corresponds to a repulsive force which is expected for unlike boundary conditions . to leading order the energy scales as @xmath326 since the first two coefficients vanish . this behavior can be understood from the absence of low - frequency @xmath327-wave scattering for a sphere with neumann boundary conditions . _ neumann mirror _ in this case we have @xmath328 . for a dirichlet sphere we obtain the coefficients @xmath329 and hence the force is again repulsive . the modulus of the coefficients are smaller than those in eq . for a dirichlet mirror with the exception of the leading one , @xmath330 and the one with inverted sign , @xmath331 . notice that the sum of the casimir energies for the dirichlet and neumann mirror , both opposite to a dirichlet sphere , is identical to the casimir energy between two dirichlet spheres at a center - to - center distance @xmath332 @xcite . the reason for this is that any field configuration can be decomposed into symmetric and antisymmetric modes with respect to the mirror plane which obey neumann and dirichlet boundary conditions on that plane , respectively . since the energy between two spheres scales as @xmath333 for asymptotically large separations , the modulus of the coefficients @xmath330 in eqs . ( [ eq : coeff_d - d ] ) and ( [ eq : coeff_n - d ] ) must be identical so that the contribution @xmath334 cancels . it is easily seen that the higher order coefficients of eqs . ( [ eq : coeff_d - d ] ) and ( [ eq : coeff_n - d ] ) combine to the correct coefficients of the large distance expansion for two dirichlet spheres @xcite . for a neumann sphere we get @xmath335 for the same reason as before , the first two coefficients vanish . notice that all given coefficients , with the exception of @xmath331 , are equal to minus the corresponding coefficients for a neumann sphere and a dirichlet mirror in eq . ( [ eq : coeff_d - n ] ) . this result is consistent with the observation that the casimir energy for two neumann spheres at a center - to - center distance @xmath332 is given by the sum of the energies of a neumann sphere at distance @xmath290 from a dirichlet mirror and a neumann mirror , respectively . since two neumann spheres interact at large distance with an energy @xmath336 with the next - to - leading term @xmath337 , the sum of the corresponding coefficients , excepting @xmath331 , must vanish . the coefficients @xmath331 in eqs . ( [ eq : coeff_d - n ] ) , ( [ eq : coeff_n - n ] ) combine to the correct value for two neumann spheres that was obtained in ref . @xcite . we consider a dielectric sphere at a center - to - surface distance @xmath290 from a perfectly conducting mirror . due to spherical symmetry , the electric and magnetic multipoles for all @xmath148 , @xmath149 are decoupled so that the t - matrix is diagonal @xcite , @xmath338 - n i_{l+{1\over 2}}(nz ) \left[i_{l+{1\over 2}}(z)+2z i'_{l+{1\over 2}}(z)\right ] } { \eta k_{l+{1\over 2}}(z ) \left[i_{l+{1\over 2}}(nz)+2nzi'_{l+{1\over 2}}(nz)\right ] - n i_{l+{1\over 2}}(nz ) \left[k_{l+{1\over 2}}(z)+2z k'_{l+{1\over 2}}(z)\right ] } \ , , \ ] ] where the sphere radius is @xmath289 , @xmath311 , @xmath339 , @xmath340 . @xmath341 is obtained from eq . ( [ eq : t - matrix - elem - sphere ] ) by interchanging @xmath342 and @xmath343 . the limit of a perfectly conducting sphere is obtained by taking @xmath344 at an arbitrarily fixed @xmath345 which can also vanish . then the matrix elements become independent of @xmath343 , @xmath346 for all partial waves , the _ leading _ low frequency contribution is determined by the _ static _ electric multipole polarizability , @xmath347r^{2l+1}$ ] , and the corresponding magnetic polarizability , @xmath348r^{2l+1}$ ] . including the next to leading terms , the t - matrix has the structure @xmath349 \ , , \nonumber\ ] ] and @xmath341 is obtained by @xmath350 , @xmath351 . the first terms are @xmath352/[5(\mu+2)^2]r^5 $ ] , @xmath353 ^ 2r^6 $ ] , and @xmath354 , @xmath355 are obtained again by interchanging @xmath343 and @xmath342 . higher order terms can be easily obtained by expanding eq . ( [ eq : t - matrix - elem - sphere ] ) for small @xmath32 . the translation matrix elements @xmath356 are obtained from eqs . ( [ eq : def - em - u - matrix - mirror ] ) , ( [ eq : em - trans - mat - elements - z - b ] ) and ( [ eq : em - trans - mat - elements - z - c ] ) . using @xmath121 we get @xmath357 \sqrt{\frac{(2l+1)(2l'+1)}{l(l+1)l'(l'+1 ) } } ( 2l''+1)\nonumber\\ & & \times\begin{pmatrix}l&l'&l''\\0&0&0\end{pmatrix } \begin{pmatrix}l&l'&l''\\m&-m&0\end{pmatrix } \frac{k_{l''+1/2}(2\kappa l)}{\sqrt{\pi \kappa l } } \ , , \\ \label{eq : em - trans - mat - elements - z - c - wk } c_{l'm'lm}(-2l\hat{{\bf z } } ) & = & \delta_{m'm } 2 \kappa l m ( -1)^m i^{l - l ' } \sum_{l '' } ( -1)^{l '' } \sqrt{\frac{(2l+1)(2l'+1)}{l(l+1)l'(l'+1 ) } } ( 2l''+1)\nonumber\\ & & \times\begin{pmatrix}l&l'&l''\\0&0&0\end{pmatrix } \begin{pmatrix}l&l'&l''\\m&-m&0\end{pmatrix } \frac{k_{l''+1/2}(2\kappa l)}{\sqrt{\pi \kappa l } } \ , .\end{aligned}\ ] ] now we can employ the series representation of the casimir energy in eq . ( [ eq : energy - scatt - exp ] ) with @xmath358 . since the t - matrix is diagonal in @xmath148 , partial waves of order @xmath148 start to contribute to the energy at order @xmath316 . also , the @xmath318 power of @xmath298 becomes important only at order @xmath359 . notice the stronger increase of the exponent with @xmath317 compared to the scalar case where the exponent is @xmath360 . this can be understood from the absence of @xmath327-waves for the electromagnetic field so that each reflection contributes a factor @xmath361 due to @xmath317-waves . matrix operations and the expansion in @xmath293 are performed with mathematica . from this we obtain for a dielectric sphere in front of perfectly conducting mirror plane the casimir energy @xmath362 \frac{1}{l^7 } + \frac{7}{7200 } \left [ 572 ( \alpha_3^\textsc{e}-\alpha_3^\textsc{m } ) + 675 \left ( 9 ( \gamma_{15}^\textsc{e } - \gamma_{15}^\textsc{m } ) -55 ( \gamma_{23}^\textsc{e } - \gamma_{23}^\textsc{m } ) \right)\right ] \frac{1}{l^8 } + \dots \right\ } \ , .\end{aligned}\ ] ] the electric contribution to the leading term , @xmath363 , was obtained by casimir and polder for the interaction of an atom with static polarizability @xmath364 and a metallic surface @xcite . later boyer has generalized the leading order result to include magnetic effects described by @xmath365 @xcite . the higher order terms are new . they show how higher order polarizabilities and frequency corrections to the static parameters influence the interaction . there is no @xmath366 term . notice also that the first three terms of the contribution at order @xmath367 have precisely the structure of the casimir - polder interaction between two atoms with static dipole polarizabilities @xmath365 and @xmath364 but it is reduced by a factor of @xmath368 . this factor and the distance dependence @xmath369 of this term suggests that it arises from the interaction of the dipole fluctuations inside the sphere with those inside its image at a distance @xmath332 . the additional coefficient of @xmath370 in the reduction factor @xmath371 can be traced back to the fact that the forces involved in bringing the dipole in from infinity act only on the dipole and not on its image @xcite . for a perfectly conducting sphere the coefficients of the expansion of the casimir energy in @xmath293 become universal numbers . these coefficients can be either obtained from eq . ( [ eq : energy - eps - mu - sphere ] ) by using the appropriate values of the parameters for a prefect metal or it can be computed directly from the t - matrix elements given in eqs . ( [ eq : t - matrix - elem - cond - sphere - m ] ) , ( [ eq : t - matrix - elem - cond - sphere - e ] ) . following the latter route , we get the series @xmath372 where the coefficients up to order @xmath373 are @xmath374 this and the corresponding results for a scalar field appear to be asymptotic series . therefore , the series can not be summed to obtain the interaction at small separations . in the next section we use a numerical implementation of eq . ( [ eq : em - casimir - energy-1-surface ] ) to compute the interaction at all separations . to obtain the casimir interaction over a broad range of distances , eq . ( [ eq : em - casimir - energy-1-surface ] ) has to be evaluated numerically . as we employ a partial wave expansion , the computational work increases with decreasing separation between the objects . however , we shall see below that even at small separations our method yields sufficient precision to obtain the leading corrections to the proximity force approximation ( pfa ) . the numerical approach is based on the technique presented in refs . @xcite . using the analytic expressions for the matrix elements of the translation and transition matrices , we compute the determinant and the integral over frequency @xmath32 in eq . ( [ eq : em - casimir - energy-1-surface ] ) numerically . the matrices are truncated at a finite multipole order @xmath148 which yields a series of estimates @xmath375 for the casimir energy . the exact casimir energy is then obtained by extrapolating the series to @xmath376 . for the geometry considered here , we observe an exponentially fast convergence that allows us to obtain accurate results even at small separations from a moderate multipole order . we compare our results for the interaction energy to the estimate that follows from pfa . this is important since pfa is often used over a range of separations although its accuracy is unknown even at small distances for most geometries , including the electromagnetic casimir interaction between a plane and a sphere . the pfa estimate for the latter geometry is given by @xmath377 where @xmath378 is the surface - to - surface distance . the amplitudes follow from the result for two parallel plates and are given by @xmath379 for a scalar field with like ( @xmath52 ) and unlike ( @xmath53 ) boundary conditions and the electromagnetic field , respectively . in the following we present our results for the plane - sphere interaction over a wide range of separations for scalar fields with dirichlet and neumann boundary conditions and for the electromagnetic field . ) . the dashed curve represents the large distance expansion of eq . ( [ eq : spheres - energy - large - l ] ) with the coefficients of eq . ( [ eq : coeff_d - d ] ) . inset : corrections to the pfa at small separations as function of the surface - to - surface distance @xmath378 . the dashed curve corresponds to the lowest order correction to the pfa obtained by bordag @xcite . ] for neumann boundary conditions at the sphere and the plane . the dashed curve represents the large distance expansion of eq . ( [ eq : spheres - energy - large - l ] ) with the coefficients of eq . ( [ eq : coeff_n - n ] ) . ] for neumann boundary conditions at the sphere and dirichlet boundary conditions at the plane . the dashed curve represents the large distance expansion of eq . ( [ eq : spheres - energy - large - l ] ) with the coefficients of eq . ( [ eq : coeff_d - n ] ) . ] for dirichlet boundary conditions at the sphere and neumann boundary conditions at the plane . the dashed curve represents the large distance expansion of eq . ( [ eq : spheres - energy - large - l ] ) with the coefficients of eq . ( [ eq : coeff_n - d ] ) . ] we consider the four cases that correspond to dirichlet or neumann boundary conditions at the sphere and the mirror plane . the results for the casimir energy are shown in figs . [ fig : dd]-[fig : dn ] . at large separations the results of the numerical evaluation of the determinant agree nicely with the large distance expansions of the previous section . for a sphere with dirichlet boundary conditions , the energy scales both at large and small separations as @xmath380 so that the curves for @xmath381 in figs . [ fig : dd ] and [ fig : dn ] tend also to a constant for @xmath382 . the pfa underestimates the actual interaction energy at all separations . for a neumann sphere the interaction scales at large separations as @xmath326 so that the curves for @xmath381 in figs . [ fig : nn ] and [ fig : nd ] tend to zero for @xmath382 . here the pfa overestimates the actual interaction energy at all separations . the non - perturbative approach allows us to study also the case of small separations . in that limit our results can be fitted to a power law of the form @xmath383 \ , .\ ] ] the coefficients @xmath384 obtained from a fit of the function of eq . ( [ eq : e - pfa+corrections ] ) to the data points for the four smallest studied separations are summarized in tab . [ tab : thetas ] . the fitted curves are shown as insets in figs . [ fig : dd]-[fig : dn ] . the result for @xmath385 for the case of dirichlet boundary conditions at the sphere and the mirror agrees with the analytical result @xmath386 presented in ref . again for dirichlet conditions , for the second order coefficient a considerably larger numerical estimate of @xmath387 has been obtained from world - line monte carlo sampling @xcite . for the other combinations of boundary conditions our findings represent the first results for the corrections to the pfa . .coefficients describing corrections to pfa , see eq . ( [ eq : e - pfa+corrections ] ) . d = dirichlet , n = neumann ( scalar field ) and em = perfectly conducting boundary conditions ( electromagnetic field ) . [ cols="^,^,^ " , ] we focus on a perfectly conducting sphere and mirror plane . the casimir energy resulting from a numerical computation of the determinant of eq . ( [ eq : em - casimir - energy-1-surface ] ) is shown for a wide range of separations in fig . [ fig : em ] . at large separations the interaction is described by the asymptotic result of eq . ( [ eq : em - energy - series ] ) . at small distances the casimir energy approaches to pfa estimate and corrections to pfa can be described again by eq . ( [ eq : e - pfa+corrections ] ) . a corresponding fit to the data points for the four smallest separations is shown as inset in fig . [ fig : em ] . the corresponding amplitudes of the correction terms are listed in tab . [ tab : thetas ] . ) . the asymptotic expansion of eq . ( [ eq : em - energy - series ] ) to order @xmath388 is shown as dashed line . inset : corrections to the pfa at small distances as function of @xmath378 . ] this result is important for a number of recent measurements of the casimir force between almost flat surfaces . these experiments have been performed for a plane mirror and a sphere with a radius that is much larger than the distance between the surfaces in order to avoid difficulties from parallelism control . our results at short distances indicate that for a sphere of radius @xmath389 m the corrections to pfa are below @xmath390 only for surface - to - surface distances @xmath391 that are smaller than @xmath392 nm . for a sphere of that size , pfa fails already by @xmath393 at a distance of @xmath394 m . from this we conclude that the currently achieved experimental accuracy for measurements of casimir forces in the sphere - plate geometry with a sphere of @xmath389 m at distances below @xmath395 m is within the range or slightly less than the corrections to pfa . however , deviations from pfa become severe when smaller objects interact with surfaces . hence it is important to understand the crossover of the casimir interaction between macroscopic objects and the eventual casimir - polder interaction between single atoms and a surface . a description of this crossover is provided by our results in fig . [ fig : em ] . corresponding results can be also obtained for a dielectric sphere or less symmetric objects by the methods presented here . finally , we note that it should be also possible to compute correction amplitudes like the @xmath384 in eq . analytically by applying methods similar to those used in ref . however , the validity range of the corresponding series would be limited to small separations and for an overall description of the interaction one should resort to the device of a numerical evaluation along the lines presented here . this work emerged from a collaboration with r. l. jaffe , m. kardar , and n. graham on related problems . support by the heisenberg program of the deutsche forschungsgemeinschaft is acknowledged . a. rodriguez , m. ibanescu , d. iannuzzi , f. capasso , j. d. joannopoulos and s. g. johnson , phys . lett . * 99 * , 080401 ( 2007 ) ; a. rodriguez , m. ibanescu , d. iannuzzi , j. d. joannopoulos , and s. g. johnson , phys . a * 76 * , 032106 ( 2007 ) .

the aim of this paper is to investigate an optimal stopping problem under partial observation for piecewise - deterministic markov processes ( pdmp ) both from the theoretical and numerical points of view . pdmp s have been introduced by davis @xcite as a general class of stochastic models . they form a family of markov processes involving deterministic motion punctuated by random jumps . the motion depends on three local characteristics , the flow @xmath1 , the jump rate @xmath2 and the transition measure @xmath3 , which selects the post - jump location . starting from the point @xmath4 , the motion of the process @xmath5 follows the flow @xmath6 until the first jump time @xmath7 , which occurs either spontaneously in a poisson - like fashion with rate @xmath8 or when the flow hits the boundary of the state space . in either case , the location of the process at @xmath7 is selected by the transition measure @xmath9 and the motion restarts from @xmath10 . we define similarly the time until the next jump and the next post - jump location and so on . one important property of a pdmp , relevant for the approach developed in this paper , is that its distribution is completely characterized by the discrete time markov chain @xmath11 where @xmath12 is the @xmath13-th post - jump location and @xmath14 is the @xmath13-th inter - jump time . a suitable choice of the state space and local characteristics provides stochastic models covering a large number of applications such as operations research @xcite , reliability @xcite , neurosciences @xcite , internet traffic @xcite , finance @xcite . this list of examples and references is of course not exhaustive . in this paper , we consider an optimal stopping problem for a partially observed pdmp @xmath5 . roughly speaking , the observation process @xmath15 is a point process defined through the embedded discrete time markov chain @xmath11 . the inter - arrival times are given by @xmath16 and the marks by a noisy function of @xmath17 . for a given reward function @xmath18 and a computation horizon @xmath19 , we study the following optimal stopping problem @xmath20,\ ] ] where @xmath21 is the @xmath22-th jump time of the pdmp @xmath5 , @xmath23 is a stopping time with respect to the natural filtration @xmath24 generated by the observations @xmath15 . in some applications , it may be more appropriate to consider a fixed optimization horizon @xmath25 rather than the random horizon @xmath21 . this is a difficult problem with few references in the literature , see for instance @xcite where the underlying process is not piecewise deterministic . regarding pdmp s , this problem could be addressed using the same ideas as in @xcite . it involves the time - augmented process @xmath26 . although this process is still a pdmp , its local characteristics may not have the same good properties as those of the original process leading to several new technical difficulties . a general methodology to solve such a problem is to split it into two sub - problems . the first one consists in deriving the filter process given by the conditional expectation of @xmath27 with respect to the observed information @xmath28 . its main objective is to transform the initial problem into a completely observed optimal stopping problem where the new state variable is the filter process . the second step consists in solving this reformulated problem , the new difficulty being its infinite dimension . indeed , the filter process takes values in a set of probability measures . our work is inspired by @xcite which deals with an optimal stopping problem under partial observation for a markov chain with finite state space . the authors study the optimal filtering and convert their original problem into a standard optimal stopping problem for a continuous state space markov chain . then they propose a discretization method based on a quantization technique to approximate the value function . however , their method can not be directly applied to our problem for the following main reasons related to the specificities of pdmps . firstly , pdmps are continuous time processes . although the dynamics can be described by the discrete - time markov chain @xmath11 , this optimization problem remains intrinsically a _ continuous - time _ optimization problem . indeed , the performance criterion is maximized over the set of stopping times defined with respect to the _ continuous - time _ filtration @xmath29 . consequently , our problem can not be converted into a fully discrete time problem . secondly , the distribution of a pdmp combines both absolutely continuous and singular components . this is due to the existence of forced jumps when the process hits the boundary of the state space . as a consequence the derivation of the filter process is not straightforward . in particular , the absolute continuity hypothesis * ( h ) * of @xcite does not hold . thirdly , in our context the reformulated optimization problem is not standard , unlike in @xcite . as already explained , this reformulated optimization problem combines _ continuous - time _ and _ discrete - time _ features . consequently , this problem does not correspond to the classical optimal stopping problem of a discrete - time markov chain . moreover , it is different from the optimal stopping problem of a pdmp under complete observation mainly because the new state variables given by the markov chain @xmath30 are not the underlying markov chain of some pdmp . therefore the results of the literature @xcite can not be used . finally , a natural way to proceed with the numerical approximation is then to follow the ideas developed in @xcite namely to replace the filter @xmath31 and the inter - jump time @xmath14 by some finite state space approximations in the dynamic programming equation . however , a noticeable difference from @xcite lies in the fact that the dynamic programming operators therein were lipschitz continuous whereas our new operators are only lipschitz continuous between some points of discontinuity . we overcome this drawback by splitting the operators into their restrictions onto their continuity sets . this way , we obtain not only an approximation of the value function of the optimal stopping problem but also an @xmath0-optimal stopping time with respect to the filtration @xmath29 that can be computed in practice . our approximation procedure for random variables is based on quantization . there exists an extensive literature on this method . the interested reader may for instance consult @xcite and the references within . the quantization of a random variable @xmath32 consists in finding a finite grid such that the projection @xmath33 of @xmath32 on this grid minimizes some @xmath34 norm of the difference @xmath35 . roughly speaking , such a grid will have more points in the areas of high density of @xmath32 . as explained for instance in @xcite , under some lipschitz - continuity conditions , bounds for the rate of convergence of functionals of the quantized process towards the original process are available , which makes this technique especially appealing . quantization methods have been developed recently in numerical probability or optimal stochastic control with applications in finance , see e.g. @xcite . the paper is organized as follows . section [ section - def ] introduces the notation , recalls the definition of a pdmp , presents our assumptions and defines the optimal stopping problem we are interested in , especially the observation process . the recursive formulation of the filter process is derived in section [ section - filtre ] . in section [ section - dynamic ] , we reduce our partially observed problem for the pdmp @xmath5 to a completely observed one involving the process @xmath36 for which we provide the dynamic programming equation and construct a family of @xmath0-optimal stopping times . then , our numerical methods to compute the value function and an @xmath0-optimal stopping time are presented in section [ section - quantif ] where we also prove the convergence of our algorithms after having recalled the main features of quantization . finally , an academic example is discussed in section [ section - example ] while technical results are postponed to the appendices . in this first section , let us define a piecewise - deterministic markov process ( pdmp ) and introduce some general assumptions . for any metric space @xmath37 , we denote @xmath38 its borel @xmath23-field , @xmath39 the set of real - valued , bounded and measurable functions defined on @xmath37 and @xmath40 the subset of functions of @xmath39 that are lipschitz continuous . for @xmath41 , denote @xmath42 and @xmath43 . let @xmath37 be an open subset of @xmath44 . let @xmath45 be its boundary and @xmath46 its closure and for any subset @xmath47 of @xmath37 , @xmath48 denotes its complement . a pdmp is defined by its local characteristics @xmath49 . * the flow @xmath50 is continuous . for all @xmath51 , @xmath52 is an homeomorphism and @xmath53 is a semi - group : for all @xmath54 , @xmath55 . for all @xmath56 , define the deterministic exit time from @xmath37 : @xmath57 we use here and throughout the convention @xmath58 . * the jump rate @xmath59 is measurable and satisfies : @xmath60 * finally , @xmath3 is a markov kernel on @xmath61 which satisfies : @xmath62 from these characteristics , it can be shown @xcite that there exists a filtered probability space @xmath63 on which a process @xmath64 is defined . its motion , starting from a point @xmath65 , may be constructed as follows . let @xmath66 be a nonnegative random variable with survival function : @xmath67 where for @xmath65 and @xmath68 $ ] , @xmath69 one then chooses an @xmath37-valued random variable @xmath70 with distribution @xmath71 . the trajectory of @xmath72 for @xmath73 is : @xmath74 starting from the point @xmath75 , one selects in a similar way @xmath76 the time between @xmath7 and the next jump time @xmath77 , as well as @xmath78 the next post - jump location and so on . davis showed @xcite that the process so defined is a strong markov process @xmath79 with jump times @xmath80 ( @xmath81 ) . the process @xmath82 where @xmath83 is the @xmath13-th post - jump location and @xmath84 ( @xmath85 ) is the @xmath13-th inter - jump time is clearly a discrete - time markov chain . the following non explosion assumption about the jump - times is standard ( see for example @xcite ) . [ hyp - tk_goes_to_infty ] for all @xmath86 , @xmath87<+\infty$ ] . it implies that @xmath88 a.s . when @xmath89 . moreover , we make the following assumption about the transition kernel @xmath3 . we assume that there exists a finite set @xmath90 such that for all @xmath65 , one has @xmath91 . in other words , for all @xmath92 , @xmath12 may only take its values in the finite set @xmath93 . this assumption ensures that the filter process , defined in the next section , has finite dimension . this is required to derive a tractable numerical method in section [ section - quantif ] . when this assumption does not hold , one may consider a preliminary discretization of the transition kernel to introduce it . [ hyp - ts - bounded ] we assume that the function @xmath94 is bounded on @xmath93 i.e. for all @xmath95 , we assume that @xmath96 . [ def - ts - order ] for all @xmath95 , denote @xmath97 and assume that @xmath98 , , @xmath99 are numbered such that @xmath100 @xmath101 . moreover , let @xmath102 . for any function @xmath103 in @xmath39 , introduce the following notation @xmath104 for any lipschitz continuous function @xmath103 in @xmath40 , denote @xmath105 $ ] its lipschitz constant @xmath106=\sup_{x\neq y \in e}\frac{|w(x)-w(y)|}{|x - y|}.\ ] ] the jump rate @xmath2 is in @xmath107 i.e. is bounded by @xmath108 . denote @xmath109 the set of finite signed measures on @xmath93 and @xmath110 the subset of probability measures on @xmath93 . we equip @xmath109 with the norm @xmath111 given by @xmath112 where @xmath113 denotes @xmath114 . we consider from now on a pdmp @xmath5 of which the initial state @xmath115 is a fixed point @xmath116 . we assume that this pdmp is observed through a noise and we now turn to the description of our observation procedure . for all @xmath92 , we assume that @xmath14 is perfectly observed but that @xmath12 is not ( except for the initial state @xmath117 ) . in some examples , it seems reasonable to consider that the jump times of the process are observed ( for instance , if the jumps correspond to changes of environment ) and that , when a jump occurs , the actual post - jump location is measured with a noise . the _ observation _ process of @xmath12 , denoted by @xmath118 is assumed to be of the following form : @xmath119 ( deterministic ) and for @xmath120 , @xmath121 where @xmath122 and where the _ noise _ @xmath123 is a sequence of @xmath44-valued , i.i.d . random variables with bounded density function @xmath124 that are also independent from @xmath11 . in order to define real - valued stopping times adapted to the observation process , we need to consider a continuous time version of the observation process . we therefore define the piecewise - constant process @xmath125 with a slight abuse of notation represents the value of the process @xmath15 at time @xmath126 and must not be confused with the value of the process at time @xmath127 . ] as @xmath128 let @xmath24 be the filtration generated by @xmath15 ( the _ observed _ filtration ) and @xmath129 be the filtration generated by @xmath130 ( the _ total _ filtration ) . without changing the notation , we then complete these filtrations with all the @xmath131-null sets . this leads us to the following definition . denote @xmath132 the set of @xmath29-stopping times that are a.s . finite and for @xmath92 , define @xmath133 for all @xmath92 , we define the filter @xmath134 . the quantity @xmath135 , denoted by @xmath136 , represents the probability of the event @xmath137 given the information available until time @xmath138 i.e. @xmath139.\ ] ] finally , let @xmath19 be the _ horizon _ and @xmath140 the _ reward function _ , we are interested in maximizing the following _ performance criterion _ @xmath141\ ] ] with respect to the stopping times @xmath142 . the _ value function _ associated to this partially observed optimal stopping problem is given by @xmath143,\ ] ] where @xmath144 is a probability measure in @xmath110 . the solution of our problem is then obtained by setting @xmath145 . for some applications , it would be interesting to consider a more general form for the reward function such as an integral term also possibly depending on the observation process , see for instance @xcite . however , this new setup would lead to several technical difficulties . in particular , the dynamic programming would be more complex . thus the derivation of the error bounds for the numerical approximation would be possibly intractable . we will also need the following assumption about the reward function @xmath18 associated with the optimal stopping problem . [ hyp - g - lip ] the function @xmath18 is in @xmath107 i.e. bounded by @xmath146 and there exists @xmath147_{2}\in \mathbb{r}^+$ ] such that for all @xmath148 and @xmath149 $ ] , one has : @xmath150_{2}|t - u|.\ ] ] now , the aims of this paper are first to explicit the filter process @xmath151 ( section [ section - filtre ] ) ; second to rewrite the partially observed optimal stopping problem as a totally observed one for a suitable markov chain on @xmath152 ( section [ sec - opt - stop - complete ] ) ; third to derive a dynamic programming equation and construct a family of @xmath0-optimal stopping times ( section [ section - dynamic - eq ] ) ; and finally to propose a numerical method to compute an approximation of the value function and an @xmath0-optimal stopping time ( section [ section - quantif ] ) . as a starting point , we will derive , in the next section , a recursive construction of the optimal filter that is the key point of our approach . the goal of this section is to obtain a recursive formulation of the filter @xmath31 . as far as we know , there is no result concerning the filter process for generic pdmps . we may however refer to @xcite for a recursive formulation of the filter for point processes , that can be seen as a sub - class of pdmp s . for all @xmath92 , we denote @xmath153 . the continuous - time observation process @xmath15 being a point process in the sense developed in @xcite , one has @xmath154 ( see ( * ? ? ? * , theorem t2 ) ) . moreover , @xmath155 . concerning the filter @xmath31 , first notice that , since it is an @xmath156-measurable random variable , there exists for all @xmath92 a measurable function @xmath157 such that @xmath158 as in the case of the kalman - bucy filter , the iteration leading from @xmath159 to @xmath31 can be split into two steps : prediction and correction . for all @xmath120 , let @xmath160 be the conditional distribution of @xmath161 given @xmath162 . thus , @xmath160 is a transition kernel defined on @xmath163 for all @xmath164 and @xmath165 by @xmath166 [ lemme - filtre - zys ] for all @xmath165 , we have the following equality of probability measures on @xmath167 , for all @xmath164 , @xmath168 [ [ proof ] ] proof + + + + + set @xmath169 in @xmath170 , using eq . that defines @xmath118 , one has @xmath171}\\ & = & \sum_{j=1}^{q}\int h(x_{j},\varphi(x_{j})+w , s){\mathbf{p}}(z_{n}=x_{j},s_{n}\in ds , w_{n}\in dw|\mathcal{g}_{n-1}=\gamma_{n-1}).\end{aligned}\ ] ] moreover , @xmath172 is independent from @xmath173 and admits the density function @xmath124 . consequently , one easily obtains the result by using the change of variable @xmath174 . @xmath175 + + integrating w.r.t . to the first variable in the previous lemma ( i.e. summing w.r.t . @xmath176 ) yields the following result . [ lemme - filtre - ys ] for all @xmath165 , we have the following equality of probability measures on @xmath177 , @xmath178dy.\ ] ] [ lemme - filtre - mu ] for all @xmath120 , @xmath165 and @xmath179 , the distribution @xmath160 , defined by eq . , satisfies @xmath180{t^{*}}_{m};{t^{*}}_{m+1}[\}}\left(\sum_{i = m+1}^{q}\pi_{n-1}^{i}(\gamma_{n-1})\lambda(\phi(x_i , s))e^{-\lambda(x_i , s)}q(\phi(x_i , s),x_j)\right)ds\\ & & + \sum_{m=1}^{q}\left(\pi_{n-1}^{m}(\gamma_{n-1})e^{-\lambda(x_m,{t^{*}}_m)}q(\phi(x_m,{t^{*}}_m),x_j)\right)\delta_{{t^{*}}_m}(ds).\end{aligned}\ ] ] [ [ proof-1 ] ] proof + + + + + let @xmath169 be a function of @xmath181 . since @xmath182 , the law of iterated conditional expectations yields @xmath183={\mathbf{e}}\left[{\mathbf{e}}\left[h(z_{n},s_{n})\big|{\mathfrak{f}}_{t_{n-1}}\right]\big|\mathcal{g}_{n-1}=\gamma_{n-1}\right].\ ] ] besides , @xmath184 so that @xmath185={\mathbf{e}}\left[h(z_{n},s_{n})\big|z_{0},s_{0},\ldots , z_{n-1},s_{n-1}\right],\ ] ] by independence of the sequences @xmath186 and @xmath11 . now , we apply the markov property of @xmath11 and a well - known special feature of the transition kernel of the underlying markov chain of a pdmp to obtain @xmath185={\mathbf{e}}\left[h(z_{n},s_{n})\big|z_{n-1},s_{n-1}\right]={\mathbf{e}}\left[h(z_{n},s_{n})\big|z_{n-1}\right].\ ] ] moreover , the transition kernel can be explicitly expressed in terms of the local characteristics of the pdmp , and this yields the next equations @xmath187}\\ & = & { \mathbf{e}}\big[\sum_{i=1}^{q}{\mathbbm{1}}_{\{z_{n-1}=x_{i}\}}{\mathbf{e}}[h(z_{n},s_{n})|z_{n-1}=x_{i}]\big|\mathcal{g}_{n-1}=\gamma_{n-1}\big]\\ & = & { \mathbf{e}}\big[\sum_{i=1}^{q}{\mathbbm{1}}_{\{z_{n-1}=x_{i}\}}\sum_{j=1}^{q}\big[\int_{{\mathbb{r}}^{+ } } h(x_{j},s)\lambda(\phi(x_i , s))e^{-\lambda(x_i , s)}{\mathbbm{1}}_{\{s<{t^{*}}_i\}}q(\phi(x_i , s),x_j)ds\\ & & + h(x_{j},{t^{*}}_i)e^{-\lambda(x_i,{t^{*}}_i)}q(\phi(x_i,{t^{*}}_i),x_j)\big]\big|\mathcal{g}_{n-1}=\gamma_{n-1}\big]\\ & = & \sum_{j=1}^{q}\big(\int_{{\mathbb{r}}^{+ } } h(x_{j},s)\sum_{i=1}^{q}\pi_{n-1}^{i}(\gamma_{n-1})\lambda(\phi(x_i , s))e^{-\lambda(x_i , s)}{\mathbbm{1}}_{\{s<{t^{*}}_i\}}q(\phi(x_i , s),x_j)ds\\ & & + \sum_{i=1}^{q}h(x_{j},{t^{*}}_i)\pi_{n-1}^{i}(\gamma_{n-1})e^{-\lambda(x_i,{t^{*}}_i)}q(\phi(x_i,{t^{*}}_i),x_j)\big).\end{aligned}\ ] ] this can be written equivalently as @xmath188}\\ & = & \sum_{j=1}^{q}\bigg(\sum_{m=0}^{q-1}\bigg(\int_{{t^{*}}_{m}}^{{t^{*}}_{m+1 } } h(x_{j},s)\sum_{i = m+1}^{q}\pi_{n-1}^{i}(\gamma_{n-1})\lambda(\phi(x_i , s))e^{-\lambda(x_i , s)}q(\phi(x_i , s),x_j)\bigg)ds\\ & & + \sum_{i=1}^{q}h(x_{j},{t^{*}}_i)\pi_{n-1}^{i}(\gamma_{n-1})e^{-\lambda(x_i,{t^{*}}_i)}q(\phi(x_i,{t^{*}}_i),x_j)\bigg).\end{aligned}\ ] ] hence the result . @xmath175 + + we now state the main result of this section , namely the recursive formulation of the filter sequence @xmath151 . [ prop - rec - filtre ] let @xmath189 be defined as follows : for all @xmath179 , @xmath190{t^{*}}_{m};{t^{*}}_{m+1}[\}}\frac{\psi^{j}_{m}(\pi , y , s)}{\overline{\psi}_{m}(\pi , y , s)}+\sum_{m=1}^{q}{\mathbbm{1}}_{\{s={t^{*}}_{m}\}}\frac{\psi^{*j}_{m}(y)}{\overline{\psi}^{*}_{m}(y)},\ ] ] where @xmath191 then , the filter , defined in eq . , satisfies @xmath192 and the following recursion : for all @xmath120 , @xmath193 [ [ proof-2 ] ] proof + + + + + fix @xmath194 in @xmath195 . bayes formula yields for all @xmath179 , @xmath196 lemma [ lemme - filtre - zys ] and corollary [ lemme - filtre - ys ] yield @xmath197dy.\end{gathered}\ ] ] with respect to @xmath198 , one recognizes the equality of two absolutely continuous measures which implies the equality a.e . of the density functions . thus , one has for almost all @xmath199 w.r.t . the lebesgue measure , @xmath200.\nonumber\end{aligned}\ ] ] eq . states the equality of two measures of the variable @xmath201 that contain both an absolutely continuous part and some weighted dirac measures . denote @xmath202 ( respectively @xmath203 ) the left - hand ( resp . right - hand ) side term of the previous equality . eq . means that for all function @xmath204 and for almost all @xmath199 w.r.t . the lebesgue measure , one has @xmath205 recall that , from lemma [ lemme - filtre - mu ] , the distribution @xmath206 has a density on the interval @xmath207{t^{*}}_{m};{t^{*}}_{m+1}[$ ] denoted by @xmath208 and given by @xmath209 first , take @xmath210{t^{*}}_{m};{t^{*}}_{m+1}[\}}$ ] in equation with @xmath211 . one has from equation ( [ eq - egal - mesures ] ) @xmath212 and thus on @xmath207{t^{*}}_{m};{t^{*}}_{m+1}[$ ] , almost surely w.r.t . the lebesgue measure , one has @xmath213 finally , for @xmath214 , choosing @xmath215 in eq . yields the equality of the weights at the point @xmath216 thus , using lemma [ lemme - filtre - mu ] , @xmath217 thus there exists two measurable sets @xmath218 and @xmath219 , negligible w.r.t . the lebesgue measures on @xmath44 and @xmath220 respectively , such that for all @xmath221 , @xmath222 , @xmath223 , one has @xmath224 on the one hand , one has @xmath225 by absolute continuity of the distribution of @xmath172 . on the other hand , @xmath226 because the distribution of @xmath14 is absolutely continuous on @xmath227 and one has @xmath228 . we therefore conclude from eq . that @xmath131-a.s . , one has @xmath229 the result follows since @xmath131-a.s . , one has @xmath230 and @xmath231 . @xmath175 + + this proposition will play a crucial part in the sequel . on the one hand , this result will enable us to prove the markov property of the sequence @xmath30 w.r.t . the observed filtration . on the other hand , the recursive formulation allows for simulation of the process @xmath232 which is crucial to obtain numerical approximations . finally , notice that the specific structure of the pdmp appears in the recursive formulation of the filter which contains both an absolutely continuous part and some weighted points . the main objective of this section is to derive the dynamic programming equation for the value function of the partially observed optimal stopping problem . the proof of this result can be roughly speaking decomposed into two steps . the first point consists in converting the partially observed optimal stopping problem into an optimal stopping problem under complete observation where the state variables are described by the _ discrete - time _ markov chain @xmath30 ( see section [ sec - opt - stop - complete ] ) . it is important to remark that under this new formulation , the optimization problem remains intrinsically a _ continuous - time _ optimization problem because the performance criterion is maximized over the set of stopping times with respect to the _ continuous - time _ filtration @xmath29 . we show in the second step ( see section [ section - dynamic - eq ] ) that the value function associated to the optimal stopping problem can be calculated by iterating a functional operator , labelled @xmath233 ( see definition [ def - g - h - i - j - l ] ) . as a by - product , we also provide a family of @xmath0-optimal stopping times . we would like to emphasize that the results obtained in this section are not straightforward to obtain due to the specific structure of this optimization problem . indeed , as already explained , it combines _ continuous - time _ and _ discrete - time _ features . consequently , this problem does not correspond to the classical optimal stopping problem of a discrete - time markov chain . moreover , it is different from the optimal stopping problem of a pdmp under complete observation mainly because the new state variables given by the markov chain @xmath30 are not the underlying markov chain of some pdmp . therefore the results of the literature @xcite can not be used . these derivations require some technical results about the structure of the stopping times in @xmath234 . for the sake of clarity in exposition , they are presented in the appendix [ stop - time ] . we start with a technical preliminary result required in the sequel , investigating the markov property of the filter process . [ prop - markov ] the sequences @xmath235 , @xmath236 and @xmath237 are @xmath238-markov chains . [ [ proof-3 ] ] proof + + + + + let @xmath239 . the law of iterated conditional expectations yields @xmath240 = { \mathbf{e}}\big[{\mathbf{e}}[h(\pi_{n},y_{n},s_{n})| { \mathfrak{f}}_{t_{n-1}}]\big|{\mathfrak{f}^{y}}_{t_{n-1}}\big].\end{aligned}\ ] ] from proposition [ prop - rec - filtre ] and eq . which defines @xmath118 one obtains @xmath241}\\ & = & { \mathbf{e}}\big[h\big(\psi(\pi_{n-1},\varphi(z_{n})+w_{n},s_{n}),\varphi(z_{n})+w_{n},s_{n}\big)\big| { \mathfrak{f}}_{t_{n-1}}\big]\\ & = & \sum_{j=1}^{q}\int h\big(\psi(\pi_{n-1},\varphi(x_{j})+w , s),\varphi(x_{j})+w , s\big)\\ & & \times { \mathbf{p}}(z_{n}=x_{j},w_{n}\in dw , s_{n}\in ds|{\mathfrak{f}}_{t_{n-1}}).\end{aligned}\ ] ] yet , @xmath172 is independent from @xmath242 and admits the density function @xmath124 . as in the proof of lemma [ lemme - filtre - zys ] one thus obtains @xmath241}\\ & = & \sum_{j=1}^{q}\int h\big(\psi(\pi_{n-1},y , s),y , s\big){\mathbf{p}}(z_{n}=x_{j},s_{n}\in ds|{\mathfrak{f}}_{t_{n-1}})f_{w}(y-\varphi(x_{j}))dy.\end{aligned}\ ] ] besides , we have @xmath243 as in the proof of lemma [ lemme - filtre - mu ] , so that one has @xmath241}\\ & = & \sum_{i=1}^{q}{\mathbbm{1}}_{\{z_{n-1}=x_{i}\}}\sum_{j=1}^{q}\int \big(\int_{0}^{{t^{*}}_{i } } h\big(\psi(\pi_{n-1},y , s),y , s\big)\lambda\big(\phi(x_i , s)\big)e^{-\lambda(x_i , s)}q\big(\phi(x_{i},s),x_{j}\big)ds\\ & & + h\big(\psi(\pi_{n-1},y,{t^{*}}_{i}),y,{t^{*}}_{i}\big)e^{-\lambda(x_i,{t^{*}}_{i})}q\big(\phi(x_{i},{t^{*}}_{i}),x_{j}\big)\big)f_{w}(y-\varphi(x_{j}))dy.\end{aligned}\ ] ] take now the conditional expectation w.r.t . @xmath162 , to obtain @xmath241}\\ & = & \sum_{i=1}^{q}\pi_{n-1}^{i}\sum_{j=1}^{q}\int \big(\int_{0}^{{t^{*}}_{i } } h\big(\psi(\pi_{n-1},y , s),y , s\big)\lambda\big(\phi(x_i , s)\big)e^{-\lambda(x_i , s)}q\big(\phi(x_{i},s),x_{j}\big)ds\\ & & + h\big(\psi(\pi_{n-1},y,{t^{*}}_{i}),y,{t^{*}}_{i}\big)e^{-\lambda(x_i,{t^{*}}_{i})}q\big(\phi(x_{i},{t^{*}}_{i}),x_{j}\big)\big)f_{w}(y-\varphi(x_{j}))dy.\end{aligned}\ ] ] hence @xmath244 $ ] is merely a function of @xmath159 yielding the result for the three processes . @xmath175 + in this section , we show how our optimal stopping problem under partial observation for the process @xmath5 can be converted into an optimal stopping problem under complete observation involving the markov chain @xmath245 . more precisely , for a fixed stopping time @xmath246 , we show in proposition [ lemme value fonction ] that the performance criterion @xmath247 $ ] can be expressed in terms of the discrete - time markov chain @xmath245 . we would like to emphasize the following important fact . although the performance criterion can be written in terms of _ discrete - time _ process , the optimization problem remains intrinsically a _ continuous - time _ optimization problem . indeed , the performance criterion is maximized over the set of stopping times with respect to the _ continuous - time _ filtration @xmath29 . [ lemme value fonction ] let @xmath248 and @xmath120 . for all @xmath249 one has @xmath250}\\ & = & \sum_{k=0}^{n-1}\sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{k}\leq \sigma\}}{\mathbbm{1}}_{\{r_{k}<{t^{*}}_{i}\}}g\circ\phi(x_{i},r_{k})e^{-\lambda(x_{i},r_{k})}\pi_{k}^{i}|\pi_0=\pi]\\ & & + \sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{n}\leq \sigma\}}g(x_{i})\pi_{n}^{i}|\pi_0=\pi],\end{aligned}\ ] ] where @xmath251 is the sequence of non negative random variables associated to @xmath23 as introduced in theorem [ theo - bremaud - adapted ] . [ [ proof-4 ] ] proof + + + + + we split @xmath252 $ ] into several terms depending on the position of @xmath23 w.r.t . the jump times @xmath253 @xmath254 } & = & \sum_{k=0}^{n-1}\sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{k}\leq \sigma < t_{k+1}\}}{\mathbbm{1}}_{\{z_{k}=x_{i}\}}g\circ\phi(x_{i},r_{k})|\pi_0=\pi]\\ & & + \sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{n}\leq \sigma\}}{\mathbbm{1}}_{\{z_{n}=x_{i}\}}g(x_{i})|\pi_0=\pi].\end{aligned}\ ] ] for notational convenience , consider @xmath255 on the one hand , one has @xmath256=g(x_{i}){\mathbbm{1}}_{\{t_{n}\leq \sigma\}}\pi_{n}^{i}$ ] since @xmath257 ( see for instance ( * ? ? ? * , theorem t7 ) ) . on the other hand , to compute @xmath258 $ ] , we use lemma [ lemme - tech - event ] to obtain @xmath259 & = { \mathbbm{1}}_{\{t_{k}\leq \sigma\}}g\circ\phi(x_{i},r_{k}){\mathbf{e}}[{\mathbbm{1}}_{\{s_{k+1}>r_{k}\}}{\mathbbm{1}}_{\{z_{k}=x_{i}\}}|{\mathfrak{f}^{y}}_{t_{k}}]\\ & = { \mathbbm{1}}_{\{t_{k}\leq \sigma\}}g\circ\phi(x_{i},r_{k}){\mathbf{e}}\big[{\mathbbm{1}}_{\{z_{k}=x_{i}\}}{\mathbf{e}}[{\mathbbm{1}}_{\{s_{k+1}>r_{k}\}}|{\mathfrak{f}}_{t_{k}}]\big|{\mathfrak{f}^{y}}_{t_{k}}\big]\\ & = { \mathbbm{1}}_{\{t_{k}\leq \sigma\}}g\circ\phi(x_{i},r_{k}){\mathbf{e}}[{\mathbbm{1}}_{\{z_{k}=x_{i}\}}{\mathbbm{1}}_{\{r_{k}<{t^{*}}(z_{k})\}}e^{-\lambda(z_{k},r_{k})}|{\mathfrak{f}^{y}}_{t_{k}}]\\ & = { \mathbbm{1}}_{\{t_{k}\leq \sigma\}}g\circ\phi(x_{i},r_{k}){\mathbbm{1}}_{\{r_{k}<{t^{*}}_{i}\}}e^{-\lambda(x_{i},r_{k})}\pi_{k}^{i}.\end{aligned}\ ] ] details to obtain the third line in the above computations are provided by lemma [ lemme - tech - esp - cond ] . the result follows . @xmath175 + based on the new formulation , the main objective of this section is to derive the backward dynamic programming equation . it involves some operators introduced in definition [ def - g - h - i - j - l ] . by iterating the operator , labelled @xmath233 , we define a sequence of real valued functions @xmath260 in definition [ def - vn ] . theorem [ value - fonction ] establishes that @xmath261 is the value function of our partially observed optimal stopping problem with horizon @xmath262 and in particular that @xmath263 is the value function of problem defined in equation . another important result of this section is given by theorem [ theo - sn - epsilon ] which constructs a sequence of @xmath0-optimal stopping times . [ def - g - h - i - j - l ] the operators @xmath264 , @xmath265 , @xmath266 , and @xmath267 are defined for all @xmath268 and @xmath269 by @xmath270,\\ hh(\pi , u)&=&{\mathbf{e}}\big[\sum_{i=1}^qh\circ\phi(x_i , u)\pi_0^i{\mathbbm{1}}_{\{u < t^*_i\}}{\mathbbm{1}}_{\{s_{1 } > u\}}| \pi_0=\pi\big],\\j(v , h)(\pi , u)&=&hh(\pi , u)+gv(\pi , u),\\ l(v , h)(\pi)&=&\sup_{u\geq 0 } j(v , h)(\pi , u).\end{aligned}\ ] ] [ def - vn ] the sequence @xmath260 of real - valued functions is defined on @xmath110 by @xmath271 the following theorem is the main result of this section showing that the operator @xmath233 is the dynamic programming operator associated to the initial optimization problem . [ value - fonction ] for all @xmath272 and @xmath273 , one has @xmath274 = v_{n - n}(\pi).\ ] ] [ [ proof-5 ] ] proof + + + + + the proof of this result is based on proposition [ theo - value - fonction ] and theorem [ theo - sn - epsilon ] . proposition [ theo - value - fonction ] proves that @xmath275 is an upper bound for the value function of the problem with horizon @xmath138 . the reverse inequality is derived in theorem [ theo - sn - epsilon ] by constructing a sequence of @xmath0-optimal stopping times . @xmath175 + + [ theo - value - fonction ] for all @xmath272 and @xmath273 , one has @xmath274\leq v_{n - n}(\pi).\ ] ] [ [ proof-6 ] ] proof + + + + + let @xmath276 . consider @xmath251 the sequence associated to @xmath23 as introduced in theorem [ theo - bremaud - adapted ] . we prove the theorem by induction on @xmath13 . for @xmath277 , proposition [ lemme value fonction ] yields @xmath278 & = & { \sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{r_{0}<{t^{*}}_{i}\}}g\circ\phi(x_{i},r_{0})e^{-\lambda(x_{i},r_{0})}\pi_{0}^{i}|\pi_0=\pi ] } \nonumber \\ & & + { \sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{1}\leq\sigma\}}g(x_{i})\pi_{1}^{i}|\pi_0=\pi]}. \label{beben}\end{aligned}\ ] ] since @xmath279 is deterministic and by using lemma [ lemme - def - op - h ] , we recognize that the first term of the right hand side of equation ( [ beben ] ) is @xmath280 . we now turn to the second term of the right hand side of equation ( [ beben ] ) which is given by @xmath281 & = & { \mathbf{e}}[v_{n}(\pi_{1}){\mathbbm{1}}_{\{s_{1}\leq r_{0}\}}|\pi_0=\pi ] \\ & = & \ gv_{n}(\pi , r_{0}),\end{aligned}\ ] ] from lemma [ lemme - tech - event ] and the definition of @xmath282 . recall that from definition [ def - g - h - i - j - l ] one has @xmath283 thus , one obtains @xmath278 \ = \ j(v_{n},g)(\pi , r_{0 } ) & \leq & \sup_{u\geq 0}j(v_{n},g)(\pi , u)\\ & = & l(v_{n},g)(\pi)= v_{n-1}(\pi).\end{aligned}\ ] ] set now @xmath284 and assume that @xmath285\leq v_{n-(n-1)}(\pi)$ ] , for all @xmath286 . proposition [ lemme value fonction ] yields @xmath287}\\ & = & \sum_{k=0}^{n-1}\sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\ { t_{k}\leq \sigma\}}{\mathbbm{1}}_{\{r_{k}<{t^{*}}_{i}\}}g\circ\phi(x_{i},r_{k})e^{-\lambda(x_{i},r_{k})}\pi_{k}^{i}|\pi_0=\pi]\\ & & + \sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{n}\leq \sigma\}}g(x_{i})\pi_{n}^{i}|\pi_0=\pi].\end{aligned}\ ] ] as in the case @xmath277 , the term for @xmath288 equals @xmath280 . notice that for @xmath289 , @xmath290 and that @xmath291 is @xmath292-measurable . by taking the conditional expectation w.r.t . @xmath292 it follows that @xmath293={\mathbf{e}}[\xi |\pi_0=\pi]$ ] where @xmath294 is defined by @xmath295.\end{aligned}\ ] ] therefore , we obtain @xmath296= hg(\pi , r_{0})+{\mathbf{e}}[\xi{\mathbbm{1}}_{\{s_{1}\leq r_{0}\}}|\pi_0=\pi].\ ] ] we now use the markov property of the chain @xmath297 . indeed , for @xmath289 , one has @xmath298 , where @xmath299 is the translation operator of the @xmath238-markov chain @xmath235 . moreover , when @xmath300 , one has , from proposition [ propb4 ] , @xmath301 ( indeed , we pointed out in remark [ rq - rn - rnbar ] that @xmath302 can be replaced by @xmath303 defined in lemma [ lemmeb2 ] ) and @xmath304 where @xmath305 and @xmath306 are defined in definition [ defb3 ] and proposition [ propb4 ] ( with @xmath307 in the present case ) . since for @xmath289 , @xmath308 , one has @xmath309 . finally , combining the markov property of the chain @xmath297 and proposition [ lemme value fonction ] we have @xmath310 with @xmath311 $ ] . moreover , one has @xmath312 from the induction assumption since @xmath313 ( indeed , both @xmath306 and @xmath314 are @xmath29-stopping times from corollary [ corb6 ] and lemma [ prop - tn - stop - time ] respectively ) . one has then @xmath315 finally , combining eq . and , one has @xmath316 \leq hg(\pi , r_{0})+{\mathbf{e}}[v_{n-(n-1)}(\pi_{1}){\mathbbm{1}}_{\{s_{1}\leq r_{0}\}}|\pi_0=\pi].\ ] ] in the second term , we recognize the operator @xmath282 and one has @xmath317 & \leq & hg(\pi , r_{0})+gv_{n-(n-1)}(\pi , r_{0})\\ & = & j(v_{n-(n-1)},g)(\pi , r_{0})\\ & \leq & \sup_{u\geq 0 } j(v_{n-(n-1)},g)(\pi , u)\\ & = & l(v_{n-(n-1)},g)(\pi ) \ = \ v_{n - n}(\pi),\end{aligned}\ ] ] that proves the induction . @xmath175 + + we now prove the reverse inequality by constructing a sequence of @xmath0-optimal stopping times . [ def - sn - epsilon]for @xmath318 , @xmath272 and for @xmath319 , we define @xmath320 consider @xmath321 and for @xmath284 , @xmath322 and finally set @xmath323 the following lemma describes the effect of the translation operator @xmath299 on the sequence @xmath324 . [ rnk - epsilon - theta ] for @xmath325 and @xmath326 , on the set @xmath327 , one has @xmath328 [ [ proof-7 ] ] proof + + + + + for @xmath329 , one just has to prove that on the event @xmath330 , one has @xmath331 . yet , from the definition of the sequence @xmath324 , one has @xmath332 and @xmath333 . the result follows since we are on the event @xmath334 . for a fixed @xmath335 , we prove the lemma by induction on @xmath326 . set @xmath336 . one has from the definition on the sequence @xmath324 , @xmath337 and @xmath338 . we obtain @xmath339 because we have assumed that we are on the event @xmath340 . the propagation of the induction is similar to the case @xmath336 . @xmath175 + + equipped with this preliminary result , we may now prove that @xmath341 is a sequence of @xmath0-optimal stopping times with respect to the filtration . generated by the observations . [ theo - sn - epsilon ] for all @xmath272 and @xmath318 , one has @xmath342 and @xmath343\geq v_{n - n}(\pi)-\epsilon.\ ] ] [ [ proof-8 ] ] proof + + + + + let @xmath344 . first notice that , as a direct consequence of proposition [ propb5 ] , @xmath345 is an @xmath29-stopping time since , by construction , the @xmath346 are @xmath347-measurable and satisfy the condition @xmath348 on the event @xmath349 . it is also clear that @xmath350 . thus , one has @xmath342 . let us now prove the second assessment by induction . set @xmath277 . let @xmath319 , we denote @xmath351 . since @xmath352 is deterministic , one has clearly @xmath353 . consequently , by using the same arguments as in the proof of proposition [ theo - value - fonction ] , we obtain @xmath354= & hg(\pi , r_{0}^{\epsilon})+gv_{n}(\pi , r_{0}^{\epsilon})=j(v_{n},g)(\pi , r_{0}^{\epsilon}).\end{aligned}\ ] ] finally , the definition of @xmath355 yields @xmath356 thus one has @xmath357\geq v_{n-1}(\pi)-\epsilon.\ ] ] now set @xmath284 and assume that @xmath358\geq v_{n-(n-1)}(\pi)-\epsilon$ ] , for all @xmath318 . proposition [ lemme value fonction ] yields @xmath359}\\ & = & \sum_{k=0}^{n-1}\sum_{i=1}^{q}{\mathbf{e}}\left[{\mathbbm{1}}_{\{t_{k}\leq u_{n}^{2\epsilon}\}}{\mathbbm{1}}_{\{r_{n , k}^{2\epsilon}<{t^{*}}_{i}\}}g\circ\phi(x_{i},r_{n , k}^{2\epsilon})e^{-\lambda(x_{i},r_{n , k}^{2\epsilon})}\pi_{k}^{i}\big|\pi_0=\pi\right]\\ & & + \sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{n}\leq u_{n}^{2\epsilon}\}}g(x_{i})\pi_{n}^{i}|\pi_0=\pi].\end{aligned}\ ] ] denote @xmath360 . as in the case @xmath277 , the term for @xmath288 equals @xmath361 since @xmath362 . take the conditional expectation w.r.t . @xmath292 in the other terms . one has then , @xmath363= hg(\pi , r_{n-1}^{\epsilon})+{\mathbf{e}}[\xi'{\mathbbm{1}}_{\{t_{1}\leq u_{n}^{2\epsilon}\}}|\pi_0=\pi],\ ] ] with @xmath364.\end{aligned}\ ] ] our objective is to apply the markov property of @xmath365 in the term @xmath366 . recall that , from lemma [ rnk - epsilon - theta ] , one has @xmath367 for @xmath325 and @xmath326 on the event @xmath368 ( the equality of these events stems from lemma [ lemme - tech - event ] ) . thus , on this set one has @xmath369 besides , recall that @xmath370 , for @xmath371 . consequently , on the set @xmath327 , one has @xmath372 and thus , combining the markov property of the chain @xmath297 and proposition [ lemme value fonction ] , we have @xmath373 with @xmath374 $ ] . moreover , thanks to the induction assumption , one has @xmath375 so that one obtains @xmath376 finally , combining equation and and noticing that , according to lemma [ lemme - tech - event ] , @xmath377 , one obtains @xmath378 } & \geq & hg(\pi , r_{n-1}^{\epsilon})+{\mathbf{e}}[v_{n-(n-1)}(\pi_{1}){\mathbbm{1}}_{\{s_{1}\leq r_{n-1}^{\epsilon}\}}|\pi_0=\pi]-\epsilon\\ & = & j(v_{n-(n-1)},g)(\pi , r_{n-1}^{\epsilon})-\epsilon\\ & \geq & v_{n - n}(\pi)-2\epsilon,\end{aligned}\ ] ] from the definition of @xmath379 , showing the result . in this section , we are interested in the computational issue for our optimal stopping problem under partial observation . indeed , we want to compute a numerical approximation of the value function ( [ opt - stop - pb ] ) and propose a computable @xmath0-optimal stopping time . as we have seen in the previous section , the value function @xmath380 can be obtained by iterating the dynamic programming operator @xmath233 . however , the operator @xmath233 involves conditional expectations that are in essence difficult to compute and iterate numerically . we manage to overcome this difficulty by combining two special properties of our problem . on the one hand , the underlying process @xmath381 in the expression of the operator @xmath233 is a markov chain . therefore , it can be discretized using a quantization technique which is a powerful method suitable for numerical computation and iteration of conditional expectations . on the other hand , the recursion on the functions @xmath382 involving the operator @xmath233 can be transformed into a recursion on suitably defined random variables . thus they are easier to iterate numerically as we do not need to compute an approximation of each @xmath383 on the whole state space . this section is organized as follows . we first explain how the recursion on the functions @xmath382 can be transformed into a recurrence on random variables involving only the markov chain @xmath381 . then , we present a quantization technique to discretize this markov chain . afterwards , we construct a discretized version of the main operators in definition [ def op chapeau ] that is used to build an approximation of the value function in definition [ def v chap ] , and a computable @xmath0-optimal stopping time . the main results of this section are theorems [ theo - conv ] and [ th arret chap ] that prove the convergence of our approximation scheme and provide a rate of convergence . we first explain how the dynamic programming equations on the functions @xmath382 yield a recursion on the random variables @xmath384 . introduce now the sequence @xmath385 of random variables defined by @xmath386 in other words , one has @xmath387,\nonumber\end{aligned}\ ] ] for @xmath388 . notice that @xmath389 is known and the expression of @xmath390 involves only @xmath391 and the markov chain @xmath381 . thus , the sequence @xmath392 is completely characterized by the system ( [ def vn ] ) . in addition , @xmath393 . thus to approximate the value function @xmath380 at the initial point of our process , it is sufficient to provide an approximation of the sequence of random variables @xmath394 . there exists an extensive literature on quantization methods for random variables and processes . we do not pretend to present here an exhaustive panorama of these methods . however , the interested reader may for instance , consult the following works @xcite and references therein . consider @xmath32 an @xmath395-valued random variable such that @xmath396 where @xmath397 denotes the @xmath34-nom of @xmath32 : @xmath398)^{1/p}$ ] . let @xmath399 be a fixed integer , the optimal @xmath34-quantization of the random variable @xmath32 consists in finding the best possible @xmath34-approximation of @xmath32 by a random vector @xmath33 taking at most @xmath399 values : @xmath400 . this procedure consists in the following two steps : 1 . find a finite weighted grid @xmath401 with @xmath402 . 2 . set @xmath403 where @xmath404 with @xmath405 denotes the closest neighbour projection on @xmath406 . the asymptotic properties of the @xmath34-quantization are given by the following result , see e.g. @xcite . [ theore ] if @xmath407<+\infty$ ] for some @xmath408 then one has @xmath409 where the distribution of @xmath32 is @xmath410 with @xmath411 , @xmath412 a constant and @xmath413 the lebesgue measure in @xmath414 . there exists a similar procedure for the optimal quantization of a markov chain . our approximation method is based on the quantization of the markov chain @xmath415 . thus , from now on , we will denote , for @xmath416 , @xmath417 . the clvq ( competitive learning vector quantization ) algorithm ( * ? ? ? * section 3 ) provides for each time step @xmath416 a finite grid @xmath418 of @xmath419 as well as the transition matrices @xmath420 from @xmath418 to @xmath421 . let @xmath422 such that for all @xmath423 , @xmath424 and @xmath425 have finite moments at least up to order @xmath426 and let @xmath427 be the nearest - neighbor projection from @xmath419 onto @xmath418 . the quantized process @xmath428 with value for each @xmath429 in the finite grid @xmath418 of @xmath419 is then defined by @xmath430 we will also denote by @xmath431 , the projection of @xmath418 on @xmath110 , and by @xmath432 , the projection of @xmath418 on @xmath433 . some important remarks must be made concerning the quantization . on the one hand , the optimal quantization has nice convergence properties stated by theorem [ theore ] . indeed , the @xmath434-quantization error @xmath435 goes to zero when the number of points in the grids goes to infinity . however , on the other hand , the markov property is not maintained by the algorithm and the quantized process is generally not markovian . although the quantized process can be easily transformed into a markov chain , this chain will not be homogeneous . it must be pointed out that the quantized process @xmath436 depends on the starting point @xmath437 of the process . in practice , we begin with the computation of the quantization grids , which merely requires to be able to simulate the process . notice that in our case , what is actually simulated is the sequence of observation @xmath438 . we are then able to compute the filter @xmath439 thanks to the recursive equation provided by proposition [ prop - rec - filtre ] . the grids are only computed once and for all and may be stored off - line . our schemes are then based on the following simple idea : we replace the process by its quantized approximation within the different recursions . the computation is thus carried out in a very simple way since the quantized process has finite state space . our approximation scheme of the sequence @xmath385 follows the same lines as in @xcite , but once more , the results therein can not be applied directly as the markov chain @xmath440 is not the underlying markov chain of some pdmp . our approach decomposes in two steps . the first one will be to discretize the time - continuous maximization of the operator @xmath233 to obtain a maximization over a finite set . the second step consists in replacing the markov chain @xmath441 by its quantized approximation @xmath442 within the dynamic programming equation . thus , the conditional expectations will become easily tractable finite sums . let us first build a finite time grid to discretize the continuous - time maximization in the expression of the operator @xmath233 . the maximum is originally taken over the set @xmath443 . however , it can be seen from definition [ def - g - h - i - j - l ] that @xmath444 for all @xmath445 . indeed , the random variable @xmath446 is bounded by the greatest deterministic exit time @xmath447 that is finite thanks to assumption [ hyp - ts - bounded ] . therefore , the maximization set can be reduced to the compact set @xmath448 $ ] . instead of directly discretizing the set @xmath448 $ ] , we will actually discretize the subsets @xmath207t^*_m , t^*_{m+1}[$ ] . the reason why we want to exclude the points @xmath449 from our grid is technical and will be explained with lemma [ xi3-indicator ] . now , it seems natural to distinguish wether @xmath450 or @xmath451 . let @xmath452 be the set of indices @xmath453 such that @xmath451 . notice that @xmath454 is not empty because it contains at least the index 0 since we assumed that @xmath455 . we can now build our approximation grid . let @xmath456 be such that @xmath457 for all @xmath458 , let @xmath459 be the finite grid on @xmath207{t^{*}}_{m};{t^{*}}_{m+1}[$ ] defined as follows @xmath460 where @xmath461 . we also denote @xmath462 [ rq - grm]let @xmath458 . notice that , thanks to eq . , @xmath459 is not empty . moreover , it satisfies two properties that will be crucial in the sequel : a. : : for all @xmath463 $ ] , there exists @xmath464 such that @xmath465 , b. : : for all @xmath464 and @xmath466 , one has @xmath467\subset ] { t^{*}}_{m};{t^{*}}_{m+1}[$ ] . a discretized maximization operator @xmath468 is then defined as follows . [ def k ] let @xmath468 : @xmath469 be defined for all @xmath319 by @xmath470 with @xmath471.$ ] we now proceed to our second step : replacing the markov chain @xmath441 by its quantized approximation @xmath442 within the operators involved in the construction of the value function . [ def op chapeau ] we define the _ quantized operators _ @xmath472 , @xmath473 , @xmath474 , @xmath475 and @xmath476 for @xmath477 , @xmath478 , @xmath479 , @xmath480 and @xmath481 as follows @xmath482,\\ \widehat{h}_{n}h(\pi , u)=&\hspace{0.3cm}\sum_{i=1}^{q}\pi^{i}{\mathbbm{1}}_{\{u < { t^{*}}_{i}\}}h\circ\phi(x_{i},u){\mathbf{e}}[{\mathbbm{1}}_{\{\widehat{s}_{n } > u\}}| \widehat \pi_{n-1}=\pi],\\ \widehat{j}_{n}(v , h)(\pi , u)=&\hspace{0.3cm}\widehat{h}_{n}h(\pi , u)+\widehat{g}_{n}v(\pi , u),\\ \widehat{k}_{n}v(\pi)=&\hspace{0.3cm}\widehat{j}_{n}(v , h)(\pi , t^*_q)=\hspace{0.3cm}{\mathbf{e}}[v(\widehat{\pi}_{n})| \widehat \pi_{n-1}=\pi],\\ \widehat l^{d}_{n}(v , h)(\pi)=&\hspace{0.3cm}\max_{m\in m}\big\{\max_{u\in gr_{m}(\delta)}\ { \widehat j_{n}(v , h)(\pi , u)\}\big\}\vee \widehat k_{n}v(\pi).\end{aligned}\ ] ] the quantized approximation of the value functions naturally follows . [ def v chap ] for @xmath483 , define the functions @xmath484 on @xmath485 as follows @xmath486 for @xmath483 , let @xmath487 we may now state our main result for the numerical approximation . [ theo - conv ] suppose that for all @xmath388 , @xmath488 then , one has the following bound for the approximation error @xmath489\|\pi_{n+1}-\widehat \pi_{n+1}\|_{p},\end{aligned}\ ] ] where @xmath490_{2}+2c_{g}c_{\lambda}$ ] , @xmath491 and @xmath492 + 4c_{g}+2[v_{n+1}]$ ] with @xmath493 $ ] , @xmath494 $ ] defined in proposition [ prop - lip - v ] and @xmath147_{2}$ ] defined in assumption [ hyp - g - lip ] . theorem [ theo - conv ] establishes the convergence of our approximation scheme and provides a bound for the rate of convergence . more precisely , it gives a rate for the @xmath34 convergence of @xmath495 towards @xmath496 . indeed , one has @xmath497 , so by virtue of theorem [ theo - conv ] @xmath498 can be made arbitrarily small when the quantization errors @xmath499 go to zero i.e. when the number of points in the quantization grids goes to infinity . in order to prove theorem [ theo - conv ] , we proceed similarly to @xcite and split the approximation error into four terms @xmath500 , with @xmath501 to obtain bounds for each of these terms , one needs to study the regularity of the operators and the value functions @xmath383 . the results are detailed in appendix [ apx lip ] . in particular , we establish in proposition [ prop - lip - v ] that the value functions @xmath383 are lipschitz continuous , yielding a bound for the first term . the first term @xmath502 is bounded as follows @xmath503\|\pi_{n}-\widehat \pi_{n}\|_{p}.\ ] ] the other error terms are studied separately in the following sections . for the second error term , we investigate the consequences of replacing the continuous maximization in operator @xmath233 by a discrete one on @xmath504 . for all @xmath458 , @xmath505 and @xmath319 one has @xmath506_{2}+c_{g}c_{\lambda}+c_{v}c_{\lambda}\right)\delta.\ ] ] [ [ proof-9 ] ] proof + + + + + we use definition [ def - op - j - l ] to split operator @xmath507 into a sum of continuous operators @xmath508 . thus , one has @xmath509 } j^m(v , g)(\pi , u).\ ] ] the function @xmath510 being continuous , there exists @xmath511 $ ] such that @xmath512 } j^{m}(v , h)(\pi , u)=j^{m}(v , h)(\pi,\overline{t})$ ] . moreover , from remark [ rq - grm].a , one may chose @xmath513 so that @xmath514 . propositions [ prop - lip - h ] and [ prop - lip - g ] stating the lipschitz continuity of @xmath515 then yield @xmath516 } j^{m}(v , h)(\pi , u)-\max_{u\in gr_{m}(\delta ) } j^{m}(v , h)(\pi , u)\\ & \leq j^{m}(v , h)(\pi,\overline{t})-j^{m}(v , h)(\pi,\overline{u})\\ & \leq \left([g]_{2}+c_{g}c_{\lambda}+c_{v}c_{\lambda}\right)|\overline{t}-\overline{u}|\leq \left([g]_{2}+c_{g}c_{\lambda}+c_{v}c_{\lambda}\right)\delta,\end{aligned}\ ] ] showing the result . @xmath175 + the second term @xmath517 is bounded as follows @xmath518_{2}+2c_{g}c_{\lambda}\right)\delta.\ ] ] [ [ proof-10 ] ] proof + + + + + this is a straightforward consequence of the previous lemma once it has been noticed that for all @xmath519 , @xmath520 , @xmath521 , @xmath522 , one has @xmath523 . notice also that proposition [ prop - lip - v ] provides @xmath524 . @xmath175 + to investigate the third error term , we use the properties of quantization to bound the error made by replacing an operator by its quantized approximation . as in @xcite , we must first deal with non - continuous indicator functions . the fact that the @xmath449 and a small neighborhood around them do not belong to the discretization grid @xmath504 is crucial to obtain the following lemma . [ xi3-indicator ] for all @xmath388 , @xmath458 and @xmath466 , one has @xmath525\big\|_{p } \leq \eta^{-1}{\|s_{n+1}-\widehat s_{n+1}\|_{p}}+2\eta c_{\lambda}.\ ] ] [ [ proof-11 ] ] proof + + + + + let @xmath466 . the difference of the indicator functions equals 1 if and only if @xmath526 and @xmath527 are on different sides of @xmath528 . therefore , if the difference of the indicator functions equals 1 , either @xmath529 , or @xmath530 and in the latter case @xmath531 too since @xmath532 . one has @xmath533 leading to @xmath534\big\|_{p}\\ \leq \|{\mathbbm{1}}_{\{|s_{n+1}-\widehat s_{n+1}| > \eta\}}\|_{p } + \big\|\max_{u\in gr_{m}(\delta)}{\mathbf{e } } [ { \mathbbm{1}}_{\{|s_{n+1}-u|\leq \eta\ } } |\widehat \pi_{n}]\big\|_{p}.\end{gathered}\ ] ] on the one hand , markov inequality yields @xmath535 on the other hand , since @xmath464 , one has @xmath467\subset ] { t^{*}}_{m};{t^{*}}_{m+1}[$ ] from remark [ rq - grm].b , thus @xmath526 has an absolutely continuous distribution on the interval @xmath467 $ ] since it does not contain any of the @xmath536 . besides , recall that @xmath537 , hence , the following inclusions of @xmath23-fields @xmath538 . we also have @xmath539 , the law of iterated conditional expectations provides @xmath540 & = & { \mathbf{e}}\big[{\mathbf{e}}\big[{\mathbf{e } } [ { \mathbbm{1}}_{\{|s_{n+1}-u|\leq \eta\ } } | { \mathfrak{f}}_{t_{n}}]\big| { \mathfrak{f}^{y}}_{t_{n}}\big]\big|\widehat \pi_{n}\big]\\ & \leq & { \mathbf{e}}\big[{\mathbf{e } } [ \int_{u-\eta}^{u+\eta}\lambda\big(\phi(z_{n},s)\big)ds\big| { \mathfrak{f}^{y}}_{t_{n}}]\big|\widehat \pi_{n}\big]\\ & = & { \mathbf{e}}\big [ \sum_{i=1}^{q } \pi_{n}^{i}\int_{u-\eta}^{u+\eta}\lambda\big(\phi(x_{i},s)\big)ds\big|\widehat \pi_{n}\big].\end{aligned}\ ] ] finally , one obtains @xmath541\leq 2\eta c_{\lambda},$ ] showing the result . @xmath175 + [ xi3-k ] for all @xmath388 , one has @xmath542{\mathbf{e}}\big[|\pi_{n+1}-\widehat \pi_{n+1}|\big|\widehat \pi_{n}\big]+(2c_{g}+2[v_{n+1}]){\mathbf{e}}\left[|\pi_{n}-\widehat \pi_{n}|\big|\widehat \pi_{n}\right].\end{gathered}\ ] ] [ [ proof-12 ] ] proof + + + + + by the definitions of operators @xmath543 and @xmath544 , one has @xmath545-{\mathbf{e}}[v_{n+1}(\widehat \pi_{n+1})| \widehat \pi_{n } ] |\nonumber\\ & \leq&{|{\mathbf{e}}[v_{n+1}(\pi_{n+1})| \pi_{n}=\widehat \pi_{n } ] -{\mathbf{e}}[v_{n+1}(\pi_{n+1})| \widehat \pi_{n } ] |}\nonumber\\ & & + { |{\mathbf{e}}[v_{n+1}(\pi_{n+1})-v_{n+1}(\widehat \pi_{n+1})| \widehat \pi_{n } ] |}.\label{eq dif k}\end{aligned}\ ] ] the second term in the right - hand side of eq . ( [ eq dif k ] ) is readily bounded by using proposition [ prop - lip - v ] stating that @xmath546 is lipschitz continuous @xmath547|\leq[v_{n+1}]{\mathbf{e}}\big[|\pi_{n+1}-\widehat \pi_{n+1}|\big|\widehat \pi_{n}\big].\ ] ] to deal with the first term in the right - hand side of eq . ( [ eq dif k ] ) , we need to use the special properties of quantization . indeed , one has @xmath548 so that we have the inclusion of @xmath23-fields @xmath549 . the law of iterated conditional expectations gives @xmath550={\mathbf{e}}\big[{\mathbf{e}}[v_{n+1}(\pi_{n+1})|(\pi_{n},s_{n})]\big| \widehat \pi_{n } \big].\ ] ] moreover , proposition [ prop - markov ] yields @xmath551={\mathbf{e}}[v_{n+1}(\pi_{n+1})|\pi_{n}]$ ] , as the conditional distribution of @xmath552 w.r.t . @xmath553 merely depends on @xmath31 . in addition , @xmath554 $ ] is @xmath555-measurable . one has then @xmath556-{\mathbf{e}}[v_{n+1}(\pi_{n+1})| \widehat \pi_{n } ] |}\\ & = & \left|{\mathbf{e}}\big[{\mathbf{e}}[v_{n+1}(\pi_{n+1})| \pi_{n}=\widehat \pi_{n } ] -{\mathbf{e}}[v_{n+1}(\pi_{n+1})| \pi_{n } ] \big|\widehat \pi_{n}\big]\right|\\ & = & |{\mathbf{e}}[kv_{n+1}(\widehat \pi_{n})-kv_{n+1}(\pi_{n})|\widehat \pi_{n}]|,\end{aligned}\ ] ] by definition of @xmath543 . finally , one has @xmath557-{\mathbf{e}}[v_{n+1}(\pi_{n+1})| \widehat \pi_{n } ] |}}\\ & \leq&2(c_g+[v_{n+1}]){\mathbf{e}}\left[|\pi_{n}-\widehat \pi_{n}|\big|\widehat \pi_{n}\right],\end{aligned}\ ] ] thanks to propositions [ prop - lip - k ] and [ prop - lip - v ] stating the lipschitz continuity of operator @xmath543 and function @xmath546 . @xmath175 + [ lemme - xi3 ] if @xmath558 satisfies condition ( [ conditiondelta ] ) , a upper bound for the third term @xmath559 is @xmath560\|\pi_{n+1}-\widehat \pi_{n+1}\|_{p}+(4c_{g}+2[v_{n+1}])\|\pi_{n}-\widehat \pi_{n}\|_{p}\\ & & + 2c_{g } ( 2 c_{\lambda})^{1/2}{\|s_{n+1}-\widehat s_{n+1}\|_{p}}^{1/2}.\end{aligned}\ ] ] [ [ proof-13 ] ] proof + + + + + one has @xmath561 the term involving operator @xmath543 was studied in the previous lemma . let us now study the term involving operator @xmath507 . set @xmath453 in @xmath454 , @xmath528 in @xmath459 and define @xmath562 . one has then @xmath563-{\mathbf{e}}[\alpha(\widehat \pi_{n},\widehat \pi_{n+1},\widehat s_{n+1})|\widehat \pi_{n}]\right|\leq a+b,\end{aligned}\ ] ] where@xmath564-{\mathbf{e}}[\alpha(\pi_{n},\pi_{n+1},s_{n+1})|\widehat\pi_{n}]\right|,\\ b=&|{\mathbf{e}}[\alpha(\pi_{n},\pi_{n+1},s_{n+1})-\alpha(\widehat \pi_{n},\widehat \pi_{n+1},\widehat s_{n+1})\big|\widehat \pi_{n}]| . \ ] ] using the boundedness of @xmath18 and @xmath546 as well as the lipschitz continuity of @xmath546 given in proposition [ prop - lip - v ] , we get a upper bound for the second term @xmath565 + [ v_{n+1}]{\mathbf{e}}\big[|\pi_{n+1}-\widehat \pi_{n+1}|\big|\widehat \pi_{n}\big]\nonumber\\ & & + 2c_{g}{\mathbf{e}}\left[|{\mathbbm{1}}_{\{s_{n+1}\leq u\}}-{\mathbbm{1}}_{\{\widehat s_{n+1}\leq u\}}|\big|\widehat \pi_{n}\right].\end{aligned}\ ] ] for the first term , we use the properties of quantization as in the previous proof to obtain @xmath566-{\mathbf{e}}[\alpha(\pi_{n},\pi_{n+1},s_{n+1})|\pi_{n}]\big| \widehat \pi_{n}\big]\right|.\end{aligned}\ ] ] we now recognize operator @xmath515 , and from propositions [ prop - lip - h ] and [ prop - lip - g ] , one has @xmath567\nonumber\\ & \leq & ( 3c_{g}+2[v_{n+1}]){\mathbf{e}}\left[|\widehat{\pi}_{n}-\pi_{n}\big|\widehat \pi_{n}\right].\end{aligned}\ ] ] we gather the bounds provided by eq . and to obtain @xmath568){\mathbf{e}}\big[|\pi_{n}-\widehat \pi_{n}\big ] + [ v_{n+1}]{\mathbf{e}}\big[|\pi_{n+1}-\widehat \pi_{n+1}|\big|\widehat \pi_{n}\big]\nonumber\\ & & + 2c_{g}{\mathbf{e}}\left[|{\mathbbm{1}}_{\{s_{n+1}\leq u\}}-{\mathbbm{1}}_{\{\widehat s_{n+1}\leq u\}}|\big|\widehat \pi_{n}\right].\end{aligned}\ ] ] finally , combining the result for operators @xmath507 and lemma [ xi3-k ] , we obtain @xmath569{\mathbf{e}}\left[|\pi_{n+1}-\widehat \pi_{n+1}|\big|\widehat \pi_{n}\right]+(4c_{g}+2[v_{n+1}]){\mathbf{e}}\left[|\pi_{n}-\widehat \pi_{n}|\big|\widehat \pi_{n}\right]\\ & & + 2c_{g}\max_{u\in gr(\delta)}{\mathbf{e}}\left[|{\mathbbm{1}}_{\{s_{n+1}\leq u\}}-{\mathbbm{1}}_{\{\widehat s_{n+1}\leq u\}}|\big|\widehat \pi_{n}\right].\end{aligned}\ ] ] we conclude by taking the @xmath34 norm in the equation above and using lemma [ xi3-indicator ] to bound the last term @xmath570\|\pi_{n+1}-\widehat \pi_{n+1}\|_p+(4c_{g}+2[v_{n+1}])\|\pi_{n}-\widehat \pi_{n}\|_p\\ & & + 2c_{g}(\eta^{-1}{\|s_{n+1}-\widehat s_{n+1}\|_{p}}+2\eta c_{\lambda}),\end{aligned}\ ] ] for some @xmath466 . the best choice for @xmath571 minimizing the error is when @xmath571 satisfies @xmath572 which yields @xmath573 . if @xmath558 satisfies condition ( [ conditiondelta ] ) , one has @xmath574 as required for this optimal choice . @xmath175 + finally , the fourth error term is bounded using lipschitz properties . [ lemme - xi4 ] the fourth term @xmath575 is bounded as follows @xmath576\|\pi_{n+1}-\widehat \pi_{n+1}\|_{p}+\|v_{n+1}-\widehat v_{n+1}\|_{p}.\end{aligned}\ ] ] [ [ proof-14 ] ] proof + + + + + one has @xmath577\nonumber\\ & & \vee { \mathbf{e}}[v_{n+1}(\widehat{\pi}_{n+1})-\widehat v_{n+1}(\widehat{\pi}_{n+1})| \widehat{\pi}_{n } ] \big\|_{p}\nonumber\\ & \leq&\|v_{n+1}(\widehat{\pi}_{n+1})-\widehat v_{n+1}(\widehat{\pi}_{n+1})\|_{p}.\label{jvchap}\end{aligned}\ ] ] we now introduce @xmath578 to split this term into two differences . the lipschitz continuity of @xmath546 stated by proposition [ prop - lip - v ] allows us to bound the first term while we recognize @xmath391 and @xmath579 in the second one . @xmath580\left\|\pi_{n+1}-\widehat \pi_{n+1}\right\|_{p}+\|v_{n+1}-\widehat v_{n+1}\|_{p}.\end{aligned}\ ] ] hence , the result . @xmath175 + as in the previous section , we follow the idea of @xcite and we use both the markov chain @xmath581 and its quantized approximation @xmath582 to approximate the expression of the @xmath0-optimal stopping time introduced in definition [ def - sn - epsilon ] . we check that we thus obtain actual stopping times for the observed filtration @xmath29 and that the expected reward when stopping then is a good approximation of the value function @xmath496 . for all @xmath583 and @xmath483 , we denote @xmath584 . let @xmath585 for @xmath272 and @xmath319 , we define @xmath586 let now for @xmath120 , @xmath587 and set @xmath588 the following result is a direct consequence of proposition [ propb5 ] . it is a very strong result as it states that the numerically computable random variables @xmath589 are actual @xmath29-stopping times . for @xmath483 , @xmath589 is an @xmath29-stopping time . we now intend to prove that stopping at time @xmath590 provides a good approximation of the value function @xmath496 . for all @xmath591 and @xmath483 we therefore introduce the performance when abiding by the stopping rule @xmath592 and the corresponding random variables @xmath593,\qquad \overline{v}_{n}=\overline{v}_{n}(\pi_{n}).\ ] ] [ th arret chap ] suppose that for all @xmath388 , @xmath594 one has then the following bound for the error between the expected reward when stopping at time @xmath589 and the value function @xmath595\|\pi_{n+1}-\widehat \pi_{n+1}\|_{p}\\ & & + b\|s_{n+1}-\widehat s_{n+1}\|_{p}^{1/2},\end{aligned}\ ] ] where @xmath596 , @xmath597 $ ] , @xmath494 $ ] defined in proposition [ prop - lip - v ] . it is important to notice that @xmath598 and thus @xmath599 . therefore , the previous theorem proves that @xmath600 goes to zero when the quantization errors @xmath601 go to zero . in other words , the expected reward @xmath602 when stopping at the random time @xmath590 can be made arbitrarily close to the value function @xmath496 of the partially observed optimal stopping problem and hence @xmath590 is an @xmath0-optimal stopping time . [ [ proof-15 ] ] proof + + + + + the first step consists in finding a recursion satisfied by the sequence @xmath603 in order to compare it with the dynamic programming equation giving @xmath604 . let @xmath388 . first of all , proposition [ lemme value fonction ] gives @xmath605}\\ & = & \sum_{k=0}^{n - n-1}\sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{k}\leq \widehat{u}_{n - n}\}}{\mathbbm{1}}_{\{\widehat{r}_{n - n , k}<{t^{*}}_{i}\}}g\circ\phi(x_{i},\widehat{r}_{n - n , k})e^{-\lambda(x_{i},\widehat{r}_{n - n , k})}\pi_{k}^{i}|\pi_0]\\ & & + \sum_{i=1}^{q}{\mathbf{e}}[{\mathbbm{1}}_{\{t_{n - n}\leq \widehat{u}_{n - n}\}}g(x_{i})\pi_{n}^{i}|\pi_0].\end{aligned}\ ] ] the term corresponding to @xmath288 in the above sum equals @xmath606 . taking the conditional expectation w.r.t . @xmath292 in the other terms and noticing that one has @xmath607 yield @xmath608=hg(\pi_{0},\widehat{r}_{n - n,0})+{\mathbf{e}}[\xi''{\mathbbm{1}}_{\{s_{1}\leq \widehat{r}_{n - n,0}\}}|\pi_0],\ ] ] with @xmath609.\end{aligned}\ ] ] we now make use of the markov property of the sequence @xmath151 in the term @xmath610 . similarly to lemma [ rnk - epsilon - theta ] , for @xmath120 , on the set @xmath611 , one has @xmath612 for all @xmath326 . thus , on the set @xmath611 , one has @xmath613 . recall that @xmath614 . we may therefore apply the markov property . using proposition [ lemme value fonction ] , we now obtain @xmath615 . finally , we have @xmath616 recall that @xmath617 and apply the translation operator @xmath618 to obtain the following recursion @xmath619 we are now able to study the error between @xmath620 and @xmath621 . let us recall that , from its definition , @xmath622 equals either @xmath623 or @xmath624 . in the latter case , notice that @xmath625 . eventually , one has @xmath626 to bound the first term @xmath47 , we introduce the function @xmath546 . one has @xmath627 let us study these four terms one by one . by definition of @xmath543 , the first term @xmath628 is bounded by @xmath629 $ ] . for the second term @xmath630 , we use proposition [ prop - lip - k ] stating the lipschitz continuity of the operator @xmath543 . the term third term @xmath631 is bounded by lemma [ xi3-k ] and a upper bound of the fourth term @xmath632 is given by eq . ( [ jvchap ] ) . thus , one obtains @xmath633)\|\pi_{n}-\widehat \pi_{n}\|_{p}\\ & & + 2[v_{n+1}]\|\pi_{n+1}-\widehat \pi_{n+1}\|_{p}. \end{aligned}\ ] ] we now turn to the second term @xmath634 . in the following computations , denote @xmath635 . its definition yields we split this expression into four differences again . on the set @xmath637 , one has the equality @xmath638 . hence , one this set , one obtains from eq . ( [ prog dyn vbar ] ) @xmath639 for the other terms , we use propositions [ prop - lip - h ] and [ prop - lip - g ] for the lipschitz continuity of @xmath507 and eq . ( [ eq j - jchap ] ) and ( [ jvchap ] ) to obtain @xmath640)\|\pi_{n}-\widehat \pi_{n}\|_{p}+2[v_{n+1}]\|\pi_{n+1}-\widehat \pi_{n+1}\|_{p}\\ & & + 2c_g(2c_{\lambda})^{1/2}\|s_{n+1}-\widehat s_{n+1}\|_{p}^{1/2 } , \end{aligned}\ ] ] after optimizing @xmath571 . the result is obtained by taking the maximum between @xmath47 and @xmath634 . we apply our procedure to a simple pdmp similar to the one studied in @xcite . let @xmath641 . for @xmath65 and @xmath642 , the flow is defined by @xmath643 so that @xmath644 . we set the jump rate to @xmath645 for some @xmath646 and the transition kernel @xmath647 to the uniform distribution on a finite set @xmath648 . thus , the process evolves toward 1 and the closer it gets to 1 , the more likely it will jump back to some point of @xmath93 . a trajectory is represented in figure [ plotecg ] . jump time with @xmath649 , @xmath650 and @xmath651 . the dotted lines represent the possible post - jump values.,width=302 ] the observation process is @xmath652 where @xmath653 and @xmath654 for some @xmath655 . finally , we choose the reward function @xmath656 . our assumptions thus clearly hold . simulations are run with @xmath649 , @xmath650 , @xmath657 , @xmath658 and @xmath659 . the numerical approximation is implemented as follows . first , we make an exact simulator for the sequence @xmath660 . from the values of @xmath661 , one builds the observation sequence @xmath662 that allows for a recursive computation of the filter process thanks to proposition [ prop - rec - filtre ] . thus , we can simulate trajectories of the markov chain @xmath381 that we feed into the clvq algorithm to obtain quantization grids . by monte carlo simulations , we can also estimate the quantization errors . to run our numerical procedure , one then needs to choose the parameter @xmath558 satisfying conditions ( [ eq - condition - delta ] ) and ( [ conditiondelta ] ) . in this special case , they boil down to @xmath663 we have chosen @xmath558 just above the monte carlo approximation of the lower bound . the values are given in the second column of table [ ecg_table ] for different grids sizes . then , we recursively compute the approximated value functions @xmath664 on the quantization grids . the conditional expectations are now merely weighted sums . the approximation we obtain for the value function of the partially observed optimal stopping problem are given in the fourth column of table [ ecg_table ] . finally , we implemented the construction of our @xmath0-optimal stopping time and ran @xmath665 monte carlo simulations to compute its mean performance . the results are given in the third column of table [ ecg_table ] . the exact value of @xmath496 is unknown but one has as in @xcite , @xmath666 \leq v_{0}=\sup_{\sigma\in\sigma_{n}^{y}}{\mathbf{e}}[g(x_{\sigma } ) ] \leq { \mathbf{e}}\big [ \sup_{0\leq t \leq t_{n}}g(x_{t})\big].\ ] ] both the first and the last term may be estimated by monte carlo simulations . one has thus , with @xmath667 trajectories , @xmath668=0.9944 $ ] . the theoretical bound @xmath669 of the error @xmath670 provided by theorem [ theo - conv ] is computed using the approximated quantization errors . this bound decreases as the number of points in the quantization grids increases , as expected . moreover , we computed the empirical bound given by eq . @xmath671-\widehat v_{0}| \big\}$ ] . .simulation results . the terms @xmath672 and @xmath669 respectively denote an empirical bound and the theoretical bound provided by theorem [ theo - conv ] for the error @xmath670 . [ cols="^,^,^,^,^,^",options="header " , ] in this section , we study the special structure of @xmath29-stopping times . [ prop - tn - stop - time ] for all @xmath673 , @xmath138 is an @xmath29-stopping time . [ [ proof-16 ] ] proof + + + + + notice that for all @xmath92 , @xmath674 . this stems from the absolute continuity of the distribution of the random variables @xmath186 since @xmath675 hence , for all @xmath92 and @xmath676 , one has @xmath131 a.s . @xmath677 where we denote @xmath678 . the process @xmath679 is @xmath680-adapted thus @xmath681 and since the filtration @xmath680 contains the @xmath131-null sets , one has @xmath682 . for all @xmath92 , @xmath138 is therefore an @xmath29-stopping time . @xmath175 + + we now recall theorem a2 t33 from @xcite concerning the structure of the stopping times for point processes and apply it in our case . define the filtration @xmath683 as follows @xmath684 [ theo - bremaud ] let @xmath23 be an @xmath683-stopping time . for all @xmath92 , there exists a @xmath685-measurable non negative random variable @xmath686 , such that one has @xmath687 our observation process @xmath15 being a point process that fits the framework developed in @xcite , we apply this theorem to @xmath29-stopping times . for all @xmath642 , one has @xmath688 . [ [ proof-17 ] ] proof + + + + + first prove that @xmath689 . let @xmath690 and @xmath691 , one has @xmath692 indeed , in the above equation , we used that @xmath693 and @xmath694 are assumed to be deterministic . for the reverse inclusion , let @xmath690 , @xmath120 and @xmath691 . recall that @xmath695 . one has @xmath696 since @xmath15 is @xmath680-adapted and @xmath138 is an @xmath29-stopping time from lemma [ prop - tn - stop - time ] . therefore , one has @xmath697 , showing the result . @xmath175 + + we may therefore apply theorem [ theo - bremaud ] to @xmath29-stopping times . [ theo - bremaud - adapted ] let @xmath23 be an @xmath29-stopping time . for all @xmath92 , there exists a non negative random variable @xmath686 , @xmath156-measurable such that one has @xmath687 we outline the following result , which is a direct consequence of the above theorem , because it will be used several times in our derivation . [ lemme - tech - event ] let @xmath23 be an @xmath29-stopping time and @xmath698 be the sequence of random variables associated to @xmath23 as introduced in theorem [ theo - bremaud - adapted ] . for all @xmath673 , @xmath699 [ [ proof-18 ] ] proof + + + + + theorem [ theo - bremaud - adapted ] states that on the event @xmath700 , on has @xmath701 so that , still on the event @xmath700 , one has @xmath702 . we deduce the result from this observation . @xmath175 + + we now investigate the effect of the translation operator of the markov chain @xmath703 on the @xmath29-stopping times . proposition [ prop - markov ] states that @xmath703 is a @xmath238-markov chain . let us consider its canonical space @xmath704 . thus , for @xmath705 , one has @xmath706 . besides , we define the _ translation operator _ @xmath707 we then define @xmath708 and recursively for @xmath709 , @xmath710 . thus , for all @xmath711 , one has @xmath712 . as @xmath713 , one has @xmath714 the next results of this section are given without proof because their proofs follow the very same lines as in @xcite from which they are adapted . however , notice that the results from @xcite can not be applied directly to our case because the sequence @xmath703 , although it is a markov chain , is not the underlying markov chain of some pdmp . set now @xmath276 . from theorem [ theo - bremaud - adapted ] , for all @xmath92 , there exists a non negative @xmath156-measurable random variable @xmath686 , such that , on the event @xmath715 , one has @xmath716 [ lemmeb2 ] let @xmath23 be an @xmath29-stopping time and @xmath698 be the sequence of random variables associated to @xmath23 as introduced in theorem [ theo - bremaud - adapted ] . let @xmath717 and for @xmath289 , @xmath718 . one has then @xmath719 [ rq - rn - rnbar ] this lemma proves that in theorem [ theo - bremaud - adapted ] , the sequence @xmath720 can be replaced by @xmath721 . therefore , we can assume , without loss of generality that the sequence @xmath720 satisfies the following condition : for all @xmath92 , @xmath722 on the event @xmath723 . since @xmath724 and @xmath302 is @xmath347-measurable , there exists a sequence of real - valued measurable functions @xmath725 defined on @xmath726 such that @xmath727 , where @xmath728 . [ defb3 ] let @xmath23 be an @xmath29-stopping time and @xmath729 be the sequence of functions associated to @xmath23 as introduced in remark [ rq - rn - rnbar ] . let @xmath730 and @xmath731 be a sequence of functions defined on @xmath732 by @xmath733 and for @xmath289 , @xmath734 [ propb4 ] let @xmath23 be an @xmath29-stopping time and @xmath735 ( respectively , @xmath731 ) be the sequence of functions associated to @xmath23 as introduced in lemma [ lemmeb2 ] ( respectively , in definition [ defb3 ] ) . assume that @xmath736 . for all @xmath737 , one has then @xmath738 and @xmath739 , with @xmath306 : @xmath740 defined as @xmath741 [ propb5 ] let @xmath742 be a sequence of non negative random variables such that for all @xmath13 , @xmath743 is @xmath156-measurable and @xmath744 on @xmath745 . we define @xmath746 then @xmath747 is an @xmath29-stopping time . [ corb6 ] let @xmath23 be an @xmath29-stopping time and @xmath306 be the mapping associated to @xmath23 introduced in proposition [ propb4 ] . for all @xmath748 , @xmath749 is a @xmath29-stopping time . the objective of this section is to prove the technical lemma [ lemme - tech - esp - cond ] used in the proof of proposition [ lemme value fonction ] . [ lemme - tech - esp - cond ] for all @xmath750 , one has @xmath751={\mathbbm{1}}_{\{r_{k}<{t^{*}}(z_{k})\}}e^{-\lambda(z_{k},r_{k})}.$ ] [ [ proof-19 ] ] proof + + + + + first recall some results concerning the random variables @xmath752 , details may be found in @xcite . after a jump of the process to the point @xmath753 , the survival function of the time until the next jump is @xmath754 define its generalized inverse @xmath755 . then , for all @xmath737 , one has @xmath756 , where @xmath757 are i.i.d . random variables with uniform distribution on @xmath758 $ ] independent from @xmath759 . thus , one has @xmath751={\mathbf{e}}[f(\upsilon_{k},z_{k},r_{k})|{\mathfrak{f}}_{t_{k}}]$ ] where @xmath760 . as @xmath761 is @xmath759-measurable , @xmath757 is independent from @xmath759 and @xmath762={\mathbbm{1}}_{\{r<{t^{*}}(z)\}}e^{-\lambda(z , r)}$ ] , ( * ? ? ? * proposition 11.2 ) yields the result . in this section , we derive the lipschitz properties of our operators in order to obtain them for the value functions @xmath260 . similarly to the proof of proposition [ prop - markov ] , we first derive the integral form of operators @xmath282 and @xmath763 . now , notice that the functions @xmath767 and @xmath768 are not continuous . however , they are cdlg with a finite number of jumps . therefore , they can be rewritten as sums of continuous functions as follows . [ rq - continu - hm ] for all @xmath769 and for all @xmath479 , @xmath505 and @xmath764 , the functions @xmath780 , @xmath781 and @xmath510 are continuous . moreover , they are constant on @xmath782 $ ] and on @xmath783 and one has @xmath784 since the function @xmath781 is constant on the intervals @xmath782 $ ] and @xmath783 , we may assume that @xmath528 , @xmath787 $ ] so that one has @xmath788 and similarly for @xmath789 . then , on the one hand , one has @xmath790 on the other hand , lemma a.1 in @xcite yields @xmath791_{2}+c_{g}c_{\lambda})|u-\tilde u|,\ ] ] showing the result . @xmath175 + + the following technical lemma will be useful to derive the lipschitz properties of the operator @xmath792 . the first part of its proof is adapted from @xcite . let @xmath795{t^{*}}_{m};{t^{*}}_{m+1}[$ ] and @xmath796 . in the following computation , we denote @xmath797 and @xmath798 , one has @xmath799 notice that @xmath800 so that the second sum above reduces to @xmath801 . finally , one has @xmath802 as @xmath803 and @xmath804 , one obtains on the one hand , one clearly has @xmath810 on the other hand , one has @xmath811 besides , we have assumed that @xmath380 is lipschitz continuous so that one has @xmath812\big|\psi(\pi , y',s')-\psi(\tilde \pi , y',s')\big|.\ ] ] thus , one has @xmath813\sum_{i=1}^{q}\pi^{i}\int_{0}^{{t^{*}}_{i}}\int_{{\mathbb{r}}^{d}}\big|\psi(\pi , y',s')-\psi(\tilde \pi , y',s')\big|\\ & & \sum_{j=1}^{q}q\big(\phi(x_{i},s'),x_{j}\big)f_{w}(y'-\varphi(x_{j}))\lambda\circ\phi(x_{i},s')e^{-\lambda(x_{i},s')}dy'ds'\\ & \leq & c_{v}|\pi-\tilde \pi|+[v]\sum_{m=0}^{q-1}\sum_{i = m+1}^{q}\pi^{i}\int_{{t^{*}}_{m}}^{{t^{*}}_{m+1}}\int_{{\mathbb{r}}^{d}}\big|\psi(\pi , y',s')-\psi(\tilde \pi , y',s')\big|\\ & & \times\sum_{j=1}^{q}q\big(\phi(x_{i},s'),x_{j}\big)f_{w}(y'-\varphi(x_{j}))\lambda\circ\phi(x_{i},s')e^{-\lambda(x_{i},s')}dy'ds'\\ & \leq & c_{v}|\pi-\tilde \pi|+[v]\sum_{m=0}^{q-1}\int_{{t^{*}}_{m}}^{{t^{*}}_{m+1}}\int_{{\mathbb{r}}^{d}}\big|\psi(\pi , y',s')-\psi(\tilde \pi , y',s')\big|\overline{\psi}_{m}(\pi , y',s')dy'ds'.\end{aligned}\ ] ] the previous lemma provides the result . @xmath175 + as in the proof of proposition [ prop - lip - h ] , we may assume without loss of generality that @xmath528 , @xmath787 $ ] so that one has @xmath815 and similarly for @xmath816 . the second term does not depend on @xmath528 thus @xmath817 as @xmath818 by proposition [ prop - rec - filtre ] . this yields the result . @xmath175 + we proved that @xmath261 is the value function of the optimal stopping problem with horizon @xmath262 thus one has @xmath828\leq c_{g}.$ ] therefore @xmath261 is bounded and @xmath827 . the second assessment is proved by backward induction . let @xmath144 , @xmath829 . one has @xmath830 therefore , we have the result for @xmath831 with @xmath832\leq c_{g}$ ] . moreover , since @xmath833 for @xmath388 , proposition [ prop - lip - l ] yields @xmath493\leq 3c_{g}+2[v_{n+1}]$ ] which proves the propagation of the induction . @xmath175 + g. pags , h. pham , and j. printems . optimal quantization methods and applications to numerical problems in finance . in _ handbook of computational and numerical methods in finance _ , pages 253297 . birkhuser boston , boston , ma , 2004 .

stars form in dense and cool molecular clouds . when the local density is high enough , the matter can gravitationally collapse and form a young stellar object ( yso ) . in the early phases , the thick envelope dominates the emission from the yso and hides what is going on within ( class i ) . eventually , the envelope flattens out to a circumstellar accretion disk . this disk still causes an infrared ( ir ) excess above the level of a stellar photosphere ( class ii or classical t tauri star - ctts ) , which can be used to distinguish those objects from main - sequence stars , for example using the _ spitzer _ space telescope @xcite infrared array camera ( irac , * ? ? ? * ) . when the disk is cleared , the ir colors of the yso match those of main - sequence stars ( class iii or weak - lined t tauri star - wtts ) . in addition to the circumstellar absorption , many ysos are embedded in the molecular cloud , so that even class iii objects can appear reddened . the accretion disk does not reach down to the central star . instead , the inner edge of the gas disk is truncated by the stellar magnetic field . the inner radius of the optically thick dust in the disk is larger than the inner radius of the gas disk and mostly given by the dust - sublimation temperature . some of the mass in the circumstellar disk condenses into planets , some is blown out by accretion - driven disk and stellar winds , and is accreted onto the central star . this accretion can happen via magnetically confined accretion funnels ( e.g. , * ? ? ? * ) or via some magneto - hydrodynamical instability ( e.g. , * ? ? ? t tauri stars ( tts ) were originally identified by their variability @xcite long before anybody realized that tts are indeed pre - main sequence stars . the dominant timescale in the optical is the stellar rotation period , typically a few days to a week or more . ysos can have cool spots caused by magnetic activity similar to our sun and also hot spots which mark the impact points of the accretion funnels onto the stellar surface ( see , e.g. , review by * ? ? ? this impact happens at free - fall velocities up to 500 km s@xmath1 ; thus , the accretion shock heats the accreted mass to x - ray emitting temperatures . in the optical , the accretion region appears as emission that often is approximated as a blackbody with temperature @xmath2 k ( @xcite , but see also @xcite who argue that line emission contributes to the veiling in addition to a continuum ) . variability in the mass accretion rate can lead to changes in the hot spot signatures . the dynamical timescale that controls the accretion is the keplerian period of the inner disk where the accretion funnels start . the inner disk radius is found close to the co - rotation radius leading to a typical timescale of a few hours for typical masses and rotation periods of ysos . indeed optical variability with amplitudes around 0.1 mag is often observed in ctts on this timescale @xcite . another source of variability related to the accretion could be oscillations of the accretion shock on timescales of seconds . this has been predicted theoretically ( e.g. , * ? ? ? * ) , but is not observed so far , possibly because the accretion spot separates into many small funnels that oscillate independently at different phases and frequencies . however , @xcite find indications that strong accretion in v1647 ori could excite radial pulsations of the star itself . one of the largest classes of short timescale ( @xmath3 days ) optical and ir variability in ysos is that due to variable extinction events @xcite . these come in three categories - aa tau - type variables ( stars with broad , periodic flux dips , whose amplitudes can be up to a magnitude or more in the optical ) , presumably due to our line of sight passing through a warp in the inner circumstellar disk ; stars with similar or narrower flux dips that have no obvious periodicity - presumably due to stars where our line of sight passes close to the disk and where disk instabilities can levitate dust high enough above the plane to intersect our line of sight briefly ; and stars with narrow , periodic flux dips - perhaps where our line of sight is being intersected by dust entrained in material accreting onto the star in a funnel flow . about 20% of the ysos in ngc 2264 fall into one of these categories in the sample of @xcite . ysos can also vary on much longer timescales . variability on the timescales of years could be caused by changing circumstellar extinction for a keplarian disk around a solar - mass yso this timescale translates to a radius of a few au ) or by massive accretion events when a significant fraction of the disk mass drains onto the yso @xcite . in this case the accretion luminosity can outshine the yso by orders of magnitudes and it takes months to years ( in the case of exor outbursts , * ? ? ? * ) or even centuries until the accretion decays back down to the original level . in any lightcurve , several of the processes dicussed above can contibute to the observed variability at the same time and it depends on the properties of each object which one dominates and if secondary effects can be detected in the lightcurve . for example , cool spots , hot spots , absorption , and massive accretion events can all influence the same optical light curve . another case are x - rays , where the flux and the spectrum can change due to periodic absorption , variability in the accretion rate ( tw hya , * ? ? ? * ) or coronal activity similar to what is seen on the sun . in most ysos , the last point is dominant and x - ray lightcurves often show the fast rise in flux and temperature and a slower decay characteristic of coronal activity ( see , e.g. , the _ chandra _ monitoring of the orion nebular cluster , * ? ? ? * ; * ? ? ? * ) . the spectral energy distribution ( sed ) of ysos in the optical is dominated by the stellar photosphere and the accretion spot . thus , optical monitoring is very effective for understanding the stellar rotation and the accreting spot . however , the disk radiates mostly at longer wavelengths , which are probed in the ir observations presented in this article . depending on the mass of the disk and the size of the inner hole , the disk will start to dominate the sed at the @xmath4 band or in the irac bands at @xmath5 m and @xmath6 m . simple disk models still treat the disk as a static and axisymmetric structure , but observationally it now seems that the disk is in fact `` a bubbling , boiling , wrinkled , dented , warped mass of gas and dust '' @xcite , see also @xcite . this paper is part of the ysovar ( young stellar object variability ) project , which has monitored the ( onc ) and eleven smaller star forming regions with irac in @xmath5 m and @xmath6 m to understand the mid - ir variability of ysos . first results on the onc are published in @xcite . more details of the observing strategy and an overview of the data can be found in @xcite ( from now on `` paper i '' ) . comparing data from all clusters , paper i defines certain cut - off values for the data reduction , e.g. how much variability in a lightcurve is required to reliably identify an object as variable . in the analysis , paper i concentrates on variability in the ir on timescales of years . in this article , we present a _ spitzer_/irac monitoring campaign of the star forming region l1688 in the mid - ir to characterize the variability timescales and amplitudes as well as the color changes in the mid - ir in much more detail for the objects in l1688 than paper i on timescales up to two years . the structure of this paper is as follows : first , we introduce l1688 , the star forming region targeted by these observations ( section [ sect : thestarformingregionl1688 ] ) . in section [ sect : observationsdatareductionandauxiliarydata ] we introduce the data reduction and discuss source lists and stellar properties obtained from the literature . section [ sect : midirvariability ] classifies all sources according to their variability . section [ sect : resultsanddiscussion ] presents our results and discusses physical models to explain the observed features in the lightcurves . we end with a summary and some conclusions in section [ sect : summary ] . lynds 1688 ( l1688 ) is a sub - cloud of the @xmath7 ophiuchus star forming region , one of the best - studied young clusters in the sky ( see e.g. review by * ? ? ? the central region of l1688 is very dense and deeply embedded ( @xmath8 mag , see figure [ fig : map ] ) . thus , all surveys of the regions necessarily miss some cluster members . an extinction limited , spectroscopic survey @xcite finds an average age of 3.1 myr for a 6.8 pc@xmath9 region centered on l1688 and no significant deviation from the initial mass function . earlier studies concentrated on the deeper embedded core and found a much younger age of 0.3 myr . some , but not all , of this difference is due to the specific reddening laws or pre - main sequence evolutionary tracks used in these studies ( see discussion in * ? ? ? @xcite present photometric and spectroscopic monitoring for five ysos in l1688 nearly simultaneous with the _ spitzer _ observations discussed here . they do not see any correlation between the hydrogen emission lines that are usually considered accretion indicators and the features in the ir lightcurves of their targets , indicating that the relatively modest variability they observed is not caused by changes in the accretion rate . additional notable objects with well - sampled nir lightcurves are @xcite and , which show eclipses with periods of 131 and 93 days , respectively . these sources can be interpreted as multiple systems , where one or more components are eclipsed by a warped circumstellar disk . thus far , the most comprehensive study of near - ir variability of ysos in l1688 is @xcite ( see references therein for other ir variability studies ) , who make use of a two micron all sky survey ( 2mass * ? ? ? * ) calibration field that overlaps l1688 so each source has up to 1584 datapoints in @xmath10 , @xmath11 , and @xmath4 spanning 2.5 years . they find 79% of the known ysos to be variable . in total , 32 sources are periodic ( including cool starspots , hot accretion spots and 6 systems with eclipses ) , 31 sources show a long - term trend and 40 sources vary aperiodically on shorter timescales . the new data presented in this article complements the @xcite study with observations at longer wavelengths . in this section we briefly describe the data reduction for the _ spitzer _ and _ chandra _ observations . a detailed account of the observations , data processing and the source extraction is given in paper i. we also give an overview of auxiliary data on the stellar properties retrieved from the literature , which we need to test if the variability characteristics depend on the central star . we then assess the cluster membership and sed class for every source with a usable lightcurve . ccc 29267200 & 2010 - 04 - 12 11:00:41 & 2010 - 04 - 12 11:18:44 + 29266688 & 2010 - 04 - 12 17:02:37 & 2010 - 04 - 12 17:28:01 + 29266176 & 2010 - 04 - 12 22:48:49 & 2010 - 04 - 12 23:09:28 + 29265664 & 2010 - 04 - 13 08:44:31 & 2010 - 04 - 13 09:05:35 + 29265408 & 2010 - 04 - 13 20:09:17 & 2010 - 04 - 13 20:26:33 + three fields in l1688 were observed with _ spitzer _ in four observing windows from 2010 - 04 - 12 to 2010 - 05 - 16 ( visibility window 1 ) , 2010 - 09 - 22 to 2010 - 10 - 27 ( visibility window 2 ) , 2011 - 04 - 20 to 2011 - 05 - 23 ( visibility window 3 ) , and 2011 - 10 - 01 to 2011 - 11 - 06 ( visibility window 4 ) . these windows are consecutive visibility periods dictated by the _ spitzer _ orbit @xcite . they are shown visually in figure [ fig : cadence ] . in the first visibility window , the sampling is much denser in time than in the later visibility windows . for most sources , about 70 observations with irregular time intervals to aid period detection were obtained in visibility window 1 . in the first visibility window , we use a repeating pattern of 8 observations every 3.5 days . within the 3.5 day period the time - step increases from 2 to 16 hours . in the last three visibility windows , the time steps increase linearly with step lengths of roughly 1 , 2 , 3 , ... days to again sample multiple variability frequencies equally . this irregular sampling minimizes aliasing with the observation frequency . less than ten datapoints per source are taken in each of the later three visibility windows . in total , there are 108 observations with a total mapping time of 30.7 hours . table [ tab : aor ] lists the time of each observation . they can be found under program identification number ( pid ) 61024 in the spitzer heritage archive . each observation consists of six dithers in irac mapping mode using the high - dynamic - range ( hdr ) data acquisition mode which obtains a 0.4 s and a 10.4 s exposure for each pointing . the three fields chosen were observed with the irac 1 and irac 2 channels ( effective wavelengths 3.6@xmath12 m and 4.5@xmath12 m ) . both channels operate simultaneously , but their fields - of - view are non - overlapping . thus , each target field is observed in two consecutive pointings , one for each channel . a secondary field is observed in the secondary channel while the primary channel is observing the target field . in visibility windows 1 and 3 , sources south of the main fields have only irac 1 data , while those to the north only have irac 2 data . in visibility windows 2 and 4 , the situation is reversed . not all sources in the central fields have usable data in both bands , because they might be too bright or too faint in one channel , or fall on the edge of the map . additional sources with two band coverage are found where the northern side field of one target field overlaps with the southern side field of another target field . here we summarize the main data reduction and processing steps described in detail in paper i. basic calibrated data ( bcd ) are obtained from the _ spitzer _ archive . further data reduction is performed with the idl package cluster grinder @xcite , that treats each bcd image for bright source artifacts . aperture photometry is performed on individual bcds with an aperture radius of @xmath13 . to increase the signal - to - noise ratio and to reject cosmic rays , the photometry from all bcds in each observation is combined . the reported value is the average brightness of all bcds within that observation that contain the source in question , after rejecting outliers . the photometric uncertainties obtained from the aperture photometry are , particularly for faint sources , only lower limits to the total uncertainty , since distributed nebulosity often found in star forming regions can contribute to the noise . to improve these estimates , paper i introduces an error floor value that is added in quadrature to the uncertainties of individual photometric points . the value of the error floor is 0.01 mag for irac 1 and 0.007 mag for irac 2 . we cross - match sources from individual observations with a matching radius of @xmath14 with each other and with the 2mass catalog , which is used as a coordinate reference . all photometric measurements performed in the context of the ysovar project are collected in a central database , which we intend to deliver to the infrared science archive ( irsa ) for general distribution . data for this article were retrieved from the ysovar database on 2013 - 10 - 31 and further processed using custom routines in python available at https://github.com/ysovar . we visually checked all frames for lightcurves that are classified as variable in section [ sect : midirvariability ] and removed datapoints visibly affected by instrumental artifacts ( cosmic rays , read - out streaks for bright neighbors ) . figure [ fig : map ] shows the distribution of the sources with lightcurves in our input catalog overlayed on a larger irac 1 map observed during the cryogenic mission . in this article , we consider only objects that have at least five datapoints in our irac 1 or irac 2 lightcurves . a stricter definition is employed in paper i , where only sources with more than five datapoints _ in the fast - cadence data ( first visibility window ) _ are used . in l1688 , 822 of the total list of 882 sources fullfill this stricter condition . in table [ tab : tab2 ] they are marked in the column ` standardset ` . despite the careful data reduction described above , some residual artifacts remain in the _ spitzer _ lightcurves . in this section , we search for artifacts that are related to the position on the detector . compared with other clusters in the ysovar project , l1688 is particularly suited to discover these kinds of effects because the fields observed in l1688 have almost no rotation within one visibility window . the spacecraft orientation flips between visibility windows , so that most instrumental artifacts produce lightcurves that have one level in visibility window 1 and 3 and another level in visibility window 2 and 4 . figure [ fig : instrumentallcs ] shows sources with a magnitude between 8 and 15 in @xmath6 m , that might fall in this category . the relative difference in magnitude is smaller for brighter sources , but since the photometric errors are also smaller these instrumental effects can still be significant . we visually inspected every frame for a sample of sources with lightcurves similar to those in figure [ fig : instrumentallcs ] to identify the cause for the artifacts . we found that several different effects can cause these steps in the lightcurve . some sources are close to a detector edge , such that the background is not well - determined , some are in the wings of the point - spread function ( psf ) of a bright neighbor , and some show residuals from hot pixels . proximity to a detector edge or hot pixel affects only one of the spacecraft orientations ; even the psf wings change with the orientation since the psf is not circular . sufficiently strong intrinsic variability can mask this offset between visibility windows and for any individual source , this instrumental effect can not be distinguished from intrinsic variability with a period of one year . thus , we use a statistical approach to quantify the number of lightcurves that suffer from this problem . for each channel , we have about 200 lightcurves with datapoints in all four observing visibility windows ( the total number of lightcurves is larger , but only for sources in the three target fields do we have data for all four visibility windows ) . if the mean magnitude in a visibility window depends on the detector position of the source , this source will on average be brighter in visibility window 1 and 3 and fainter in visibility window 2 and 4 ( or vice - versa ) . we calculate the mean magnitude in each visibility window and test whether the two brightest mean magnitudes belong to visibility window [ 1,3 ] or [ 2,4 ] or any other combination ( [ 1,2 ] , [ 1,4 ] , [ 2,3 ] , [ 3,4 ] ) . we use the following abbreviated notation : when we calculate the mean magnitudes in each visibility window for a sample of @xmath15 sources , then @xmath16:@xmath17:@xmath18 means that @xmath16 sources have the brightest mean magnitudes in visibility windows [ 1,3 ] , @xmath17 sources in visibility windows [ 2,4 ] and @xmath18 sources have their two brightest mean magnitudes in any other combination of visibility windows . if the difference in the mean magnitude between visibility windows is unrelated to the position on the detector , we expect the ratio @xmath1917%:17%:67% . in contrast , the observed lightcurves have the ratio 47:66:75=25%:35%:40% for irac 1 and 52:59:95=25%:29%:46% for irac 2 . this is incompatible with the expected multinomial distribution ( the probability to observe a distribution at least as far from the expected @xmath20 by chance is @xmath21 ) . in each case , about 50 lightcurves , a quarter of the sample , need to be shifted from the first two bins to the last bin to make the observed distribution compatible with the expected distribution . this implies that about a quarter of all lightcurves suffer from the artifacts described above . however , both irac channels use independent detectors and thus the chance that both channels are affected for the same source and that the effect goes the same way ( bright - faint , vs. faint - bright ) is low . below in section [ sect : midirvariability ] we identify variable sources using a stetson and a @xmath22 test . the limits in those tests are designed to be conservative and indeed we find a distribution of 5:7:24=14%:19%:67% for irac 1 and 5:5:27=14%:14%:73% for irac 2 , when we restrict the sample to those lightcurves that will be classified as variable below . both ratios are fully compatible with the expected multinomial distribution @xmath20 . this shows that the limits we apply are conservative enough that the sample of stars we identify as variable has no or only few sources where the variability is not due to intrinsic source variability . therefore it is not necessary to remove any source based on magnitude jumps between visibility windows . only one of the four lightcurves shown in figure [ fig : instrumentallcs ] ( sstysv j162727.53 - 242611.2 ) will be identified as variable below . we visually inspected all lightcurves that are marked as variable and , apart form the example shown above , we did not see lightcurves where the variability seems to be due to the pattern discussed in this section . in summary , about a quarter of all lightcurves are affected by detector position dependent artifacts . in extreme cases , the associated jumps reach 0.5 mag for a 14 mag source , and up to 0.05 mag in bright sources around 8 mag . however , we show statistically that the definition of variability we use is so conservative that this effect does not contribute a significant number of objects to our sample of variable sources . disk - bearing ysos can be identified from _ spitzer _ data alone , but information in other spectral bands is required to find the other cluster members . x - ray observations are one way to identify diskless ( class iii ) ysos . in the ir the seds of those sources are indistinguishable from a main - sequence field star , but due to their rapid rotation , ysos are much brighter in x - rays than field stars . l1688 was observed by _ chandra _ on 2000 - 04 - 13 for 100 ks exposure time in the ` faint ` mode with the acis instrument ( ) . we reprocessed this exposure with the anchors pipeline @xcite using a recent calibration ; see discussion in paper i. these data has been analyzed in detail to study the distribution of x - ray properties in ctts and to identify brown dwarfs in this star forming region @xcite . during the observation , five acis chips were operational , four from the central acis - i imaging array , as well as one acis - s chip . the point - spread - function ( psf ) degrades significantly for sources located off - axis , and thus the coordinates of the outer sources are less reliable . to cross - match x - ray sources with _ sources we used a matching radius of 1 for x - ray sources within 3 of the optical axis of _ chandra _ , 1.5 for sources between 3 and 6 away from the optical axis and 2 for all sources located further than 6 from the optical axis . the observed _ chandra _ field overlaps about two thirds of the area covered in the _ spitzer _ monitoring . however , the variable psf leads to a sensitivity that varies over the observed field , thus the absence of an x - ray detection for _ spitzer _ sources does not necessarily imply the absence of x - ray emission . the acis detector has an intrinsic energy resolution , and we use the net flux to characterize the x - ray properties . for sources with more than 20 counts , we also fit an absorbed single - temperature apec model @xcite with abundances fixed at 0.3 times the solar value from @xcite . sources are extracted down to a very low significance . in total , there are 315 detected x - ray sources , but only 31 of them match an object with a _ spitzer _ we disregard all unmatched sources ; for sources with a lightcurve and an x - ray counterpart , the x - ray properties are given in table [ tab : tab2 ] . to estimate the number of spurious matches , we multiply the fraction of the total survey area that is included in the positional error circles of the x - ray sources with the number of _ spitzer _ sources with lightcurves . the result is the average number of spurious matches . we expect at most 2 - 4 _ spitzer _ sources to be matched to a spurious x - ray source . x - ray sources that are cross - matched successfully are marked in figure [ fig : map ] . the star forming region l1688 has been the target of intense study over the past decades and a wealth of additional information exists in the literature . in particular , we refer the reader to two reviews @xcite . the latter review compiles a list of objects with a high probability of membership from a variety of published sources . the membership criteria employed are ( i ) x - ray emission , which at the distance of l1688 is detectable only from young , and thus active stars ; ( ii ) optical spectroscopy , with h@xmath23 in emission or li in absorption ; ( iii ) a location above the main - sequence in the hr diagram ; or ( iv ) ir emission that is indicative of a circumstellar disk . l1688 was also observed with _ spitzer _ in the cryogenic mission phase with all four irac channels and the 24@xmath24 m channel of the multiband imaging photometer for spitzer ( mips , * ? ? ? objects classified as ysos from these data @xcite are already contained in the membership list of @xcite . we augment our own _ spitzer _ data reduction with values from the catalog published by the c2d project ( c2d = from cores to disks ; * ? ? ? if we did not obtain a photometric value for an irac band or the @xmath25 m mips , but a value with the quality specifier a , b or c is present in the c2d catalog then we use that value . the data are given in table [ tab : tab2 ] , which specifies if a datapoint is taken from our own data reduction ( g09 : using the pipeline from * ? ? ? * ) or the c2d database . near - ir data is taken from 2mass @xcite and cross - matched by the cluster grinder pipeline @xcite . additionally , we take detections from the ukirt infrared deep sky survey ( ukidss ) galactic cluster survey , data release 9 . the ukidss project is defined in @xcite . ukidss uses the united kingdom infrared telescope ( ukirt ) wide field camera and a photometric system described in @xcite . the pipeline processing and science archive are described in @xcite . we only retain detections with a ` mergedclass ` flag between -3 and 0 . sources brighter than @xmath26 mag can be saturated and we discard the ukidss data for any source that lies above this threshold either in ukidss or 2mass . this leaves us with 2mass data in the range @xmath27 mag and ukidss @xmath28 mag . the luminosity function for both dataset is almost identical from @xmath26 to @xmath29 . for fainter sources , 2mass is incomplete ; ukidss is incomplete for @xmath30 mag . the ysovar data is also cross - matched with data from the simbad service to provide an identification with known objects from the literature . in all cases the matching radius is set to @xmath14 . if a catalog contains multiple entries within @xmath14 of a ysovar source , we match it to the closest catalog entry . in some cases the best cross - match is not obvious . those sources are discussed in appendix [ sect : remarksaboutcrossmatchingindividualsources ] . ukidss has a better spatial resolution than our irac data . there are nine sources where more than one ukidss source is found within the size of the aperture we use for irac photometry . in six cases ( wsb 52 , iso - oph 152 , iso - oph 131 , sstysv j162728.13 - 243719.6 , sstysv j162718.11 - 244814.1 , sstysv j162718.25 - 244955.8 ) the second source is visible in the @xmath31 band , the ukidss band that is closest in wavelength to the irac data , so it is likely that both sources contribute to the observed irac emission . in the remaining three cases ( iso - oph 28 , iso - oph 57 , sstysv j162741.14 - 242038.3 ) the second source is not visible at @xmath31 band . ccccc 1 & ra & deg & & j2000.0 right ascension + 2 & dec & deg & & j2000.0 declination + 3 & name & & & identifier for object + 4 & iau_name & & & j2000.0 iau designation within the ysovar program + 5 & other_names & & & alternative identifiers for object + 6 & c2d_id & & & + 7 & wil08_id & & & wilking et al . ( 2008 ) + 8 & adoc08_aoc & & & j2000.0 iau designation ( jhhmmss.ss+ddmmss.s ) + 9 & ukidss_sourceid & & & + 10 & jmag & mag & @xmath10 & + 11 & e_jmag & mag & @xmath10 & observational uncertainty + 12 & r_jmag & & @xmath10 & data source + 13 & hmag & mag & @xmath11 & + 14 & e_hmag & mag & @xmath11 & observational uncertainty + 15 & r_hmag & & @xmath11 & data source + 16 & kmag & mag & @xmath31 & + 17 & e_kmag & mag & @xmath31 & observational uncertainty + 18 & r_kmag & & @xmath31 & data source + 19 & 3.6mag & mag & @xmath32 m & + 20 & e_3.6mag & mag & @xmath32 m & observational uncertainty + 21 & r_3.6mag & & @xmath32 m & data source + 22 & 4.5mag & mag & @xmath33 m & + 23 & e_4.5mag & mag & @xmath33 m & observational uncertainty + 24 & r_4.5mag & & @xmath33 m & data source + 25 & 5.8mag & mag & @xmath34 m & + 26 & e_5.8mag & mag & @xmath34 m & observational uncertainty + 27 & r_5.8mag & & @xmath34 m & data source + 28 & 8.0mag & mag & @xmath35 m & + 29 & e_8.0mag & mag & @xmath35 m & observational uncertainty + 30 & r_8.0mag & & @xmath35 m & data source + 31 & 24mag & mag & @xmath36 m & + 32 & e_24mag & mag & @xmath36 m & observational uncertainty + 33 & r_24mag & & @xmath36 m & data source + 34 & sedclass & & & ir class according to sed slope + 35 & s1_sedclass & & & ir class according to sed slope ( visibility window 1 ) + 36 & member(ysovar ) & & & cluster membership according to ysovar standard + 37 & standardset & & & source in ysovar standard set ? + 38 & ns1_36 & ct & @xmath32 m & number of datapoints ( visibility window 1 ) + 39 & ns1_45 & ct & @xmath33 m & number of datapoints ( visibility window 1 ) + 40 & maxs1_36 & mag & @xmath32 m & maximum magnitude in lightcurve ( visibility window 1 ) + 41 & mins1_36 & mag & @xmath32 m & minimum magnitude in lightcurve ( visibility window 1 ) + 42 & maxs1_45 & mag & @xmath33 m & maximum magnitude in lightcurve ( visibility window 1 ) + 43 & mins1_45 & mag & @xmath33 m & minimum magnitude in lightcurve ( visibility window 1 ) + 44 & means1_36 & mag & @xmath32 m & mean magnitude ( visibility window 1 ) + 45 & stddevs1_36 & mag & @xmath32 m & standard deviation calculated from non - biased variance ( visibility window 1 ) + 46 & deltas1_36 & mag & @xmath32 m & width of distribution from 10% to 90% ( visibility window 1 ) + 47 & means1_45 & mag & @xmath33 m & mean magnitude ( visibility window 1 ) + 48 & stddevs1_45 & mag & @xmath33 m & standard deviation calculated from non - biased variance ( visibility window 1 ) + 49 & deltas1_45 & mag & @xmath33 m & width of distribution from 10% to 90% ( visibility window 1 ) + 50 & redchi2tomeans1_36 & & @xmath32 m & reduced @xmath22 to mean ( visibility window 1 ) + 51 & redchi2tomeans1_45 & & @xmath33 m & reduced @xmath22 to mean ( visibility window 1 ) + 52 & coherence_time_36 & d & @xmath32 m & decay time of acf ( visibility window 1 ) + 53 & coherence_time_45 & d & @xmath33 m & decay time of acf ( visibility window 1 ) + 54 & s1_stetson_36_45 & & @xmath32 m , @xmath33 m & stetson index for a two - band lightcurve . ( visibility window 1 ) + 55 & s1_cmd_alpha_36_45 & rad & @xmath32 m , @xmath33 m & angle of best - fit line in cmd ( visibility window 1 ) + 56 & s1_cmd_alpha_error_36_45 & rad & @xmath32 m , @xmath33 m & uncertainty on angle ( visibility window 1 ) + 57 & n_36 & ct & @xmath32 m & number of datapoints + 58 & n_45 & ct & @xmath33 m & number of datapoints + 59 & max_36 & mag & @xmath32 m & maximum magnitude in lightcurve + 60 & min_36 & mag & @xmath32 m & minimum magnitude in lightcurve + 61 & max_45 & mag & @xmath33 m & maximum magnitude in lightcurve + 62 & min_45 & mag & @xmath33 m & minimum magnitude in lightcurve + 63 & mean_36 & mag & @xmath32 m & mean magnitude + 64 & stddev_36 & mag & @xmath32 m & standard deviation calculated from non - biased variance + 65 & delta_36 & mag & @xmath32 m & width of distribution from 10% to 90% + 66 & mean_45 & mag & @xmath33 m & mean magnitude + 67 & stddev_45 & mag & @xmath33 m & standard deviation calculated from non - biased variance + 68 & delta_45 & mag & @xmath33 m & width of distribution from 10% to 90% + 69 & redchi2tomean_36 & & @xmath32 m & reduced @xmath22 to mean + 70 & redchi2tomean_45 & & @xmath33 m & reduced @xmath22 to mean + 71 & stetson_36_45 & & @xmath32 m , @xmath33 m & stetson index for a two - band lightcurve . + 72 & cmd_alpha_36_45 & rad & @xmath32 m , @xmath33 m & angle of best - fit line in cmd + 73 & cmd_alpha_error_36_45 & rad & @xmath32 m , @xmath33 m & uncertainty on angle + 74 & teff & k & & effective temperature from literature + 75 & r_teff & & & reference for teff + 76 & x - ray & & & chandra counterpart ? + table [ tab : tab2 ] contains the position , the designation , the flux densities of each source and properties of their lightcurves . the properties of the lightcurve will be discussed in detail in the remainder of this article . most properties of the lightcurve , e.g. mean , minimum and maximum , appear twice . they are calculated once over the entire available lightcurve and once for the first visibility window only ; the fast - cadence sampling is available uniformly for all clusters in the ysovar project and thus values calculated over the fast - cadence only can be compared between clusters ( see , e.g. paper i ) . a subset of the properties of lightcurves with a mean magnitude @xmath37 in irac 1 or irac 2 is shown in table [ tab : tab2small ] . cccccccc cfhtwir - oph 29 & f & yes & 0.57 & 0.55 & 368.58 & 18.61 & 3.20 + [ edj2009 ] 809 & ii & yes & 0.17 & 0.18 & 41.13 & & + wl 6 & i & yes & 0.58 & & 440.74 & & 4.70 + cfhtwir - oph 16 & ii & yes & 0.05 & 0.05 & 2.42 & & + iso - oph 138 & ii & yes & 0.11 & 0.12 & 15.90 & & 1.00 + iso - oph 53 & ii & yes & 0.06 & 0.08 & 5.56 & 1.72 & 1.10 + wsb 52 & ii & yes & 0.20 & 0.24 & 69.45 & 7.81 & 7.80 + wl 4 & ii & yes & 0.46 & 0.38 & 366.61 & 17.97 & 5.80 + iso - oph 137 & i & yes & 0.20 & 0.17 & 65.12 & 5.60 & 1.40 + wl 3 & i & yes & 0.23 & 0.25 & 61.16 & 8.67 & 8.50 + iso - oph 139 & f & yes & 0.12 & 0.14 & 17.48 & 4.10 & 1.60 + iso - oph 51 & f & no & 0.40 & 0.30 & 239.84 & & 4.20 + iso - oph 122 & f & yes & 0.19 & & 45.78 & & 1.10 + wsb 49 & ii & no & 0.23 & 0.18 & 84.90 & & + iso - oph 161 & i & yes & 0.34 & 0.34 & 241.40 & 13.16 & 5.40 + rox 25 & ii & yes & & 0.20 & & & + iso - oph 140 & ii & yes & 0.17 & 0.19 & 42.95 & 5.99 & 4.70 + iso - oph 120 & f & yes & 0.18 & 0.24 & 69.78 & 6.44 & 2.20 + sstysv j162636.08 - 242404.2 & i & no & 0.09 & 0.07 & 6.14 & 1.28 & 0.90 + iso - oph 152 & ii & yes & 0.08 & 0.07 & 5.45 & 0.73 & 1.00 + iso - oph 21 & i & yes & 0.52 & 0.35 & 329.65 & 13.48 & 7.70 + roxn 44 & ii & no & 0.06 & 0.05 & 5.13 & 1.86 & 3.20 + ylw 15 & i & yes & 0.13 & & 34.38 & & 7.20 + [ gy92 ] 30 & i & yes & 0.13 & 0.10 & 12.90 & 2.03 & 3.60 + sstysv j162721.82 - 241842.4 & ii & no & 0.03 & 0.06 & 0.76 & & 1.30 + wl 11 & ii & yes & 0.21 & 0.25 & 28.38 & & + ylw 47 & ii & yes & 0.16 & 0.13 & 49.72 & & 3.10 + iso - oph 35 & ii & yes & 0.06 & 0.06 & 3.68 & & + [ gy92 ] 264 & ii & yes & 0.20 & 0.26 & 62.16 & 7.17 & 2.10 + 2mass j16271881 - 2448523 & iii & no & 0.10 & 0.09 & 4.79 & & + iso - oph 153 & ii & no & 0.29 & 0.23 & 118.17 & 8.42 & 0.70 + sstysv j162622.19 - 242352.2 & iii & no & 0.02 & 0.02 & 0.68 & 0.32 & 0.70 + cfhtwir - oph 74 & ii & no & 0.21 & 0.38 & 3.56 & 1.08 & 5.00 + iso - oph 34 & f & yes & 0.05 & 0.05 & 2.26 & 1.03 & 0.50 + crbr 2322.3 - 1143 & ii & yes & 0.07 & 0.12 & 4.53 & & 2.00 + iso - oph 33 & f & yes & 0.10 & 0.12 & 6.25 & 1.90 & 4.70 + iso - oph 145 & f & yes & 0.36 & 0.41 & 216.30 & 13.03 & 1.80 + roxn 41 & ii & no & 0.05 & 0.03 & 3.02 & 1.04 & 1.30 + iso - oph 144 & f & yes & 0.13 & 0.08 & 21.70 & 3.32 & 1.50 + sstysv j162617.46 - 242314.3 & ii & no & 0.35 & 0.11 & 21.64 & 6.37 & + iso - oph 50 & i & yes & 0.99 & 1.90 & 821.89 & & 5.60 + [ gmm2009 ] oph l1688 30 & i & yes & 0.24 & 0.20 & 42.64 & 5.27 & 3.70 + iso - oph 165 & i & yes & 0.24 & 0.26 & 73.32 & 6.23 & 6.90 + [ edj2009 ] 892 & f & yes & 0.95 & 1.13 & 1084.07 & 33.56 & 7.60 + iso - oph 26 & f & yes & 0.11 & 0.12 & 16.35 & 2.65 & 3.00 + iso - oph 154 & ii & yes & 0.15 & 0.24 & 22.39 & 4.74 & 1.30 + cfhtwir - oph 21 & f & no & 0.06 & 0.05 & 5.10 & & 3.00 + iso - oph 124 & i & yes & 0.17 & 0.19 & 33.31 & 4.76 & 1.20 + wl 13 & ii & yes & 0.11 & & 14.34 & & 1.70 + rox 26 & i & yes & 0.29 & & 120.65 & & 8.40 + iso - oph 37 & i & yes & 0.46 & 0.36 & 228.62 & 11.54 & 6.40 + [ edj2009 ] 824 & f & yes & 0.08 & 0.07 & 5.94 & 2.19 & 3.70 + iso - oph 118 & f & yes & 0.27 & 0.29 & 52.56 & 7.06 & + sstysv j162727.53 - 242611.2 & i & no & 0.31 & 0.27 & 5.03 & 1.46 & 1.10 + iso - oph 52 & f & yes & 0.30 & 0.27 & 110.30 & 9.03 & 4.90 + sstysv j162621.66 - 241820.1 & f & no & 0.23 & 0.28 & 7.15 & & 0.70 + iso - oph 19 & ii & yes & 0.09 & 0.12 & 13.24 & 3.00 & + 2mass j16263046 - 2422571 & f & yes & 0.40 & 0.44 & 185.94 & 12.69 & 4.20 + sstysv j162728.30 - 244029.5 & i & no & 0.52 & 0.54 & 4.72 & 1.97 & 6.20 + we build two l1688 membership lists based on different criteria . the first is defined in paper i and is applied uniformly for all clusters in the ysovar project . sources are treated as cluster members if they fulfill at least one of the following criteria : ( i ) they are classified as ysos by @xcite based on their ir excess in cryogenic mission _ spitzer _ data or ( ii ) they are detected as x - ray sources in _ chandra _ imaging and have a spectral slope compatible with a stellar photosphere ( sed class iii , see section [ sect : spectralslope ] ) . at the distance of l1688 ( we use 120 pc from , but see also discussion in paper i ) , cluster members that are young and thus still magnetically active stars can be detected in x - rays . a total of 57 sources fulfill one or both conditions . in table [ tab : tab2 ] , these objects are marked as `` member ( ysovar ) '' . the main biases in this sample are that the ir criterion selects only those members with disks and not class iii sources , while the _ chandra _ criterion suffers from incomplete spatial coverage and it may include late - type foreground stars . the different biases are a common problem in multiwavelength studies of star forming regions ( see , e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the second membership list is taken from @xcite . on this list , 74 of our 884 sources with _ lightcurves are cluster members ; 51 of those 74 are also included in the ysovar membership list due to their x - ray or ir emission . when we compare properties of members and non - members below without referring to a specific set , then we mean membership as defined by the standard ysovar criteria . six sources in the ysovar standard member set are not part of the @xcite list . more details on those source are given in appendix [ sect : g09notwil ] . on the other hand , all of the sources in @xcite that are not part of the ysovar standard set were selected based on an x - ray detection from observations other than _ chandra _ ( for the specific references see * ? ? ? figure [ fig : hist36 ] shows a histogram of the observed mean magnitudes in @xmath32 m . almost all bright sources are cluster members . the fraction of members drops below 11th mag and no cluster member is found below 14th mag . we expect background sources to be fainter than cluster members because of their larger distance and because they are seen through the cloud , but also fainter sources can not be reliably classified . the thickness of the cloud is inhomogenous . the three primary fields cover roughly @xmath38 mag in the extinction maps from the c2d project @xcite . this correspondes to @xmath39}=1.3 - 2.2 $ ] mag @xcite . at the distance of l1688 , a k5 star with little or no extinction has @xmath40 } = 12.2 $ ] mag according to the evolutionary tracks of and the yso colors from @xcite . thus , even late - k or early - m star members in the cloud are bright enough to be detected . just over 10% of the 884 sources with lightcurves are classified by @xcite at all , because this classification scheme is conservative , representing a set of sources that can be dereddened and classified to a high degree of confidence . however , for many sources , we lack the required spectral coverage . of those sources that can be classified in this way , 16 are class i sources , 36 are class ii sources and 59 are class iii candidates , i.e. , sources with a weak or absent ir excess in their sed ( in this case the ir sed does not provide sufficient information to decide if a star is a yso or a field star ) . ten of those 59 have an x - ray counterpart in our _ chandra _ data . in order to classify all sources , including those not classified in @xcite , paper i defines a simpler approach , which uses the observed colors only . in this scheme , we fit the spectral slope @xmath41 , where @xmath42 is the wavelength and @xmath43 the flux density per unit wavelength interval at that wavelength . we make use of all measured flux densities ( no upper limits ) in the 2 - 24@xmath12 m range , which corresponds to the range from the @xmath4 or @xmath31 filter , which we take from 2mass and/or ukidss , respectively , to the mips 24@xmath12 m channel . we use the flux densities for all _ spitzer _ bands from the observations presented in @xcite ( and we include all sources extracted using this pipeline including those that are not published in @xcite because they are not classified as yso in that work ) or c2d and summarize the new lightcurves by calculating the mean and standard deviation for the new irac1 and irac2 data . thus , sources detected both in the cryogenic mission and in the new dataset will have two datapoints for irac1 and irac2 . if a source is detected in both 2mass and ukidss we use both values for the fit , since they are independent measurements . we perform a least - squares fit and call sources with @xmath44 class i , @xmath45 flat - spectrum ( f ) , @xmath46 class ii sources and @xmath47 class iii candidates . the classifications are given in table [ tab : tab2 ] in two columns . ` s1_sedclass ` presents the derived sed class using all available literature data as described above and the mean of the lightcurves from visibility window 1 for comparison with paper i ; ` sedclass ` uses the same literature data but the mean flux density for the lightcurves calculated for all visibility windows . in this scheme a significantly reddened main - sequence star , which has an intrinsic slope @xmath48 , may appear as class ii object , so this observational classification can not be translated directly into the evolutionary stage of an object . of the remaining sources we find 110 class i , 78 flat - spectrum and 455 class ii sources and 221 class iii candidates . we can not classify 20 sources , because they are seen in one band only . not only the evolutionary status but also other properties of the individual source such as inclination or stellar mass influence the value of @xmath23 . massive stars emit more energetic radiation and can thus change the structure of the accretion disk . however , l1688 is a region without massive ysos @xcite and we define @xmath23 using wavelengths longward of the @xmath31 band far from the peak of the stellar sed , so the shape of the photospheric sed has only negligible influence on the total sed . @xcite simulated different ysos of low mass . their results show that seds depend mostly on the evolutionary stage except for very extreme cases such as stars with an edge - on disk . thus , @xmath23 provides a good proxy for the evolutionary state of the ysos in l1688 . comparing with @xcite , the resulting classification is very similar , particularly for the class i and flat - spectrum sources . most sources with a spectral slope @xmath49 according to our slope - fitting are class i sources in both classification schemes . with two exceptions our flat - spectrum sources are either class i or ii in @xcite . of the 52 sources with a spectral slope between -1.6 and -0.3 , 21 are also called class ii by @xcite , but 30 are class iii in that paper , indicating that a significant fraction of what we call class ii might indeed be reddened background stars . the general agreement between the more complex classification scheme and the observed spectral slope is also found for other star forming regions ( paper i ) . we use three different methods to detect variability in all lightcurves , independent of their membership status or sed slope . sources are considered variable , if they fullfill at least one of the following conditions : ( i ) a two - band lightcurve exisits and their stetson index is larger than 0.9 ( sect . [ sect : stetsonindex ] ) ; ( ii ) only a one - band lightcurve exisits and @xmath50 ( sect . [ sect : chi2 ] ) ; ( iii ) the lightcurve is periodic ( sect . [ sect : periodicity ] ) . sources in the target fields are observed in irac 1 and irac 2 almost simultaneously ( within a few minutes ) . for those sources , we calculate the stetson index @xmath51 with points weighted evenly @xcite : @xmath52 where the sum is taken over all @xmath53 pairs of observations in irac1 and irac2 with observed magnitudes @xmath54 and @xmath55 and uncertainties @xmath56 and @xmath57 ; @xmath58 denotes the sign of @xmath59 . @xmath59 is the product of the normalized residuals in both bands : @xmath60 here @xmath61 denotes the error - weighted mean of all irac 1 magnitudes and @xmath62 the error - weighted mean of all irac 2 magnitudes . the stetson index is very robust to observational errors since those are unlikely to affect both bands in the same way . following paper i , we define a source as variable if @xmath63 . in paper i , this is calculated over the first visibility window only ; in this article we use the lightcurve from all four visibility windows . the stetson index calculated for the first visibility window only can be found in table [ tab : tab2 ] in column ` s1_stetson_36_45 ` and for the entire lightcurve in column ` stetson_36_45 ` . figure [ fig : stetsonlcs ] shows examples of lightcurves that are classified as variable according to the stetson index . in the first visibility window , we find 34 sources to be variable , that is 18% of all lightcurves where we can calculate the stetson index . the number is similar ( 38 sources , 18% ) if we consider all observing visibility windows . this is not surprising , given that the stetson index is designed to be a `` robust '' statistic that is little influenced by a small number of datapoints , and compared to the first visibility window , the other visibility windows only contribute a small number of datapoints . we caution that equation [ eqn : stetson ] will introduce a systematic bias in such a way that a source that is variable in only one band will not be recognized as variable . [ sect : chi2 ] for sources outside the primary target fields or for sources which are too bright or too faint in one irac channel , the stetson index can not be calculated . for those sources we rely on the reduced chi - squared value to detect variability : @xmath64 instrumental uncertainties lead to a non - gaussian error distribution , so we use a conservative cut - off and mark sources as variable only if @xmath65 ( paper i ) . using this metric , we find 22 sources that exhibit variability in irac1 and 18 that exhibit variability in irac2 . for comparison , we note that of the 38 sources classified as variable according to their stetson index in the last section , 34 are also variable according to the @xmath22 test for their irac1 lightcurve and 37 according to their irac2 lightcurve . the @xmath66 calculated for the first visibility window only can be found in table [ tab : tab2 ] in column ` redchi2tomeans1_36 ` and ` redchi2tomeans1_45 ` for irac i and irac 2 , respectively , and for the entire lightcurve in columns ` redchi2tomean_36 ` and ` redchi2tomean_45 ` , again for irac 1 and irac 2 , respectively . examples are shown in figure [ fig : chi2lc ] . if we fit a linear slope instead of comparing to the mean ( i.e. a constant ) , the results are the same . many sources have significantly larger values of @xmath66 for the whole observation series than within one visibility window , indicating a constant luminosity over 40 days , but a change over timescales between six months and two years . ccccc cfhtwir - oph 29 & [ 3.6]-[4.5 ] & 2.6 & 0.03 & f + iso - oph 138 & [ 3.6 ] & 3.3 & 0.00 & ii + sstysv j162636.08 - 242404.2 & [ 3.6 ] & 4.1 & 0.00 & i + iso - oph 152 & [ 3.6 ] & 4.7 & 0.03 & ii + [ gy92 ] 30 & [ 3.6 ] & 14.5 & 0.00 & i + wl 11 & [ 4.5 ] & 3.0 & 0.01 & ii + 2mass j16271881 - 2448523 & [ 4.5 ] & 6.4 & 0.00 & iii + sstysv j162622.19 - 242352.2 & [ 4.5 ] & 6.1 & 0.00 & iii + iso - oph 34 & [ 3.6 ] & 2.2 & 0.00 & f + iso - oph 33 & [ 4.5 ] & 2.4 & 0.02 & f + roxn 41 & [ 3.6 ] & 6.5 & 0.00 & ii + iso - oph 154 & [ 3.6 ] & 5.6 & 0.00 & ii + iso - oph 124 & [ 3.6 ] & 3.4 & 0.00 & i + wl 13 & [ 3.6 ] & 10.7 & 0.00 & ii + lastly , we search for periodicity in the fast - cadence ( visibility window 1 ) lightcurves using a lomb - scargle periodogram ; again , see paper i for details . we require a false alarm probability @xmath67 and to further reduce the number of false positives , we additionally run a period detection of all lightcurves on the nasa exoplanet archive periodogram service . this service employs several algorithms because each algorithm is particularly suited to a different signal shape ( see paper i or the website for details on the other algorithms ) . we also calculate the autocorrelation function of each signal . we find that the lomb - scargle periodogram is advantageous for lightcurves with @xmath68 points as is the case for most of our data . we require that at least one of three supplementary algorithms that we run retrieves a periodicity with a similar timescale ( see paper i for details ) . we only search for periods between 0.1 and 14.5 days , so that at least three periods fit in each visibility window . allowing longer periods leads to the detection of many long - term trends , where the data do not show that these trends are actually periodic . finally , all algorithmically detected periodicity is vetted by eye . if periods are found in multiple bands , we report the period in irac 1 , which is generally the most reliable due to the lower measurement uncertainties . if irac 1 does not reveal a periodicity , we report the value for the irac 2 lightcurve , and , as a last resort the period found in the [ 3.6]-[4.5 ] color to include objects where a periodic signal is overlayed by a long - term trend so that it is undetectable in each individual channel , but might be visible in the color . the final list of adopted periods is given in table [ tab : period ] . it contains sources from all sed classes . we find 14 sources that show a periodic behavior in the first visibility window , but this periodicity is not stable over more than one visibility window . the datapoints taken in the first visibility window follow the folded period much better than the data from the later visibility windows , indicating that the period is not stable for longer than a few months . in roxn 41 , the datapoints of the later visibility windows , folded with the same period , seem to follow a different , yet clearly defined lightcurve with a similar period . ten periodic sources have information in both irac bands . eight of those are already classified as variable by the stetson index . three out of four sources with data in one band only are variable according to the @xmath66 test . the remaining two sources with information in both irac bands show periodicty , but fail the stetson index test because a larger variability amplitude is required for a significant detection in stetson index , which does not make any assumption about the form of the variability compared with the lomb - scargle periodigram , that only detects periodic signals . equally , the remaining source with data in only one band fails the @xmath66 test , because the @xmath66 test also requires a larger variability amplitude for a significant detection than the lomb - scarge periodogram . the number of datapoints in visibility windows 2 , 3 , and 4 is too low to search for periodicity in those visibility windows alone . figure [ fig : periodlc ] shows the phase - folded lightcurve in one band for six periodic sources where the periodicity is significant in a single band . figure [ fig : periodcolorlc ] shows the phased lightcurve for , where the period is seen only in the color term [ 3.6]-[4.5 ] . most of the periods found in table [ tab : period ] are in the range 3 - 7 days . only two sources have longer periods of 11 and 14 days . the largest amplitude is around 0.3 mag , and the smallest around 0.05 mag . all periodic sources except roxn 41 , sstysv j162636.08 - 242404.2 , 2mass j16271881 - 2448523 , and sstysv j162622.19 - 242352.2 are cluster members according to our membership criteria . roxn 41 and sstysv j162636.08 - 242404.2 have class i and ii seds , respectively . all but cfhtwir - oph 29 ( see appendix [ sect : g09notwil ] ) , sstysv j162636.08 - 242404.2 ( see appendix [ sect : g09notwil ] ) , and 2mass j16271881 - 2448523 ( no information beyond the @xmath69 magnitudes is available in the literature ) are cluster members according to @xcite . ten of the 56 cluster members are periodic ; in total there are 14 periodic sources out of 60 sources that are variable in the first visibility window . in comparison , @xcite find 18% of all variable stars to be periodic in @xmath70 monitoring of the orion a molecular cloud with a similar time coverage as we have for l1688 . @xcite find a third of all variable stars in l1688 to be periodic ; due to their longer time baseline , @xcite are sensitive to different periods , but only 4 out of their 32 periods have values that are outside of the range we could detect see their table . we are sensitive to periods of up to 14.5 days here . the largest group of lightcurves in the sample of l1688 does not have any significant variability . the lower limit where variability is detected depends on the brightness of the object - fainter sources need a stronger relative variability due to the larger measurement uncertainties . paper i presents monte - carlo simulations to show that the stetson test finds variability when the amplitude is a few times larger than the average uncertainty ; the exact number depends on the signal shape . for example , the variability in a source that switches between two discrete levels will be detected when the amplitude is at least twice the uncertainty ( @xmath71 mag for a star with magnitude 14 ) . if data from only one band are available , the step size must be more than four times the uncertainty to be found by the @xmath22-test ( @xmath72 mag for a star with magnitude 14 . ) . given the observing cadence , the monte - carlo simulations show that we are sensitive to periods between 0.1 and 14.5 days ( paper i ) . in summary , we call a source variable if it is either periodic with a low false alarm probability ( 14 sources ) or fulfills one of the following conditions : sources with simultaneous data in two bands need to have a stetson index @xmath73 ( 34 sources in the ysovar standard set fast cadence data and 38 sources in total ) and sources without simultaneous data need to have @xmath65 ( 22 sources in the ysovar standard set fast cadence data and 29 sources in total ) . we consider the lightcurves for all 882 distinct sources with at least five datapoints in at least one irac band . of those lightcurves , 70 are classified as variable ; 56 sources are cluster members according to the ysovar criteria ( 44 of them are variable ) and 73 sources are cluster members according to @xcite ( 47 of them are variable ) . both membership samples have considerable overlap . in the following subsections , we compare properties of the lightcurves between different sed classes . in most cases , the lightcurves of irac1 ( 3.6@xmath12 m ) and irac2 ( @xmath6 m ) have very similar properties . figure [ fig : probvarsed ] contains the variability fraction sorted by sed class ( see section [ sect : spectralslope ] ) . within each class , the population is sub - divided into previously identified members , bright ( @xmath74 mag in @xmath32 m or @xmath33 m ) stars not previously identified as members , and faint ( @xmath75 mag ) stars not previously identified as members . sources are considered variable if they have a low false alarm probability for periodicity or a stetson index @xmath73 ( if simultaneous data in irac 1 and irac 2 exist ) or @xmath76 ( if no simultaneous data exist ) . note that the classes shown here are based on the observed sed and thus background non - member stars seen through the cloud might appear red ( like a class ii source ) even if their intrinsic sed slope is compatible with sed class iii . no star classified as a member is fainter than 14 mag . the observed fraction of variable stars as a function of sed class is the best guess for the probability to find variability for a star of a given sed class . we calculate the uncertainty for the probability based on the observed number of variable stars @xmath77 in a bin with @xmath53 total sources as follows : if the true probability for a source in the bin to be variable is @xmath78 , then the probability to observe @xmath77 variables in @xmath53 stars is given by @xmath79 , which follows a binomial distribution : @xmath80 the peak of this distribution is @xmath81 . we then calculate the boundaries of the confidence interval @xmath82 for @xmath78 such that @xmath83 and the area under the curve includes 68% of the probability . those uncertainty ranges are shown in figure [ fig : probvarsed ] . almost all known cluster members are variable ; the fraction of variables is decreasing slightly from class i to ii . there are five class iii members and none of them shows variability . this is compatible with the variability fraction observed for class iii non - members . by definition , sources with a class iii sed do not have a large ir excess over a stellar photosphere , thus they are not expected to harbor a substantial disk or show disk - related variability . however , even field stars can show significant variability , at least in the near - ir . @xcite find 1.6% of all field stars to be variable in @xmath69 . this includes eclipsing systems , stars with unusually large photospheric spots and other unidentified variables . all effects that influence the @xmath69 lightcurve are likely also visible in the mid - ir . our fraction of variable objects with a class iii sed is similar ( 1.8% ) to what wolk et al . found . in the optical and near - ir , the fraction of variable field stars is lower than that value . _ spitzer _ observations of other young clusters also find that younger stars are more variable , but with a lower variability fraction . @xcite find only 70% of all stars with disks in their onc sample to have detectable variability . @xcite find 60% in ic 348 and @xcite find @xmath84% of all members in ngc 2264 to be variable in the irac bands . all three clusters are located at larger distances and thus observations are less sensitive to small variations , the observations of @xcite are more densely sampled and thus provide a better signal . in l1688 , a k5 star with little extinction will have a magnitude of 12.2 in @xmath5 m using the evolutionary tracks of and the yso colors from @xcite . in this magnitude range , variability down to 0.05 mag can be detected ( paper i ) . in contrast , for the same star in ic 348 , the variability has to be about twice as large to be detected and three times as large to the detected in the onc . of course , even our observations miss the faint end of the distribution . thus , it seems likely that essentially all ysos show substantial mid - ir variability on timescales of days to weeks . sources with a class iii sed are not contaminated by disk - bearing stars , since reddening by the cloud can only increase the spectral slope @xmath23 but never hide an existing ir excess . the larger fraction of non - member variable objects in the other classes in figure [ fig : probvarsed ] thus shows that the membership lists are incomplete . one caveat here is that the class iii sources in the sample of members are selected in a different way than the other classes . if true class iii cluster members with x - ray emission ( that are included in our member sample ) are systematically less variable than class iii members where we do not detect x - ray emission ( that are therefore not included in the member sample ) , that would also lead to a lower observed variability fraction in class iii member sources . however , given the size of the observed effect in variability such a bias seems to be unlikely to be the sole reason for the observed distribution . the observed probability that a source is variable for all classes of bright non - members ( blue bars ) is consistent with around 5% the fact that this barely changes with the sed class , quite unlike the distribution for the cluster members , indicates that unidentified cluster members can not make up a large fraction of the non - member sample since they would bias the observed variability fraction to higher values for the earlier classes . the vastly different number of sources in the different bins makes it highly unlikely that each bin is contaminated by the same fraction of class i - ii cluster members . the sample of faint stars has almost the same fraction of variable sources in every sed class . the low fraction can be explained by two effects . first , variability of faint sources can not be reliably detected unless the amplitude is exceptionally large . second , as most cluster members are brighter than 14 mag , the faint sample contains fewer unrecognized cluster members than the sample of brighter sources . figure [ fig : meanampsed ] shows that there is a wide distribution of amplitudes in ysos of class i , f and ii . the error bars in this figure represent the standard deviation of the mean amplitudes within one class . amplitude and standard deviation in the figure are measured in magnitudes ; a median amplitude of 0.1 mag means that the flux in @xmath33 m varies by 10% for a typical source . for members , the variability amplitude is larger in class i sources than in flat - spectrum and class ii sources , where the amplitude is still larger than in class iii sources . also , the spread of the amplitudes in class i and f is much larger than for class ii and iii . for non - members , the observed amplitudes seem to follow a similar , but less pronounced trend . this suggests that the non - member category still includes not only reddened background objects with an intrinsic class iii sed , but also some unidentified cluster members . indeed , we propose that all variable class i , f and ii sources are members and table [ tab : newmembers ] lists those new members that are neither contained in our ysovar standard membership set ( see table [ tab : tab2 ] ) nor in @xcite . however , for consistency , we do not modify our sample of members at this point and continue to treat the sources in the table as non - members for the remainder of the analysis . two of the objects in the table , and were suggested as a @xmath7 oph substellar candidate members by , the remaining objects have not been studied before . in summary , the probability for any given source to be observed as a mid - ir variable decreases little between class i and ii sources , but the mean amplitude as well as the differences within a class are much larger for sources which have more circumstellar material . cccc 246.88244 & -24.79978 & sstysv j162731.78 - 244759.2 & ii + 246.64060 & -24.31667 & sstysv j162633.74 - 241900.0 & ii + 246.84236 & -24.78363 & sstysv j162722.16 - 244701.0 & ii + 246.59001 & -24.49297 & sstysv j162621.60 - 242934.7 & ii + 246.65036 & -24.40117 & sstysv j162636.08 - 242404.2 & i + 246.84092 & -24.31180 & sstysv j162721.82 - 241842.4 & ii + 246.84135 & -24.78713 & sstysv j162721.92 - 244713.6 & ii + 246.84336 & -24.64383 & cfhtwir - oph 74 & ii + 246.90225 & -24.64854 & sstysv j162736.53 - 243854.7 & ii + 246.57278 & -24.38732 & sstysv j162617.46 - 242314.3 & ii + 246.67446 & -24.25616 & sstysv j162641.86 - 241522.1 & ii + 246.91134 & -24.55516 & sstysv j162738.72 - 243318.5 & ii + 246.60121 & -24.26383 & cfhtwir - oph 21 & f + 246.88281 & -24.34855 & sstysv j162731.87 - 242054.7 & f + 246.86474 & -24.43647 & sstysv j162727.53 - 242611.2 & i + 246.59027 & -24.30560 & sstysv j162621.66 - 241820.1 & f + 246.86794 & -24.67487 & sstysv j162728.30 - 244029.5 & i + next , we will look at the timescales of the lightcurves . the list of periodic sources contains objects of all evolutionary stages , but is too small to recognize any trends in the period with evolutionary stage . we calculate the discrete auto - correlation function @xmath85 for each fast - cadence ( visibility window 1 ) lightcurve to characterise the time - scale in all sources , not just the periodic ones . one complication here is that our data are unevenly sampled . in order to calculate the @xmath86 we linearly interpolate the lightcurve on a grid with time steps of 0.1 days . this process can change the properties of the lightcurves on short timescales . however , for most lightcurves the relevant timescales are longer than the distance between two observations ( 0.1 to 0.8 days ) . we use the following definition of the @xmath86 @xmath87 where @xmath53 is the total number of points in the lightcurve @xmath88 with a mean of @xmath61 and a standard deviation of @xmath89 . in the discrete @xmath86 @xmath90 is the number of timesteps . by definition @xmath91 and the @xmath86 decays for longer time - lags @xmath90 . we take the first value of @xmath90 with @xmath92 as the characteristic time - scale for a lightcurve . a more common definition would be to use the position of the first local maximum in the @xmath86 , but due to the low number of datapoints in our lightcurves , the noise in the @xmath86 is large and this value is often not well defined . while our definition might not give us the timescale of the physical processes at work , the @xmath86 timescale still provides relative comparisons . we find that the average @xmath86 timescale decreases from 4 days for class i sources to one day for class iii sources ( figure [ fig : acf ] ) . a more detailed analysis of stochastic and quasi - periodic properties of the lightcurves is beyond the scope of the current paper ( for a discussion of these properties in @xmath70 lightcurves , see * ? ? ? the majority of the stars included in our data show no significant variability , and we exclude them from further discussion . the remaining stars are definitely variable , with light curve shapes that display a variety of morphologies . visual inspection , and some quantitative analysis , allows us to group these stars into sets with similar light curve morphologies - which can be the first step in attaching physical mechanisms as the cause of the observed variability . based on a near - ir ( @xmath69 ) monitoring campaign of similar cadence and duration to ours , @xcite identified four light curve morphological classes : ( a ) periodic ; ( b ) quasi - periodic - which they defined as stars with cyclic brightening and fading , but where the frequency and amplitude of the variations varied from cycle to cycle ; ( c ) long - duration - stars with relatively long term monotonic changes in brightness over weeks or months , with eventual changes in sign of the variability ; ( d ) stochastic - which indicated all other variability types , where no obvious pattern to the variability was present . in another recent paper , @xcite analysed a @xmath93 day monitoring campaign for the star - forming region ngc 2264 , using optical data from corot and ir data from irac , to assign light curve morphology classes to their young star set . their proposed light curve morphologies were : ( i ) periodic - stars whose light curves show periodic waveforms whose amplitude and shape are unchanging or only change in minor ways over the 30 day observing window . these light curves were ascribed generally to cold spots on the stellar photosphere ; ( ii ) dippers - stars showing a well - defined maximum brightness , upon which are superposed flux dips of variable shape and amplitude . these are sub - divided into stars where the flux dips occur at an approximately constant period - designated as quasi - periodic systems - and stars with no obvious periodicity to the dips - designated aperiodic . in previous literature , these two sub - classes , or portions of them , have been referred to as aa tau systems and ux ori systems , respectively ; ( iii ) short duration bursters - stars with relatively well - defined minimum light curve levels , superposed on which are brief ( hours to day ) flux increases , attributed to accretion bursts . see @xcite for further discussion of this set ; ( iv ) quasi - periodic variables - stars lacking a well - defined maximum brightness but showing periodic variability whose waveform changes shape and amplitude from cycle to cycle ; ( v ) stochastic - stars with prominent luminosity changes on a variety of timescales , with no preference for `` up '' or `` down '' . ( vi ) long timescale variability - stars with slow ( weeks to months ) changes in brightness . while there are some clear similarities in the two schemes , their usage of the terms quasi - periodic and stochastic are not the same , and light curves described as belonging to those classes in one scheme would not necessarily be so classified in the other scheme . it will be important to resolve these nomenclature issues in the future , perhaps in the same manner as was done for sorting out similar issues on classifying pre - main sequence disk sed morphologies ( evans et al . 2009 ) . in the meantime , we must choose which scheme to adopt and be clear that is what we have done . for this paper , we adopt the @xcite scheme . periodic lightcurves are described in section [ sect : midirvariability ] and examples can be seen in figure [ fig : periodlc ] . examples of quasi - periodic lightcurves are shown in figure [ fig : morelcs ] . shows variability with a timescale @xmath94 days in the first half of the first visibility window , but around mjd 55316 , the flux increases significantly and this rise masks out any periodicity . the dip around mjd 55325 again has a similar duration as those observed in the beginning of the visibility window . the other two sources shown might be similar , but the signal - to - noise is not as good . other sources are variable , but no preferred timescale for the variability is discernible . and 51 ( figure [ fig : morelcs2 ] ) are examples of this aperiodic behavior . in yet other cases , the timescale of the variability is so long that we can not decide if a feature is a singular event or part of a recurring pattern . good examples of this are wl 4 and , which are shown in figure [ fig : cmd ] . in some cases , we see short duration features in the lightcurve in addition to a longer trend . this can either be a brightening ( e.g. mjd 55303 and 55311 in , figure [ fig : morelcs2 ] ) or short dips in the lightcurve that last between one and ten days ( in the same figure ) . like @xcite , we find that even strongly periodic sources are not perfect clocks in the mid - ir in that the amplitude can vary from cycle to cycle . there might also be phase shifts from visibility window to visibility window . in a very similar study to the one we present here , @xcite observed ic 348 over a 40 day window with irac1 and 2 and find 25% of the variable stars are likely periodic . here we find a similar division ; 10 of 52 members are strongly periodic . we present the first mid - ir monitoring of l1688 , but the field has been monitored before in the near - ir . figure [ fig : amplitudekvs36 ] compares the @xmath4 band amplitude found in 2mass data taken between 1997 and 2001 @xcite and the @xmath31 band amplitude observed at ukirt between 2005 and 2006 with the amplitude of our @xmath32 m lightcurves . given the different bands , it is not surprising that the absolute value of the amplitude differs , but there is a good correlation such that the sources with the largest @xmath31 or @xmath4 band amplitudes in earlier observations also have the largest amplitudes in the irac bands five to ten years later . in most objects , color changes are small or happen in parallel with luminosity changes ( section [ sect : colorchangesandreddening ] ) , so the @xmath31 or @xmath4 band and the [ 3.6 ] variability should be strongly correlated . figure [ fig : amplitudekvs36 ] shows that the amplitude of the variability is relatively stable over at least one decade . @xcite used a larger number of observations than and observe a larger spread between a typical yso s brightest and faintest @xmath31-band magnitude , indicating that the longest timescale of variability is longer than the time span of the observations . @xcite noted that roughly 6% of the stars in ic 348 had data which indicated significant changes in the source s mid - ir flux over the three year interval since the last irac observation . @xcite also displayed numerous examples of stars which showed continuous change over the course of at least one observing visibility window . all this shows that the amplitude of variability of a given source may evolve over very long timescales , but is consistent over at least one decade . ysos apparently do not switch between highly variable and much less variable states . of time - variable extinction . the largest @xmath95 observed is 1 , but for clearity only the region to @xmath96 mag is shown . _ bottom _ : in this panel the length of each line gives the sed class of that object . only cluster members have slopes where the source bluens as it fades . [ fig : cmdslope],title="fig:",scaledwidth=40.0% ] of time - variable extinction . the largest @xmath95 observed is 1 , but for clearity only the region to @xmath96 mag is shown . _ bottom _ : in this panel the length of each line gives the sed class of that object . only cluster members have slopes where the source bluens as it fades . [ fig : cmdslope],title="fig:",scaledwidth=40.0% ] for sources in the primary target fields , the observations in @xmath5 m and @xmath6 m are separated by only a few minutes . for those sources , we compare the color and the magnitude in the @xmath5 m band in a color - magnitude diagram ( cmd ) ( figures [ fig : morelcs ] and [ fig : cmd ] ) . in some cases , the color is fairly stable within one visibility window , but changes between visibility windows ; in others the timescale of variability is much shorter and color changes are seen within one visibility window as well . for non - variable sources , the cmd forms a point cloud with the size set by the photometric uncertainties , but for variable sources , the shape of the cmd can reveal the physical cause of the variability for example , if a disk warp or accretion funnel passes in front of a yso , we expect its color to become redder as it becomes fainter . if the absorber has the same gas and dust properties as the inter - stellar medium ( ism ) and absorbs star and inner disk at the same time , then the datapoints in the cmd follow a line with the slope of the interstellar reddening law ( e.g. * ? ? ? * ) . in each cmd , the observational data ( i1 , i1-i2 ) are fit to a line segment using an orthogonal distance regression method that takes the errors in both the x and y directions into account . we calculate the length in magnitudes of the line segment excluding the 10% of the data that are outliers on either side . the cmd slopes of the 37 sources with a well - defined slope are shown in figure [ fig : cmdslope ] . the x and the y axis in the cmd are correlated , because they both depend on the @xmath5 m magnitude so sources which are noise - dominated would show a slope of @xmath97 . no source is seen in this region . from a total of 42 variable sources with cmds with at least ten datapoints , 37 can be fitted with a formal uncertainty on the best fit slope below @xmath98 . in four of the five sources with a statistical error on the slope in the cmd @xmath99 , the color variability is dominated by observational uncertainties ; the remaining source iso - oph 140 is discussed below . as can be seen in figure [ fig : cmd ] , even in those sources with a well defined slope in the cmd , the scatter around the best linear fit is larger than the measurement uncertainties ; individual visibility windows are systematically above or below the fitted line . the slopes shown in figure [ fig : cmdslope ] can be separated ( somewhat arbitrarily ) in two groups . one group becomes redder when the sources are fainter with slopes comparable to the ism reddening . this group contains almost all variables that are not classified as cluster members in section [ sect : l1688membership ] . most sources in this group have class i or flat - spectrum seds ( bottom panel ) . sources that bluen ( we use the term `` bluen '' as a verb to mean that a source comes bluer similar to the common expressions `` redden '' or `` reddening '' ) when they are fainter are mostly class ii and flat spectrum sources . figure [ fig : morelcs ] shows the lightcurves and cmds for two of the bluening sources , and . the top panel in figure [ fig : cmdslope ] quantifies the magnitude of the reddening or bluening observed in each source . for a source where the slope in the cmd is compatible with ism reddening , this can be interpreted as the @xmath95 of the intervening material . a range of values is observed , but in most sources it is @xmath49 mag . no significant difference in vector length is seen between sources that bluen or redden when they dim . we discuss the physical mechanisms that could cause these slopes in section [ sect : scenariosforcolorchanges ] . comparing the amplitudes of all sources in figure [ fig : cmdslope ] that either redden or bluen over the full time span of the observations , we do not find significant differences in value or distribution of the variability amplitudes . similar to our results , find reddening and bluening slopes in the cmd in their @xmath100 monitoring , but compared to figure [ fig : cmdslope ] they see a higher fraction of stars that bluen as they dim ( 33/49 ) . only six sources have well defined slopes in both their study and ours . in the data of , all of them belong to the group that bluen with lower fluxes . in five cases , our data agree . in contrast , roxn 44 reddens in our observations , so its behavior changed over the time interval of five years , or it is behaving differently in the near and mid - ir . is the only source with a statistical error on the slope in the cmd @xmath99 where the color variability is not dominated by observational uncertainties , but where the slope varies substantially over time . its lightcurve and cmd for the first visibility window are shown in figure [ fig : cmdsquare ] . it has a class ii sed slope and is a low mass yso ( spectral type m1 ; * ? ? ? there is no single slope in the cmd ; instead the source seems to switch between different modes . initially , the luminosity drops sharply , while the color becomes slightly bluer . the behavior changes around mjd 55305 . for the next 15 days , the source becomes redder and dimmer , but the slope is less steep than the ism reddening law . the evolution is not monotonic , but a brightening ( e.g. around mjd 55313 ) corresponds to a bluer color and the source reddens as it dims . at the end of this period , the brightness increases again sharply and the slope in the cmd is comparable to the period mjd 55295 to 55305 : the source becomes noticeably redder with higher luminosity . in the last 10 days of the monitoring , the source again has a slope close to the ism reddening law . its luminosity continues to increase and the color bluens until it has a similar color and luminosity as it did at the beginning of the monitoring . this closes the circle in the cmd . this is the only source in our sample where we observe multiple changes of the reddening slope over time . in this case , variability similar to ism reddening corresponds to slow changes in luminosity , while the other direction of the slope in the cmd is associated with faster luminosity changes . now we discuss the lightcurves with the largest changes in magnitude in our sample . wl 4 , shown in figure [ fig : cmd ] , has an outburst event consisting of a brightness increase by about 0.6 mag and bluening at the same time . the rise and fall take only a few days and the entire outburst lasts about a month . [ edj2009 ] 892 ( shown in the same figure ) presents a slow rise by about 0.2 mag in @xmath33 m and smooth decay over two years where the source gradually bluens ( although more activity between visibility windows can not be excluded ) . there is an indication of a slow and smooth increase in brightness again in the last observing visibility window , pointing to a recurring phenomenon . these timescales are much longer than the dynamical time in the inner disk , so they are presumably driven by disk phenomena that originate at larger radii . @xcite identified a few such bursting or fading events in their analysis of ysovar data for the orion nebula cluster ( onc ) and they have also been observed in the optical @xcite . the physical cause is not known yet . @xcite find a few lightcurves that match theoretical predictions for short mass accretion events , but without simultaneous spectroscopy this is hard to prove . the best example presented here , wl 4 , becomes bluer during the burst , but the opposite happens for wl 3 , which otherwise has a similar lightcurve . thus , it is likely that different mechanisms cause these burst events . the reddening sources in figure [ fig : cmdslope ] have slopes that are roughly compatible with the slope of an interstellar reddening law . small deviations can be explained by modifications of the dust - to - gas ratio or the grain size and composition . this is commonly observed in individual young stars and star forming regions as a whole @xcite . the most prominent example of a class ii source where the extinction changes on timescales of days is . in this case , the absorption dip is periodic and caused by an inner disk warp partially occulting the star . many examples of comparable lightcurves have been found in the onc by @xcite and in ngc 2264 by @xcite . these authors call objects with absorption events in the lightcuve `` dippers '' . we observe reddening on timescales of months to years in about half of all sources with a well - defined slope in the cmd . an example of that is shown in the top right panel of figure [ fig : cmd ] . we do not know the inclination of the objects in our sample , but if it is close to edge - on , this might be caused by an asymmetric disk similar to aa tau . the required warp or local change in scale height that lets some gas and dust protrude above the average disk height , could be caused by a low - mass star , brown dwarf or planetary mass object orbiting in the disk ( e.g. * ? ? ? * ; * ? ? ? alternatively , vortices in the disk or dust traps can also change the local scale height . changes in the optical brightness and color of a yso are often attributed to spots , either hot accretion spots or cold magnetic spots , rotating in and out of view or to time - variable extinction ( e.g. * ? ? ? * ; * ? ? ? if the reddening increases and , at the same time , the visible fraction of the hot spot decreases , then the resulting slope in the cmd would be intermediate between ism reddening and colorless . since spots on the star can not explain sources that bluen as they dim , this observed phenomenon must be related to the disk . several parameters of the inner disk could be time variable . the most obvious one is the accretion rate . however , the direct effect of increased accretion is a larger accretion spot , which would make the star bluer and brighter , not fainter . also , @xcite find no relation between mid - ir lightcurves and spectroscopic accretion tracers in their limited sample . this indicates that the relevant parameter can change without affecting the accretion rate . in the mid - ir , we see the optically thick dust at the inner disk edge . because the disk is heated over its entire vertical height at the inner edge while only the disk surface absorbs radiation at larger radii , it can form a puffed - up inner rim that reaches above the usual disk scale height and casts a shadow on the remaining disk , reducing the luminosity at longer wavelengths ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? if this rim grows , it intercepts more stellar light . thus , the emission from the inner wall increases and the emission from the outer disk decreases , which results in bluer colors . since the total emitting area also decreases , the source can become bluer and fainter . @xcite also present radiative transfer models with hot spots and warped disks that reproduce features of the observed lightcurves . alternatively , @xcite suggested a non - axisymmetric model . if the emission from the stellar photosphere is stronger at some longitudes due to a strong accretion spot or a spot from magnetic activity on the star , it irradiates the inner disk like a searchlight beam and causes the dust at this longitude to retreat . the result is a disk where the inner rim is broken at one ( or several ) positions . this would reduce the emission in @xmath5 m , but cause more energy to be emitted at a larger radius , where the dust is cooler on average and thus emits at longer wavelengths such as @xmath6 m . in this way , the source can become redder as it brightens . there is one group of ysos , the so - called ux ori ( uxor ) variables where several members have been observed to bluen when they fade in optical observations . this can be explained by a larger fraction of scattered light that contributes to the observed sed in their faint state . a similar scenario for the ysos discussed here requires , first , scattering by relatively large grains ( @xmath101 mm ) , because only for large grains radiation at @xmath32 m will be scattered more than radiation at @xmath33 m . grains this large will be present in the disk , but not at all radii and all disk heights . second , for this mechanism to work , a large fraction of the ir radiation must be scattered light and not intrinsic emission . in contrast , many bluening sources are class i to flat - spectrum sources , where the ir luminosity is comparable to the total stellar irradiation . none of the simple models presented so far offers a convincing explanation of the observed lightcurves and cmds . however , detailed simulations that take into account turbulent transport processes in the disk and different dust species with a complete chemical network have not yet been performed . the real situation is likely to be much more complex than sketched above . since different dust species form and sublimate with different speeds and at different temperatures , time variable irradiation can actually cause a very complex mixture of species with different opacities , which will not react linearly to increased irradiation . such a complex network might cause a delayed and non - linear response of the disk , which masks the underlying relation between accretion and disk emission . the characteristics of the time variability in the lightcurves could be related to the spectral type of the central star as hotter stars irradiate the disk with harder spectra . we compared the 90% quantiles of the observed @xmath6 m magnitudes for those sources in our sample that have spectroscopically determined spectral types from @xcite . we use the relation between spectral type and @xmath102 from @xcite . this gives values for ten objects in our sample and we do not find a correlation in this set . similarly , @xcite do not identify any trends in infarared variability versus effective temperature among few myr old ngc 2264 stars . in apparent contrast , @xcite find that the variability in [ 3.6 ] and [ 4.5 ] increases with increasing @xmath102 , but they show that this trend probably does not reflect a change in disk properties , but is instead due to the lower relative contribution of the photosphere compared to the disk for hotter stars . in this section , we analyze the subsample of sources with x - ray counterparts . since the subsample is not selected for its ir properties , it provides a clean sample to calculate the variability fraction . there are 31 sources with detected x - ray emission , of which 20 are variable in the ir . again , we find high variability fractions in class i ( 6/7 ) , f ( 3/4 ) and ii ( 11/15 ) , while none of the four class iii sources is variable . the sample size is small , but this is consistent with our analysis of the ir selected sample in section [ sect : evolutionarytrendsofthevariability ] . we searched for correlations between the parameters of the x - ray emission ( median energy , fitted temperature and absorbing column density ) and the amplitude of the mid - ir variability . no such correlation is apparent within each sed class . for a given absorbing column density , soft x - ray emission is more strongly absorbed than hard x - ray emission . thus , the observed x - ray spectrum from more embedded sources always appears harder . since class ii sources have more circumstellar matter than class iii sources , it is not surprising that they appear harder . @xcite already discuss this for the l1688 x - ray data ; figure [ fig : probvarsed ] and [ fig : meanampsed ] show that class ii sources are more variable than class iii sources . we see no additional effect of the x - ray emission on the variability characteristics in the mid - ir . we present _ spitzer _ observations of ysos in the star forming region l1688 . observations were taken in four visibility windows in spring and fall of 2010 and 2011 , with about 70 observations in spring 2010 and about ten observations in the remaining visibility windows . the cadence of the observations is non - uniform to avoid bias in the period detection . our sample consists of 882 sources with lightcurves in irac1 and irac2 with at least 5 datapoints . of those 882 sources , we classify 70 sources as variable using the stetson test , the @xmath22 test and the lomb - scagle periodogram . the faintest sources in the sample have @xmath103 mag , but naturally the measurement uncertainties are larger for fainter sources . the algorithms detect variability if the amplitude is larger than @xmath104 mag for a source of @xmath105 mag . we define a sample of cluster members , including sources with an ir excess due to a circumstellar disk or with x - ray emission . for cluster members , there is a clear correlation between evolutionary status and ir variability . more embedded sources are more often detected to be variable , and they have on average larger variability amplitudes . overall , the data are consistent with the idea that all ysos are variable at @xmath5 m and @xmath6 m , and we thus propose that all variable sources in our sample are members of the l1688 star forming region . qualitatively different morphological types of lightcurves can be distinguished : 14 lightcurves are detected to be periodic ; beyond that , we find quasi - periodic lightcurves , where the variability has an apparent timescale , but is not regular enough to be detected as periodic ; aperiodic lightcurves without a preferred timescale ; and long - term variable lightcurves where variability is apparent , but no periodicity or time - scale can be determined within the observational window . in addition , there are lightcurves with short , non - repeating bursts or dips . roughly half of all sources become redder when they are fainter ; the other half becomes bluer . the reddening values of the first group are compatible with ism reddenig . the color changes in the second group require variability in the inner disk structure as proposed by @xcite or @xcite . this work is based on observations made with the spitzer space telescope , which is operated by the jet propulsion laboratory , california institute of technology under a contract with nasa . support for this work was provided by nasa through an award issued by jpl / caltech . this research made use of astropy , a community - developed core python package for astronomy @xcite . this research has made use of the simbad database and the vizier catalogue access tool , both operated at cds , strasbourg , france and of data products from the two micron all sky survey , which is a joint project of the university of massachusetts and the infrared processing and analysis center / california institute of technology , funded by the national aeronautics and space administration and the national science foundation . acknowledges spitzer grant 1490851 . h.y.a.m . and p.p . acknowledge support by the ipac visiting graduate fellowship program at caltech / ipac . p.p . also acknowledges the jpl research and technology development and exoplanet exploration programs . rag gratefully acknowledges funding support from nasa adap grants nnx11ad14 g and nnx13af08 g and caltech / jpl awards 1373081 , 1424329 , and 1440160 in support of spitzer space telescope observing programs . s.j.w . was supported by nasa contract nas8 - 03060 . _ facilities : _ : : we identify 2mass j16272802 - 2439335 with because they match within the positional accuracy and the observed flux densities fit nicely together in a sed . : : after a visual inspection of the 2mass images , which form the basis of @xcite , we also identify 2mass j16273288 - 2428116 with . : : based on the position and the expected shape of the sed for a yso , we also identify [ gpj2008 ] source 3 with . : : we propose that [ amd2002 ] j162724 - 242850 and are the same source , because their positions match and the fluxes form an sed that is fully consistent with a yso in l1688 . and : : [ sect : misidentified ] in comparison with @xcite , we find that is erroneously identified with source oph l1688 3 in simbad . the distance between the positions on the sky is 13 , much larger than the positional uncertainty . therefore , the name is not used in our source table . ylw 16 : : in one case , the simbad database does contain a reference to a multiple source ( ) , that is resolved in the ir . we associate our lightcurves with , which is the dominant component . : : the ukidss @xmath31-band image shows at least two sources within 1 of iso - oph 152 with partially overlapping point - spread functions ( sourceids 442426789990 and 442426789990 ) . the position of the second source is uncertain and fitted differently in the other bands ( where it is called source i d 442426789991 ) . all of this is not resolved in the irac data ; thus , in each band , we assign the brightest magnitude of either of those sources to iso - oph 152 . [ sect : g09notwil ] : : this source detected in several near - ir observations and is known to be variable in that wavelength range . this source is part of the triple system that is well resolved in sub - mm observations . based on its sed stretching out to 6 cm , this is a class i source , which matches the classification we derive from the ir data . most likely , this souce contributes to the seizable outflows observed from the triple system , which would further confirm its youth and thus its status as a cluster member @xcite . : : this source is detected at @xmath106 m , but not at @xmath107 m by @xcite but no further information is available in the literature . : : in @xcite , this objects is listed as a yso and potential member of l1688 based on the g09 sed classification and a match to a scuba source at @xmath108 m . however , the next peak of the scuba flux is 29 from the position of [ gmm2009 ] oph l1688 30 , just below the maximum distance for a match that is accepted in that work . : : this source is also listed as a yso with a scuba flux peak 20 away @xcite . : : we find that this source is mismatched in simbad ( see section [ sect : misidentified ] ) . there are several radio observations in molecular lines and in the continuum which indicate substantial circumstellar material and thus confirm the status as a yso , but if sstc2d j162621.7 - 242250 and [ gmm2009 ] oph l1688 3 are different sources either of them could be the source of the radio signatures . , a. , whitney , b. , wood , k. , plavchan , p. , terebey , s. , stauffer , j. r. , morales - calderon , m. , & ysovar . 2013 , in american astronomical society meeting abstracts , vol . 221 , american astronomical society meeting abstracts , # 256.10

chaos in ordinary language means disorder , randomness , absence of law , and unpredictability . it played the major role in philosophy of ancient greeks . already empedocles ( cca.490430 b.c . ) viewed the real world , , as a combination of the world of perfect order , ( today we may tentatively translate it as ) , and the world of complete disorder , . since chaos seemed to be too elusive for any kind of rigorous description , the attention of science has for long been focused only to sphairos . this has changed over the past decades when numerous explorations have shown that chaos is not as lawless as originally thought . at present , the term _ chaos theory _ belongs to the common speak @xcite . in both classical and quantum physics @xcite is built on very sophisticated mathematics . it sticks on the original idea of randomness , but at the same time it creates a new type of universality in the representation of reality . the universality of chaos is based on ergodic character of motions and implies high efficiency of the statistically oriented description . since majority of systems in nature are chaotic , at least to a considerable extent , this type of description is often the ultimate one . the routes to more deterministic descriptions may exist but they are usually impassable . chaos thus belongs to very fundamental subjects in physics . among the physical systems that materialize the signatures of chaos in nature , an important place is held by atomic nuclei . already the fact that even more than 80 years after the discovery of their composition we are not able to present a satisfactory microscopic theory of nuclear structure is a good reason to think that nuclei are indeed chaotic objects par excellence . it is not an accident that nuclei were at dawn of the field named quantum chaos . however , atomic nuclei not only show well recognized signs of chaos , they also provide a variety of models that can be used to better understand the rules of chaos in a more general context . such a path is followed in this text . our aim is not to draw an exhaustive map of the overlap between physics of chaos and physics of nuclei such reviews exist @xcite and we may only recommend to read them . instead , we want to make a passage through selected topics of classical and quantum chaos using models taken from the description of nuclear collective dynamics . before diving into the ocean of chaos , we have to briefly introduce the models we will be playing with . these are the old ( and the good ) geometric collective model ( below abbreviated as gcm ) and the newer and more sophisticated interacting boson model ( ibm ) of nuclei . both these models attempt to describe nuclear collective motions vibrations and rotations assuming only the quadrupole type of nuclear deformations . the number of degrees of freedom associated with both gcm and ibm is five . two degrees of freedom correspond to the quadrupole deformation of the nucleus in the body - fixed frame ( one can think of an ellipsoid with a fixed volume ) ; their dynamics therefore represents vibrational motions . the other three degrees of freedom correspond to the orientation of the whole nucleus in the laboratory frame they describe rotations . in the following we will mostly consider only the non - rotational case , i.e. , fix the angular momentum ( which is an integral of motions ) to zero . in this case , we will be left just with two vibrational degrees of freedom , opening a direct link to other @xmath0 systems famously studied in the context of chaos . we will see that nuclear vibrations represent a rather rich specimen of chaos in two dimensions . the gcm , introduced already in 1952 by a.bohr ( for the review see @xcite ) , treats the nucleus as structureless drop described by quadrupole shape variables @xmath1 and rotation euler angles @xmath2 . these 5 generalized coordinates are associated with a particular parametrization of the rank-2 tensor @xmath3 ( real , symmetric , and traceless in the cartesian form ) , describing the quadrupole deformation of the nucleus in the laboratory frame . the deformation tensor @xmath3 can be introduced via the expansion of the nuclear radius @xmath4 to spherical harmonics @xmath5 , or alternatively via multipole moments of the nuclear mass ( charge ) distribution @xmath6 . in the first case , the @xmath7 spherical component of the deformation tensor ( @xmath8,@xmath9,@xmath10 ) is associated with the coefficient ( up to complex conjugation ) at the @xmath11 term of the radius expansion . in the second case , the deformation tensor is obtained via the following integral ( up to a scaling factor ) of the mass ( charge ) density : @xmath12 . any variables describing the shape of the nucleus must be invariant under rotations of the nucleus as a whole . there are just two independent building blocks for all rotational invariants made of the quadrupole deformation tensor namely the quadratic and cubic couplings of the quadrupole tensor to zero total angular momentum . the shape variables @xmath13 and @xmath14 can be defined as a certain parameterization of these two elementary couplings:^{(\lambda)}$ ] of rank-@xmath15 and -@xmath16 spherical tensors @xmath17 and @xmath18 to the resulting spherical tensor of rank @xmath19 ( the definition is analogous to the elementary angular momentum coupling ) . scalar product of spherical tensors is defined for @xmath20 as @xmath21^{(0)}$ ] . ] @xmath22^{(0)}=\tfrac{1}{\sqrt{5}}\ \beta^2 \ , \qquad [ [ \alpha^{(2)}\times\alpha^{(2)}]^{(2)}\times\alpha^{(2)}]^{(0)}=-\sqrt{7}\ \beta^3\cos 3\gamma \label{bega } \,.\ ] ] alternatively , @xmath13 and @xmath14 can be derived from the eigenvalues of @xmath3 , which read as @xmath23 $ ] , @xmath24 , @xmath25\bigr)$ ] . these values are the only components of the deformation tensor in the intrinsic frame ( if ) of the nucleus ( the frame connected with the principal axes of the quadrupole deformation ) , where the deformation tensor is diagonal . the transformation between the if and laboratory frame ( lf ) is expressed in terms of the euler angles @xmath2 , describing the orientation of the deformed nucleus in lf . ) with the same choice of parameters . adapted from ref.@xcite . ] note that @xmath1 can be considered as radial coordinates in the cartesian plane @xmath26 . while the radius @xmath13 quantifies the degree of deformation of the nucleus ( @xmath27 for spherical shape ) , the angle @xmath14 parameterizes the type of the deformed shape . the plane is divided into six equivalent angular sectors , which follow from an inherent discrete symmetry of the problem , given by the dynamical equivalence of three possible orientations of the deformed shape in the if ( along @xmath28 ) and two deformation types ( prolate , oblate ) . the @xmath29 , @xmath30 , and @xmath31 axially symmetric prolate or oblates shapes are located at @xmath14 equal to @xmath32 , @xmath33 , and @xmath34 , respectively , while the intermediate @xmath14 values correspond to the related triaxial shapes . note that these conclusions are consistent with the above - given form of the deformation tensor in the intrinsic frame . to become more familiar with the formalism ( which , as one has to agree , looks rather tough as the first sight ) , the reader is encouraged to consult specialized literature , e.g. , ref.@xcite . the plane of deformation coordinates with the basic sample of shapes is depicted in fig.[begafi ] . the momenta associated with the coordinates @xmath13 and @xmath14 are denoted as @xmath35 and @xmath36 , respectively . in the non - rotational case , as noted above , the euler angles @xmath2 are fixed , rendering zero values of the associated rotational momenta . the resulting vibrational hamiltonian therefore contains just the shape variables @xmath1 and the corresponding momenta @xmath37 . our analysis is performed mostly with the hamiltonian which contains purely quadratic momentum term of the kinetic energy , and quadratic , cubic and quartic coordinate terms of the potential energy : @xmath38}_{t_{\rm vib}}+\underbrace{a\beta^2+b\beta^3\cos 3\gamma+c\beta^4}_{v_{\rm vib}}\ , . \label{hgcm}\ ] ] @xmath39 and @xmath40 are adjustable parameters satisfying physical constraints @xmath41 ( kinetic energy is positive ) and @xmath42 ( the potential is confining ) . the geometric hamiltonian ( [ hgcm ] ) is closely related to the famous hnon - heiles hamiltonian , which was originally introduced in the context of stellar dynamics in galaxies and became a paradigmatic example of chaos in the classical domain @xcite . the hnon - heiles hamiltonian is a special case of eq.([hgcm ] ) with @xmath43 . in this contribution ( as in ref.@xcite ) we claim that the inclusion of the confining @xmath44 term generates even much richer structures than those obtained in the @xmath43 case . the coexistence of regular and chaotic types of classical motions generated by the full geometric hamiltonian at high energies is illustrated in fig.[begafi ] . the two trajectories shown in this figure were calculated for the same model parameters and energy , but represent very dissimilar species of collective motions . the regular orbit corresponds to a moderate , highly organized vibration , localized in a relatively small region around an axially symmetric equilibrium shape ( above the potential well to which the nucleus with given parameters drops in the low energy limit ) . in contrast , the chaotic orbit represents some large - amplitude vibrations , which rather disorderly wander through small , medium and large oblate deformations with all three intrinsic orientations . a generalization of the simple hamiltonian ( [ hgcm ] ) can be achieved by considering higher - order terms in kinetic or potential energy . we consider two extensions of the kinetic energy , particularly @xcite : @xmath45 \,,\qquad t_{\rm vib}^{(ii)}=\frac{1}{2m(1+\kappa\beta^2)}\left[p_\beta^2+\left(\frac{p_\gamma}{\beta}\right)^2\right ] \ , , \label{halt}\ ] ] both based on some specific deformation - dependent effective mass ( @xmath46 is an additional model parameter ) . such or similar extensions of the effective mass are relevant for the description of nuclear rotational bands since they modify the behavior of nuclear moments of inertia . here we consider them to probe the sensitivity of the vibrational measures of chaos to the general form of the hamiltonian kinetic term . another extension of the above hamiltonian is the inclusion of rotational terms , containing the intrinsic components of angular momentum and the corresponding moments of inertia . this however leads beyond the 2d description and makes the analysis much more difficult . we follow this line only in rather a restricted manner considering only rotations around a fixed axis @xmath31 @xcite . this can be done by introducing a third collective coordinate , angle @xmath47 , and the associate momentum @xmath48 , which coincides with the angular momentum of the nucleus around @xmath31 in both laboratory and intrinsic frames . the quadrupole deformation tensor @xmath49 in the laboratory frame is obtained from the diagonal form @xmath50 in the intrinsic frame ( see above ) via rotation by angle @xmath51 around the @xmath31-axis . this identifies @xmath47 as a variable connected with the if @xmath52 lf transition . the rotational energy reads as : @xmath53 which has to be added to the vibrational hamiltonian in eq.([hgcm ] ) to get the total hamiltonian @xmath54 . details can be found in ref.@xcite . an important feature of the geometric hamiltonian is related to its scaling properties the fact that dynamics of the system must be invariant under scale transformations affecting the units of relevant quantities , i.e. , energy , coordinate , and momentum . let us return now to the bare vibrational hamiltonian ( [ hgcm ] ) . since the scale transformations of this form can be absorbed in parameters @xmath55 , we deduce that some choices of these parameters must be dynamically equivalent . a more detailed treatment @xcite shows that the quantity @xmath56 is invariant under all scale transformation and represents the only essential parameter of the bare vibrational model in the classical case . the value of the mass parameter @xmath40 is relevant only in the quantum case , weighting the absolute density of quantum states in the discrete energy spectrum ( for @xmath57 we get the classical limit with infinite state density ) . this effective parameter reduction implies a crucial simplification of the numerical analysis of the geometric model . the quantization of the nuclear geometric model is achieved by replacing individual terms in the gcm hamiltonian by the corresponding operators in the hilbert space of wavefunctions @xmath58 . this task is mathematically rather involved and was completed with a substantial contribution of the patron of this school , marcos moshinsky see , e.g. , refs.@xcite and many others . the basic form of the vibrational kinetic energy reads as follows : @xmath59 note that this operator acts in the space where the scalar product @xmath60 is given by an integration over the collective coordinates with a specific measure @xcite . the expression of the potential @xmath61 is the same as in the classical case , see eq.([hgcm ] ) . since we are mostly interested in the non - rotational motions , the rotational energy will not be included into the quantized hamiltonian . examining eq.([tqgcm ] ) one immediately notices the differences from the kinetic energy of a particle in a 2d space , which in polar coordinates @xmath62 reads as @xmath63 . the absence of euler angles in the description of vibrational motions suggests a possibility to quantize the vibrational energy in the standard 2d manner , i.e. , using the formula : @xmath64 note that this implies a change of the definition of scalar product and the necessity to impose the condition @xmath65 to keep link with the 5d quantization , where this condition is naturally satisfied . in addition , we choose one of two possible angle - reflection symmetries of the wavefunctions , namely @xmath66 . we therefore construct three different quantizations of the vibrational collective hamiltonian , all of them yielding the same classical limit : the 2d - even , 2d - odd , and the 5d quantizations ( in the 5d case , the reflection symmetry is fixed ) . from the viewpoint of nuclear theory , only the 5d option is correct , but from the viewpoint of chaos theory ( which is in our focus here ) we have the freedom to probe all three possibilities . we will use this freedom to check the influence of the quantization scheme on the signatures of quantum chaos . for details , see ref.@xcite . let us move to the alternative description of nuclear collective motions with the aid of the interacting boson model . the ibm , introduced in 1970 s by f.iachello and a.arima ( for a review see @xcite ) , does a similar job as the gcm , but in terms of an ensemble of bosons with angular momentum 0 ( @xmath67-bosons ) and 2 ( @xmath68-bosons ) . these bosons are commonly interpreted as an approximation for nucleon pairs in an even - even nucleus ( their total number @xmath69 is usually fixed at a half of the number of valence nucleons ) , but at the same time they give rise to quadrupole degrees of freedom needed to describe basic collective motions of nuclei . the @xmath67- and @xmath68-bosons live in a hilbert space associated solely with their spin degree of freedom ( such space has a finite dimension ) and interact with each other through two - body interactions . for instance , a popular form of the hamiltonian is composed of the @xmath68-boson number operator @xmath70 and a bosonic quadrupole operator @xmath71 as follows : @xmath72)^{(2)}}_{\hat{q}_\chi}\cdot\underbrace{(d^\dag s+s^\dag\tilde{d}+\chi[d^\dag\times\tilde{d}])^{(2)}}_{\hat{q}_\chi } \ , , \label{hibm}\ ] ] where we used @xmath68-boson annihilation operators @xmath73 having proper transformation properties of a rank-2 tensor . the parameters @xmath74 and @xmath75 , weighting both terms in eq.([hibm ] ) , are more or less arbitrary , while @xmath76 from the quadrupole operator varies within interval @xmath77 for prolate shapes and @xmath78 $ ] for oblate shapes . note that the @xmath70 term of the above hamiltonian is of the 1-body nature and the term @xmath79 generates specific 2-body interactions . the link between the bosonic hamiltonian la ( [ hibm ] ) and the geometric hamiltonian of the type ( [ hgcm ] ) goes via the use of coherent states in the ibm . for instance , the glauber type of coherent states with an average total number of boson equal to @xmath69 is given by : ) onto a subspace with a sharp boson number , yielding a bosonic condensate state . ] @xmath80\ket{0 } \label{glau } \,,\ ] ] where @xmath81 stands for a normalization factor to ensure @xmath82 . the complex variables @xmath83 ( with @xmath84 ) which parameterize the coherent state , have transformation properties of a rank-2 spherical tensor and can be associated with a linear combination of the deformation tensor @xmath3 and the tensor of associated momenta @xmath85 . performing the same gymnastics as in the gcm case ( for details see ref.@xcite ) , one obtains the five generalizad collective coordinates @xmath86 and the corresponding momenta . the general bosonic hamiltonian @xmath87 is then translated into a function of these canonical variables by evaluating its expectation value in the respective coherent state @xmath88 . in the @xmath89 limit , this function coincides with the classical hamiltonian @xmath90 associated with @xmath87 . and @xmath91 , using two truncation dimensions of the hamltonian matrix expressed in the 2d oscillator basis , namely @xmath9239905 and 59080 . panel ( a ) shows a quantum measure of chaos @xmath93 ( where @xmath94 is so - called brody parameter of the nearest - neighbor spacing distribution ; for explanation see below ) calculated for spectra determined by a numerical diagonalization in both truncated bases . panels ( b ) and ( c ) show the absolute and relative differences between individual level energies , @xmath95 and @xmath96 , respectively . it is seen that at a certain energy the two calculations yield diverging results . only the fraction of the spectrum below the vertical dashed line ( corresponding to less than a half of the smaller dimension ) is reproduced correctly . adapted from ref.@xcite . ] for instance , hamiltonian ( [ hibm ] ) yields the following classical limit @xcite : @xmath97\sqrt{1-\frac{t+\beta^2}{2 } } \nonumber\\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\ ! -\frac{2\chi^2 b}{7}\left[\left(\frac{t+\beta^2}{2}\right)^2-p_\gamma^2\right ] \ , , \qquad\qquad\qquad\qquad\qquad\qquad t = p_\beta^2+\left(\frac{p_\gamma}{\beta}\right)^2 \label{hibmclas } \,.\end{aligned}\ ] ] it needs to be stressed , with regard to the boundedness of the ibm energy spectrum , that the coordinates and momenta appearing in the coherent - state parameterization are restricted only to finite domains , namely : @xmath98 $ ] , @xmath99 $ ] , and @xmath100 $ ] . thus the square root in eq.([hibmclas ] ) is always well defined . we may conclude that the ibm classical hamiltonians are closely related to the gcm hamiltonians , discussed above , though by far not identical with them . the most important difference between both models lies in the kinetic energy terms , which have a much more complex form in the ibm case . on the other hand , the potential energy terms , which can be extracted from the hamiltonian by setting all momenta to zero , are rather similar and yield analogous dynamical features . in conclusion of this section we say a few words on numerical computations . classical trajectories for both gcm and ibm are determined with the aid of the fourth - order runge - kutta integration of the classical equations of motions . some details can be found in ref.@xcite . quantum spectra are obtained through a numerical diagonalization of the respective quantum hamiltonian . calculations within the ibm ( which has just a finite dimensional hilbert space ) are naturally done in the complete basis composed of states with different numbers @xmath67- and @xmath68-bosons , coupled to definite angular momenta . the calculations in gcm ( infinite dimensional space ) has to be performed in a truncated basis . the results of these must be checked for convergence . let us streess that due to the restriction to the subset of the quantum spectrum with angular momentum @xmath101 , we perform all gcm calculations in two dimensions . in general , only a certain fraction of the gcm states obtained by the numerical diagonalization within a truncated 2d basis can be employed in the analyses . as illustrated in fig.[egcm ] , these converged states yield unambiguous and reliable results . in physics , we like most to deal with integrable systems . these are the systems which roughly speaking have a sufficiently large degree of symmetry to make the dynamical equations solvable . more precisely @xcite , the full integrability of a classical hamiltonian system with @xmath102 degrees of freedom requires the following three conditions to be simultaneously satisfied : ( a ) there exists @xmath102 integrals of motions @xmath103 ( @xmath104 ) , represented as functions in the @xmath105-dimensional phase space which have zero poisson bracket with the hamiltonian , @xmath106 , ( b ) all these integrals are mutually , i.e. , have vanishing poisson brackets with each other , @xmath107 @xmath108 ( in quantum mechanics , we would say : they are compatible ) , and ( c ) the @xmath105-dimensional gradients @xmath109 are all linearly independent in every point of the phase space . these conditions guarantee that the hamiltonian ( as well as any other conserved quantity ) can be written as a function of the given integrals , @xmath110 , and that there exists a canonical transformation @xmath111 to some new variables ( so - called action - angle variables ) in terms of which the solution of the hamilton equations becomes rather trivial ( the transformation itself being usually highly nontrivial ) . for example , the dynamics of the geometric collective model and the interacting boson model is integrable if the corresponding hamiltonian does not depend on variable @xmath14 . then the momentum @xmath36 becomes an integral of motion , in addition to the hamiltonian @xmath112 itself . it is easy to verify that all three conditions for integrability in dimension @xmath0 are fulfilled in this situation : ( a ) and ( b ) are obvious , and ( c ) follows from the fact that the gradient of @xmath112 is a nonvanishing normal to the energy surface , while the gradient of @xmath36 is a nonvanishing tangent therefore , they always define two perpendicular directions . as we see from eqs.([hgcm ] ) and ( [ hibmclas ] ) , the gcm and ibm hamiltonians are independent of @xmath14 ( hence integrable ) for @xmath113 and @xmath114 , respectively . the @xmath36-dependent integral of motion is usually written in a less trivial form @xcite : @xmath115 ( where @xmath116 are components of the angular momentum @xmath117 in the if ) , which defines a quantity in the nuclear context called . note that the rigorous proof of integrability in the full dimension @xmath118 ( including rotational degrees of freedom ) requires a construction of the fifth integral of motions , which is additional to @xmath112 , @xmath119 , @xmath120 , and @xmath121 . the ibm hamiltonian ( [ hibm ] ) has an extra realization of integrability , situated at the points @xmath122 of the parameter space ( @xmath75 remains an arbitrary scaling parameter ) . it can be shown that this case of integrability of the ibm hamiltonian is connected with its underlying su(3 ) dynamical symmetry @xcite . ( red and blue dots ) and @xmath123 ( black dot ) . the two cases , differing just by a small displacement of the lighter body , show rather different sets of trajectories . exponentially diverging distance between both alternative positions of the lighter body is shown in the inset . adapted from ref.@xcite . ] for 3-body system ) . panel ( a ) shows evolution of initially close trajectories for times @xmath124 units , in which the deviation is not large . panel ( b ) displays the evolution for times @xmath125 units , the two final positions represented by the blue and red dots ( with the corresponding shapes drawn aside ) . note that long - time deviation between the two vibrational orbits ( given in the inset ) reaches a saturation regime , which follows from the boundedness of the system for any finite energy . ] already from these examples it must be obvious that integrability is a rather rare spice , therefore that by far not all the systems are integrable . the full realization of this simple fact and its essential consequences came surprisingly late only in the last decade of the 19th century . it happened independently and almost simultaneously in the works of h.poincar ' e and a.lyapunov , who can undoubtedly be named the founders of chaos . poincar ' e s story , in particular , can be used as an exemplary illustration of the random ways in which science makes its progress @xcite . poincar ' e , who in the preceding years spent a lot of effort to solve the three body problem , submitted an account of his results to a scientific competition , organized in 1885 to celebrate the birthday anniversary of king oscar ii of sweden . his work won the competition , but an assistant of one of the famous mathematicians in the jury noticed some unclear points in the derivations . the thing was even much more complex than poincar ' e thought . in 1890 , he submitted a new version of his treatise , in which the mathematical notion of chaos defined in the sense of instability of motions was born . in one of the later writings poincar ' e explains : _ _ an example of this behavior is given in fig.[body3 ] . the new treatise was published at poincar ' e own expenses , exceeding the reward of the competition , which may serve as a warning to those who think that science must always be profitable . lyapunov came to similar conclusions ( concerning the instability of motions ) in 1892 in his doctoral thesis @xcite . the description of chaos remained for long a rather exotic domain of abstract mathematics . in integrable systems , the transformation to action - angle variables @xmath126 ensures that motions in the phase - space , described by hamilton equations @xmath127 and @xmath128 , are localized on manifolds , which are topologically equivalent to tori . these motions are therefore highly organized . poincar ' e himself developed a method how the tori can be visualized for systems with @xmath0 . the method is based on cutting the phase space by a planar section and marking passages of individual trajectories through the section today we call these pictures poincar maps . systems results from the requirement that each point of the planar section is crossed by a single trajectory . indeed , the point in the section is characterized by 2 variables ( usually a coordinate @xmath129 and the associated momentum @xmath130 ) , another variable is fixed by the section constraint ( often of the form @xmath131 ) , and the fourth variable ( e.g. , the momentum @xmath132 ) can for hamiltonian systems be computed from energy conservation . therefore , we have determined all 4 variables in the 4-dimensional phase space . for @xmath133 systems with no integrals of motions additional to @xmath112 the poincar ' e map would have to be more than 2-dimensional , which is rather impractical . ] but what happens when the integrability is destroyed ? certainly , some of the tori must die , but what is the exact scenario and an underlying mechanism ? plane in the phase space ; details of the hamiltonian can be found in ref.@xcite . the concentric curves correspond to deformed phase - space tori . all panels show examples of the poincar ' e - birkhoff fixed - point theorem . courtesy of m.macek . ] in 1912 , poincar ' e conjectured a theorem , in which he partly attacked this problem . the theorem was proven in 1913 and finalized in 1925 by g. birkhoff and became known as the poincar ' e - birkhoff fixed - point theorem . consider an @xmath0 torus characterized by a pair of frequencies @xmath134 with ratio @xmath135 . if @xmath136 is an irrational number , the trajectory never closes a loop the motion is only quasi - periodic . in contrast , if @xmath136 is rational , all trajectories on the given torus are periodic after @xmath137 crossings of the poincar ' e surface they return to the initial point . the theorem of poncar ' e and birkhoff is focused on such a rational tori , or more precisely , on the remnants of these tori after the destruction of integrability by a general perturbation . they showed that each periodic torus leaves an even number of fixed points in the poincar ' e map a half of these fixed points is stable ( elliptic ) , the other half is unstable ( hyperbolic ) . this was the first indication that the transition from the full regularity of an integrable system to a complete chaos of a strongly perturbed system is a highly non - trivial ( and usually rather fascinating ! ) process . figure [ ibmsec ] shows some examples of poncar ' e maps generated by the ibm hamiltonian with @xmath101 ( hence @xmath0 ) . alternating islands of elliptic and hyperbolic fixed points are clearly visible . several decades later , an essential and famous result related to the order - to - chaos transition in classical hamiltonian systems was formulated and proven by a.kolmogorov ( 1954 ) , v.arnold ( 1963 ) , and j.moser ( 1962 ) . it is nowadays known as the kam theorem @xcite . what does it tell us ? it attacks the problem of what exactly happens if an integrable system is subject to an increasingly strong non - integrable perturbation . which of the tori of the original system die , for a given strength of perturbation , and which survive in a deformed form ? it turned out that the rational tori , such as those considered above , disappear immediately , leaving only the above described remnants . on the other hand , some of the irrational tori are rather they may survive ( in a deformed form ) even rather strong perturbations . the clue for the torus to survive is that it must be . this superficial statement can be quantified by means of the number theory , through the convergence properties of rational approximations of the number @xmath136 ( the ratio of torus frequencies for an @xmath0 system ) . the tori characterized by irrational ratios @xmath136 ( slowly converging rational sequences ) are more resistant than the tori with ratios ( fast converging rational sequences ) . some people say that this law demonstrates the higher immunity of ugliness in the world , but sometimes even the most terrible irrational orbits look really pretty . ) with @xmath138 . value of @xmath75 is given in each column . the maps are constructed for energy @xmath139 of the potential local maximum at @xmath27 ( here , all sections are contracted to a single point with @xmath140 ) . the upper and lower rows show the @xmath141 and @xmath142 sections , respectively . each map visualizes 1000 intersections of 100 trajectories with the plane of the section . adopted from ref.@xcite . ] several examples of complex poncar ' e maps generated by the gcm vibrational hamiltonian ( [ hgcm ] ) are shown in fig.[gcmsec ] . particularly the maps in the middle column ( @xmath143 ) show numerous fine structures , which must be caused by rather sophisticated regular orbits . they appear in the system in spite of a large strength @xmath75 of the hamiltonian term violating the full integrability at @xmath113 . however , a surprising observation is that most of these orbits can not be identified with the survived tori from the @xmath113 system since the maps for lower values of @xmath75 ( c.f . , @xmath144 ) contain no trace of such orbits . one may perhaps think of a of the given hamiltonian to an unknown integrable hamiltonian beyond the parameter space of the gcm . note that also the @xmath145 maps contain some suspicious fine structures , whose origin can not be explained via the only known integrable regime of the model . nuclear vibrations even in their most simplistic description are highly nontrivial type of motion ! to classify the overall degree of chaos for transitional systems such as those exemplified in fig.[gcmsec ] , one needs to divide the whole phase space into the part filled with regular orbits , and the part filled with chaotic orbits . since each point of the phase space is crossed by a single trajectory , this division is unique , and one can introduce the degree of classical chaos through the following quantity , below shortened as the : @xmath146 \label{freg } \,.\ ] ] here , @xmath147 stands for the total of the fixed - energy hypersurface @xmath148 in the @xmath105-dimensional phase space , while @xmath149 is the occupied by regular orbits . the fraction @xmath150 is therefore a quantity between 0 ( for fully chaotic systems ) and 1 ( for completely regular systems ) . corresponding to the given orbits is shown in panel ( e ) , where we notice a clear difference between regular and chaotic cases . this is verified more systematically in panel ( d ) , which shows @xmath151 at three different times for 2000 orbits initiating along the line @xmath152 of section ( b ) . we observe that chaotic domains yield @xmath153 decreasing to negligible values , while in regular domains @xmath153 fluctuates around much higher values . adapted from ref.@xcite . ] it is clear that any numerical estimate of the regular fraction ( [ freg ] ) for a system between order and chaos must be imperfect . its precision depends primarily on the sampling density of the phase space by test trajectories . each of these trajectories is either regular ( stable ) or chaotic ( unstable ) , and the regular fraction can be directly approximated by the fraction @xmath154 . the convergence of this estimate to the exact value of @xmath150 is guaranteed ( assuming a uniform coverage of the phase space by initial points ) in the limit @xmath155 . however , a truly hard computational problem is how to unmistakably determine the stability or instability of each individual orbit . this requires to compute simultaneously the evolution of several orbits @xmath156 $ ] ( both @xmath29 and @xmath157 stand for @xmath102-dimensional vectors ) which initiate in a close vicinity of the given reference orbit @xmath158 $ ] . if the maximal lyapunov exponent for the deviation @xmath159 of these orbits is positive , the reference orbit is chaotic , if the lyapunov exponent is zero , the orbit is regular . is defined through the long - time phase - space distance @xmath160 between a given orbit and an arbitrary accompanying orbit : @xmath161 . this yields an exponential growth of the maximal distance for long times : @xmath162 . fof instance , the lyapunov exponent for the trajectory in fig.[ginst ] , determined however only from a finite time interval of the linear growth of @xmath163 ( where @xmath47 stands for the distance in the coordinate plane ) , is @xmath164 . ] applicability of this asymptotic - time criterion is however hindered in bounded systems , where the deviation of different orbits has an upper limit for any finite energy @xmath165 , yielding therefore all lyapunov exponents formally zero ( cf . the time dependence of the deviation of the vibrational orbits in fig.[ginst ] ) . several ways to bypass this computational problem has been proposed in the literature . one of these ways the one based on so - called alignment indices @xcite was used in our gcm and ibm calculations . the procedure resorts to evaluating only the relative phase - space deviations of trajectories instead of absolute ones : @xmath166 , where @xmath167 . the relative deviation is clearly a unity vector in the phase space , @xmath168 , and only the mutual orientation of two or more such vectors is meaningful . let us take a pair of randomly selected orbits , both of them accompanying the reference orbit @xmath169 , and calculate relative deviations @xmath170 and @xmath171 . if the reference orbit is unstable , the relative deviations tend to align in either parallel or antiparallel directions along the line of the maximal lyapunov exponent . this means that the smaller of the lengths @xmath172 ( this quantity is called and denoted below as @xmath153 ) converges to zero for long enough times . in contrast , for a stable reference orbit @xmath153 keeps oscillating within the definition interval @xmath173 $ ] . an illustration of the efficiency of this method in the gcm is presented in fig.[sali ] . for the gcm hamiltonian ( [ hgcm ] ) at @xmath139 as a function of control parameter @xmath74 and/or @xmath75 for @xmath174 . three computation methods are compared by distinct curves . the methods are distingished by initiating the trajectories on the @xmath141 or @xmath142 sections ( two curves denoted as @xmath150 ) , or in the whole phase space ( curve @xmath175 ) . the horizontal axis is split into two parts : the first with @xmath176 and @xmath177 $ ] ( the scale invariant parameter @xmath178 $ ] ) , the second with @xmath179 and @xmath180 $ ] ( hence @xmath181 $ ] ) . adopted from ref.@xcite . ] our procedure for calculating the regular fraction @xmath182 for a given gcm or ibm vibrational hamiltonian ( @xmath0 ) at fixed energy @xmath165 proceeds in the following steps : ( i ) we choose either the @xmath141 or @xmath142 section of the phase space and calculate the domain in this section that can be reached by trajectories with the given energy . ( ii ) we cover the accessible domain with tiny rectangular cells of the same area ; total number of these cells is @xmath183 . ( iii ) we perform a repeated computation in which an arbitrary point inside any cell of the accessible domain becomes an initial point of a calculated orbit ( note that all 4 canonical variables are fixed by selecting a point in the section and energy ) . the stability of this orbit is determined by the alignment index technique . if the orbit is stable , the initial cell as well as all the cells visited by the orbit during the evaluated time interval are counted and the resulting number is incremented to variable @xmath184 ( initially set to zero ) . it can happen that a single cell is visited by both regular and chaotic orbits ; in that case the value incremented to @xmath184 for this cell is the properly weighted average between 0 and 1 . ( iv ) the calculation is repeated until all cells of the accessible domain are either initial for a trajectory or are visited by at least one of these trajectories . ( v ) we estimate the regular fraction as @xmath185 . this method has an advantage of a systematic coverage of the phase space or more precisely , its 2d section with trajectories . on the other hand , it is clear that the resulting value of the regular fraction may depend on the phase - space section selected . the efficiency of our procedure and its inherent ambiguities in the gcm are illustrated by fig.[freg0 ] , where we compare the values of @xmath182 ( for @xmath139 and variable control parameter @xmath75 and/or @xmath74 ) calculated for both @xmath141 and @xmath142 sections , as well as by directly counting the stable and unstable orbits generated without reference to any phase - space section . it is seen that all three methods give approximately the same results . the differences between the three curves in fig.[freg0 ] can be declared to be an internal uncertainty of the method at the selected level of resolution ( number @xmath186 ) . figure [ freg0 ] , apart from its methodological aspect , has also a very interesting physical content . we see that for moderate values of parameter @xmath75 the regular fraction quickly drops from value @xmath187 at @xmath113 to @xmath188 at @xmath189 . in view of the kam scenario , this is a perfectly expectable behavior since @xmath75 weights the nonintegrable perturbation in the hamiltonian ( [ hgcm ] ) . however , at @xmath190 the regularity starts growing again ! in particular , in the subsequent region up to @xmath191 we observe a series of fine - structured peaks in the @xmath150 dependence . or @xmath192 is limited to energies @xmath193 and results from a specific shape of the potential . ] the increased regularity in this parameter domain is connected with rather complex poincar maps , such as those shown in the middle column of fig.[gcmsec ] . quite surprisingly , the peaks in fig.[freg0 ] follow a certain if expressed in scale invariant parameter @xmath194 ( the four local peaks appear at integer multiples of the value @xmath194 corresponding to the first peak ) . the origin of this behavior is still unknown . the complete dependence of @xmath150 on the gcm hamiltonian parameters and energy can be found in ref.@xcite . for gcm hamiltonians with extended kinetic term @xmath195 ( panel a ) and @xmath196 ( panel b ) , see eq.([halt ] ) . dependences for several values of parameter @xmath46 are given for both types of kinetic term . the other parameters are the same as in fig.[freg0 ] ( which corresponds to the @xmath197 case in both panels ) . adapted from ref.@xcite . ] in fig.[freg0ext ] , we show the regularity plots for generalized gcm hamiltonians with the kinetic energy terms according to eq.([halt ] ) . one can point out the opposite prevailing influence of type i and ii extensions of the kinetic energy on the regular fraction : while the type i extension with growing @xmath46 tends to reduce @xmath150 , the type ii extension seems to rather increase it . note that the influence of the rotational term ( [ trot ] ) on the regularity plot was analyzed in ref.@xcite . let us note that similar structures as in figs.[freg0 ] and [ freg0ext ] are observed also in the interacting boson model , where people often discuss about a so - called @xcite . it needs to be stressed that the general ibm hamiltonian is more sophisticated than the gcm one and therefore can not be characterized by a single scale invariant parameter such as @xmath194 . in particular , the common parameterization ( [ hibm ] ) yields two essential parameters , namely @xmath76 and the proportion between @xmath74 and @xmath75 , so that the peak series from fig.[freg0 ] becomes a kind of in the regularity landscape . the ridge follows an arc shape in the parameter plane and , as in the gcm , it is most pronounced for energies just above the @xmath27 local maximum of the potential . the physics behind both phenomena is probably rather similar . there is a fundamental problem to define chaos in quantum mechanics since linearity of the theory excludes any chance to introduce an exponential sensitivity to initial conditions . consider two normalized initial states @xmath198 and @xmath199 differing by @xmath200 with @xmath201 and @xmath202 , @xmath203 . the evolution of both states reads : @xmath204 where all scalar products remain conserved due to unitarity of the evolution operator . as a consequence , the squared distance of both states , @xmath205 , is independent of time . quantum hilbert space does not permit any kind of butterfly - wing effect . a different type of instability , the one which remains relevant even in the quantum domain , was pointed out by a.peres in 1984 @xcite . instead of the dependence on initial conditions he proposed to consider the dependence of quantum dynamics on details of the hamiltonian ( tiny changes of interaction parameters or small external perturbations ) . although the resulting concept of so - called plays an important role in many branches of quantum physics , including quantum information applications , it does not provide unfortunately!a selfsufficient definition of chaos in general quantum systems @xcite . according to present understanding , chaos is firmly anchored in classical physics . quantum physics can only explore a multitude of its distinct consequences ( signatures ) on the quantum level of description . quantum chaos in this sense is therefore not a , but rather a research field located somewhere in the highlands along the border between quantum and classical physics . m.berry , who undertook some pioneering expeditions to these wild territories , speaks about instead of quantum chaos @xcite . for bounded quantum systems those with discrete energy levels the most important signatures of chaos were identified in statistical properties of energy spectra . the relevant mathematics was created already in the 1950 s , long before the quest for quantum chaos started @xcite . it was when e.wigner looked for an achievable type of physical description related to long sequences of neutron resonances plentiful and rather complex excited states of compound atomic nuclei . wigner decided to give up the exact description prediction of the properties of all individual resonances and proposed to strictly separate smooth ( predictable ) and fluctuating ( statistical ) parts of the corresponding quantities . while the smoothened dependences ( averages , which reflect only sort of properties of the system ) can be described in terms of some reasonable simplifying models , the fluctuations ( that depend on all details of the dynamics ) represents just random , unpredictable component of the problem . in the 1980 s it became clear that quantum signatures of chaos lie right in these fluctuating properties @xcite . as an example , consider a discrete energy spectrum described by the state density @xmath206 . to predict the position of every individual level @xmath207 might be too hard ( or impossible especially if the system is as complex as a nucleus . however , one can perhaps estimate a smoothened density @xmath208 , where @xmath209 stands for a suitable smoothening function ( e.g. , the gaussian of width @xmath210 ) . indeed , the smoothed density is closely related to the size @xmath211 of the energy shell in the multidimensional phase space in units associated with the planck constant , which , in principle , can be evaluated with the aid of suitable techniques ( like the fermi gas approximation in the case of a nucleus ) . in contrast , the remaining component of the state density , its fluctuating part @xmath212 , depends on tiny details of the hamiltonian and can not be predicted by simple models . nevertheless , statistical features of @xmath212 reflect the degree of chaos in the classical limit of the system . to extract the chaos - related information , one usually performs a transformation of the spectrum called @xcite . it can be written in the following way : @xmath213 the unfolded sequence @xmath214 has a constant average density ( equal to one ) and therefore materializes statistical properties of the spectrum in the most explicit form . it turns out that mutual correlations between the members of this sequence show some universal behavior , which depends on whether the classical counterpart of the system is regular or chaotic . surprisingly , in regular ( integrable ) systems the unfolded sequences of energies with the same quantum numbers ( such as angular momentum or parity ) exhibit only weak or virtually no correlations , making the visual appearance of spectra rather disordered . in contrast , chaotic systems yield strong correlations and produce seemingly more regular spectra . ( classical regular fraction ) , @xmath93 ( the brody parameter adjunct ) , and @xmath215 ( a shifted exponent of the power spectrum ) for the gcm . panels ( a ) and ( b ) show results for @xmath216 and for @xmath217 , respectively . the mass parameter was chosen such that the displayed energy intervals contain 30 ( panel a ) and 40 ( panel b ) thousand of quantum levels , calculated with the 2d quantization option ( see the text ) . panels ( c ) and ( d ) show separately @xmath93 and @xmath215 for all three different quantization schemes ( the hamiltonian parameters are the same as in the respective left panel ) . we conclude that within the error bars ( given by the fitting methods of the respective quantities in finite samples of levels in a vicinity each energy)the correlation measures show no dependence on the quantization . ] the correlations in unfolded spectra of chaotic systems are of both short- and long - range types . the short range correlations are usually expressed through the statistical distribution of nearest neighbor spacing @xmath218 constructed for sequences of levels with the same quantum numbers . this distribution reveals whether the system exhibits the so - called , a phenomenon indicating the absence of additional integrals of motions . we use the following formulae @xcite : @xmath219 where the brody form with @xmath220 interpolates between the limiting cases @xmath221 ( poisson ) and @xmath222 ( wigner ) . note that the symbol @xmath223 stands for the normalization factor and @xmath224 for a coefficient ensuring that @xmath225 . the poissonian distribution , which describes an uncorrelated spectrum , is typical for integrable systems . it implies the absence of level repulsion ( since @xmath226 for @xmath227 ) , which is related to the existence of additional integrals of motions that eliminate the corresponding offdiagonal matrix elements of the hamiltonian . the wigner formula includes the level repulsion given by @xmath228 for small spacings and captures properties of chaotic systems ( with generally nonvanishing offdiagonal hamiltonian matrix elements ) . ) is only an approximation of the distribution predicted by the random matrix theory . it applies in system symmetric under the time reversal . for other systems , the formula is modified to include even stronger ( @xmath229 ) level repulsion @xcite . ] system in between order and chaos are characterized by a transitional brody form with the parameter @xmath94 somehow related to the overall degree of chaos . to be more specific , we expect a correlation ( generally nonlinear ) between values @xmath230 . associated with the gcm unfolded energy spectrum at a single energy @xmath165 . model parameters : @xmath231 , @xmath232 . panel ( a ) displays the number @xmath137 of levels in the energy interval @xmath233 corresponding to the shortest periodic orbit ( period @xmath234 ) as a function of energy @xmath165 . the value @xmath235 sets the minimal frequency @xmath236 for the evaluation of the power spectrum . panel ( b ) shows in the log - log scale the power spectrum @xmath237 for a sample of 2048 levels in a vicinity of the energy indicated in panel ( a ) by the vertical line . the slope of the cloud of points in the above @xmath238 ( vertical line in panel b ) yields the exponent in the @xmath239 dependence . the power spectrum is determined by two different methods : ( i ) by considering all 2048 levels ( black dots ) and ( ii ) by grouping the levels into 32 bunches of 64 levels and calculating an average power spectrum over the bunches ( green dots ) . the corresponding fits of the slope , indicated by the full ( i ) and dashed ( ii ) tilted lines , yield values @xmath240 and @xmath241 . note that in fig.[qchagcm ] we used the method ( ii ) . ] long range correlations in spectra of chaotic systems are also very strong . they can be expressed in different ways , for instance via the variance of the number of unfolded levels in an interval of length @xmath242 , or equivalently via a so - called @xmath243 statistics @xcite . an alternative approach @xcite is based on the fourier analysis of a sequence of statistical quantities @xmath244 . the sequence apparently satisfies @xmath245 , expressing fluctuations of the cumulative number @xmath246 of unfolded levels around the straight line @xmath247 . the fourier transformed sequence reads as @xmath248 $ ] , with @xmath40 being the number of levels included into the analysis , and yields a power spectrum @xmath249 . the power spectrum contains information on the frequencies @xmath102 present in the @xmath250 ( number @xmath137 being identified with a discrete time ) and represents a statistical quantity associated with @xmath250 . it turns out that the average of the power spectrum can be well approximated by the formula @xmath239 , where @xmath251 for strongly correlated spectra of chaotic systems , and @xmath252 for the uncorrelated poissonian spectra of regular systems @xcite . hence we establish a link @xmath253 . statistical properties of spectra were extensively studied for both the geometric collective model and the interacting boson model @xcite . figure [ qchagcm ] shows some of the gcm results , namely a comparison of classical regular fraction with the corresponding measures of short - range and long - range spectral correlations for two choices of the hamiltonian parameters . a clear correspondence between @xmath150 and the dependences of @xmath254 and @xmath255 is observed , which confirms the above given statement concerning the link between chaos , on the classical level , and statistical properties of spectra , on the quantum level . this , as some authors call it , has been already tested in numerous systems , but mostly in the regime of a weak energy dependence of the regular fraction @xcite . in particular , the most influential analyses have been performed with 2d billiard systems , which show a constant value of @xmath150 , totally independent of @xmath165 . here we verify the bohigas conjecture in a more complex form ( though still based on a 2d model ) , demonstrating that even rapid changes of the classical regularity are closely followed by both short- and long - range quantum correlation measures . moreover , the comparison is performed for different quantization schemes and shows that the correlation properties are quantization - independent . these properties must therefore reflect only the features related to the classical limit of the system , in agreement with the bohigas conjecture . we want to emphasize that the rapid variability of the gcm regular fraction with energy not only creates a desirable framework for testing the quantum chaos assumptions , but also represents a considerable computational challenge ! the statistical analysis of the spectrum can only be performed piece by piece on samples of levels situated in a close vicinity of energy @xmath165 , where @xmath165 varies along the whole spectrum . on the one hand , the samples have to be sufficiently large to allow for a reliable statistical evaluation , on the other hand , the energy interval @xmath95 spanned by each sample needs to be sufficiently narrow with respect to the scale that determines local variations of @xmath182 . as an example , we illustrate in fig.[onef ] the fourier analysis of a single sample of levels around one particular energy @xmath165 , outlining the procedures capable to determine the local power - spectrum exponent @xmath256 . in the rest of this section we describe an interesting visual approach to quantum chaos , which will complement the cute poincar ' e maps presented above in connection with classical chaos . the method was proposed by a.peres @xcite ( also in 1984 , which seems to be _ annus mirabilis _ for quantum chaos ) and its practical application is limited to systems with two degrees of freedom . it is based on the fact that the time average @xmath257 of an arbitrary quantity @xmath258 represents an automatic integral of motion , regardless of any specific dynamical properties of the system . classically , the average is defined through infinite - time integration over the whole classical orbit , and therefore assigns a unique value @xmath259 to any point of the phase space ( the same value is assigned to all points visited by a single trajectory ) . the integrals of motions @xmath257 do not generally satisfy the conditions for integrability ( any system has an infinite number of such integrals ) , but , as shown below , they provide a valuable analyzing tool for systems between order and chaos . ) with @xmath260 [ the o(6 ) dynamical symmetry limit ] for @xmath261 bosons . symbol @xmath194 stands for the o(5 ) seniority , while the other symbols denote casimir invariants of the su(3 ) and u(5 ) algebras , which are involved in the overall u(6 ) dynamical algebra of the ibm . the lower row shows the lattices @xmath262 of mean values of the respective quantities @xmath258 , while the upper row shows the corresponding lattices of the variances @xmath263 . courtesy of m. macek . ] in the 5d quantization with @xmath264 ( upper rows ) and @xmath265 ( lower rows ) . parameters @xmath74 and @xmath75 of the potential are indicated above the panels , @xmath174 , and @xmath266 . ] in the quantum case , the time average is defined by the following formula : @xmath267\,\hat{p}\,\exp[-\tfrac{i}{\hbar}\hat{h}t ] \,,\qquad [ \hat{\bar{p}},\hat{h}]=0 \label{per } \,,\ ] ] where @xmath268 is an operator corresponding to the original observable . it is easy to verify that the commutator of the time - average operator @xmath269 with the hamiltonian @xmath87 vanishes in the @xmath270 limit , irrespective of the concrete choice of @xmath268 . quantum expectation values of both observables @xmath268 and @xmath269 are the same for the energy eigenstates of the system , @xmath271 , and it is therefore rather straightforward to create a 2d picture in which the discrete energies @xmath272 are plotted against the corresponding expectation values @xmath273 . for the resulting lattice of points we use the term . it can be constructed for any observable @xmath268 . peres lattices for systems with @xmath0 can be seen as quantum counterparts of the classical poincar ' e maps . it turns out that geometric regularity ( irregularity ) of the lattice signals regular ( chaotic ) dynamics of the system on the classical level . in particular , if the system is classically integrable , any integral of motion must be a function ( suppose a function ) of the canonical actions , hence @xmath274 and @xmath275 . in transition to the quantum regime , @xmath276 and @xmath277 become approximate multiples of the planck action and form a rectangular grid in the @xmath278 plane . as the peres lattice in the @xmath279 plane is related to this @xmath278 grid by a sooth mapping , it will certainly be somehow distorted , but one expects that it will still look orderly . for non - integrable systems , on the other hand , there is no reason for the peres lattice to be regular . for given energy @xmath165 tend to be contracted to a single value due to ergodicity of the system . ] in this case , any partial arrangement of the lattice is a signature of fractional regularity of the dynamics . it should be noted that various peres lattices can in general be combined with each other or smoothly transformed without affecting regularity or chaoticity of the patterns present in the lattice . for instance , the lattice of @xmath272 versus @xmath263 , which displays the variance of the quantity @xmath258 in the hamiltonian eigenstates , is just a combination of lattices for @xmath280 and @xmath258 . some examples of peres lattices for the above ibm and gcm hamiltonians are given in figs.[peribm ] and [ pergcm ] . the first of them , fig.[peribm ] , shows several incompatible lattices calculated for an integrable ibm hamiltonian with the o(6 ) dynamical symmetry . the o(5 ) seniority , cf . eq.([senior ] ) , whose average and variance lattices are drawn in the first column , represents a conserved quantity for any ibm hamiltonian ( [ hibm ] ) with @xmath114 , as can be directly verified through the null variance of the seniority in all hamiltonian eigenstates . the other quantities are not integrals of motions , they nevertheless generate perfectly regular lattices in both average values and dispersions . this is in agreement with integrability of the hamiltonian employed . figure [ pergcm ] depicts several peres lattices for the gcm , using eigenstates with @xmath101 in the 5d quantization . the lattices were constructed using only the average values of the respective peres operator , the corresponding variances are not given this time . two choices of the peres operator are presented : the first one is the @xmath101 seniority @xmath281 , which expresses an operator @xmath282 in the plane @xmath283 of the deformation coordinates ( in the 5d quantization ) . the corresponding averages are denoted as @xmath284 and their lattices are shown in the first row of each panel . the second choice of the peres operator coincides with the cubic term of the gcm potential , @xmath265 , which represents the nonintegrable perturbation of the @xmath113 integrable hamiltonian . the averages , denoted @xmath285 , are seen in the second row of each panel . peres lattices in fig.[pergcm ] are given for several points in the parameter plane @xmath286 , following a path which crosses all inequivalent gcm potentials differing by the scale invariant parameter @xmath194 . at first , we fix @xmath176 and change @xmath287 , visualizing the onset of chaotic vibrations in deformed nuclei and their partial regularization at larger values of @xmath75 . next we keep @xmath179 and change @xmath288 . in this way , we make a transition between two qualitatively different types of the potential the one corresponding to deformed equilibrium shapes of the nucleus and the one associated with spherical nuclei ( the @xmath289 potentials have three degenerate minima at @xmath290 , while those with @xmath74 above a certain critical value @xmath291 have just a single global minimum at @xmath292 ) . and @xmath290 minima of the potential become degenerate and swap is at @xmath293 . ] finally , we fix @xmath294 and decrease @xmath295 , coming back to the integrable regime , but on the spherical side of the parameter plane . the most chaotic lattices are the ones in panels ( c ) and ( e ) . the first corresponds to @xmath296 , which coincides with the first minimum of @xmath150 at @xmath139 in fig.[freg0 ] . both peres lattices in this panel can be compared with the energy dependences of @xmath182 and with the other measures of chaos shown in panels ( a ) and ( c ) of fig.[qchagcm ] . we may notice that the regions with increased regularity in fig.[qchagcm ] correspond to the occurrence of regular parts of the lattice in fig.[pergcm](c ) . the chaotic lattice in panel ( e ) corresponds to @xmath297 , which is a point situated near the between spherical and deformed shapes . many other examples of peres lattices and their detailed comparison with the dependence @xmath182 can be found in refs.@xcite and @xcite . from the examples shown here and in the literature it can be inferred that peres lattices significantly complement the standard methods used to investigate chaotic properties of quantum systems . in fact , they even enable one to go beyond the standard methods . this is so because quantum states can be _ individually assigned _ to distinctly regular or distinctly chaotic parts of the lattice , which is in contrast to the statistical approach that can only attribute some averaged regular / chaotic attributes to sufficiently large ensembles of states . constructing a suitable peres lattice for a given quantum system with known spectral correlations is therefore similar to analyzing the classical dynamics with a given @xmath150 by means of poincar ' e sections . two remarks on the applicability of the method are in order : first , to unambiguously deduce whether a given state is in a regular or chaotic domain of the spectrum , it is usually useful to construct peres lattices for several operators . a comparison of several lattices helps particularly in situations when an overlap of two or more regular parts in a single lattice produces a seemingly chaotic mesh . second , the method in the above - presented form is applicable only in systems with @xmath0 . consider , for example , an integrable system with @xmath298 . to ensure a one - to - one mapping from the space of canonical actions @xmath299 , one would have to consider a @xmath300-dimensional lattice ( @xmath102-dimensional in general ) . this is , however , rather difficult with only 2d physical literature being available . these unalterable limitations of us , poor 3d creatures , will probably always hinder our understanding of higher than 2d systems . we are arriving to the end of our short tour . its purpose was to show some known concepts of chaos theory by means of simple models taken from the nuclear context . was the marriage of chaos with nuclei successful ? we believe it was . we think that the models like those used here provide an excellent environment for general investigations of chaos in both classical and quantum incarnations environment , which is sufficiently complex to embrace diverse nontrivial phenomena , but at the same time simple enough to allow an intelligible description . in analogy with the billiard models , that have been extensively used for such investigations in the past decades , the nuclear models can be used as models with two degrees of freedom . in contrast to the billiard models , however , they exhibit very rich dependence of chaos measures on both energy and hamiltonian parameters , which opens some new directions of potential research . * geometrical 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popularity plays a major role in the dynamics of online systems . public attention can suddenly concentrate on a web page or application @xcite , a youtube video @xcite , a trending topic in twitter @xcite , or on a story in the news media @xcite , sometimes even in absence of an apparent reason . typically , after an initial increase of attention , the focus will move elsewhere leaving as a trace a characteristic activity profile . such popularity peaks are not only of great relevance for the monetization of online content , but also pose scientific challenges related to understanding the mechanisms ruling their dynamics @xcite . in particular , specific features of the popular item under consideration can now be related to its activity profile by means of semantic analysis and natural language processing of the messages exchanged by the users @xcite . here we use data from the twitter micro - blogging system to investigate the relation between activity profiles over time and content . there are several reasons for selecting twitter : it is one of the most popular online social networks , part of its message stream is programmatically accessible to the public @xcite , and the content of the messages is short , making it amenable to automated processing . twitter is used as an hybrid between a communication media and an online social network @xcite and hosts real - time discussion of current topics of popular interest . we take advantage of the practice introduced by twitter users of attaching `` hashtags '' to their messages as a way of explicitly marking the relevant topics . twitter has incentivated this practice by supporting hashtags in their web interface and in their programmatic api , turning them into lightweight social annotations of the information streams users consume . here we focus our analysis on those hashtags that exhibited a popularity peak during our observation period , and systematically analyze the corresponding messages ( `` tweets '' ) by grounding the words they contain in a semantic lexicon . this paper is structured as follows : section [ sec : background ] reviews the literature on twitter and in particular the literature on temporal patterns of twitter activity . section [ sec : data ] describes the twitter dataset we used and the techniques we applied to select popular hashtags and their usage patterns . in section [ sec : classes ] we identify dynamical classes of hashtag usage and relate them to the semantics of the corresponding tweets . in section [ sec : spreading ] we relate the same dynamical classes to the spreading properties of hashtags over the underlying social network . section [ sec : discussion ] summarizes our findings and points to applications and further research directions . several aspects of twitter have been extensively investigated in the literature , including its network topology @xcite , the relations and types of messages between users @xcite , the internal information propagation @xcite , the credibility of information @xcite , and even its potential as an indicator of the state of mind of a population @xcite . the possibility that popular trends or hashtags could be classified in groups have been discussed in refs . @xcite , and the effect of semantic differences on the persistence of a hashtag have also been considered @xcite . the shape of peaks in popularity profiles has been used to classify the events in groups @xcite . the hypothesis that both the increase and decrease of public attention follow a power - law - like functional shape whose exponents define universality classes , in parallel to what occurs with phase transitions in critical phenomena , has been explored @xcite . this approach , however , is difficult to apply to twitter : the fast timescales involved and the highly reactive nature of twitter make the time series very noisy and pose the challenge of characterizing activity dynamics in a way which is both robust and scalable . the causes that underlie the existence of distinct classes of popularity are thought to be a combination of all the mechanisms that drive public attention . news regarding a popular item can propagate either over the social network of the users of a given system a so - called endogenous process or it can be injected through mass media ( exogenous driving ) . the duality between exogenous and endogenous information propagation has permeated the analysis of popularity in several recent studies @xcite , even though it is not always clear how to distinguish between them based solely on the shape of the respective popularity profiles @xcite . our dataset comprises about @xmath0 million twitter messages or _ tweets _ posted between november @xmath1 , @xmath2 and may @xmath3 , @xmath4 . the data were collected at indiana university thanks to their temporary privileged access to the twitter data stream @xcite . each tweet includes textual content , an author , the time at which it was posted , whether or not it was in reply to another tweet , and additional metadata . the collected tweets come from about @xmath5 million unique user accounts . in order to build a representation of the social network over which hashtag diffusion takes place , we queried the twitter rest api for the complete list of followers and friends of @xmath6 million users . we collected neighbor information for @xmath7 million of them , the discrepancy being accounted for by users with a private profile . using this information we constructed a directed follower network , where each edge takes on the direction in which information flows : if user a follows user b , the respective social link points from b to a , as a can see b s status updates . for the identification of topics , we extracted all the hashtags contained in the twitter messages ( by matching the tweet content to the pattern # [ a - za - z0 - 9 _ ] * ) . our dataset includes about @xmath8 distinct hashtags ( see table [ tab : data ] ) . we selected the most popular topics by restricting our data to the hashtags used by at least @xmath9 distinct users and to the messages containing at least one of such hashtags . based on this selection , we used for the following analysis about @xmath10 million tweets and @xmath11 popular hashtags . .general statistics about the dataset [ tab : data ] [ cols= " < , > " , ] like most systems driven by human actions , twitter exhibits bursty activity , circadian rhythms , and in general the full temporal complexity of a large - scale social aggregate . because of this , there is no single natural scale for investigating its temporal behavior , and the choice of a time scale is not neutral with respect to the phenomena one can study at that scale . here we choose to investigate activity at the scale of days , i.e. , we do not study human dynamics at the level of minutes and seconds , nor phenomena driven by the circadian cycle , nor slower trends that develop over several weeks of months . we analyze daily activity levels , and focus on events that are meaningful at that scale , such as the wait for a scheduled social event . at the daily scale the popularity profile of hashtags can look very different . on visual inspection the individual temporal profiles of hashtag usage display behaviors that typically fall into one of the following three categories : continuous activity , periodic activity , or activity concentrated around an isolated peak . continuous - activity profiles are those for which a rather constant level of daily activity is maintained by the user community ( e.g. , ` music ` ) . hashtags with periodic activity profiles display series of spikes spaced by one or more weeks , or months ( e.g. , ` followfriday ` ) . finally , activity profiles with an isolated peak are characteristic of hashtags associated with a unique event to which a user community pays attention for a limited span of time ( e.g. , ` oscars ` ) . in the following we will concentrate on this class of hashtags . to identify activity peaks , for every hashtag @xmath12 we compute the time series of daily activity , where the activity @xmath13 on day @xmath14 is defined as the number of tweets containing @xmath12 . in the following we will write @xmath15 to indicate the activity level of a generic hashtag . we use a sliding window of @xmath16 days ( @xmath17 ) centered on day @xmath18 , @xmath19 $ ] , and let @xmath18 slide along the activity time series for the hashtag . within this window we evaluate the baseline hashtag activity as the median @xmath20 of @xmath21 . then , we define the outlier fraction @xmath22 of the central day @xmath18 as the relative difference of the hashtag activity @xmath23 with respect to the median baseline @xmath20 : @xmath24 / \max(n_b , n_{\mbox{\tiny min}})$ ] . here @xmath25 is a mininum activity level used to regularize the definition of @xmath22 for low activity values . we say that there is an activity peak at @xmath18 if @xmath26 , where @xmath27 is an arbitrary threshold value that in the following we set as @xmath28 . we checked that different values of the threshold do not change significantly our results , and that the same peaks can be identified by using different peak - detection techniques . of course it may happen that for a given hashtag @xmath12 the time series @xmath13 exhibits more than one peak . since we are interested in isolated popularity bursts , we ignore all peaks that are separated from other peaks by less than one week . finally , for every hashtag we select the peak ( if any ) with the highest @xmath22 and we offset the day index so that for all hashtags the activity peak occurs on day @xmath29 , as shown in fig . [ activity ] . using this method we select @xmath30 peaks of daily hashtag activity : the corresponding hashtags are listed in appendix [ appendix - usage ] , together with manual annotations about their meaning and a coarse classification . to correlate the temporal activity patterns with content , we perform a simple semantic grounding of the tweets by using the wordnet @xcite semantic lexicon . for each tweet , we pre - process the text by removing user mentions ( ` @username ` ) , hashtags , urls and a standard set of english stop words . then , for each word we perform stemming ( with the standard porter algorithm ) , lemmatization , and we finally attempt to look up in wordnet the corresponding ( i.e. , the basic node of the wordnet lexicon , a set of synonyms that refer to a single concept ) . from now on we will refer to wordnet synsets as _ concepts_. words for which no concept can be looked up in wordnet are ignored . if few or no terms are successfully looked up in wordnet as english words , we attempt to identify the tweet language : we run the textcat @xcite language categorization algorithm on the text and we discard the tweet if english is not included in the top @xmath31 most likely languages identified by textcat . overall , the above analysis identifies about @xmath32 distinct concepts that are associated with the hashtags under study . typical examples of the activity profiles for the selected hashtags are shown in figure [ activity ] . the curves are centered around the day on which the popularity reaches its maximum ( day @xmath29 ) . the displayed time window spans one week before and after the peak . in the top plots of figure [ activity ] the activity of four sample hashtags is reported as a function of time in days after the peak . the bars on the top right display the percentage of activity before , at and after the peak . the four hashtags exhibit different behaviors in terms of approach to the peak ( dark blue bars ) and relaxation after the peak ( light blue bars ) . the hashtag ` masters ` exhibits an anticipatory pattern , with a gradual build - up of activity before the peak . the hashtag ` winnenden ` , conversely , corresponds to an unexpected event , with a sudden onset of activity followed by a gradual relaxation . the hashtag ` watchmen ` displays both a gradual build - up of attention and a gradual relaxation after the peak . finally , the hashtag ` nsotu ` concentrates almost all of its activity during the single day of the peak . in the middle plot row we show the activity of individual users as a function of time . users who have posted the hashtag at least once ( within the observation interval ) are ranked according to the time of first usage of the hashtag ( rank along the ordinate axis ) : early adopters lie at the bottom and late adopters are at the top . for each user , colored segments mark the times at which the hashtag under consideration was used . the inset bar plots show the fraction of users who used the hashtag more than once during the selected time window . finally , in the bottom plot row we visualize the content of tweets as word clouds . each word cloud contains the @xmath33 most frequent words , with font sizes proportional to word frequencies . the patterns displayed by these hashtags are representatives of the four classes of activity peaks found in our analysis . and @xmath34 axes ( vvi model of the mclust implementation ) . [ em - clusters ] ] the possibility of classifying online popularity peaks in a few discrete classes has been discussed in the literature @xcite . typically the classification is done according to the different shapes or functional forms of the increasing and decreasing parts of the popularity profiles . the origin of these few classes has been linked in the literature to two mechanisms that , to some extent , are present in most online social systems : endogenous propagation of information over the social network , and the injection into the system of information from exogenous online or offline sources . this scenario was tested for the evolution of popularity of youtube videos @xcite and has also been discussed for trending topics or memes in twitter @xcite . the lack of a clear distinction between endogenous and exogenous information flow in twitter means that the number of classes , the possible functional shapes of the popularity profiles , and even the importance of the endogenous / exogenous distinction are all far from clear @xcite . here we take a different approach and attempt to simplify the possible scenarios by shifting emphasis from the detailed time series of popularity to coarse - grained information on the balance of activity before , during , and after the popularity peak . to achieve this , for each hashtag exhibiting a popularity peak we summarize the hashtag usage timeline with the triple @xmath35 of the fractions of tweets posted before ( @xmath34 ) , during ( @xmath36 ) and after the peak ( @xmath37 ) . by definition these fractions satisfy @xmath38 . we restrict the computation to a two - week period centered on the peak time , as shown in the examples of fig . [ activity ] . we identify hashtag clusters in the @xmath39 space of independent parameters using a standard implementation of the expectation maximization ( em ) algorithm @xcite to learn an optimal gaussian mixture model . the number of components ( clusters ) of the mixture is set by using the bayesian information criterion , as well as by means of a @xmath31-fold cross - validation , yielding in both cases the @xmath40 clusters shown in fig [ em - clusters ] . the clusters are robust with respect to the initial conditions and parameters of the em algorithms ( provided that care is taken to deal with the points on the @xmath41 axis ) : @xmath42 of the hashtags have a classification accuracy below @xmath43 , and only @xmath44 of them have a classification accuracy in excess of @xmath45 . figure [ simplex ] shows the identified clusters in the @xmath46-simplex @xmath47 . the marker representing each of the @xmath30 selected hashtag is colored and shaped according to the group it has been classified into . the hatched area is the parametric space excluded by the constraint that hashtags should have a peak - day activity of at least @xmath31 times the baseline daily activity ( i.e. , the excluded parametric space is due to our selection of hashtags that exhibit a peak in their activity timeline ) . the four groups of fig . [ simplex ] correspond to different temporal patterns of collective attention , as illustrated below in relation to the hashtags of fig . [ activity ] . * activity concentrated _ before and during _ the peak ( orange triangles ) . these hashtags correspond , by definition , to anticipatory behavior , with users posting increasing amount of content as the date of the event approaches , followed by a sharp drop in attention right after the event . see for example the hashtag ` # masters ` ( underlined in the figure ) which was used to discuss the @xmath4 golf masters . * activity concentrated _ during and after _ the peak ( purple circles ) . in this class we find hashtags indicating unexpected events that make an impact , such as the ` # winnenden ` school shooting . the sudden onset of activity is a reaction to the unexpected event , and it is likely to be driven by exogenous sources such as communication in mass media . * activity concentrated _ symmetrically _ around the peak ( red squares ) . these hashtags have neither the purely anticipatory nor the purely reactive behaviors illustrated above , and this may indicate a mix of exogenous and endogenous factors building up collective attention to a peak intensity , as a specific day approaches , and then away from it as user attention shifts away . see for example the case of the hashtag ` # watchmen ` , used to discuss a blockbuster movie . the peak occurs on the day of the movie release in theatres . * activity almost totally concentrated _ on the single day of the peak _ ( green rounded square ) . these hashtags correspond to transient collective attention associated with events that are highly discussed only while they happen , such as the @xmath4 state of the union address ( ` # nsotu ` ) , or the transient large - scale malfunctions of widely used google services ( ` # gfail ` ) . these patterns are somehow expected , in the sense that these are the only possibilities for the coarse - grained temporal profile of a hashtag with a popularity peak . however , the existence of well defined hashtag clusters , as well as their stability , are far from trivial and indicate that coarse graining the temporal dynamics of collective attention as shown here can expose robust indicators of the social semantics associated with hashtags . the presence of clearly separated clusters may also be deeply linked to the diverse nature of the mechanisms driving popularity in online social systems . details on the usage and origin of the hashtags shown in fig . [ simplex ] are available in appendix [ appendix - usage ] . the examples discussed above , such as those of fig . [ activity ] , point to important differences in the social semantics of the different classes of hashtags . in order to shed light on this aspect , we systematically analyze the content of the tweets associated with each group of hashtags , using the semantic grounding described in section [ semanticanalysis ] . wordnet provides hierarchical structures of concepts that can be made into a single directed acyclic graph by adding a root `` entity '' node as parent of the wordnet taxonomies . thus , wordnet can be used to coarse - grain the semantics of the looked - up terms by focusing on a given ( high enough ) level of the subsumption hierarchy . our interest here is to provide a semantic fingerprint of the content associated with the different hashtag classes , in order to expose differences in their social semantics . the concepts at depth @xmath40 of the wordnet hierarchy were identified as appropriate for this purpose , as that hierarchical level provides a good enough semantic diversity while featuring a small number of generic subsuming categories . we restricted our analysis to the concepts at depth @xmath40 that occur most frequently in the text associated with the hashtags under study : the right - hand side of fig . [ semantic ] lists the @xmath48 selected wordnet concepts , together with sample terms that are subsumed by them . to expose the semantic differences between hashtag classes we proceed as follows : for each hashtag we compute a normalized feature vector of the frequencies of occurrence of the selected wordnet concepts . we then average this vector over all hashtag belonging to a given class and obtain the class feature vectors of fig . [ semantic ] , where the radius of discs is proportional to the normalized frequency of the corresponding concept in a given class of hashtags . clearly , different dynamic classes correspond to different semantics of the corresponding tweets . the content of hashtags with activity concentrated before the peak has a stronger prevalence of concepts like `` social events '' and `` time period '' ( e.g. , ` easter ` ) , consistent with the social anticipation of a known event . conversely , hashtags whose activity is concentrated after the peak , usually associated to unexpected events , include several marketing campaigns such as ` macheist ` , and this is reflected in the prevalence of concepts like `` free '' and `` evidence '' . tags with the activity concentrated mostly on the peak day correspond to events that attract the users attention for short periods of time , such as sport events and media events ( e.g. , concepts associated with ` oscar ` , subsumed by the `` symbol '' concept ) . the detailed annotations of appendix [ appendix - usage ] allow to make contact between specific hashtags or hashtag classes and the information of figs . [ simplex ] and [ semantic ] . notice that the observed selectivity between content and activity profiles may open the door to content tagging techniques based on popularity dynamics and on other behavioral cues . ] having identified classes of popular hashtags that differ in activity profiles and semantics , we now turn to investigating whether such classes are also associated with distinct patterns of information propagation . similarly to the approach of ref . @xcite , we regard information spreading as an epidemic process , where the behavior of using a given hashtag spreads from one user to another . the relevant social network for this epidemic process is twitter s _ follower network _ : whenever a user posts a given hashtag , her followers are exposed to the hashtag and can decide to adopt it in turn . of course , users can also start using the hashtag spontaneously , as a result of exposure to external events ( elections , sport matches , disasters , etc . ) or to exogenous information sources . the first feature we analyze is the fraction of retweets to total tweets in the messages associated with each hashtag under study . retweets are forwarding actions in which a tweet from a followed user is delivered to one s followers together with a reference to the source . because of their nature , retweets have been investigated as a mechanism for information diffusion in twitter @xcite . the fraction of retweets is an indicator of how many ( forwarded ) copies are present in the tweets associated with a hashtag , and provides information on the spreading attitude of the corresponding topic . retweets were identified both by checking for an initial `` rt '' marker or through tweet metadata . the top - left panel of figure [ epidemic ] reports the fraction of retweets for the four hashtag classes . a box plot is used to provide information on the dispersion of parameter values inside each hashtag class . hashtags with the activity distributed symmetrically around the peak or concentrated at the peak day have a higher fraction of retweets . this supports the idea that those hashtags are associated with a higher level of endogenous activity , similarly to what happens for some youtube videos @xcite . conversely , hashtags characterized by activity before the peak are associated to anticipatory behaviors and appear less prone to viral spreading . the box - plot in the top - right panel of fig . [ epidemic ] reports the fraction @xmath49 of users who adopt the hashtag when none of the users they follow have used it before . in other words , @xmath49 estimates the fraction of `` seeders '' that inject the information related to the hashtag into the social network . although the level of heterogeneity inside the four groups is high , we see that the hashtags with activity concentrated after the peak tend to have more seeders . this indicates that the propagation is probably fueled by exogenous factors , such as publicity campaigns or mass media communication . a further corroboration is provided by the semantic analysis of fig . [ semantic ] , as these hashtags contain concepts such as `` sign '' ( sign - up for a service ) , `` account '' ( create an account ) or `` free '' that are usually associated with commercial campaigns that are heavily diffused in traditional media . . bottom left : fraction @xmath50 of followers that adopt the hashtag after seeing it . bottom right : average time @xmath51 between the first tweet with the hashtag and the last one . [ epidemic ] ] the box - plot in the bottom - left panel of fig . [ epidemic ] reports the average fraction @xmath50 of a user s followers who adopt the hashtag after he or she has posted a tweet containing it . in modeling epidemic processes , @xmath50 is a measure of infectiousness . in this context , it bears information about the capacity of a behavior or meme to propagate from a user to her followers . the box - plot shows that @xmath50 does not depend strongly on the hashtag class and its median value is about @xmath52 . this might suggest the existence of a generic mechanism controlling the propagation of the information over the twitter social network independently of the content or popularity profile of the hashtags . the estimation of both @xmath49 and @xmath50 depends on the sampling of the social network at hand . however , an analysis made using sub - samplings of the follower network obtained by cutting edges has showed that @xmath50 is relatively stable to the level of sampling , while @xmath49 is more sensitive . nevertheless , since our sampling of the network is fixed it is legitimate to compare the results obtained for different hashtags even in the case of @xmath49 . finally , in the bottom - right panel of fig . [ epidemic ] we report the average time @xmath51 , in hours , between the first tweet and the last tweet with the same hashtag posted by each user ( we set @xmath53 for those users who post the hashtag only once ) . that is , @xmath51 indicates the time during which users are likely to spread their use of the hashtag to followers . the four hashtags classes display similar values of @xmath51 except for the case of activity concentrated on the peak day . in that case , hashtags have the lowest @xmath51 value , since activity is concentrated in a small period of time corresponding , for example , to a short - term disruption of online services . in summary , we performed an extensive analysis of the twitter hashtags that exhibit a popularity peak . previous work found that popularity peaks in online systems can be clustered in a few prototypical classes according to the temporal features of their popularity dynamics . here we introduce a simple way of coarse - graining the temporal usage patterns of hashtags that exposes discrete dynamical classes . the clusters we find correspond to the four possible ways of distributing the hashtag activity with respect to the day of peak usage . clusters are well defined and the classification of hashtags is stable with respect to small perturbations . we ground in a semantic lexicon the contents of tweets associated with popular hashtags , and find insightful correlations between the class a hashtag belongs to and the ( social ) semantics of the associated content . in particular , hashtags that are mostly active before reaching a peak usually deal with scheduled social events or specific moments in time , indicating an anticipatory collective behavior . hashtag with symmetric activity patterns across the peak seem to be associated with endogenous propagation over the social network . hashtags that only exhibit a tail of activity after the peak correspond to unexpected events or exogenous driving . furthermore , we measure standard parameters of epidemic propagation over the on - line social network and relate these parameter values to the different hashtag classes , to unveil patterns of injection or propagation of information . the balance between internal propagation ( endogenous ) and external injection of information was assumed so far to be the main explanation for the existence of different clusters of online popular events . our results indicate that the content type is also very important . for instance , the hashtags used to discuss the `` swine flu '' pandemic ( top of fig . simplex ) or a popular event such as the oscars ceremony ( bottom - left of the simplex ) show markedly different popularity profiles despite the fact that both attract a high level of attention from the media . both hashtags display high levels of external seeding , as well as relatively low levels of endogenous propagation . thus , the different social semantics of these hashtags is likely the cause underlying the observed differences in activity dynamics . we remark that a robust classification into dynamical classes of user attention was obtained by using very simple parameters computed on time series of daily popularity . contrary to other methods , which require the estimation of power - law exponents for popularity growth , or the computation of expensive correlations between high - resolution activity time series , the parameters introduced here can be easily computed in a scalable way . while they lack predictive power , as they need a record of past activity to be computed , they can support the discovery of specific behavioral patterns in large - scale records of user activity . the robustness of the proposed approach , if confirmed in other settings , could support implicit temporal tagging of the twitter data stream , where for example anticipatory behavior associated to a given date points to that date as a focus of collective expectation . the specific semantics that can be linked to a given temporal profile may be used to mine collective attention in order to construct implicit annotations of timelines on the basis of social media streams . of course , this requires an extensive work of validation that falls outside the scope of the present work . progress in this direction will requires more refined content analysis by means of natural language processing and sentiment analysis , as well as validation in user studies or crowd - sourced settings . the authors thank the pis of the truthy project , fil menczer , alessandro flammini , johan bollen and alessandro vespignani , for their support and for many inspiring discussions . cc and jl thank andr panisson for interesting discussions and technical help . cc thanks yamir moreno for stimulating discussion . this work was carried out while jl was at the isi foundation with support from the leonardo da vinci scholarship . jl acknowledges support from the spanish ministry of science through the project tin2009 - 14560-c03 - 01 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macworld conference & expo + masters & sport & golf tournament ( masters cup ) + mrtweet & twitter application & introduction of a new twitter service to find people + myfirstjob & twitter game & sharing of first job experiences + nfl & sport & super bowl : cardinals vs. steelers + oneword & twitter game & tweeting of a word that s in the mind of twitter user + plurk & twitter application & integration of plurk into twitter ( service similar to twitter ) + poynterday & holiday / honor & honoring of dougie poynter + rncchair & political & rnc chairmanship election + sxswi & convention & set of film , interactive and music festivals ( south by southwest ) + teaparty & political & protests across the united states + therescue & awareness / charity & event from the organization `` invisible children '' against child soldiers in northern uganda + tweepme & twitter game & contest for the twitter application tweepme + twestival & awareness / charity & charity event of cities to raise money for clean water + wbc & sport & japan s world baseball classic + & + + amazonfail & disruption & demonstration against the new ranking of books in amazon + americanidol & media & television competition to find new singing talents + blogger & twitter application & introduction of a new twitter directory ( wefollow ) + bsg & media & finale of battlestar galactica + contest & marketing / contest & competition to win the album `` cardinology '' from ryan adams + cricket & sport & cricket game : india vs. england + earthday & awareness / charity & celebration of the earth day + evernoteclarifigiveaway & marketing / contest & competition to win iphone 3 g cases + free & marketing / contest & see # macheist + fridayfollow & twitter game & unusual tag for # followfriday + g20 & political & g-20 summit + happy09 & holiday / honor & congratulations to new year s eve + hoppusday & holiday / honor & honoring of mark hoppus of the band blink182 + inaug09 & political & inauguration of barrack obama 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in our previous paper @xcite , we extended the notion of vassiliev invariant and other invariants associated with chern simons theory to the context of spin networks @xcite . we showed that these invariants are loop differentiable in the sense of distributions and we defined an infinitesimal generator of diffeomorphisms in terms of the loop derivative . the generator correctly annihilates diffeomorphism invariant states . in this paper we will introduce a hamiltonian constraint based on the loop derivative . we will limit the discussion to invariants of trivalent vertices and will only concentrate on the `` euclidean '' part of the hamiltonian constraint in the sense of barbero @xcite and thiemann @xcite . limiting ourselves to trivalent intersections is clearly unphysical since the volume operator identically vanishes on such states . however , the essence of most calculations is already present with trivalent intersections and the calculational difficulty is significantly lower than with four or higher valent intersections . we will come back to this issue in the discussion section where we will highlight which results are quite plausibly going to survive when one goes to higher valent intersections . we will then extend the space of invariants through the construction of `` habitats '' such that the resulting states are not diffeomorphism invariant . based on these states we will check the consistency of the quantum commutator algebra of diffeomorphisms and hamiltonian . we will show that there are no anomalies , therefore constituting a consistent theory of canonical quantum gravity . we will start in the next section by introducing the hamiltonian constraint of quantum gravity in terms of the loop derivative . this derivation will be `` generic '' in the sense that we will motivate it through formal manipulations of the loop transform and therefore will not make specific assumptions about the quantum state in the connection representation on which the operator is acting upon . without being more precise about the space of states upon which it acts , we can not guarantee that the operator is well defined . we will show that on several spaces related to the vassiliev invariants the operator is indeed well defined . the results obtained will be later applied in specific `` habitats '' in which we will show that the operators exist and we can compute correctly the commutators . this we will discuss in section iii . the need to introduce new ( non - diffeomorphism - invariant ) `` habitats '' is based on the fact that the spaces we have considered in the companion paper are all diffeomorphism invariant and therefore one can not explore in a non - trivial way commutators involving the diffeomorphisms . we will introduce an explicit example of such a habitat in detail , the `` functions with marked vertices '' . in the end we will see that we will be able to recover the classical poisson algebra at the level of quantum commutators , but there will be subtle issues involved in the definition of the right hand side of the commutator of two hamiltonians , which we will discuss in detail . the last section will be devoted to discussing the implications of the level of consistency achieved by the theory . we wish to introduce a quantum version of the hamiltonian constraint of canonical general relativity . following thiemann @xcite , one can construct this operator introducing the real version of the ashtekar variables first discussed by barbero @xcite . in this formulation , the canonical pair consists of a set of ( densitized ) triads @xmath0 and as conjugate variables real - valued @xmath1 connections @xmath2 . the hamiltonian constraint of canonical , real , lorentzian general relativity in this framework consists of two terms , @xmath3 where @xmath4 in these expressions , @xmath5 is the newton constant and in ( [ k ] ) @xmath6 means that the smearing function @xmath7 is unity . this remarkable formulation implies that the hamiltonian constraint is composed by two terms . the first one , @xmath8 , coincides with the hamiltonian constraint one would obtain in a canonical formulation of general relativity on a manifold of euclidean signature . the lorentzian theory is therefore attained through the addition of an extra term . both terms are given as expressions involving the poisson bracket of a connection with the volume of the space @xmath9 and with the function @xmath10 . the latter can be in turn obtained as a poisson bracket of the euclidean part of the hamiltonian constraint with the volume . in this paper we will only concentrate on realizing at a quantum mechanical level the euclidean part of the hamiltonian . if the reader wishes , we are considering canonical quantum gravity in the euclidean context . however , it should be noticed that the additional term in the hamiltonian is obtained by successive commutators of the operator we will consider , with the volume operator . this suggests that one could generalize our construction to the lorentzian case rather straightforwardly , but this has not been studied in detail . the work of thiemann also shows that similar structures arise if one couples the theory to matter . we will in this paper only concentrate on the vacuum case but again one does not see a priori impediments to extend our treatment to general relativity coupled to matter . to implement the hamiltonian constraint we will partially use the same procedure proposed by thiemann . we will not describe it again in full detail here , we just list some of the salient features . one starts by writing the classical expression ( [ hame ] ) as @xmath11 f_\epsilon(y , x)\ ] ] where @xmath12 is a regularization of the dirac delta function , i.e. @xmath13 . strictly speaking , this expression is only gauge invariant in the limit , so to preserve gauge invariance in the regularization procedure we will `` join '' the @xmath14 with the connection using an infinitesimal piece of holonomy @xmath15 along an arbitrary path @xmath16 going from @xmath17 to @xmath18 , which tends to the null path in the limit @xmath19 . to unclutter the notation we will therefore denote @xmath20 , @xmath21 f_\epsilon(y , x ) \label{uno}\ ] ] in order to prepare the classical expression for quantization , we triangulate the space - like hypersurface @xmath22 in terms of elementary tetrahedra @xmath23 . the triangulation has the following properties : we select a finite set of distinct points of @xmath22 , denoted as @xmath24 . at each of these points we choose three independent directions @xmath25 and construct the eight tetrahedra with vertex @xmath26 and edges @xmath27 , with @xmath28 . the eight tetrahedra saturating @xmath26 define a closed region @xmath29 of length scale @xmath30 . the remaining region @xmath31 is triangulated by arbitrary tetrahedra @xmath32 . the motivation for this peculiar discretization is that when we promote the classical expression to a quantum operator , we will adapt the triangulation to the spin network of the state in question by choosing the points @xmath26 to coincide with the vertices of the spin network . in terms of this triangulation of space we can write equation ( [ uno ] ) in the form , @xmath33 f_\diamond(y)\,,\label{hamt}\ ] ] where the regions @xmath34 indicate either a box @xmath29 or a tetrahedron @xmath32 , and @xmath35 is any point interior to @xmath34 . conventionally we will choose for the boxes @xmath36 , and for the tetrahedra @xmath37 will represent one of its vertices . to discretize the integral we have introduced the volume of each region @xmath38 , @xmath39 where @xmath40 if @xmath41 , and @xmath42 if @xmath43 . in this last case , the @xmath44 s represent the edges of the tetrahedra @xmath32 adjacent to @xmath37 . we also have adapted the regularization of the dirac delta function to the tetrahedral decomposition by defining , @xmath45 where @xmath46 is one if @xmath47 and zero otherwise . we now replace in ( [ hamt ] ) the @xmath48 using formula ( [ vol ] ) and we represent @xmath49 using a holonomy in the fundamental representation along the edge @xmath50 of the triangulation , @xmath51f_\diamond(y)\ , , \label{classham}\ ] ] and we retraced the path described by the holonomy to preserve gauge invariance . it should be noticed that this retracing does not contribute to the expression at leading order in @xmath30 ( the scale parameter of the regions @xmath34 ) . we will now proceed to study the quantization of the last expression . for that we need to `` adapt '' the triangulation we introduced to a spin network state @xmath52 @xcite . let @xmath53 be the set of vertices of the spin network and let @xmath54 be the triples of non - coplanar edges incident at @xmath55 ( @xmath56 , being @xmath57 the number of such triples ) . to adapt the triangulation to the spin network state we perform two operations : first we identify the points @xmath26 of the boxes @xmath29 with the vertices @xmath55 and second , given a triple of edges @xmath54 incident at @xmath55 , we orientate the three unit vectors @xmath58 along the tangents of the edges at @xmath55 . then we can write , @xmath59 to simplify the notation we identify from now on @xmath60 . as in the previous paper , we assume we are acting on a state given by a loop transform , @xmath61 w_a[s]\ , , \label{states}\ ] ] and we realize the hamiltonian operator over the spin network wavefunctions by promoting the classical expression ( [ classham ] ) as an operator acting on the wilson net appearing in the loop transform , very much as we did in the companion paper when we discussed the diffeomorphism operator , @xmath62 \right)f_{\box_v}(y ) w_a(s)\,.\nonumber\end{aligned}\ ] ] a first observation is that the volume operator has non - vanishing contributions only at the vertices of the spin net , so we replace the sum in the hamiltonian over all @xmath34 s by a sum over all the vertices of the spin net . the second observation is that , for an n - valent vertex , we have taken the average over all the non - coplanar triples of edges associated with this vertex . finally notice that only one of the terms in the commutator contributes , that with the volume operator on the left . the one with the volume operator on the right vanishes since it is proportional to the trace of the lie algebra element @xmath63 . let us now consider a generic @xmath64valent vertex and study the action of the operator on the wilson net . we assume that the holonomies are all outgoing from @xmath26 . schematically , we are trying to represent the action of @xmath65 , with @xmath66 a generic edge of spin @xmath67 incident at @xmath26 . an @xmath64valent vertex is characterized by an intertwiner that can be represented through a vector @xmath68 of @xmath69 of irreducible representations . the factor @xmath70 acts by adding a line of spin @xmath71 ingoing from the vertex @xmath26 and parallel to the line @xmath66 , @xmath72 notice that the insertion of the holonomy leaves a wilson net that is not gauge invariant anymore , we represent this by keeping the group indices @xmath73 in the diagram . if one was acting on an @xmath64valent vertex characterized by an intertwiner @xmath68 , this leaves an @xmath74valent vertex characterized by intertwiners @xmath75 ; @xmath67 being in the diagram the spin of the line connecting the original vertex to the point ( infinitesimally nearby ) where the holonomy was inserted . notice that @xmath67 coincides with the spin of the original line @xmath66 . we denote that the line is infinitesimally close by the dashed circle . we now act with the volume operator . for that we first have to reduce the product of holonomies associated with @xmath66 to a superposition of irreducible representations . this procedure defines a new intertwiner for the vertex @xmath26 over which the action of the volume operator is well - defined , @xmath76 the action of the volume operator can be represented by a matrix that rearranges the intertwining of the vertex . this matrix elements would depend also on the spins of the external edges of the vertex ( notice that @xmath10 is the color of one of this external edges ) . schematically , @xmath77 where in the last step we reduce the line of spin @xmath71 of the holonomy using recoupling theory . one is left with a double insertion at the vertex , which we can rearrange via recoupling , since everything is happening infinitesimally close to the vertex . we write the result in the following way , @xmath78 the above expression has now to be contracted with @xmath14 at the open strands labeled with @xmath79 and @xmath80 . one can generate @xmath14 by introducing an extension of the idea of loop derivative for spin networks . the reason why one does not simply recover the ordinary loop derivative is that the edge before and after the point of the insertion of @xmath14 are in general in different representations @xmath67 and @xmath81 . the ordinary loop derivative inserts an @xmath14 without changing the representation of the original line . this definition of loop derivative appeared as natural in the context of loops , however in terms of spin networks one expects in general to have a situation like the one we have here . one can use the same set of constructions we did for the ordinary loop derivative of the invariant @xmath82 to compute the action of this generalized loop derivative is proportional to the inverse of the coupling constant of chern - simons theory . for a definition of the invariant @xmath82 , see the companion paper , section ii.c . ] . however , we will not pursue the exploration of this generalized derivative in this paper . instead we will concentrate on the case of trivalent intersections . if the spin networks considered have trivalent intersections , when we repeat the construction we did above , the representations of the edges before and after the insertion of @xmath14 are the same . the operator we then obtain to represent @xmath14 corresponds in this case to the ordinary loop derivative . we have now to consider the action of the volume operator over a four - valent vertex with only three real edges @xcite , @xmath83 with @xmath84 where @xmath85 , and @xmath86 . using this result , the matrix elements of equation ( [ nval ] ) are reduced to the following simple expression for trivalent vertices ( @xmath87 in this case ) , @xmath88 where @xmath89 , and @xmath90 and @xmath91 are the other spins of the edges incident at the trivalent vertex . therefore the spin of the link going from the intersection to the point of insertion of @xmath14 is unchanged through the action of the volume operator . by inserting the @xmath14 we obtain the final action of the hamiltonian constraint on trivalent vertices , @xmath92 the superscript on the loop derivative denotes at which of the incident edges at the intersection it adds the extra strand @xmath16 . notice that since @xmath93 is summed by the einstein convention , the derivative ends up acting on all edges ( we have abused of the summation convention allowing the index @xmath93 to be repeated three times in the last expression ) . the vector @xmath94 labels the spins of the tree edges of the spin network incident at @xmath26 ( @xmath95 is the spin of the edge @xmath96 ) , and the group factor @xmath97 is given by the expression , @xmath98 where the indices @xmath99 take cyclic values in the set @xmath100 . we have therefore a general expression for the hamiltonian constraint acting on any loop - differentiable wavefunction with support on trivalent spin networks , that bears a relationship with a wavefunction in the connection representation @xmath101 given by ( [ states ] ) . at this point it is worthwhile comparing this action of the hamiltonian constraint we just introduced with the original proposal of a hamiltonian constraint in the loop representation ( doubly densitized ) in terms of loop derivative as introduced in @xcite . such constraint was written in @xcite as , @xmath102 the hamiltonian acted on functions of loops and the operator @xmath103 had the action of re - routing one of the lobes of the partition of the loop determined by the points @xmath104 . we see two fundamental differences with the operator we just introduced . the first one is the re - routing operator @xmath103 . this operator arose in the loop representation to account for the fact that states based on loops in the fundamental representation do not diagonalize operators like the volume . in the spin network context the operator is replaced by the group factors we discussed . more important is the difference concerning density weights . in the hamiltonian in terms of loops we see that one obtains a doubly densitized quantity by considering the product of a dirac delta times the regulator . to understand properly this difference , it is worthwhile considering on which spaces of functions are these operators meant to operate upon . in the hamiltonian in terms of loops one had in mind that the loop derivative was acting on functions of loops such that the result was a smooth function . an example of such functions would be holonomies built with a smooth connection . the resulting expression would then consist of a smooth tensor , @xmath14 , a dirac delta integrated on three dimensional space and , in the limit , a two dimensional dirac delta given by @xmath105 integrated along the two one dimensional integrals . the result is finite but is regularization dependent , since the two dimensional dirac delta has an inverse power of the three dimensional volume in it that is not compensated . if we now turn our attention to the hamiltonian we introduced in this paper , and consider its action on a state of the form @xmath82 , we will see a different behavior . given the action ( companion paper : formulas 71 and 44 ) of the loop derivative on @xmath82 we get , @xmath106 where the extra edge added by the loop derivative starts and ends in the edge @xmath107 and as before , @xmath99 take cyclic values in the set @xmath100 . the reason for this is that if it ended in any other edge incident on the vertex one would ( in the limit @xmath108 ) get zero for the action since one would have an integral repeated over one of the edges contracted with @xmath109 . the action on @xmath110 of the loop derivative that appears in the hamiltonian constraint , for trivalent vertices reduces therefore to a chord diagram . if we evaluate the chord diagram using recoupling identities , we finally get , @xmath111 at this point it is worthwhile continuing the comparison with the doubly densitized case . if we evaluate the integrals involved in the above expression we see that we have six one dimensional integrals and two three dimensional dirac delta functions . in the limit we therefore have a finite result , that is defined independent of the background structures introduced by the regulator @xmath112 . one first performs the integral in @xmath18 , which fixes @xmath113 and then one is left with three one dimensional integrals with three step functions ( two explicit , one present in the @xmath114 ) . the result of the integrals is @xmath115 , which cancels the denominator of the @xmath116 ( @xmath117 ) and one is left with a finite result , that in the limit @xmath108 gives , @xmath118 where @xmath119 is the factor group associated with the vertex @xmath26 . we have introduced a quantum version of thiemann s singly - densitized classical hamiltonian in terms of the loop derivative that has the remarkable following properties : \a ) it has a significant resemblance to the original doubly - densitized hamiltonian proposed in the loop representation @xcite , which in turn closed the constraint algebra formally @xcite . we will discuss the implications of this in the next section . \b ) the hamiltonian introduced is finite on the space of vassiliev invariants we introduced in the previous paper . this happens due to the detailed form in which the loop derivative acts on these states . \c ) the hamiltonian may be well defined on other spaces of states . for instance , on the space of functions of spin nets obtained by considering the wilson nets along smooth holonomies ( the original kind of function that was first considered as loop differentiable ) , it is straightforward that the hamiltonian identically vanishes ( essentially , since the loop derivative is smooth , the integrals that before gained contributions due to the distributional character of the loop derivative now vanish ) . \d ) the last two points show that something surprising is happening here , in the sense that _ we did not construct the hamiltonian in an ad - hoc way to obtain properties b ) and c)_. that is , we introduced a discretization of the singly - densitized hamiltonian and it naturally turned out to be well defined on the vassiliev invariants and to vanish on holonomies of smooth connections . this might be pointing to a certain `` naturalness '' of the vassiliev invariant space in the context of quantum gravity . in spite of the a priori huge ambiguity in the regularization of the singly - densitized hamiltonian , it is difficult to imagine a regularization involving the loop derivative that would not be finite on the vassiliev invariants . in addition , we have not introduced in our construction any ad - hoc `` renormalization '' of the operator to obtain finiteness ( as one had to do in the doubly - densitized case ) . we will now proceed to discuss the issue of the constraint algebra . the operators we introduced are written purely in terms of the loop derivative and integrals along edges of the spin net . one can discuss the successive action of such operators without making specific reference to a space of states on which one is acting upon . such a calculation would be `` formal '' in the sense that terms arising in it could fail to be well defined when acting on particular spaces of states . this is the underlying philosophy of the kind of ( unregulated ) formal calculations some of us undertook in reference @xcite . such an approach can be pursued with the regulated operators we introduce in this paper . the fact that the operators are regulated and that they might act on the space of diffeomorphism invariant states will imply certain departures from the specifics of the calculations of reference @xcite . if one pursues calculations at this level of generality , one can not expect to reproduce the classical poisson algebra at the level of quantum commutators without making some assumptions about the distributional behavior of the various quantities involved in the calculations . one will recover terms that correspond to the classical ones but generically there will be additional terms . when one particularizes to a certain `` habitat '' of wavefunctions , these extra terms in many cases will vanish . however , it is not excluded that on certain habitats pathologies could appear . it would be an interesting exercise to pursue the calculation in general and then particularize to various `` habitats '' . this , however , is a complex and lengthy calculation in terms of loop calculus . we will not attempt a calculation at this level of generality here . in this paper we will concentrate on computing the constraint algebra on a series of habitats on which we recover the classical constraint algebra at the level of quantum commutators . these habitats will include as particular cases spaces of diffeomorphism invariant functions on which the appropriate algebra will be recovered . a point to be addressed when studying the constraint algebra is that , given the action of the hamiltonian constraint we have introduced , if one starts with a function of spin networks , when one acts with a hamiltonian constraint one ends with a more general object . this can be explicitly seen for instance in equation ( [ nu ] ) . what we see is that the result of the action of a hamiltonian on a function of spin networks is a function of spin networks times a `` vertex function '' , that is , a function dependent on the position of the vertex where the hamiltonian acts . if one wishes to compute the constraint algebra , the action of the second operator in the calculation of the algebra therefore takes place on such a space of functions . one can view these functions as simply more general functions of spin networks , since after all the information of the position of the vertices comes with a given spin network . this is possible , but one has to exhibit that dependence -of the vertices on the spin networks- in an explicit enough way that operators like the loop derivative have an appropriate action on such a dependence . one is initially tempted to say that the loop derivative simply ignores the extra factors , since they do not appear to contain an explicit loop dependence . but the position of the vertices depends on the edges of the spin network ( more precisely , a vertex is defined by the intersection point of at least three edges of the net ) , and therefore the vertex function should be affected by the loop derivative operator . to operate with the loop derivative over this kind of functions , we need to make more explicit the implicit functional dependence @xmath120 $ ] of the vertex function . we limit the analysis to the case of trivalent vertices . given three arbitrary open paths @xmath121 ( @xmath122 ) with a common origin @xmath123 , we define the quantity , @xmath124 where @xmath125 denotes generically the three curves @xmath126 , @xmath127 is @xmath128 if @xmath129 and zero otherwise , and @xmath130 is a scalar function defined on the manifold . it is assumed that the three paths are oriented and we denote by the overbar the path in the reversed orientation . we also assume that , starting at @xmath123 , the three curves overlap in a finite segment and then they separate following disjoint paths , as it is shown in the figure . we call @xmath26 the bifurcation point of the three curves . it is immediate to show that , @xmath131 and , @xmath132 the above expressions can be used to calculate the loop derivative of a vertex function . given a vertex of the net with edges @xmath133 , we identify the portion of the curve @xmath121 going form @xmath26 to the end point with the portion of the edge @xmath96 from the vertex to some intermediate point . this automatically guarantees that the point @xmath26 defined through the family of curves @xmath121 coincides with the chosen vertex of the graph . if the end points of the edges @xmath134 lie outside of the ball of radius @xmath30 centered at @xmath26 , then one has the identity @xmath135 . we can now proceed to compute explicitly the loop derivative . for any function @xmath136 we have ( @xcite chapter 1 ) , @xmath137z } f(z).\label{loopder}\ ] ] using this result it is straightforward to calculate the loop derivative of @xmath138 , @xmath139z } m(z ) \theta_{\epsilon}(z - y)\right . \nonumber\\&&\left.+ ( m(z)+m(y ) ) \delta(\epsilon-|z - y| ) \frac{\delta^d_{[b}(y - z)_{a]}}{|z - y|}\right\}.\label{derr}\end{aligned}\ ] ] the loop derivative of @xmath140 is evaluated using the above result and the leibniz rule . now that we know how to compute the loop derivative of a vertex function , we can consider the construction of a `` habitat '' of functions of spin networks with marked points of the definite following kind : @xmath141 where @xmath142 is a group factor depending on the spins of the edges incident at @xmath26 . this type of dependence on the vertex is precisely the one that appears in the action of the hamiltonian constraint ( [ hame2 ] ) . in the last identity , we express the relationship between @xmath143 and @xmath144 through an operator @xmath145 . it is clear that , in the limit @xmath19 , one gets as a result a function of the spin network times a function of the vertices and incoming spins , @xmath146 to define the action of the constraints on these kinds of functions we will study the action of the constraints before taking the limit @xmath19 in these expressions in a suitable way . let us start by considering the diffeomorphism constraint , @xmath147\,.\end{aligned}\ ] ] in the second term the diffeomorphism acts on @xmath144 leaving all the factors unaffected . this term is immediate to evaluate ( and in the case of diffeomorphism invariant @xmath144 s vanishes ) , so we will not discuss it further . in order to analyze the first contribution it is convenient to make the following intermediate calculation , @xmath148z } m(z ) \theta_{\epsilon}(z - y)\right . \nonumber\\ & & \left.+(m(z)+m(y ) ) \delta(\epsilon-|z - y|)\frac{\delta^d_{[b}(y - z)_{a]}}{|z - y|}\right\}+ \mbox{\footnotesize{cyclic permutations in $ e_1,e_2,e_3$}},\nonumber\end{aligned}\ ] ] where we have used ( [ derr ] ) . in the evaluation of this quantity , the result depends on the order in which the limits are taken . if one takes the limit @xmath19 first , one finds that the diffeomorphism annihilates @xmath149 . this means that @xmath150 . on the other hand , if we take the limit @xmath151 first one gets the result , @xmath152 r_{c}(1,\epsilon , e ) + o(\epsilon),\ ] ] and , @xmath153 with this choice we see that the action of the diffeomorphism is what one would have expected geometrically : it lie - drags the dependence of @xmath91 on the position of the vertex . this action immediately ensures that the action of the diffeomorphism constraint closes the appropriate commutator algebra on the space of functions we are considering , provided @xmath144 is either invariant ( like in the case of the vassiliev invariants ) or is such that the diffeomorphism constraint has the natural geometric action . in the general case , @xmath154 based on these calculations for the diffeomorphism , let us now explore the action of the hamiltonian constraint on the habitat we have been discussing . from ( [ hgen ] ) and ( [ derr ] ) we get ( @xmath155 is now the scale parameter of the triangulation adapted to the spin network ) , @xmath156z } m(z ) \theta_{\epsilon}(z - y)+ ( m(z)+m(y ) ) \delta(\epsilon-|z - y| ) \frac{\delta^d_{[b}(y - z)_{a]}}{|z - y|}\right\ } + \mbox{\footnotesize{cyclic permut . in $ e_1,e_2,e_3$}},\label{hr } \nonumber\end{aligned}\ ] ] where we have written @xmath157 in order to make explicit the dependence of the triangulation with @xmath155 . notice that , @xmath158 and , @xmath159 irrespective of the order in which the limits are taken . this means that , @xmath160 and taking into account ( [ psim ] ) , @xmath161 } h(n)o(m,\omega ) \psi ( s ) = o(m,\omega ) h(n ) \psi(s).\ ] ] this is an important result . it will imply that the hamiltonian constraints will commute with each other . we will discuss this in detail in the next section . in the classical theory , the poisson bracket of two smeared hamiltonian constraints is given by , @xmath162 with the one - form @xmath163 and @xmath164 is the ( unsmeared ) diffeomorphism constraint . that is , the right hand side of this expression is a diffeomorphism generated by the vector obtained by contracting the one - form @xmath165 with a double - contravariant metric . if one wishes this expression to have a quantum counterpart , it implies realizing the product of the doubly contravariant metric times a diffeomorphism as a quantum operator . in order to do this , we need to re - express it in terms of variables that are suitable for the ashtekar formulation . this is done via the following classical identities , as discussed by thiemann @xcite , @xmath166 and we recall the relationship of the covariant ( undensitized ) triads with the more usual contravariant densitized triads , @xmath167 . we can now use the key identity , due to thiemann , @xmath168 to obtain , @xmath169 this expression can not be easily promoted in a direct way to an operator due to the denominators involving the determinant of the metric . we can achieve this noting that the poisson bracket of the connection and the volume depends only on local information of the volume operator in the surrounding of the point @xmath17 . we can therefore replace in the above expression the volume of the whole manifold @xmath9 , by a `` localized '' volume ( as discussed by thiemann @xcite ) @xmath170 , given by , @xmath171 where @xmath172 is an infinitesimal closed region around the point @xmath17 and @xmath173 is the characteristic function associated with this region . the following identity holds , @xmath174 and in addition , @xmath175 where @xmath176 is the euclidean volume of the infinitesimal region @xmath177 given by a fiducial flat metric ( we will see that the final result does not depend on the fiducial metric ) . the above expressions allow us to absorb the denominators involving the determinant of the metric into poisson brackets . using this results we write ( [ metc ] ) in the form , @xmath178 we now discretize the integral introducing the triangulation @xmath179 of space defined in section ii , @xmath180 where we have identified @xmath181 to simplify the notation . using ( [ vol ] ) we replace the @xmath182 and @xmath183 in terms of the volume of the elementary regions and the edges @xmath184 of the tetrahedra , and we join four of the @xmath184 s with the @xmath185 s to construct holonomies along the edges of the triangulation , @xmath186 where we have defined , @xmath187 { \rm tr}[\tau^k h(u_r)\{h^{-1}(u_r),\sqrt{v({\cal{r}}_v)}\ } ] \times \nonumber\\ & & { \rm tr}[\tau^l h(u_t)\{h^{-1}(u_t),\sqrt{v({\cal{r}}_v)}\ } ] { \rm tr}[\tau^m h(u_n)\{h^{-1}(u_n),\sqrt{v({\cal{r}}_v)}\}]\ , . \label{q}\end{aligned}\ ] ] we are now ready to promote the above expression as an operator acting over the spin network wavefunctions , which we assume to be the loop transform of a state in the connection representation . as in the case of the hamiltonian constraint , we adapt the triangulation to the graph of the spin network choosing the points @xmath24 of the boxes @xmath188 coincident with the vertices of the spin network , and we choose the regions @xmath189 equal to that defined by the triangulation around @xmath26 but with a length scale @xmath155 . with this prescriptions we get from ( [ metc2 ] ) , @xmath190 where we have used ( [ tangent ] ) and @xmath191 with @xmath30 the length scale of the regions @xmath29 . the action of the operator @xmath192 on the spin network wavefunctions is calculated as usual through its action on the wilson net appearing in the loop transform . it is important to notice that this operator looses all dependence with the scale parameters involved in the triangulation and the localized volume . this is due to the fact that , acting on the wilson net , the holonomies @xmath193 that appear in the quantum version of ( [ q ] ) generate ( in the limit @xmath194 ) finite recoupling coefficients and that the action of the volume operator is local ( it depends only on the vertex included in the region @xmath195 . with this result at hand it is straightforward to quantize the right hand side of the poisson bracket of two smeared hamiltonian constraints . from the classical expression we immediately write , @xmath196 f_{\delta}(x , y ) \delta_{cb}^{(e)}(\pi_y^x ) \psi(s)\ , , \nonumber\end{aligned}\ ] ] where we have used the regularized expression of the unsmeared diffeomorphism operator ( see the companion paper , section ii.b ) . introducing now the regularized metric operator we get , @xmath197 \,f_{\delta}(x , y)\times\nonumber\\ & & \int_{e_p } dw^a \theta_{\epsilon}(w;v ) \int_{e_s } dt^b\,\theta_{\epsilon}(t;v ) { q}^{ps}(v)\delta_{cb}^{(e)}(\pi_y^x)\ , \psi(s)\,.\end{aligned}\ ] ] at this point it is worthwhile mentioning that this expression has a regularization ambiguity , given by the two regularization parameters @xmath30 and @xmath155 . the latter was introduced , as we discussed above , in the definition of the localized volume operator . this ambiguity , associated with the fact that the localized volume and the ordinary one are the same on these spaces of functions was first noticed by lewandowski and discussed in @xcite . if one chooses @xmath198 one notices quickly that the expression for the right - hand - side vanishes . the powers @xmath199 cancel and one is left with two integrals along @xmath200 , @xmath201 which are of order @xmath30 each . the rest of the expression is simply the action of a diffeomorphism . assuming that the latter is finite , the expression therefore vanishes . if we trace back the origin of this cancellation , one notices that the expression ( [ metq ] ) for the contravariant metric operator vanishes identically over spin network states of any kind , in particular , the vassiliev invariants we are considering in this paper . this property is a consequence of the following two peculiarities of the quantization of @xmath202 in terms of spin network wavefunctions : the sum over the triangulation reduces to a sum over the vertices of the spin net ( which includes a finite number of terms ) , and in each term the result of @xmath203 is finite . notice that in the classical result ( [ metc ] ) one has an infinite sum of @xmath204 , each of which tends to zero in the limit @xmath205 . this limit gives in general a nonzero result . but in ( [ metq ] ) we have instead a finite sum of terms @xmath203 , which are independent of @xmath30 and @xmath155 . this fact alters the power counting of the factors @xmath30 and @xmath155 in such a way that all the terms tends naturally ( i.e. choosing @xmath198 ) to zero as the triangulation shrinks to a point . that the right - hand - side may vanish was already observed in @xcite . could one `` tune '' the limits in @xmath30 and @xmath155 so this quantity is nonvanishing ? one could , but the result would be dependent on the background structures introduced to regularize . if one made the expression non - vanishing it would be proportional to the normalized tangent vectors at the intersection , which are background dependent . this is not surprising : there is no naturally defined second order symmetric contravariant tensor defined in a manifold without metric . this result is quite strong . it implies that the commutator of two hamiltonians will have to vanish if one wishes to have consistency . we anticipated that this would happen in the previous subsection and we will now see in detail how it happens . as a final remark , we notice that if one wished to define the doubly covariant metric , it is straightforward to compute it using the identities introduced by thiemann and one finds that regularized with the same procedures we followed up to now , it diverges . should one worry about a theory of quantum gravity where the doubly contravariant metric vanishes and the doubly covariant metric diverges ? we will return to this in the discussion section . the loop derivative was originally introduced in the context of yang mills theories , where the natural functions to act upon were holonomies of smooth connections . these functions are not the most natural ones to consider in the context of diffeomorphism invariant theories like general relativity , but our constraints are well defined on them , so we can consider them to be a `` habitat '' where to test the constraint algebra , at least as a mathematical exercise . on these kinds of functions the diffeomorphism constraints that we have introduced here are known to close the appropriate algebra in the limit in which regulators are removed ( see @xcite , although the regularization is slightly different than the one we use in this paper , it is immediate to check that the same calculations go through ) . we therefore will not repeat the calculation here . the hamiltonian constraint vanishes identically on this habitat . starting from equation ( [ hgen ] ) , if the loop derivative is a smooth function , one is left with two one - dimensional integrals along the edges of a cell of a finite function . in the limit in which the triangulation is shrank , the result vanishes . the commutator of a diffeomorphism with a hamiltonian therefore immediately reproduces the classical result . the right - hand side of the commutator of two hamiltonians that we introduced in the previous subsection also vanishes , here one has three one dimensional integrals along the edges of a triangulation of a quantity that goes as @xmath206 . the final result therefore goes as @xmath207 . therefore , as long as one chooses @xmath155 shrinking to zero faster than @xmath208 , the right hand side vanishes . on this space therefore , we reproduce the classical poisson algebra at the level of the commutators . the diffeomorphisms close among themselves and the hamiltonian vanishes identically . could one simply claim that this is the `` right '' habitat to do quantum gravity ? the answer is that this is unlikely . although all states are solutions of the hamiltonian constraint , this space does not contain any solution to the diffeomorphism constraint . solutions to the diffeomorphism constraint in terms of holonomies can only be constructed as infinite superpositions , functional integrals or `` group averaging '' , and in these cases one includes connections that are not smooth . an interesting point is that in this habitat , since one does not have solutions of the diffeomorphism constraint but has solutions to the hamiltonian constraint , the only way that one could achieve consistency in the algebra is if the right hand side of the commutator of two hamiltonians vanishes , which is the case . another point to consider is that if one examines the expression of the commutator of two hamiltonians , although both members vanish in the limit in which one shrinks the triangulation given the smoothness of the loop derivative away from the limit the calculation is problematic . for instance , it is not clear what is the action of the volume operator on a state given by the loop derivative of a smooth holonomy . one might consider introducing a definition of the volume on this kind of space , but this has yet to be done in detail . remarkably , this space of invariants also leads to a reproduction of the classical poisson algebra at a trivial level only ( as usual , we limit our discussion to trivalent intersections ) . this is based on the fact proved in the appendix of the previous paper , that these invariants are annihilated by the loop derivative that appears in both the diffeomorphism and the hamiltonian constraints and of the expression of the rhs . therefore the ambient isotopic ( framing - independent ) vassiliev invariants for spin networks with trivalent intersections are annihilated by all the constraints of quantum gravity and consistently , by the right hand side of the commutator of two hamiltonian constraints . it should be pointed out that the space is in no way trivial : as we discussed in the appendix of the previous paper , the annihilation is a detailed property of the space of vassiliev invariants , related to the decomposition of the invariants in framing independent and framing dependent components and the detailed structure of chord diagrams appearing in the coefficients . hints that the vassiliev invariants for trivalent intersections were annihilated by the hamiltonian constraint were found for the lower invariants in terms of loops @xcite,@xcite , in the lattice @xcite . one could mention as a more trivial example of a space with similar property the states based on diffeomorphism invariants of spin networks with no vertices , which are also trivially annihilated by all the constraints we consider . this is an extension of results also first suggested in the context of loops @xcite . it should be remarked that this property of the loop derivative is not true for higher valence intersections . future extensions of the hamiltonian constraint to higher valence intersections could be tested in this habitat for non - trivial consistency . by this space we mean the invariants that appear in the power series expansion of the expectation value of the wilson net in a chern simons theory in terms of the inverse coupling constant @xmath209 . in particular , the whole series is an example of such an invariant . each coefficient of the series and certain portions of them are also examples . in general , these invariants are sums and products of the independent invariants that appear at each order , both framing - independent and framing - dependent ( see previous paper ) . these states have the important property that the loop derivative that appears in the expressions of the constraints ( which is evaluated on a path @xmath16 of infinitesimal length ) can be rewritten simply as , @xmath210 where @xmath211 is the edge of the spin net containing the point @xmath17 and where the invariant @xmath212 is another vassiliev invariant of one order less than @xmath213 . what happens is that because of the infinitesimal length of @xmath16 one can rearrange the action of the loop derivative in terms of the original spin network @xmath52 using recoupling identities , at the expense of some additional group factors , which we reabsorb notationally in @xmath212 . we will see that we do not need the details of the relation of @xmath213 to @xmath212 for proving the consistency . as usual , all our discussion is limited to trivalent intersections . the relation above implies that for the vassiliev invariants , one can write , @xmath214 this can be seen by considering equation ( [ nu ] ) and recalling that @xmath215 . in particular , we see that the action of the hamiltonian on these states can be written ( up to a constant factor which we will omit for simplicity ) as , @xmath216 on this space , the diffeomorphism constraint vanishes identically , as we discussed in the previous paper . so the algebra of diffeomorphisms is trivially satisfied . nontrivial commutators to be realized will be those of a diffeomorphism with a hamiltonian and that of two hamiltonians . in the case of a diffeomorphism with a hamiltonian , since the wavefunctions are diffeomorphism invariant , of the two terms of the commutator one is left with the one in which the diffeomorphism acts at the left . since the action of the hamiltonian on @xmath213 is not diffeomorphism invariant , one should recover the proper action of the diffeomorphism through such term . here the derivations of section [ marked ] become useful . we start from the expression ( [ honvas ] ) , and then use ( [ diffeoono ] ) to get , @xmath217 and from here the correct commutator follows immediately . this implies that the quantum hamiltonian transforms covariantly in a correct way . this calculation is one of the main differences of our construction with respect to the one of thiemann @xcite , since in that context one does not consider an infinitesimal generator of diffeomorphisms . as we discussed in section [ marked ] , the action of a hamiltonian on a function with marked points as one gets after acting with a hamiltonian on a vassiliev invariant was given by expression ( [ [ ho ] ] ) , which we can combine with the action of the hamiltonian ( [ honvas ] ) to get , @xmath218 and if one performs the calculation in the reverse order , one obtains the @xmath103 operators in the reverse order . however , these operators are multiplicative , so they commute . therefore one has the result , @xmath219 v_n(s)=0,\ ] ] which is the expected commutation relation on states that are invariant under diffeomorphisms . the states we have already considered , where one has a vassiliev invariant times a scalar function that depends on the position of the vertices of the spin network , are an attractive habitat where the diffeomorphism constraint does not vanish . as discussed in section ( [ marked ] ) the diffeomorphisms on these states reduce to lie dragging of the scalar functions of the vertices ( see formula ( [ diffeoono ] ) ) . since they correspond to a natural geometric action , the consistency of the algebra of diffeomorphisms is immediate . we start from the action of a hamiltonian on these kinds of states , @xmath220 where we have used the fact that @xmath221 and @xmath222 commute and that the action of the hamiltonian on these states produces a vassiliev invariant of order lowered by one unit times a vertex function . we now act with a diffeomorphism , @xmath223 and now use the fact that leibniz rule applies to the action of the diffeomorphism constraint ( stemming from the fact that it also applies to the action of the loop derivative ) , @xmath224 and we now reconstruct the hamiltonian in the first and second terms , @xmath225 and using the fact that @xmath222 and @xmath221 commute , we get , @xmath226 which we can rewrite as , @xmath227 from where we get the correct commutation relation , @xmath228 o(m,\omega ) v_n = h(l^a \partial_a n ) o(m,\omega ) v_{n}.\ ] ] this calculation proceeds along the same lines as the one in the previous subsection , we essentially rewrite the action of the hamiltonian in terms of the @xmath221 operator , and note that the @xmath221 operators commute , @xmath229 and we therefore get that @xmath230=0 $ ] , which as we discussed before , is consistent with the representation in these spaces of functions of the right hand side of the quantum commutator . we have presented a canonical quantization of the constraints of canonical general relativity . we represented the diffeomorphism and hamiltonian constraints using two novel ingredients : the loop derivative to represent the field tensor @xmath14 and the use of spaces related to the generalization of the vassiliev invariants to spin networks ( restricted to trivalent intersections ) as wavefunctions . in terms of the latter we constructed several `` habitats '' , including spaces of functions that are not invariant under diffeomorphisms and we checked that one obtained a consistent algebra of quantum commutators of the constraints . consistency in this sense implies that the quantization of the canonical poisson identities between the classical constraints is implemented correctly in the quantization . we have observed that this consistency is achieved at the price of having a vanishing right hand - side for the commutator of two hamiltonians and that we can trace back this fact to the vanishing nature of the doubly covariant metric operator in these spaces of functions . we should point out common elements and differences with the quantization presented by thiemann @xcite . in thiemann s case the hamiltonian was implemented on the space of diffeomorphism invariant cylindrical functions of spin networks . on these spaces one does not have a well defined notion of the field tensor @xmath14 and the functions are not loop - differentiable . in our case , the availability of the loop derivatives allows to have at hand a `` differential calculus '' that allows for several novel constructions . examples of them are the calculations performed in computing the constraint algebra , and the possibility of finding novel solutions to the hamiltonian constraint , based on the behavior of the vassiliev invariants under loop differentiation . another difference involves the implementation of an infinitesimal generator of diffeomorphisms . in thiemann s original construction one worked directly in terms of diffeomorphism invariant states and therefore one did not have an infinitesimal generator . the construction was extended by lewandowski and marolf @xcite to `` habitats '' that are dependent on diffeomorphisms . with that extension , thiemann s work achieves the same level of consistency that we have in this paper , in the sense that one can check non - trivial commutators of diffeomorphisms and diffeomorphisms with hamiltonians . one still has the feature of a vanishing right hand side of the commutator of two hamiltonians @xcite . as we discussed in this paper , this feature appears as inescapable in the context of wavefunctions only dependent on spin networks as the ones we considered here , since as we mentioned one does not have enough structures to construct a naturally defined symmetric metric tensor . other differences arise insofar as the space of wavefunctions considered . in the case of thiemann , one had spaces of functions with well defined inner products , which allow to discuss normalizability and study the spaces of solutions with a level of rigor that is not available currently in our approach , since we do not have an inner product on the habitats we are considering . it might not be impossible to find a suitable inner product with the same techniques that led to the construction of the measures on the spaces of cylindrical functions , but it has not been achieved yet . concerning solutions of the constraints , one has some available both in our approach and in thiemann s , that appear as quite distinct in their features . in thiemann s approach , the solutions are obtained using group averaging techniques . this leads to structures like `` tassels '' and others @xcite in which accumulation of lines and vertices take place . in our approach , one has a number of solutions of the hamiltonian constraint ( the framing independent vassiliev invariants for trivalent intersections ) that do not depend on such structures . on the other hand , they appear as `` trivial '' solutions in the sense that they may not exist if one considers intersections of higher valences . in our approach one may also construct solutions to the hamiltonian constraint with a cosmological constant , following similar ideas that led to the construction of states in terms of loops @xcite . although we have not pursued this in detail yet , it appears quite plausible that these types of solutions exist , given the structure of the extra term due to the cosmological constant in the hamiltonian constraint in terms of the spin network approach @xcite . thiemann s approach has also been studied in @xmath231 dimensions @xcite , and appears to lead to a satisfactory quantization , provided one chooses in an ad - hoc way an inner product that rules out certain infinite dimensional set of solutions . in a forthcoming paper we will discuss the quantization of @xmath231 dimensional gravity using an approach that has elements in common with the one we pursue here , in particular the requirement of loop differentiability of the states . we will see that this requirement limits us ( at least for low valence intersections ) to the correct solution space in a natural way . the reader might find unsatisfactory that the right hand side of the commutator of two hamiltonians vanishes . even more unsatisfactory may appear the fact that this is due to the vanishing of the doubly covariant metric in this approach to quantum gravity . in this subsection we will address this and other issues . the first question that one may raise is if this is not just a pathology that is introduced by our limitation to trivalent intersections for reasons of calculational convenience . after all , one knows that this subspace in in some sense degenerate since the volume operator vanishes identically . unfortunately , it appears unlikely that extending our results to higher valent intersections ( to which we do not see any technical obstruction , apart from greater calculational complexity ) will change things insofar as the commutator of two hamiltonians . the action of the hamiltonian constraint on higher valent intersections is more complicated largely because the volume operator and the action of the loop derivative can not be reduced to a simple group - dependent prefactor times the original state , but will in general involve a linear combination of states , with a non - trivial group - dependent matrix of coefficients . however , the dependence of the hamiltonian on the smearing function will remain as a multiplicative one . therefore it appears that the commutator of two hamiltonians will again vanish . moreover , the reasons we gave for the vanishing of the right hand side of the commutator are independent of the valences of the intersections of the spin networks . the expression for the doubly - covariant metric will be more complicated , but will still include a prefactor with the same dependence on the regulators as in the trivalent case and it will vanish . one therefore expects consistency at the same level we achieved for the trivalent case . can one consider satisfactory a theory with a vanishing doubly contravariant metric ( and a divergent doubly covariant metric ) ? the answer to this will only appear in a definitive way when one constructs physical predictions from the theory , which at the moment are lacking . one can get partial indications that the pathology might not be as severe as it appears at first sight from the fact that one can define reasonable quantities like areas , volumes and lengths in this framework in spite of having the ill defined metric operators . an attractive feature of the vassiliev states is that other diffeomorphism invariant operators can be defined as well . for instance , if one considers the integral @xmath232 , given that the chern simons states are such that the densitized triads are proportional to the magnetic vector constructed from the curvature , one finds that quantum representation of this integral is identical to the volume operator . an important observation concerning how satisfatory is a theory with a vanishing metric operator like the one we propose is that in @xmath231-dimensional gravity similar pathologies appear , yet the correct physical theory is recovered . if one pursues a quantization similar to thiemann s dimensional gravity along the same lines as the one proposed in this paper , and the observations we make in this subsection apply to it as well . we will also expand the discussion on the physical observables and their connection with the kinematical metric we present in this subsection . ] in @xmath231=dimensions ( this is discussed in detail in @xcite ) one notices that the hamiltonians also commute . moreover , the geometrical arguments leading to the vanishing of the @xmath202 operator still hold in this context : there is no quantity one can build out of loop states that will yield a doubly contravariant symmetric tensor . yet , it was shown in @xcite that one can recover the correct physical theory . how can these apparently contradictory elements be reconciled ? it has to do with the nature of the kinematical calculations we are performing . to check the `` off shell '' contraint algrebra one is required to operate with a kinematial space of wavefunctions that are not annihilated by the constraints . when one recovers `` physics '' one should really do it with states that are annihilated by the constraint . this can be carried out in detail in the @xmath231 dimensional case . suppose one wishes to ask questions about the `` metric '' of the theory . the first observation is that on physical states one can not define a metric tensor since the latter states are diffeomorphism invariant and the metric tensor is not . one could gauge fix a metric tensor , say by asking questions about the value of its components in a fixed coordinate system . these questions can be answered : the components of the metric in a fixed coordinate system can be related to the values of the invariant operators @xmath233 and @xmath234 in such a coordinate system ( and therefore in general , since the latter are coordinate invariant ) . one therefore has well defined operators associated with the metric that can be evaluated in the physical space of states . but such operators have little to do with the @xmath202 we introduced in the kinematical space . it is not obvious at all that the two operators will be related in any way since the former are non - vanishing whereas the latter vanishes identically . there is a certain disquieting element in the last observation , since it seems to imply that one should be careful before drawing conclusions from calculations at a kinematical level . this in particular , implies all calculations involving the constraint algebra . when we set out to work on the current paper , our expectation was that reproducing the contraint algebra would be a strong test for our quantization procedure , that perhaps would rule out all but a few of the possible theories . what we have learned is that this is probably not the correct view on the issue : one can obtain consistency in many ways at a kinematical level ( in particular with @xmath235 ) . one can construct many consistent theories of quantum gravity . and still , what the @xmath231 dimensional example shows is that the `` physics '' of all these theories is really deeply buried in the states that solve the constraints . in a sense this is good , since the theory we are proposing here differs significantly at this level from the one proposed by thiemann and therefore further adds to our options for trying to match experimental results . the observation that kinematical calculations may have quite nontrivial connections with `` real physics '' poses problems for computations that try to obtain heuristic physical insights from looking at properties of the kinematical states ( the `` weave '' approach to the semiclassical theory ) . an example of this are our own results on gamma - ray - burst light dispersion @xcite . from the @xmath231 example we see that quantum fluctuations in the `` physical metric '' ( the ones one would expect to influence the propagation of matter fields ) may not have a direct connection with fluctuations of the kinematical metric . yet it is the latter that are used in the concrete computations of @xcite . concerning the latter point , it is worthwhile mentioning explicitly that the theory presented in this paper can be coupled to matter and have well defined expressions for the hamiltonians of matter fields , which also involve the metric in a non - trivial way ( the coupling of the theory to matter can , for instance , be achieved using the same construction as thiemann @xcite ; or one could consider alternative settings in which the loop derivative is also used in the representation of certain matter fields ) . it is somewhat unfortunate that the remarkable results on black hole entropy that have been achieved within this context @xcite do not involve in a detailed way the action of the hamiltonian constraint in the `` bulk '' of the spacetime to be used as guideline for constructing the constraints , and therefore can not distinguish at the moment our proposal from thiemann s . the vanishing of the doubly contravariant metric tensor , apart from appearing as a `` robust '' feature based only on elementary notions of covariance of the elements involved in constructing the spin network states , is the way in which our approach bypasses the `` hermiticity problem '' of the canonical quantization . as we mentioned , if one were to demand that the diffeomorphism and hamiltonian constraints and the doubly covariant metric be hermitian operators ( since they correspond to real classical quantities ) , the commutator of two hamiltonians ( which is hermitian ) can not be simply equal to the product of a metric times a diffeomorphism ( which is not hermitian ) . one could fix the hermiticity of the right hand side by `` symmetrizing '' the operator , but then one could face an anomaly problem , since the diffeomorphism constraint would not act to the right of the product . this difficulty is bypassed ( in the context of non - diffeomorphism invariant states ) if the metric vanishes . it should be noticed that it is not obvious that one should promote the constraints to hermitian operators , for instance , see @xcite for counterexamples in the context of the quantization of non - unimodular gauge groups . related with the issue of the lack of guidelines to construct the theory is the problem of uniqueness . we already have a manifest non - uniqueness in that the theory presented here and the theory introduced by thiemann appear as both satisfactory yet distinct . the ambiguity is worse than this . even if one stays within one general approach , for instance considering the space of vassiliev invariants as `` arena '' for quantization , there are many ways in which one could implement the hamiltonian constraint . we have not analyzed them in any detail , but some salient features of them are worthwhile mentioning . the ambiguities arise in the various limits involved in constructing the hamiltonian . one of them is associated with the specific action of the loop derivative . the loop derivative is dependent on a path . in this paper we have taken such a path to coincide with one of the lines of the spin network . we then acted with the loop derivative and collapsed its action using recoupling identities . the final action of the hamiltonian was therefore `` ultra - local '' in the sense that it returned a wavefunction with the same vertex structure times a vertex dependent prefactor ( for valences higher than three one gets a linear combination with different spin weights for the intertwiners yet the topology of the vertex is unchanged ) . this may raise concerns that `` super - selections '' could appear in the sense that it could be easy to construct operators that commute with the hamiltonian . one could define a different hamiltonian in a straightforward way , by assuming a different topology for the path associated with the loop derivative . for instance , one could assume that the path starts at the vertex , advances along one of the edges of the spin network and then crosses towards another edge of the spin network along one of the edges of the tetrahedra introduced in the discretization . at the end of the path , the loop derivative acts . one then is left with an action of the hamiltonian resembling the one introduced by thiemann : the constraint produces a vertex dependent prefactor , but it also alters the structure of the spin network by `` adding a line '' at the vertex . a different proposal would be to have a path that starts at the vertex , advances along one of the edges of the spin network , crosses towards another edge along the tetrahedron and then continues to cross towards another edge of the spin network , as shown in figure [ fifi ] . in the first of these proposals ( as in thiemann s ) , the hamiltonian adds vertices called `` exceptional '' in the sense that they are planar vertices . the hamiltonian vanishes identically on such vertices , because the volume vanishes on planar vertices , even if they are four - valent . in the second proposal , the intermediate vertex that is created is non - planar . this might raise hopes that the commutator of two hamiltonians may be less trivial , since the second hamiltonian acts at the newly created vertex , but more careful analyses seem to indicate that the hamiltonians still commute @xcite . if it they were not to commute one would still be faced with how to reproduce a similar vertex structure with the right hand side operator in a natural way . summarizing this portion of the discussion : it appears that there is a non - trivial , possibly infinite , amount of ambiguity in the definition of the theory , that is not significantly constrained by the imposition of the correct quantum commutator algebra . related to the latter point is the fact that due to the vanishing of the metric , the consistency check provided by the commutator of two hamiltonians is less detailed than if one had to prove the equality of non - vanishing operators . we already see that we have two distinct and apparently consistent , quantizations of the constraints ( with several possible variants of each ) . it will require further study to determine if one quantization is `` better '' in the sense of reproducing expected results than the other . one possibility would be to consider the commutation of the hamiltonian with various operators , and study if inconsistencies appear . this procedure , however , might be limited by the fact that most commutators have non - trivial right hand sides which will inevitably involve quite a bit of ambiguity at the time of their quantization . another point that might be raised is that these approaches appear confined to four dimensions and to the einstein - hilbert action , and therefore one may have little hopes of making contact with other approaches , as those based on string theories . these approaches not only may present avenues to understand quantum gravity but they also have the attractive feature of unifying gravity with other interactions , a goal some may consider desirable in itself . it should be pointed out however , that progress is being made @xcite towards describing @xmath236 dimensional general relativity in terms of connections . although constructions like the ones we discussed here have not been pursued in detail in this context , they appear as plausible . to conclude , we have at the moment canonical quantizations of general relativity ( possibly coupled to matter ) that appear as mathematically consistent at the kinematical level at which they have been studied . this level of consistency had never been achieved before in other approaches . further exploration of the consequences of the quantization will be needed to determine if any of them are physically satisfactory theories of the quantum gravitational field . in particular the exploration of the space of states that solve the constraints and how physical quantities can be evaluated on them appears as a natural next step in the quantization program . we wish to thank abhay ashtekar , laurent freidel , john klauder , karel kucha and thomas thiemann for comments and discussions . this work was supported in part by the national science foundation under grants nsf - phy-9423950 , nsf - int-9811610 , nsf - phy-9407194 , research funds of the pennsylvania state university , the eberly family research fund at psu . jp acknowledges support of the alfred p. sloan and john simon guggenheim foundations . we acknowledge support of pedeciba ( uruguay ) . rg and jp wish to thank the institute for theoretical physics of the university of california at santa barbara and cdb , rg and jg the center for gravitational physics and geometry at penn state for hospitality during the completion of this work . c. di bartolo , r. gambini , j. griego , j. pullin `` canonical quantum gravity in the vassiliev invariants arena : i. kinematical structure '' , companion paper . c. rovelli , l. smolin , nucl . phys . * b442 * , 593 ( 1995 ) . j. barbero , phys . rev . * d51 * , 5507 ( 1995 ) . t. thiemann , class 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j. griego , j. pullin , nucl . phys . * b534 * , 675 - 696 ( 1998 ) . r. gambini , j. pullin , phys . rev . * d59 * , 124021 ( 1999 ) . t. thiemann , class . grav . * 15 * , 1249 ( 1998 ) . d. marolf , d. giulini , class . * 16 * 2479 ( 1999 ) . see for instance a. ashtekar , j. baez , a. corichi , k. krasnov phys . lett . * 80 * , 904 ( 1998 ) , and references therein . l. freidel , k. krasnov , r. puzio , hep - th/9901069 .

differential analysis is based on studying the time evolution of the distance between trajectories emanating from different initial conditions . a dynamical system is called contractive if any two trajectories converge to one other at an exponential rate . this implies many desirable properties including convergence to a unique attractor ( if it exists ) , and entrainment to periodic excitations @xcite . contraction theory proved to be a powerful tool for analyzing nonlinear dynamical systems , with applications in control theory @xcite , observer design @xcite , synchronization of coupled oscillators @xcite , and more . recent extensions include : the notion of partial contraction @xcite , analyzing networks of interacting agents using contraction theory @xcite , a lyapunov - like characterization of incremental stability @xcite , and a lasalle - type principle for contractive systems @xcite . a contractive system with added diffusion terms or random noise still satisfies certain asymptotic properties @xcite . in this respect , contraction is a robust property . in this paper , we introduce three forms of generalized contractive systems ( gcss ) . these are motivated by requiring to take place only after arbitrarily small transients in time and/or amplitude . indeed , contraction is usually used to prove _ asymptotic _ properties , and thus allowing ( arbitrarily small ) transients seems reasonable . in some cases as we change the parameters in a contractive system it becomes a gcs just before it looses contractivity . in this respect , a gcs is the analogue of marginal stability in lyapunov stability theory . we provide several sufficient conditions for a system to be a gcs . these conditions are checkable , and we demonstrate their usefulness using examples of systems that are _ not _ contractive with respect to any norm , yet are gcss . we begin with a brief review of some ideas from contraction theory . for more details , including the historic development of contraction theory , and the relation to other notions , see e.g. @xcite . consider the time - varying system [ eq : fdyn ] = f(t , x ) , with the state @xmath0 evolving on a convex set @xmath1 . we assume that @xmath2 is differentiable with respect to @xmath0 , and that both @xmath2 and @xmath3 are continuous in @xmath4 . let @xmath5 denote the solution of at time @xmath6 with @xmath7 ( for the sake of simplicity , we assume from here on that @xmath5 exists and is unique for all @xmath8 and all @xmath9 ) . we say that is _ contractive _ on @xmath10 with respect to a norm @xmath11 if there exists @xmath12 such that [ eq : contdef ] @xmath13 and all @xmath14 . in other words , any two trajectories contract to one another at an exponential rate . this implies in particular that the initial condition is `` quickly forgotten '' . recall that a vector norm @xmath15 induces a matrix measure @xmath16 defined by @xmath17 where @xmath18 is the matrix norm induced by @xmath19 . a standard approach for proving is based on bounding some matrix measure of the jacobian @xmath20 . indeed , it is well - known ( see , e.g. @xcite ) that if there exist a vector norm @xmath19 and @xmath12 such that the induced matrix measure @xmath21 satisfies [ eq : jtc ] ( j(t , x ) ) -c , for all @xmath13 and all @xmath22 then holds . ( this is in fact a particular case of using a lyapunov - finsler function to prove contraction @xcite . ) it is well - known ( * ? ? ? 3 ) that the matrix measure induced by the @xmath23 vector norm is [ eq : muqdef ] _ 1(a)=\{c_1(a ) , , c_n(a ) } , where [ eq : ccstac ] c_j(a):=a_jj+ _ |a_ij| , i.e. , the sum of the entries in column @xmath24 of @xmath25 , with non diagonal elements replaced by their absolute values . the matrix measure induced by the @xmath26 norm is [ eq : mat_meas_inf ] _ ( a)=\{d_1(a ) , , d_n(a ) } , where [ eq : dstac ] d_j(a):=a_jj+ _ |a_ji| , i.e. , the sum of the entries in row @xmath24 of @xmath25 , with non diagonal elements replaced by their absolute values . often it is useful to work with scaled norms . let @xmath27 be some vector norm , and let @xmath28 denote its induced matrix measure . if @xmath29 is an invertible matrix , and @xmath30 is the vector norm defined by @xmath31 then the induced matrix measure is @xmath32 one important implication of contraction is _ entrainment _ to a periodic excitation . recall that @xmath33 is called _ @xmath34-periodic _ if @xmath35 for all @xmath36 and all @xmath22 . it is well - known that if is contractive and @xmath37 is @xmath34-periodic then for any @xmath38 there exists a unique periodic solution @xmath39 of , of period @xmath34 , and every trajectory converges to @xmath40 . entrainment is important in various applications ranging from biological systems @xcite to the stability of a power grid @xcite . note that for the particular case where @xmath37 is time - invariant , this implies that if @xmath10 contains an equilibrium point @xmath41 then it is unique and all trajectories converge to @xmath41 . the remainder of this paper is organized as follows . . section [ sec : main ] details sufficient conditions for their existence , and describes their implications . the proofs of all the results are detailed in section [ sec : proofs ] . we begin by defining three generalizations of . [ def : qcont ] the time - varying system is said to be : * _ contractive after a small overshoot and short transient _ ( sost ) on @xmath10 w.r.t . a norm @xmath11 if for each @xmath42 and each @xmath43 there exists @xmath44 such that @xmath45 for all @xmath46 and all @xmath14 . * _ contractive after a small overshoot _ ( so ) on @xmath10 w.r.t . a norm @xmath11 if for each @xmath42 there exists @xmath47 such that @xmath48 for all @xmath46 and all @xmath14 . * _ contractive after a short transient _ ( st ) on @xmath10 w.r.t . a norm @xmath11 if for each @xmath43 there exists @xmath49 such that @xmath50 for all @xmath46 and all @xmath14 . the definition of sost is motivated by requiring contraction at an exponential rate , but only after an ( arbitrarily small ) time @xmath51 , and with an ( arbitrarily small ) overshoot @xmath52 . however , as we will see below when the convergence rate @xmath53 may depend on @xmath54 a somewhat richer behavior may occur . the definition of so is similar to that of sost , yet now the convergence rate @xmath55 depends only on @xmath54 , and there is no time transient @xmath51 ( i.e. , @xmath56 ) . in other words , so is a uniform ( in @xmath51 ) version of sost . the third definition , st , allows the contraction to `` kick in '' only after a time transient of length @xmath51 . it is clear that every contractive system is sost , so , and st . thus , all these notions are generalizations of contraction . also , both so and st imply sost and , as we will see below , under a mild technical condition on and sost are equivalent . figure [ fig : graphbn2 ] on p. summarizes the relations between these gcss ( as well as other notions defined below ) . the motivation for these definitions stems from the fact that important applications of contraction are in proving _ asymptotic _ properties . for example , proving that an equilibrium point is globally attracting or that the state - variables entrain to a periodic excitation . these properties describe what happens as @xmath57 , and so it seems natural to generalize contraction in a way that allows initial transients in time and/or amplitude . the next simple example demonstrates a system that does not satisfy , but is a gcs . [ exa : scalarsys ] consider the _ scalar _ time - varying system [ eq : scals ] x(t)=-(t)x(t ) , with the state @xmath0 evolving on @xmath58 $ ] , and @xmath59 is a class k function ( i.e. @xmath40 is continuous and strictly increasing , with @xmath60 ) . it is straightforward to show that this system does not satisfy w.r.t . _ any _ norm ( note that the jacobian @xmath61 satisfies @xmath62 ) , yet it is st , with @xmath63 , for any given @xmath43 . the next section presents our main results . the proofs are placed in section [ sec : proofs ] . the next three subsections study the three forms of gcss defined above . just like contraction , sost implies entrainment to a periodic excitation . to show this , assume that the vector field @xmath37 in is @xmath34 periodic . pick @xmath64 . define @xmath65 by @xmath66 . in other words , @xmath67 maps @xmath68 to the solution of at time @xmath69 for the initial condition @xmath70 . then @xmath67 is continuous and maps the convex and compact set @xmath10 to itself , so by the brouwer fixed point theorem ( see , e.g. ( * ? ? ? * ch . 6 ) ) there exists @xmath71 such that @xmath72 , i.e. @xmath73 . this implies that admits a periodic solution @xmath74 with period @xmath34 . assuming that the system is also sost , pick @xmath75 . then there exists @xmath44 such that @xmath76 for all @xmath77 and all @xmath78 . taking @xmath79 implies that every solution converges to @xmath80 . in particular , there can not be two distinct periodic solutions . thus , we proved the following . [ prop : st_entrain ] suppose that the time - varying system , with state @xmath0 evolving on a compact and convex state - space @xmath81 , is sost , and that the vector field @xmath37 is @xmath34-periodic . then for any @xmath82 it admits a unique periodic solution @xmath83 with period @xmath34 , and @xmath84 converges to @xmath80 for any @xmath85 . since both so and st imply sost , proposition [ prop : st_entrain ] holds for all three forms of gcss . [ def : nc ] system is said to be _ nested contractive _ ( nc ) on @xmath10 with respect to a norm @xmath19 if there exist convex sets @xmath86 , and norms @xmath87 , where @xmath88 $ ] , such that the following conditions hold . a. @xmath89 } \omega_\zeta=\omega$ ] , and [ eq : setsinc ] _ _ 1 _ _ 2 , _ 1 _ 2 . b. for every @xmath90 there exists @xmath91 $ ] , with @xmath92 as @xmath93 , such that for every @xmath77 and every @xmath94 [ eq : enter ] x(t , t_1,a ) _ , tt_1 + , and is contractive on @xmath95 with respect to @xmath96 . c. [ item : cc ] the norms @xmath96 converge to @xmath97 as @xmath98 , i.e. , for every @xmath99 there exists @xmath100 , with @xmath101 as @xmath102 , such that @xmath103 eq . means that after an arbitrarily short time every trajectory enters and remains in a subset @xmath95 of the state space on which we have contraction with respect to @xmath96 . [ thm : qcon ] if the system is nc w.r.t . the norm @xmath19 then it is sost w.r.t . the norm @xmath19 . the next example demonstrates theorem [ thm : qcon ] . it also shows that as we change the parameters in a contractive system , it may become a gcs when it hits the `` verge '' of contraction . this is reminiscent of an asymptotically stable system that becomes marginally stable as it looses stability . [ exa : bio_smith ] consider the system @xmath104 where @xmath105 , and @xmath106 as explained in ( * ? ? ? 4 ) this may model a simple biochemical feedback control circuit for protein synthesis in the cell . the @xmath107s represent concentrations of various macro - molecules in the cell and therefore must be non - negative . it is straightforward to verify that @xmath108 implies that @xmath109 for all @xmath36 . let @xmath110 , and for @xmath111 let @xmath112 we show in section [ sec : proofs ] that if [ eq : km1 ] k-1 < then is contractive on @xmath113 w.r.t . the scaled norm @xmath114 for all @xmath111 sufficiently small . if @xmath115 then does not satisfy , w.r.t . any norm , on @xmath113 , yet it is sost on @xmath113 w.r.t . the norm @xmath116 . note that for all @xmath117 , [ eq : gderi ] g(x_n)= = g(0 ) . thus implies that the system satisfies if and only if the `` total dissipation '' @xmath40 is strictly larger than @xmath118 . using the fact that @xmath119 for all @xmath120 it is straightforward to show that the set @xmath121\times[0 , ( \alpha_1 \alpha_2)^{-1 } ] \times\dots\times [ 0 , \alpha^{-1 } ] ) \ ] ] is an invariant set of the dynamics for all @xmath122 . thus , , with @xmath123 , admits a unique equilibrium point @xmath124 and @xmath125 this property also follows from a more general result ( * ? ? ? 4.2.1 ) that is proved using the theory of irreducible cooperative dynamical systems . yet the contraction approach leads to new insights . for example , it implies that the distance between trajectories can only decrease , and can also be used to prove entrainment to suitable generalizations of that include periodically - varying inputs . [ exa : phos : relay ] consider the system @xmath126 where @xmath127 , and @xmath128 . in the context of phosphorelay @xcite , @xmath129 is the strength at time @xmath130 of the stimulus activating the hk , @xmath131 is the concentration of the phosphorylated form of the protein at the @xmath132th layer at time @xmath130 , and @xmath133 denotes the total protein concentration at that layer . note that @xmath134 is the flow of the phosphate group to an external receptor molecule . in the particular case where @xmath135 for all @xmath132 becomes the _ ribosome flow model _ ( rfm ) @xcite . this is the mean - field approximation of an important model from non - equilibrium statistical physics called the _ totally asymmetric simple exclusion process _ ( tasep ) @xcite . in the rfm , @xmath136 $ ] is the normalized occupancy at site @xmath132 , where @xmath137 [ @xmath138 means that site @xmath132 is completely free [ full ] , and @xmath139 is the capacity of the link that connects site @xmath132 to site @xmath140 . this has been used to model mrna translation , where every site corresponds to a group of codons on the mrna strand , @xmath131 is the normalized occupancy of ribosomes at site @xmath132 at time @xmath130 , @xmath129 is the initiation rate at time @xmath130 , and @xmath139 is the elongation rate from site @xmath132 to site @xmath140 . our original motivation for generalizing was to prove entrainment in the rfm @xcite . for more results on the rfm , see @xcite . assume that there exists @xmath141 such that @xmath142 for all @xmath143 . let @xmath144\times\dots\times[0,p_n]$ ] denote the state - space of . then , as shown in section [ sec : proofs ] , does not satisfy , w.r.t . any norm , on @xmath10 , yet it is sost on @xmath10 w.r.t . the @xmath23 norm . considering theorem [ thm : qcon ] in the special case where all the sets @xmath95 in definition [ def : nc ] are equal to @xmath10 yields the following result . [ coro : new ] suppose that is contractive on @xmath10 w.r.t . a set of norms @xmath96 , @xmath88 $ ] , and that condition ( [ item : cc ] ) in definition [ def : nc ] holds . then is sost on @xmath10 w.r.t . @xmath19 . corollary [ coro : new ] may be useful in cases where some matrix measure of the jacobian @xmath20 of turns out to be non positive on @xmath10 , but not strictly negative , suggesting that the system is `` on the verge '' of satisfying . the next result demonstrates this for the time - invariant system [ eq : time_in_var_sys ] = f(x ) , and the particular case of the matrix measure @xmath145 induced by the @xmath23 norm . recall that this is given by with the @xmath146s defined in . [ prop : new_meas_zero ] consider the jacobian @xmath147 of the time - invariant system . suppose that @xmath10 is compact and that the set @xmath148 can be divided into two non - empty disjoint sets @xmath149 and @xmath150 such that the following properties hold for all @xmath22 : 1 . for any @xmath151 , @xmath152 ; [ item : s0 ] 2 . for any @xmath153 , @xmath154 ; [ item : sminus ] 3 for any @xmath155 there exists an index @xmath156 such that @xmath157 . [ item : rec ] then is sost on @xmath10 w.r.t . the @xmath23 norm . the proof of proposition [ prop : new_meas_zero ] is based on the following idea . by compactness of @xmath10 , there exists @xmath158 such that [ eq : comp_ass ] c_j(j(x))<- , j s_- x. the conditions stated in the proposition imply that there exists a diagonal matrix @xmath159 such that @xmath160 for all @xmath161 . furthermore , there exists such a @xmath159 with diagonal entries _ arbitrarily close _ to @xmath162 , so @xmath163 for all @xmath153 . thus , @xmath164 . now corollary [ coro : new ] implies sost . [ exa : new_from_russo ] consider the system : @xmath165 where @xmath166 , and @xmath167 $ ] . this is a basic model for a transcriptional module that is ubiquitous in both biology and synthetic biology ( see , e.g. , @xcite ) . here @xmath168 is the concentration at time @xmath130 of a transcriptional factor @xmath169 that regulates a downstream transcriptional module by binding to a promoter with concentration @xmath170 yielding a protein - promoter complex @xmath171 with concentration @xmath172 . the binding reaction is reversible with binding and dissociation rates @xmath173 and @xmath174 , respectively . the linear degradation rate of @xmath169 is @xmath175 , and as the promoter is not subject to decay , its total concentration , @xmath176 , is conserved , so @xmath177 . the jacobian of is @xmath178 , and all the properties in prop . [ prop : new_meas_zero ] hold with @xmath179 and @xmath180 . indeed , @xmath181 for all @xmath182 . thus , is sost on @xmath10 w.r.t . the @xmath23 norm . [ exa : sec_new_from_russo ] a more general example studied in @xcite is where the transcription factor regulates several independent downstream transcriptional modules . this leads to the following model : @xmath183 where @xmath184 is the number of regulated modules . the state - space is @xmath185\times\dots\times [ 0,e_{t , n}]$ ] . the jacobian of is @xmath186 and all the properties in prop . [ prop : new_meas_zero ] hold with @xmath179 and @xmath187 . thus , this system is sost on @xmath10 w.r.t . the @xmath23 norm . arguing as in the proof of proposition [ prop : new_meas_zero ] for the matrix measure @xmath188 induced by the @xmath26 norm ( see ) yields the following result . [ prop : new_meas_zero_infty ] consider the jacobian @xmath147 of the time - invariant system . suppose that @xmath10 is compact and that the set @xmath148 can be divided into two non - empty disjoint sets @xmath149 and @xmath150 such that the following properties hold for all @xmath189 : 1 . @xmath190 for all @xmath191 ; [ item : s0_inf ] 2 . @xmath192 for all @xmath193 ; [ item : sminus_inf ] 3 . for any @xmath191 there exists an index @xmath194 such that @xmath195 . [ item : rec_inf ] then is sost on @xmath10 w.r.t . the @xmath26 norm . a natural question is under what conditions so and sost are equivalent . to address this issue , we introduce the following definition . [ eq : defntr ] we say that is _ weakly expansive _ ( we ) if for each @xmath158 there exists @xmath196 such that for all @xmath14 and all @xmath64 [ eq : sep ] | x(t , t_0,a)-x(t , t_0,b ) [ prop : sepimp ] suppose that is we . then is sost if and only if it is so . [ rem : global_lip ] suppose that @xmath37 in is lipschitz globally in @xmath10 uniformly in @xmath130 , i.e. there exists @xmath197 such that @xmath198 then by gronwall s lemma ( see , e.g. ( * ? ? ? * appendix c ) ) @xmath199 for all @xmath8 , and this implies that holds for @xmath200 . in particular , if @xmath10 is compact and @xmath37 is periodic in @xmath130 then we holds under rather weak continuity arguments on @xmath37 . for _ time - invariant _ systems whose state evolves on a convex and compact set it is possible to give a simple sufficient condition for st . let @xmath201 [ @xmath202 denote the interior [ boundary ] of a set @xmath203 . we require the following definitions . we say that is _ non expansive _ ( ne ) w.r.t . a norm @xmath19 if for all @xmath204 and all @xmath205 [ eq : exp ] |x(s_2,s_1,a)-x(s_2,s_1,b)| with @xmath206 replaced by @xmath207 . [ def : ic ] the time - invariant system with the state @xmath0 evolving on a compact and convex set @xmath208 , is said to be _ interior contractive _ ( ic ) w.r.t . a norm @xmath209 if the following properties hold : a. for every @xmath210 , [ eq : cond_a_enu ] x(t , x_0 ) , t>0 ; b. for every @xmath211 , [ eq : mucom_inv ] ( j(x))<0 , where @xmath212 is the matrix measure induced by @xmath19 . in other words , the matrix measure is negative in the interior of @xmath10 , and the boundary of @xmath10 is `` repelling '' . note that these conditions do not necessarily imply that the system satisfies on @xmath10 , as it is possible that @xmath213 for some @xmath214 . yet , does imply that is ne on @xmath10 . [ thm : time_invar ] if the system is ic w.r.t . a norm @xmath19 then it is st w.r.t . @xmath19 . the proof of this result is based on showing that ic implies that for each @xmath215 there exists @xmath216 such that @xmath217 and then using this to conclude that for any @xmath218 all the trajectories of the system are contained in a convex and compact set @xmath219 . in this set the system is contractive with rate @xmath220 . the next example , that is a variation of a system studied in @xcite , demonstrates this reasoning . [ exa : new_from_russo_inf ] consider a transcriptional factor @xmath169 that regulates a downstream transcriptional module by irreversibly binding , at a rate @xmath221 , to a promoter @xmath222 yielding a protein - promoter complex @xmath171 . the promoter is not subject to decay , so its total concentration , denoted by @xmath223 , is conserved . assume also that @xmath169 is obtained from an inactive form @xmath224 , for example through a phosphorylation reaction that is catalyzed by a kinase with abundance @xmath225 satisfying @xmath226 for all @xmath227 . the sum of the concentrations of @xmath224 , @xmath169 , and @xmath171 is constant , denoted by @xmath228 , with @xmath229 . letting @xmath230 denote the concentrations of @xmath231 at time @xmath130 yields the model @xmath232 with the state evolving on @xmath233\times[0,e_t]$ ] . let @xmath234 , and consider the matrix measure @xmath235 . a calculation yields @xmath236 so @xmath237 , and @xmath238 letting @xmath239 $ ] , we conclude that @xmath240 for all @xmath241 . for any @xmath242 , @xmath243 , and arguing as in the proof of theorem [ thm : time_invar ] ( see section [ sec : proofs ] ) , we conclude that for any @xmath43 there exists @xmath244 such that @xmath245 in other words , after time @xmath51 all the trajectories are contained in the closed and convex set @xmath246\times[0,e_t]$ ] . letting @xmath247 yields @xmath248 and @xmath249 so is st w.r.t . @xmath250 . as noted above , the introduction of gcss is motivated by the idea that contraction is used to prove asymptotic results , so allowing initial transients should increase the class of systems that can be analyzed while still allowing to prove asymptotic results . the next result demonstrates this . [ coro : attract ] if is ic with respect to some norm then it admits a unique equilibrium point @xmath251 , and @xmath252 for all @xmath77 . the proof of corollary [ coro : attract ] , given in the appendix , is based on theorem [ thm : time_invar ] . consider the _ variational system _ ( see , e.g. , @xcite ) associated with : @xmath253 an alternative proof of corollary [ coro : attract ] is possible , using the lyapunov - finsler function @xmath254 , where @xmath209 is the vector norm corresponding to the matrix measure @xmath255 in , and the lasalle invariance principle described in @xcite . since ic implies st and this implies sost , it follows from proposition [ prop : st_entrain ] that ic implies entrainment to @xmath34-periodic vector fields . the next example demonstrates this . [ exa : st_ent_inf ] consider again the system in example [ exa : new_from_russo_inf ] , and assume that the kinase abundance @xmath225 is a strictly positive and periodic function of time with period @xmath34 . since we already showed that this system is st , it admits a unique periodic solution @xmath80 , of period @xmath34 , and any trajectory of the system converges to @xmath80 . figure [ fig : ent_inf ] depicts the solution of for @xmath256 , @xmath257 , @xmath258 , @xmath259 , @xmath260 , and initial condition @xmath261 . it may be seen that both state - variables converge to a periodic solution with period @xmath262 . ( in particular , @xmath263 converges to the constant function @xmath264 that is of course periodic with period @xmath34 . ) ( solid line ) and @xmath265 ( dashed line ) of the system in example [ exa : st_ent_inf ] as a function of @xmath130 . , height=264 ] contraction can be characterized using a lyapunov - finsler function @xcite . the next result describes a similar characterization for st . for simplicity , we state this for the time - invariant system . [ prop : equ_lyap ] the following two conditions are equivalent . the time - invariant system is st w.r.t . a norm @xmath19 . [ imp_clf : a ] b. [ imp_clf : b ] for any @xmath43 there exists @xmath49 such that for any @xmath14 and any @xmath266 on the line connecting @xmath68 and @xmath267 the solution of with @xmath268 and @xmath269 satisfies [ eq : delta_cond ] |x(t+)|(-t ) |x(0)|,t0 . note that implies that the function @xmath270 is a _ generalized _ lyapunov - finsler function in the following sense . for any @xmath43 there exists @xmath49 such that along solutions of the variational system : @xmath271 in the next section , we describe several more related notions and explore the relations between them . it is straightforward to show that each of the three generalizations of contraction in definition [ def : qcont ] implies that is ne . one may perhaps expect that any of the three generalizations of contraction in definition [ def : qcont ] also implies wc . indeed , st does imply wc , because @xmath272 for all @xmath273 if st holds ( simply apply the definition with @xmath274 , @xmath275 , and @xmath276 in ) . however , the next example shows that so does not imply wc . [ exa : eps ] consider the scalar system [ eq : shift ] = -2x , & 0|x| < 1/2 , + - , & |x|1 , with @xmath0 evolving on @xmath58 $ ] . clearly , this system is not wc . however , it is not difficult to show that it satisfies the definition of so with @xmath277 . the next result presents two conditions that are equivalent to sost . [ lem : eqdef ] the following conditions are equivalent . 1 . [ item1 ] system is sost on @xmath10 w.r.t . some vector norm @xmath278 . [ item2 ] for each @xmath279 there exists @xmath280 such that @xmath281 for all @xmath46 and all @xmath14 . [ item3 ] for each @xmath42 and each @xmath43 there exists @xmath282 such that [ eq : qcontnew ] |x(t , t_1,a)-x(t , t_1,b)|_v ( 1 + ) ( - ( t -t_1 ) _ 1 ) |a - b|_v , for all @xmath283 and all @xmath14 . [ fig : graphbn2 ] summarizes the relations between the various contraction notions . _ proof of theorem [ thm : qcon ] . _ fix arbitrary @xmath111 and @xmath38 . the function @xmath284 $ ] is as in the statement of the theorem . for each @xmath90 , let @xmath285 be a contraction constant on @xmath95 , where we write @xmath286 here and in what follows . pick @xmath14 and @xmath43 . by , @xmath287 for all @xmath288 , so @xmath289 in particular , @xmath290 from the convergence property of norms in the theorem statement , there exist @xmath291 such that @xmath292 and @xmath293 , @xmath294 as @xmath295 . combining this with yields @xmath296 note that taking @xmath297 yields @xmath298 now for @xmath288 let @xmath299 . then @xmath300 where the last inequality follows from . since @xmath293 , @xmath294 as @xmath295 , @xmath301 for @xmath43 small enough . summarizing , there exists @xmath302 such that for all @xmath303 $ ] @xmath304 for all @xmath305 and all @xmath306 . now pick @xmath307 . for any @xmath143 , let @xmath308 . then @xmath309 and this completes the proof . @xmath310 , so @xmath311 thus , @xmath312 suppose that @xmath313 . then for all @xmath117 , @xmath314 combining this with implies that there exists a sufficiently small @xmath111 such that @xmath315 for all @xmath316 , so the system is contractive on @xmath317 w.r.t . @xmath114 . we now use theorem [ thm : qcon ] to prove that is sost . since @xmath323 and @xmath324 , @xmath325 for @xmath326 $ ] , let @xmath327 it is straightforward to verify that satisfies condition ( br ) in ( * ? ? ? * lemma 1 ) , and this implies that for every @xmath43 there exists @xmath328 such that @xmath329 for all @xmath218 . then @xmath330 we already showed that this implies that there exists a @xmath99 and a norm @xmath331 such that is contractive on @xmath332 w.r.t . this norm . summarizing , all the conditions in theorem [ thm : qcon ] hold , and we conclude that is sost on @xmath113 w.r.t . @xmath116 . @xmath310 _ analysis of the system in example [ exa : phos : relay ] . _ for @xmath85 , let @xmath333 denote the solution of at time @xmath143 for the initial condition @xmath334 . pick @xmath43 . satisfies condition ( br ) in ( * ? ? ? * lemma 1 ) , and this implies that there exists @xmath335 such that for all @xmath85 , all @xmath336 , and all @xmath337 @xmath338 furthermore , if we define @xmath339 , @xmath340 , then the @xmath341 system also satisfies condition ( br ) in ( * ? ? ? * lemma 1 ) , and this implies that there exists @xmath342 such that for all @xmath85 , all @xmath336 , and all @xmath343 @xmath344 we conclude that after an arbitrarily short time @xmath345 every state - variable @xmath131 , @xmath346 , is separated from @xmath347 and from @xmath133 . this means the following . for @xmath348 $ ] , let @xmath349 note that @xmath350 , and that @xmath351 is a strict subcube of @xmath10 for all @xmath326 $ ] . then for any @xmath94 , and any @xmath43 there exists @xmath352 , with @xmath353 as @xmath354 , such that [ eq : zinomeps ] x(t , t_1,a ) _ , tt_1+a . the jacobian of satisfies @xmath355 , where @xmath356 note that @xmath357 is metzler , tridiagonal , and has zero sum columns for all @xmath358 . note also that for any @xmath359 every entry @xmath360 on the sub- and super - diagonal of @xmath361 satisfies @xmath362 , with @xmath363 . note also that there exist @xmath364 such that @xmath365 is singular ( e.g. , when @xmath366 and @xmath367 the second column of @xmath20 is all zeros ) , and this implies that the system does not satisfy on @xmath10 w.r.t . any norm . by ( * ? ? ? * theorem 4 ) , for any @xmath368 $ ] there exists @xmath369 , and a diagonal matrix @xmath370 , with @xmath371 , such that is contractive on @xmath95 w.r.t . the the scaled @xmath23 norm defined by @xmath372 . furthermore , we can choose @xmath54 such that @xmath373 as @xmath98 , and @xmath374 as @xmath375 . summarizing , all the conditions in definition [ def : nc ] hold , so is nc on @xmath10 and applying theorem [ thm : qcon ] concludes the analysis . @xmath310 _ proof of proposition [ prop : new_meas_zero ] . _ without loss of generality , assume that @xmath376 , with @xmath377 , so that @xmath378 . fix @xmath379 . let @xmath380 with the @xmath381s defined as follows . for every @xmath155 , @xmath382 and @xmath383 . all the other @xmath381s are one . let @xmath384 . then @xmath385 . we now calculate @xmath386 . fix @xmath387 . then @xmath388 , so @xmath389 where the inequality follows from the fact that @xmath390 for all @xmath391 , and for the specific value @xmath392 we have @xmath393 and @xmath394 . we conclude that for every @xmath191 , @xmath395 . it follows from property [ item : sminus ] ) in the statement of proposition [ prop : new_meas_zero ] and the compactness of @xmath10 that there exists @xmath158 such that @xmath396 for all @xmath153 and all @xmath397 , so for @xmath111 sufficiently small we have @xmath398 for all @xmath153 and all @xmath397 . we conclude that for all @xmath111 sufficiently small , @xmath399 , i.e. the system is contractive w.r.t . @xmath400 . clearly , @xmath401 as @xmath402 , and applying corollary [ coro : new ] completes the proof . _ proof of proposition [ prop : sepimp ] . _ suppose that is sost w.r.t . some norm @xmath403 . pick @xmath111 . since the system is we , there exists @xmath404 such that @xmath405 for all @xmath406 $ ] . letting @xmath407 yields [ eq : nnsep ] | x(t , t_0,a)-x(t , t_0,b ) |_v ( 1 + ) ( - ( t -t_0 ) _ 2 ) _ proof of lemma [ lem : strict ] . _ pick @xmath43 and @xmath415 . since @xmath10 is an invariant set , @xmath416 is also an invariant set ( see , e.g. , ( * ? ? ? * lemma iii.6 ) ) , so implies that @xmath417 for all @xmath418 . since @xmath419 is compact , @xmath420 . thus , there exists a neighborhood @xmath421 of @xmath422 , such that @xmath423 for all @xmath424 . cover @xmath10 by such @xmath421 sets . by compactness of @xmath10 , we can pick a finite subcover . pick smallest @xmath41 in this subcover , and denote this by @xmath425 . then @xmath426 and we have that @xmath427 for all @xmath9 . now , pick @xmath428 . let @xmath429 . then @xmath430 and this completes the proof of lemma [ lem : strict ] . @xmath310 we can now prove theorem [ thm : time_invar ] . we recall some definitions from the theory of convex sets . let @xmath431 denote the closed ball of radius @xmath432 around @xmath0 ( in the euclidean norm ) . let @xmath433 be a compact and convex set with @xmath434 . let @xmath435 denote the _ inradius _ of @xmath391 , i.e. the radius of the largest ball contained in @xmath436 . for @xmath437 $ ] the _ inner parallel set of @xmath433 at distance @xmath438 _ is @xmath439 note that @xmath440 is a compact and convex set ; in fact , @xmath440 is the intersection of all the translated support hyperplanes of @xmath433 , with each hyperplane translated `` inwards '' through a distance @xmath438 ( see ( * ? ? ? * section 17 ) ) . assume , without loss of generality , that @xmath441 . pick @xmath43 . let @xmath442 . by lemma [ lem : strict ] , @xmath443 and @xmath444 for all @xmath445 . let @xmath446 . then @xmath447 . pick @xmath448 . we claim that @xmath449 . to show this , assume that there exists @xmath450 such that @xmath451 . then there is a point @xmath452 on the line connecting @xmath453 and @xmath454 such that @xmath455 . therefore , @xmath456 and this is a contradiction as @xmath457 . we conclude that @xmath458 . let @xmath459 . then implies that @xmath460 . thus , the system is contractive on @xmath461 , and for all @xmath14 and all @xmath227 @xmath462 where @xmath97 is the vector norm corresponding to the matrix measure @xmath463 . this establishes st , and thus completes the proof of theorem [ thm : time_invar ] . @xmath310 _ proof of corollary [ coro : attract ] . _ since @xmath10 is convex , compact , and invariant , it includes an equilibrium point @xmath41 of . clearly , @xmath464 . by theorem [ thm : time_invar ] , the system is st . pick @xmath465 and @xmath43 , and let @xmath49 . applying with @xmath466 yields @xmath467 for all @xmath468 . taking @xmath469 completes the proof . @xmath310 another possible proof of corollary [ coro : attract ] is based on defining @xmath470 by @xmath471 . then for any @xmath85 , @xmath472 is nondecreasing , and the lasalle invariance principle tells us that @xmath473 converges to an invariant subset of the set @xmath474 , for some @xmath475 . if @xmath476 then we are done . otherwise , pick @xmath341 in the omega limit set of the trajectory . then @xmath477 , so implies that @xmath478 is strictly decreasing . this contradiction completes the proof . _ proof of proposition [ prop : equ_lyap ] . _ pick @xmath204 . let @xmath479\to \omega$ ] be the line @xmath480 . note that since @xmath10 is convex , @xmath481 for all @xmath482 $ ] . let @xmath483 this measures the sensitivity of the solution at time @xmath130 to a change in the initial condition along the line @xmath80 . note that @xmath484 , and @xmath485 comparing this to implies that @xmath486 is equal to the second component , @xmath487 , of the solution of the variational system with initial condition @xmath488 suppose that the time - invariant system is st . pick @xmath43 . let @xmath49 . then for any @xmath489 and any @xmath490 $ ] , @xmath491 dividing both sides of this inequality by @xmath54 and taking @xmath492 implies that [ eq : condwexp ] |w(t + , r ) | ( - t ) |b - a| , so @xmath493 this proves the implication @xmath494 . to prove the converse implication , assume that holds . then holds and thus @xmath495 so the system is st . _ proof of lemma [ lem : eqdef ] . _ if is sost then holds for the particular case @xmath496 in definition [ def : qcont ] . to prove the converse implication , assume that holds . pick @xmath497 . let [ eq : deftaumin ] : = \ { , } , and let @xmath49 . pick @xmath498 , and let @xmath499 . then @xmath500 where the last inequality follows from . thus , @xmath501 and recalling that @xmath497 were arbitrary , we conclude that condition 2 ) in lemma [ lem : eqdef ] implies sost . to prove that condition 3 ) is equivalent to sost , suppose that holds . then for any @xmath502 , @xmath503 so we have sost . conversely , suppose that is sost . pick any @xmath75 . then there exists @xmath504 such that for any @xmath505 @xmath506 thus , for any @xmath507 @xmath508 taking @xmath12 sufficiently small such that @xmath509 implies that holds for @xmath510 . this completes the proof that is equivalent to sost . @xmath310 z. aminzare and e. d. sontag , `` contraction methods for nonlinear systems : a brief introduction and some open problems , '' in _ proc . 53rd ieee conf . on decision and control _ , los angeles , ca , 2014 , pp . 38353847 . s. bonnabel , a. astolfi , and r. sepulchre , `` contraction and observer design on cones , '' in _ proc . 50th ieee conf . on decision and control and european control conference _ , orlando , florida , 2011 , pp . 71477151 . h. l. smith , _ monotone dynamical systems : an introduction to the theory of competitive and cooperative systems _ , ser . mathematical surveys and monographs.1em plus 0.5em minus 0.4emprovidence , ri : amer . soc . , 1995 , vol . 41 . y. zarai , m. margaliot , and t. tuller , `` explicit expression for the steady state translation rate in the infinite - dimensional homogeneous ribosome flow model , '' _ ieee / acm trans . computational biology and bioinformatics _ 10 , pp . 13221328 , 2013 . g. poker , y. zarai , m. margaliot , and t. tuller , `` maximizing protein translation rate in the nonhomogeneous ribosome flow model : a convex optimization approach , '' _ j. royal society interface _ , vol . 11 , no . 100 , p. 20140713

according to greek mythology the goddess of wisdom , pallas athene , emerged clad in full armor after hephaestus split open zeus s head . in much the same way the local group sprang forth suddenly , and almost complete , in chapter vi of the realm of the nebulae ( hubble 1936 , pp . 124 - 151 ) . hubble describes the local group as `` a typical small group of nebulae which is isolated in the general field '' . he assigned ( in order of decreasing luminosity ) m31 , the galaxy , m33 , the large magellanic cloud , the small magellanic cloud , m32 , ngc205 , ngc 6822 , ngc 185 , ic 1613 and ngc 147 to the local group , and regarded ic 10 as a possible member . in the 2/3 century since hubble s work , the number of known local group members has increased from 12 to 36 ( see table 1 ) by the addition of almost two dozen low - luminosity galaxies . recent detailed discussions of individual local group galaxies are given in mateo ( 1998 ) , grebel ( 2000 ) and van den bergh ( 2000a ) . hubble ( 1936 , p. 128 ) pointed out that investigations of the local group were important for two reasons : [ 1 ] `` [ t]he members have been studied individually , as the nearest and most accessible examples of their particular types , in order to determine the[ir ] internal structures and stellar contents '' . in the second place [ 2 ] , `` the [ g]roup may be examined as a sample collection of nebulae , from which criteria can be derived for further exploration '' . small galaxy groups , like the local group , are quite common . from inspection of the prints of the _ palomar sky survey _ van den bergh ( 2002a ) has estimated that 16% of nearby galaxies are located in such small groups . hubble ( 1936 , p. 128 ) emphasized that `` the groups [ such as the local group ] are aggregations drawn from the general field , and are not additional colonies superposed on the field '' . from its observed radial velocity dispersion of 61 @xmath0 8 km @xmath1 the local group is found to have a virial mass of ( 2.3 @xmath0 0.6 ) x @xmath2 m@xmath3 ( courteau & van den bergh 1999 ) . the zero - velocity surface of the local group has a radius of 1.18 @xmath0 0.15 mpc , but @xmath480% of the local group members are actually situated within 0.4 mpc of the barycenter of the local group , which is located between m31 and the galaxy . from its virial mass , and the integrated luminosity of group members of 4.2 x 10 @xmath5 l@xmath3 , the dynamical mass - to - light ratio of the local group is found to be @xmath6 = 44 @xmath0 12 in solar units . such a high m / l value is an order of magnitude larger than the mass - to - light ratio in the solar neighborhood of the galaxy . this supports the conclusion by kahn & woltjer ( 1959 ) that the mass of the local group is dominated by invisible matter . the local group is situated in the outer reaches of the virgo supercluster . the nearest neighbor of the local group is the small antlia group ( van den bergh 1999a ) . this tiny cluster is located at a distance of only 1.7 mpc from the barycenter of the local group , i.e. well beyond the zero - velocity surface of the group . the antlia group has a mean radial velocity of + 114 @xmath0 12 km s@xmath7 . the number of galaxies brighter than m@xmath8 = -11.0 in the local group is 22 , compared to only four such objects in the antlia group . however , because the antlia group contains no supergiant galaxies like m31 and the milky way system , its integrated luminosity is @xmath4150 times smaller than that of the local group . since the local group is a relatively small cluster , this result suggests that clusters of galaxies have a range in luminosities ( and masses ? ) that extends over at least four orders of magnitude . the centaurus group ( van den bergh 2000b ) , at a distance of @xmath43.9 mpc , which contains m 83 ( ngc 5236 ) and centaurus a ( ngc 5128 ) , is the nearest massive cluster . if one assumes ngc 5128 and ngc 5236 to be members of a single cluster one obtains a total virial mass of 1.4 x 10@xmath9 m@xmath3 and a zero - velocity radius of 2.3 mpc for the centaurus group . in other words the zero - velocity surfaces of the centaurus group and the local group would almost touch each other . however , karachentsev et al . ( 2002 ) conclude that the galaxies surrounding ngc5128 and m83 , respectively , actually form dynamically distinct clusterings . if that is indeed the case then the total mass of these clusters is reduced to only 3 x 10@xmath10 m@xmath3 , and the radius of the zero velocity surface around cen a is less than 1.3 mpc . a second massive nearby cluster contains ic342 and the highly obscured elliptical maffei 1 . mccall ( 1989 ) concluded that `` it is likely that ic342 and maffei 1 had a significant impact on the past dynamical evolution of the major members of the local group '' . more recently fingerhut et al . ( 2003ab ) have , however , found that the galactic absorption in front of maffei 1 is lower than was previously believed . as a result the distances to maffei 1 , and its companions , are too large for these objects to have had significant dynamical inetractions with individual local group members since the big bang . if the peculiar velocities of galaxies are induced by gravitational interactions , then one might have expect massive field galaxies to have a lower velocity dispersion than dwarfs . data on the nearby field ( whiting 2003 ) do not appear to support this expectation . whiting finds the mean radial velocity dispersion among field galaxies within 10 mpc to be 113 km s@xmath7 . a much lower dispersion of @xmath430 km s@xmath7 for the local hubble flow has , however , been found by karachentsev et al . it is noted in passing that the former value appears to be significantly larger than the radial velocity dispersion of 61 @xmath0 8 km @xmath1 that courteau & van den bergh ( 1999 ) found within the local group itself . to first approximation the local group is a binary system with massive clumps of galaxies centered on m31 and on the galaxy . van den bergh ( 2000 , p. 290 ) estimates a mass m(a ) = ( 1.15 - 1.5 ) x 10@xmath10 m@xmath3 for the andromeda subgroup of the local group , compared to m(g ) = ( 0.46 - 1.25 ) x 10@xmath10 m@xmath3 for the galactic subgroup . more recently sakamoto , chiba & beers ( 2003 ) have given somewhat higher mass estimates . if leo i is included they find m(g ) = ( 1.5 - 3.0 ) x @xmath2 m@xmath3 , compared to m(g ) = ( 1.1 - 2.2 ) x 10@xmath10 m@xmath3 if it is assumed that leo i is not a member of the galactic subgroup . recent proper motion observations by piatek et al . ( 2002 ) suggest that the fornax dwarf spheroidal galaxy may not , as had previously been thought , be a distant satellite of the galaxy . instead the data appear to indicate that fornax is a free - floating member of the local group that is presently near perigalacticon . within each of the two main local group subclusters there are additional subclumps such as the m31 + m32 + ngc205 triplet , the ngc147 + ngc185 binary , and the lmc + smc binary . for the entire local group one finds , at the 99% confidence level , that low - luminosity early - type dsph galaxies are more concentrated in subclumps than are late - type dir galaxies . in other words most dir galaxies appear to be free - floating members of the local group , whereas the majority ( but not all ) dsph galaxies seem to be directly associated with either m31 or the galaxy . it is not yet clear if the mean ages of the stellar populations in dsph galaxies are themselves a function of location . van den bergh ( 1995 ) has tentatively suggested that there is some evidence to suggest that star formation in dsph galaxies in the dense regions close to m31 and the galaxy might typically have started earlier than star formation in remote dsph galaxies . the data that are shown in table 2 show a strong correlation between the morphological types of faint galaxies with @xmath11 ( i.e. objects fainter than m32 ) and their environment . this dependence was first noticed by einasto et al . almost all sph + dsph galaxies are seen to be associated with the two dense subclusters within the local group , whereas most ir galaxies appear to be more - or - less isolated group members . it seems quite possible possible ( cf . skillman et al . 2003 ) that the faint dir and dsph galaxies have similar progenitors , and that the differences between them that are observed now are due to environmental factors that favored gas loss from those dwarfs that occurred in dense environments , i.e. near giant galaxies . within the local group the m31 , m32 , m33 subgroup has a total luminosity l@xmath8 = 3.0 x 10@xmath5 l@xmath3 , which is significantly greater than that of the subgroup centered on the milky way system which has l @xmath8 = 1.1 x @xmath12 l@xmath3 . these luminosities of the m31 and galactic subgroups account for 71% and 24% of the total local group luminosity , respectively . it should , however , be noted that some uncertainty in the luminosity ratio of m31 , to that of the galaxy , is introduced by the fact that both of these systems are viewed edge - on . as a result the internal absorption corrections ( which may be quite large ) are uncertain . nevertheless , the notion that m31 is more massive than the galaxy receives some support from the observation that m31 appears to have 450 @xmath0 100 globular clusters , compared to only 180 @xmath0 20 such clusters associated with the galaxy ( barmby et al . 2000 ) . furthermore ( freeman 1999 ) the bulge mass of m31 is 3.6 x 10@xmath5 m@xmath3 , which is almost twice as large as the 2 x 10@xmath5 m@xmath3 mass of the galactic bulge . on the basis of these results one might have expected the total mass of the m31 subgroup of the local group to be two or three times larger than that of the galaxy subgroup . surprisingly this does not appear to be the case evans & wilkinson ( 2000 ) . using radial velocity observations evans et al . ( 2000 ) conclude that `` there is no dynamical evidence for the widely held belief that m31 is more massive - it may even be less massive '' . more recently gottesman et al . ( 2002 ) have also concluded from dynamical arguments that the mass of m31 `` is unlikely to be as great as that of our own milky way '' . these authors even make the heretical suggestion that m31 might not have a massive halo at all ! either the well - known perfidity of small - number statistics has mislead us about the relative masses of m31 and the galaxy , or the mass - to - light ratio in the milky way system is much higher than that of the andromeda galaxy . if the latter conclusion is correct then one would have to accept the existence of significant galaxy - to - galaxy variations in the ratio of visible to dark matter among giant spirals . it has been known for many years ( van den bergh 1969 ) that the halo of m31 contains some globular clusters that are much more metal rich than those that occur in the galactic halo . perhaps the best know example of such an object is the luminous globular mayall ii . furthermore the color - magnitude diagrams for individual m31 halo stars ( mould & kristian 1986 , pritchet & van den bergh 1988 , durrell , harris & pritchet 1994 ) all show that ( 1 ) halo stars have a wide range in metallicity , and ( 2 ) the mean metallicity of stars in the halo of m31 is surprisingly high . the mean values of [ fe / h ] for m31 halo stars that were obtained by mould & kristian , by pritchet & van den bergh and by durrell , harris & pritchet are @xmath13>$ ] = -0.6 , -1.0 and -0.6 , respectively . [ it is noted in passing that the halo of m31 does contain a metal - poor component which includes clusters such as mayall iv , some rr lyrae variables , and non - variable horizontal - brach stars ( sarajedini & van duyne 2001 ) . the observation that the stars in the halo of m31 appear , on average , to be so much more metal - rich than than those in the galactic halo ( durrell , harris & pritchet 1994 ) suggests that these two giant spiral galaxies had quite different evolutionary histories . the higher metallicity of m31 halo stars indicates that the building blocks of the andromeda halo had much higher masses than those of the galactic halo . simulations of murali et al . ( 2002 ) show that a significant fraction of the mass that was originally in such merging ancestral galaxies will end up as intergalactic debris , and presumably also in an extended halo of the final merged object . an independent check on the metallicities of m31 giants is provided by the recent spectroscopic observations undertaken by reitzel & guhathakurta ( 2002 ) . these authors find that their spectra of stars in the halo of m31 have a mean metallicity @xmath13>~$]= -1.3 . this value is significantly lower that that derived from photometric observations of stars in the halo of the andromeda galaxy . a possible explanation for this difference is that insufficient correction was made for the fact that old metal - rich red giants are fainter ( and therefore more difficult to observe ) than are old metal - poor giants . non homogeneity of the andromeda halo might also have contributed to the observed difference in the mean metallicities derived from photometry and from spectroscopy . for example , reitzel & guhathakurta find four stars of solar metallicity in the halo of m31 . they suggest that these objects might represent metal - rich debris from an accretion event . other evidence for such accretion events is provided by ibata et al . ( 2001 ) and ferguson et al . recently yanny et al . ( 2003 ) have also found possible evidence for such a tidal stream in the galaxy . this stream is located beyond the plane of the milky way at a distance of @xmath420 kpc from the galactic center . an interesting clue regarding the origin of the difference between the andromeda and the galactic halos is provided by the observations of pritchet & van den bergh ( 1994 ) ] [ see figure 1 ] which show that m31 has an r@xmath14 profile out to @xmath4 20 kpc from its nucleus . such a structure is likely to have resulted from violent relaxation . this suggests that the overall morphology of the andromeda galaxy was determined by violent relaxation resulting from the early merger of two ( or more ) massive metal - rich ancestral objects . on the other hand the main body of the milky way system may have been assembled la eggen , lynden - bell & sandage ( 1964 ) ] , with its metal - poor halo forming la searle & zinn ( 1978 ) via late infall and capture of `` small bits and pieces '' . however , it appears that these fragments differed in a rather fundamental way from those that produced dwarf spheroidal galaxies . shetrone , ct , & sargent ( 2001 ) find that dsph galaxies are iron - rich and have 0.02 @xmath15 [ @xmath16 /fe ] @xmath15 0.13 , compared to typical galactic values of [ @xmath16/fe ] @xmath4 0.28 dex over the same range of metallicities . this shows that the bulk of the galactic halo stars can not have formed in dwarf spheroidal galaxies that subsequently disintegrated . in particular fulbright ( 2002 ) finds that less than 10% of local metal - poor stars with @xmath17 < -1.2 $ ] have alpha - to - iron abundance ratios similar to those found in dsph galaxies . more generally tolstoy et al . ( 2003 ) conclude that the observed element abundance patterns make it difficult to form a significant proportion of the stars observed in our galaxy in small galaxies that subsequently merged to form the disk , bulge , and inner halo of the milky way . bekki , harris & harris ( 2003 ) have studied the distribution of stars of various metallicities after the merger of two spirals . however , their model is not likely to be applicable to the early merger of the ancestral objects of m31 . the reason for this is that extended disks would be destroyed by frequent tidal interactions at large look - back times . _ hubble space telescope _ images show that galaxies with obvious disks mostly have @xmath18 , whereas the vast majority of the objects seen at @xmath19 appear to have either compact or chaotic morphologies ( van den bergh 2002b ) . so the bekki et al . model , in which extended disks merge , is probably inappropriate for galaxies at @xmath19 , i.e. for mergers that took place more than 9 gyr ago . in their pioneering study of the metallicities of individual stars in galactic halos mould & kristian ( 1986 ) also observed stars in the halo of the late - type spiral m33 . their color - magnitude diagram suggested that the halo of m33 was very metal - poor and had @xmath13>~= -2.2 $ ] . this value is more than an order of magnitude lower than that for stars in the halo of the andromeda galaxy . the discussion given above may be summarised by saying that m31 may have formed from the early merger of the two or three most massive galaxies in the core of the andromeda subgroup of the local group . the less - massive andromeda companions , such as m32 and ngc205 , may represent objects in the core of the andromeda subgroup which had such low masses that they managed to survive individually . due to differences in evolutionary history the halo of m31 contains relatively metal - rich globular clusters , whereas the galactic halo does not . another difference between the m31 and galactic globular cluster systems has been noted by rich et al . ( 2002 ) who find that m31 does not appear to contain globular clusters with extremely blue horizontal branches such as m92 in the galactic halo . an additional example of major differences between globular cluster systems is provided by m33 and the lmc . these two late - type disk galaxies have comparable luminosities ( m33 m@xmath8 = -18.9 , lmc m@xmath8 = -18.5 ) but radically different globular cluster systems . surprisingly the lmc globulars , which are both very old and quite metal - poor , appear to have disk kinematics ( schommer et al . 1993 ) . on the other hand the metal - poor globular clusters associated with m33 ( schommer et al . 1991 ) seem to have halo - like kinematics . sarajedini et al . ( 1998 found that eight out of 10 globular clusters in their m33 halo sample had significantly redder horizontal branches than galactic globulars of similar metallicity . this difference might be interpreted as a second parameter effect . alternatively , and perhaps more plausibly , the m33 globulars may be a few gyr younger than their galactic counterparts . on the latter interpretation the m33 halo globular clusters exhibit an unexpectedly large age dispersion of @xmath43 - 5 gyr . it is presently a mystery why the m33 halo globular clusters would have formed a few gyr later than typical galactic halo and lmc disk globulars . some light might eventually be shed on these questions by observations of the radial velocities of rr lyrae stars in the lmc , and perhaps also in the near future , of rr lyrae stars in m33 . the main differences between the globular cluster systems of m33 ( sc ) and the galaxy ( sbc ) could perhaps be understood ( van den bergh 2002b ) by assuming that late - type galaxies take significantly longer to arrive at their final morphology than do spirals of earlier morphological types . however , the great age of the lmc cluster system appears to conflict with this simple explanation . the observations of mould & kristian ( 1986 ) ] appear to show that the field stars in the halo of m33 are extremely metal - poor and have @xmath13>~= -2.2 $ ] . it would be important to confirm this result by new photometry and to compare this value with the mean metallicity of stars in the outer halo of the large magelanic cloud . not unexpectedly the majority of local group dwarf galaxies are surrounded by small families of metal - poor globular clusters . however , it is puzzling that the sagittarius dwarf has one globular cluster companion ( terzan 7 , [ fe / h ] = -0.36 ) that is quite metal - rich . how could such a relatively high metallicity have been built up within a dwarf galaxy ? the only other galactic halo ( r@xmath20 10 kpc ) globular clusters that are known to have metallicities higher than [ fe / h ] = -1.0 are pal 1 and pal 12 . the latter object is itself suspected of also being associated with the disintegrating sagittarius dwarf ( irwin 1999 ) . this speculation is supported by the observations of martnez - delgado et al . ( 2002 ) who have found that pal 12 is possibly imbedded in tidal debris of the sagittarius dwarf . rosenberg et al . ( 1998ab ) have also shown that the cluster pal 1 is significantly younger than most other galactic globular clusters . it is presently not clear which kind of evolutionary scenarios would allow halo clusters like pal 1 and pal 12 to attain such relatively high metallicities . from its present luminosity and morphological type the dwarf elliptical galaxy m32 would have been expected to be embedded in a swarm of 10 - 20 globular clusters . it is therefore puzzling that not a single globular cluster appears to be associated with this galaxy . perhaps some of the innermost m32 clusters were dragged into its compact luminous nucleus by dynamical friction . also loosely bound outer globulars , that were originally associated with m32 , might have been detached by tidal interactions with the main body of m31 . such detached m32 clusters would remain in the halo of m31 and might be recognized by being unusually compact . it would be very worthwhile to undertake a systematic search for such m32 clusters with small r@xmath21 values in the halo of m31 . for the vast majority of galaxies the specific globular cluster frequency s ( harris & van den bergh 1981 ) is less than 10 . however , the fornax dwarf , which has 5 globulars associated with it , has @xmath22 . recent work by kleyna et al . ( 2003 ) appears to indicate that the ursa minor dwarf may have an even higher s value . if a dynamically cold clustering of stars that these authors find in umi is a globular cluster ( or a disintegrated cluster ) then s @xmath4400 for the umi system . taken at face value this result suggests , that the fraction of the light of dwarf spheroidals , that is in the form of globular clusters , may be much higher in dwarf spheroidal galaxies than it is in more luminour ( massive ) systems . if the milky way system had collapsed in the fashion advocated by eggen , lynden - bell & sandage ( 1962 ) , then one would have expected the stars and globular clusters in the galactic halo to exhibit a radial metallicity gradient . on the other hand the halo of the milky way system would not be expected to have such a metallicity gradient if , as envisioned by searle & zinn ( 1978 ) , it had formed by the accretion of many `` bits and pieces '' . using the 1999 version of the globular cluster catalog of harris ( 1996 ) , van den bergh ( 2003 ) , found a possible hint for the existence of such a radial metallicity gradient among galactic halo ( r@xmath2310 kpc ) globular clusters . however , the reality of this a gradient is not supported by the more recent data contained in the 2003 version of harris s catalog . on the other hand the clusters in the main body of the galaxy , i.e. those with r@xmath2410 kpc , do appear to show a radial abundance gradient . globulars at r@xmath25 kpc are , on average , found to be more metal - rich than those having 4.0 @xmath15 r@xmath26 kpc . a kolmogorov - smirnov test shows that there is only a 4% probability that the metallicities of these inner and outer globular clusters samples were drawn from the same parent population . taken at face value the existence of a galactic metallicity gradient between 4 kpc and 10 kpc favors the suggestion that the els model provides an adequate description of the formation of the main body of the galactic halo , whereas the sz model predictions agree with the observed lack of a metallicity gradient in the region with r@xmath2310 kpc . it is a curious ( and unexplained ) fact that the distribution of flattening values of globular clusters differs significantly from galaxy to galaxy . both open and globular clusters in the lmc are , for example , on average more flattened than their galactic counterparts . furthermore ( van den bergh 1983 ) the luminous clusters in the large cloud are typically more flattened than are the less luminous ones . finally it is of interest to note that the most luminous globular in many local group galaxies also seem to be among the most flattened . examples are the globular mayall ii ( @xmath27 = 0.22 ) in m31 , @xmath28 centauri ( @xmath27 = 0.19 ) in the galaxy , and ngc1835 ( @xmath27 = 0.21 ) in the lmc . the presently known members of the local group exhibit a flat luminosity function with slope @xmath16 = -1.1 @xmath0 0.1 ( pritchet & van den bergh 1999 ) . this value is significantly lower than the slope @xmath16 = -1.8 that is predicted by the cold cold dark matter theory ( klypin et al . the low frequency of faint local group dwarfs has been confirmed in a recent hunt for new faint local group members by whiting , hau & irwin ( 2002 ) . this observed deficiency of faint lg members suggests that many of the progenitors of low mass galaxies were destroyed before they had a chance to form significant numbers of stars . alternatively it might be assumed that the missing faint galaxies can be identified with compact high - velocity clouds . however , maloney & putman ( 2003 ) have recently shown that such objects , if they were located at distances of @xmath41 mpc , would be largely ionized . these authors therefore conclude that the compact high - velocity clouds are not at cosmologically significant distances , but that they are instead associated with the galactic halo . all attempts to search for evidence of star formation in compact high - velocity clouds ( e.g. simon & blitz 2002 ) have so far remained unsuccessful . it is therefore the deficiency of faint local group members is real . figure 2 appears to show ( van den bergh 2000a , p. 281 ) that the luminosity function of dsph galaxies in the local group is steeper than that for dir galaxies . this conclusion should , however , be regarded as provisional because a kolmogorov - smirnov test shows that the difference between the luminosity distributions of local group dsph and dir galaxies is only significant at the 75% confidence level . if the luminosity function of dsph galaxies is , indeed , steeper than that for dir galaxies , then future discoveries are most likely to turn up very faint dsph ( rather than dir ) members of the local group . it would clearly be very important to undertake sky surveys in two ( or more ) colors to search for the signatures of the color - magnitude diagrams of extremely faint ( and so far undiscovered ) resolved dwarf members of the local group . three distinct explanations might be invoked to account for the apparent excess of faint dsph galaxies among presently known local group members : ( 1 ) perhaps gas was more likely to escape from faint ( low - mass ) galaxies than from more massive objects . as a result low - mass galaxies would most often end up as gas free dsph galaxies . alternatively ( 2 ) the mass spectrum with which galaxies form might depend on environmental density in such a way that high density regions ( i.e. the neighborhood of m31 and the galaxy ) form a larger fraction of low mass objects ( the majority of which end up as dsph galaxies ) . the latter assumption would be consistent with the work of trentham et al . ( 2001 ) and trentham & tully ( 2003 ) who found that the galaxian luminosity function of the dense virgo cluster is much steeper than that for less dense clusters such as the ursa major cluster and the local group . on the other hand the view that dense regions produce galaxies with steep galaxian luminosity functions appears to conflict with the result of sabatini et al . ( 2003 ) , who find that the virgo cluster luminosity function seems to be steeper in the low density outer regions of the virgo cluster than it is in the high density core of this cluster . this observation might , however , be accounted for by assuming that tides preferentially destroy fragile dwarfs in the cores of dense clusters . finally , ( 3 ) and perhaps most plausibly , the gas in the progenitors of the missing dwarfs might have been photoevaporated during reionization . it is difficult to tease out information on the morphological evolution of galaxies from the distribution of stars of various ages . the central bulges of giant spirals , such as m31 and the galaxy , are dominated by old stars . this supports the notion that these objects were built up inside out , with their bulges forming first and the disk possibly being accreted at a later time . it would be very important to establish how old the first ( presumably quite metal - poor ) generation of galactic disk stars is . this problem is made more intractable because such thin disk stars have to be disentangled from stars that are physically associated with , and embedded within , an older thick disk population . in fact , tidal interactions might pump energy into ( and hence thicken ) an initially thin disk of very metal - poor stars . _ hubble space telescope _ observations of galaxies at at large look - back times suggest ( van den bergh 2002b ) that most disk star formation occurs at @xmath291.5 , i.e. during the last 9 gyr . one reason for the paucity of disks at larger redshifts is , presumably , that such extended structures would be destroyed by tidal forces during the frequent encounters between galaxies at high redshifts . from two slightly metal - poor stars with disk kinematics liu & chaboyer ( 2000 ) find a thin disk age of 9.7 @xmath0 0.6 gyr . such an age is consistent with the ages of spiral disks that are inferred from the fact that the hst images of distant galaxies start to show obvious disks at z @xmath41.5 . since bars are generally assumed to have formed from global instabilities in disks one would not expect to see barred galaxies at @xmath19 . if bars can not form from from initially chaotic protodisks then bar formation might be delayed to even smaller look - back times . this suspicion appears to be confirmed by observations ( van den bergh et al . 1996 , 2002 ) which seem to suggest that the frequency of barred galaxies declines precipitously beyond redhifts of z @xmath40.7 . if this conclusion is correct then one would expect the bar of the lmc to be younger than 6 gyr . this conclusion is consistent with ( but not proved by ) the observation ( smecker - hane et al . 2002 ) that bursts of star formation occurred in the bar of the large cloud 4 - 6 gyr and 1 - 2 gyr ago . almost all local group galaxies are found to contain some very old stars like rr lyrae variables . this shows that these galaxies started to form stars quite early in the history of the universe , i.e. more than @xmath410 gyr ago . the best candidate for a `` young '' local group member is the dwarf irregular leo a. however , dolphin et al . ( 2002 ) have discovered a few rr lyrae variables , which have ages @xmath30 gyr , in this galaxy . furthermore , recent observations by schulte - ladbeck et al . ( 2002 ) show that this object also contains some metal - poor red horizontal branch stars . this clearly demonstrates that leo a is not a young galaxy . m32 and many of the local group dsph galaxies have not experienced recent star formation . on the other hand the gas rich spiral and irregular group members are still forming stars at a significant rate . qualitative data on the past rate of star formation in such galaxies can be obtained from both their integrated colors and from the intensity ratios of various spectral lines . however , the hope that the age distribution of star clusters in galaxies could provide more detailed information on the past rate of star formation has been shattered by the work of larsen & richtler ( 1999 , 2000 ) which appears to show that the rate of cluster formation varies as a rather high power of the rate of star formation . in other words there is not a one - to - one correspondence between the rate of cluster formation and the general rate of star formation . in the local group this phenomenon is beautifully illustrated by the difference between the quiescent dwarf irregular ic1613 , which contains few star clusters of any kind ( i.e. baade 1963 , p. 231 ; van den bergh 1979 ) , and the large magellanic cloud that is presently forming both stars and clusters quite actively . it seems likely that the present specific frequency of globular clusters in galaxies was mainly determined by their peak rates of star formation , with elevated peak rates resulting in high present specific cluster frequencies . only fragmentary information is so far available on the luminosity evolution of individual local group members . few star clusters in the lmc have ages between the 3.2 gyr age of the oldest open clusters and the @xmath413 gyr ( rich , shara & zurek 2001 ) age of the lmc globular clusters . this probably means that the large cloud experienced a quiescent period that extended for @xmath4 10 gyr . during this `` dark age '' no violent bursts of star formation ( which could have triggered the formation of star clusters ) occurred . however , it is quite likely that a trickle of star formation ( such as that which presently occurs in ic1613 ) continued during the dark ages between 3.2 gyr and 13 gyr ago . this speculation is supported by the data of da costa ( 2002 ) which seem to show that the metallicity in the lmc increased between the beginning and the end of the dark age , i.e. between the termination of globular cluster formation @xmath413 gyr ago , and the beginning of open cluster formation 3.2 gyr ago . a possibly more complicated scenario is hinted at by the work of smecker - hane et al . ( 2002 ) who conclude that star formation in the bar of the lmc was episodic , while the rate of star formation remained more or less constant within the disk of the large cloud . however , an important caveat is that the rate of star formation in lmc the disk was so low that the data do not provide strong constraints on the lmc disk star formation history . the carina dsph galaxy seems to have experienced a major burst of star formation 7 gyr ago ( hurley - keller , mateo & nemeic 1998 ) ] . however , at maximum this object probably only reached m@xmath8 @xmath4 -16 making it too faint to have become what babul & ferguson ( 1996 ) have called a `` boojum '' . there has been a long controversy among astronomers regarding the nature ( or even the existence of ) a fundamental difference between open clusters and globular clusters . the present consensus is that all clusters populations initially formed with an power law mass spectrum , and that globular clusters are simply the oldest and most massive population component that was best able to withstand the erosion caused by the destruction of lower mass clusters via evaporation , encounters with massive interstellar clouds and disk / bulge shocks . however , a different scenario has been proposed by van den bergh ( 2001 ) . he suggested that there have , in fact , been two ( perhaps quite distinct ) epochs of cluster formation . during the first of these globular clusters might have formed as halo gas was being compressed by shocks that were driven inwards by ionization fronts generated during cosmic reionization at @xmath31 . such effects would presumably be greatest in the halos of small protogalaxies that are relatively easy to ionize . a second generation of massive clusters might have formed by the compression ( and subsequent collapse ) of giant molecular clouds that was triggered by heating of the interstellar medium induced by collisions between gas rich protogalaxies . a similar view has recently been expressed by schweizer ( 2003 ) who also argues that the first generation of globular clusters formed nearly simultaneously from pristine molecular clouds that were heated and shocked by the strong pressure increase that accompanied cosmological reionization . schweizer argues that this hypothesis might also account for the similarity of metal - poor globular clusters in all types of galaxies and environments . both van den bergh ( 2001 ) and schweizer ( 2003 ) argue that second generation globular clusters were mainly formed during subsequent collisions and mergers between galaxies . from dynamical arguments kahn & woltjer ( 1959 ) ] first showed that the local group can only be stable if it contains a significant amount of invisible matter . using radial velocity observations of local group members courteau & van den bergh ( 1999 ) have estimated that the local group has a total mass of ( 2.3 @xmath0 0.6 ) x 10@xmath10 m@xmath3 , from which the mass - to - light ratio ( in solar units ) is found to be m / l @xmath8 = 44 @xmath0 14 . this high value shows that the total mass of the local group exceeds that of the visible parts of its constituent galaxies by an order of magnitude . in their 1959 paper kahn & woltjer suggested that this `` missing mass '' in the local group might be in the form of hot ( 5 x 10@xmath32 degrees ) low density ( 1 x 10@xmath33 protons @xmath34 ) gas , which would be difficult to detect observationally . hui & haiman ( 2003 ) have recently shown that the thermal history of such gas has probably been quite complex . in recent years the notion that hot gas is responsible for the missing mass in the local group has been overshadowed by the idea that this missing mass is actually in the form of cold dark matter ( blumenthal et al . however , recent _ far ultraviolet explorer _ satellite observations ( nicastro et al . 2003 ) of the absorption lines of o vi ( which have radial velocities of only a few hundred km s @xmath7 ) suggest that hot gas may provide a non - negligible contribution to the missing mass in the local group . alternatively sternburg 2003 the hot gas clouds observed by nicastro et al . might just be reprocessed metal - enriched gas , that was ejected from the neighborhood of the galactic plane in supernova driven fountains . if the latter suggestion is correct , then one might expect that the hot clouds would be relatively metal -rich . on the other hand they would be expected to be metal - poor if they are composed of hot primordial gas . the final fate of the local group has been discussed by forbes et al . ( 2000 ) who conclude that dynamic friction will eventually result in the merger of m31 and the galaxy . this merged object will resemble an elliptical galaxy with m@xmath8 @xmath4 -21 that contains @xmath4 700 globular clusters . in a universe that continues to expand for ever ( bennett et al . 2003 , spergel et al . 2003 ) this object will , in the distant future , be the only remaining visible object in the universe . * both the high metallicity of the m31 halo , and the r@xmath14 luminosity profile of the andromeda galaxy , suggest that this object might have formed from the early merger , and subsequent violent relaxation , of two ( or more ) relatively massive metal - rich ancestral objects . * the main body of the galaxy may have formed in the manner suggested by eggen , lynden - bell & sandage ( 1962 ) , whereas its halo is more likely to have been assembled by accretion of `` bits and pieces '' in the manner first suggested by searle & zinn ( 1978 ) . * it is profoundly puzzling that the old metel - poor globular clusters in the lmc appear to have been formed in an early disk , whereas the globulars associated with m33 seem to have originated in a slightly younger halo . * it is speculated that the oldest generation of globular clusters in the universe might have formed as halo gas was compressed and heated in shocks that were driven inwards by ionization fronts generated during cosmic reionization . on the other hand second generation globular clusters formed as a result of the heating of molecular clouds during collisions between gas - rich galaxies . it is emphasized that the history of star formation in galaxies is often very different from the history of cluster formation . * it is presently not understood how globular clusters like terzan 7 ( which is associated with the sagittarius dwarf ) were able to attain a relatively high metallicity . neither do we know why the globular clusters associated with some galaxies are so much more flattened than are those in others . * it is suggested that the specific globular frequency of galaxies was mainly determined by the peak rate of star formation during evolution . * all local group galaxies appear to contain a very old population component , i.e. all nearby galaxies started to form stars just after they were formed . in other words there are no truly young galaxies in the local group . * the local group has a mass of ( 2.3 @xmath0 0.6 ) x 10@xmath10 m@xmath3 , a luminosity l@xmath8 = 4.2 x 10@xmath5 l@xmath3 , and a zero - velocity radius of 1.18 @xmath0 0.16 mpc . most of the mass and luminosity of the local group appears to be concentrated in two subgroups that are centered on m31 and the galaxy , respectively . there is presently a lively controversy about which of these two subgroups is the most massive . if the galactic subgroup turns out to be more massive than the m31 group , then the ratio of dark to visible matter must differ significantly from group to group . * it is not yet clear if hot low density gas provides a significant contribution to the total mass of local group galaxies . rich , r. m. , corsi , c. e. , ballazzini , m. , frederici , l. , cacciari , c. & fusi pecci , f. 2002 in extragalactic star clusters (= iau symposium no.207 ) , eds . d. geisler , e. k. grebel , and d. minniti ( san francisco : asp ) , p.140 m31 & ngc 224 & sb i - ii & 760 & -21.2 + milky way & galaxy & sbc i - ii : & 8 & -20.9 + m33 & ngc 598 & sc ii - iii & 795 & -18.9 + lmc & ... & ir iii - iv & 50 & -18.5 + smc & ... & ir iv / iv - v & 59 & -17.1 + m32 & ngc 221 & e2 & 760 & -16.5 + ngc 205 & ... & sph & 760 & -16.4 + ic 10 & ... & ir iv : & 660 & -16.3 + ngc 6822 & ... & ir iv - v & 500 & -16.0 + ngc 185 & ... & sph & 660 & -15.6 + ic 1613 & ... & ir v & 725 & -15.3 + ngc 147 & ... & sph & 660 & -15.1 + wlm & ddo 221 & ir iv - v & 925 & -14.4 + sagittarius & ... & dsph(t ) & 24 & -13.3 + fornax & ... & dsph & 138 & -13.1 + pegasus & ddo 216 & ir v & 760 & -12.3 + leo a & ddo69 & ir v & 800 & -12.2 + sagdig & ... & ir v & 1180 & -12.0 + leo i & regulus & dsph & 250 & -11.9 + and i & ... & dsph & 810 & -11.8 + and ii & ... & dsph & 700 & -11.8 + aquarius & ddo 210 & v & 1025 & -11.3 + pegasus ii & and vi & dsph & 815 & -10.5 + and v & ... & dsph & 810 & -10.2 + and iii & ... & dsph & 760 & -10.2 + cetus & ... & dsph & 775 & -10.2 + leo ii & ... & dsph & 210 & -10.1 + pisces & lgs 3 & dir / dsph & 620 & -9.8 + phoenix & ... & dir / dsph & 395 & -9.8 + sculptor & ... & dsph & 87 & -9.8 + tucana & ... & dsph & 895 & -9.6 + cassiopeia & and vii & dsph & 690 & -9.5 + sextans & ... & dsph & 86 & -9.5 + carina & ... & dsph & 100 & -9.4 + draco & ... & dsph & 79 & -8.6 + ursa minor & ... & dsph & 63 & -8.5 +

interactions between neutral dielectric bodies are traditionally viewed as being due predominantly to electromagnetic field fluctuations , or equivalently , dipole fluctuations that give rise to van der waals ( vdw ) fluctuation - induced interactions between them @xcite . these interactions are attractive between identical bodies in vacuum or in a polarizable medium , such as water or an aqueous electrolyte ( or coulomb fluid ) . they contribute one of the two key ingredients of the classical derjaguin - landau - verwey - overbeek ( dlvo ) theory of colloidal stability , the other one being the mean - field electrostatic interaction , which is repulsive for like - charged colloidal surfaces @xcite . recent works have , however , highlighted the role of image - induced , ion - depletion effects in this scenario ( see , e.g. , refs . @xcite and references therein ) , leading to depletion of mobile solution ions from the vicinity of dielectric interfaces and , therefore , to an additional attractive force between apposed dielectric boundaries . this is because most dielectric surfaces in the context of bio- and soft materials have a lower ( static ) dielectric constant than that of water and , therefore , solution ions experience repulsion from their same - sign image charges in the proximity of dielectric boundaries @xcite . the recent advances in the study of image - induced , ion - depletion effects follow on the trail of the onsager - samaras framework formulated originally in the context of the surface tension problem of electrolytes @xcite . such non - mean - field effects , which belong to the general class of depletion interactions @xcite , arise due to the discrete nature of mobile ions neglected in the collective mean - field description based on the standard poisson - boltzmann theory @xcite . the studies of image - induced , ion - depletion effects have been focused exclusively on the case of strictly neutral ( charge - free ) dielectric surfaces . in this case , the ion - depletion interactions can be amplified in the presence of mobile multivalent ions in the solution @xcite due to stronger ion - image repulsions for these ions , even when the multivalent ions are present at small bulk concentrations around just a few mm @xcite . in this paper , we revisit the problem of interaction between neutral dielectric surfaces in a coulomb fluid by adding to it a novel feature : we relax the constraint of _ strict _ electroneutrality of surface boundaries , considered so far in the literature , by assuming that the surfaces are neutral only _ on the average _ , while microscopically they carry a _ quenched _ ( fixed ) random distribution of positive and negative charges . we show that this seemingly simple generalization leads to significant qualitative changes in the distribution of ions and , consequently , also in the effective interactions between dielectric surfaces , especially when the intervening coulomb fluid contains mobile multivalent ions . disordered charged systems are abundant in soft matter with examples ranging from polycrystalline surfaces with patchy surface potentials @xcite , dielectric contact surfaces @xcite , vapor - deposited amorphous films on solid substrates @xcite , surfactant - coated surfaces @xcite , dna microarrays @xcite , intrinsically disordered proteins @xcite , patchy colloids @xcite and random polyelectrolytes and polyampholytes @xcite . surface charge disorder in these examples can exhibit highly random distributions as well as patchy and heterogeneous patterns , originating from different sources including specific electronic and/or structural properties of materials involved , surface grafting or adsorption of charged molecules and/or contaminants , synthetic and fabrication processes , etc . the surface charge disorder can be highly sample specific and , at the same time , can depend strongly on the method of preparation . it may be set and quenched for each sample ( which is the case of interest in this paper ) , annealed ( in which case the surface charges are mobile and in thermal equilibrium with the rest of the system ) , or partially quenched or partially annealed @xcite . motivated by these examples , study of charge disorder and , in particular , effective interactions between random charge distributions has witnessed growing attention from theoreticians over the last several years @xcite , as well as from the simulation side where initial steps have been taken to include the effects of charge heterogeneity and disorder @xcite . disorder effects have been studied extensively in the context of electromagnetic fluctuation - induced interactions between two apposed , randomly charged surfaces in vacuum @xcite , where they have been associated with anomalously long - ranged surface interactions observed in recent experiments @xcite . they have also been studied in situations where a weakly coupled coulomb fluid , containing , e.g. , monovalent cation and anions , intervenes between the bounding surfaces @xcite . this however leaves out the case of asymmetric coulomb fluids containing both monovalent ions and multivalent ( counter- ) ions . these kinds of systems are in fact quite common in experiments within the biological context as in the case of viruses , dna condensates or other charged biopolymer aggregates @xcite and are expected to behave very differently since multivalent ions are known to couple strongly with fixed surface charges . strong - coupling behavior of multivalent counterions at uniformly charged surfaces or surfaces with regular charge patterns has been studied extensively over the last decade and its connection to exotic phenomena such as like - charged attraction has been throughly discussed ( for recent reviews and a more exhaustive list of references , see refs . @xcite ) . in disordered charged systems , such strong - coupling phenomena have been considered only in a few cases so far @xcite by assuming that bounding surfaces carry a _ finite _ mean surface charge density , to which multivalent counterions can couple strongly , in the same sense as considered in the context of uniformly charged surfaces as noted above . yet , the presence of charge randomness on bounding surfaces was shown to give rise to novel phenomena such as strong surface attraction of multivalent counterions , characterized by a density profile that diverges at the surface , in clear violation of the contact - value theorem established for uniformly charged surfaces . this kind of behavior originates from a singular , attractive , single - ion potential , which is created by the randomness in the distribution of surface charges and , as such , depends on the surface charge variance . consequently , one can also show that the thermal entropy of counterions is diminished upon introducing a finite degree of charge randomness on the boundaries . in other words , the system becomes more ` ordered ' as a direct outcome of the interplay between the thermal entropy of ions and the configurational entropy of charge randomness , entering through an ensemble average over various realizations of the disordered charge distribution . this peculiar disorder - induced effect , which stands at odds with what one may expect intuitively , has been referred to as _ antifragility _ @xcite . our analysis in this paper is focused on yet another facet of the disorder - induced effects by assuming , in a system of two plane - parallel dielectric slabs , that the randomly charged surfaces of the slabs bear _ no net charge_. this eliminates the direct strong coupling between multivalent ions and the mean surface charge and , thereby , also the ensuing strong - coupling interactions , considered in our previous papers @xcite , that would otherwise completely mask the vdw and image - induced , ion - depletion interactions between the slabs in an asymmetric coulomb fluid . this allows us to address the question of how the presence of surface charge disorder affects the standard picture for the interaction of neutral bodies based on the vdw and ion - depletion effects . the surface charge disorder has several different implications : first , it contributes a _ repulsive _ interaction between the slabs , which comes from self - interactions of disorder charges on the bounding surfaces with their image charges ; this contribution counteracts the _ attractive _ vdw interaction as discussed thoroughly elsewhere @xcite . then , as noted above , the individual ions in the coulomb fluid experience an attractive disorder - induced potential , which strongly attracts them toward the bounding surfaces . this effect counteracts the ion - image repulsions that tend to deplete ions from the slit region between the slabs and form the basis of the image - induced , ion - depletion mechanism for attraction between strictly neutral slabs . thus , randomly charged bounding surfaces tend to accumulate more mono- and multivalent ions in the slit . the system , however , responds differently to the increased number of ions : while the osmotic pressure due to monovalent ions increases and even becomes _ repulsive _ , consistent with the standard mean - field picture , the osmotic pressure due to multivalent ions , becomes ever more _ attractive _ ! this behavior is rooted in the combined effect of the surface charge disorder and the presence of mobile multivalent ions , with the latter creating strong inter - surface attractions upon further accumulation in the slit . as a result , the effective total interaction pressure between randomly charged , net - neutral dielectric surfaces can differ qualitatively from what one expects based on the standard vdw @xcite and image - induced , ion - depletion interactions in the case of strictly neutral surfaces ( see , e.g. , refs . @xcite and references therein ) . the net effect due to the interplay between disorder , mono- and multivalent - ion contributions yields a qualitatively different behavior for the effective surface - surface interactions , depending on the strength of disorder as quantified by the _ disorder coupling ( or strength ) parameter _ @xcite . this feature bears some conceptual resemblance to the strong - weak coupling dichotomy that exists for net - charged surfaces , dependent on the _ electrostatic coupling parameter _ in that case @xcite . in weakly disordered systems , corresponding to a small disorder coupling parameter , one then discerns a distinct non - monotonic behavior for the interaction pressure as a function of the separation between the slabs , with a pronounced repulsive hump at intermediate separations ; conversely , in strongly disordered systems , corresponding to a large disorder coupling parameter , the effective interaction pressure becomes strongly attractive , with a range and magnitude larger than that of the vdw or the image - induced , ion - depletion interaction pressure as found between strictly neutral surfaces . the organization of the paper is as follows : in section [ sec : model ] , we introduce our model and , in section [ sec : general ] , we briefly discuss the theoretical background and present the general results for the two - slab system . the results of our analysis for the density profile of multivalent ions and the effective interactions between the slabs are presented in section [ sec : results ] . we conclude our discussion in section [ sec : conclusion ] . and randomly charged inner surfaces are immersed in a bathing ionic solution of dielectric constant @xmath0 . the solution contains a mixture of monovalent and multivalent salts . multivalent ions are confined in the slit region and are shown by large spheres ; monovalent salt anions and cations are shown by small red and blue spheres . the slabs are assumed to be neutral on the average and their thickness is taken to be infinite.,width=321 ] our model consists of two plane - parallel dielectric slabs of infinite surface area @xmath1 and dielectric constant @xmath2 with inner surfaces placed perpendicular to the @xmath3 axis at a separation distance of @xmath4 ( see fig . [ f : schematic ] ) . the slabs are immersed in an asymmetric coulomb fluid of dielectric constant @xmath0 at room temperature @xmath5 . the coulomb fluid is a mixture of a monovalent @xmath6 salt of bulk ( reservoir ) concentration @xmath7 and a multivalent @xmath8 salt of bulk concentration @xmath9 with multivalent ions having charge valency of @xmath10 . the multivalent ions are confined within the slit region @xmath11 , while the monovalent ions are dispersed throughout the space except in the region occupied by the dielectric slabs that are assumed to be impermeable to mobile ions @xcite . we are interested only in the cases where the slab thickness is much larger than the debye ( or salt ) screening length and , thus , in the calculations to be presented later , we shall assume that the slab thickness is infinite . the inner surfaces of the slabs are assumed to bear a _ quenched _ , random charge distribution @xmath12 , while they remain electrically _ net - neutral_. we assume that the random surface charges are distributed according to a gaussian probability distribution function , which is determined fully by its two moments , @xmath13 , and @xmath14 with @xmath15 , \label{eq : g}\ ] ] where @xmath16 is the surface disorder variance . by assumption , thus , the disordered charge distributions on the two slabs are statistically uncorrelated . the effects of surface charge correlations ( or patchiness " ) and the internal structure of multivalent ions , which are assumed to be point - like here , will be considered elsewhere @xcite . in the present model , we have two types of mobile ions , namely , mono- and multivalent ions , that are expected to behave very differently in the presence of dielectric boundaries . on general grounds , and if the surfaces are assumed to be strictly neutral ( charge free ) , mobile ions interact only with their image charges . this interaction scales with the second power of the ionic valency , being thus much larger in the case of multivalent ions with @xmath17 ( e.g. , trivalent and tetravalent ions ) . in this latter case , the ion - image interactions are dominated by the self - image interaction . note that in most realistic examples , ions are dissolved in water , which is a highly polarizable medium with @xmath18 at room temperature ( @xmath19 k ) and , thus , we often have @xmath20 , giving image charges of the same sign and , hence , repulsive self - image interactions for each ion in the slit . such repulsive interactions lead to statistical correlations ( manifested as ion depletion ) between the mobile ions and the bounding surfaces that are expected to be weak for monovalent ions , but quite strong for multivalent ions . this leads to a complicated problem in which different ionic species show distinctly different couplings to local electrostatic fields generated by the boundaries . while the monovalent ions can be therefore described appropriately by mean - field - type theories , such as the poisson - boltzmann or the debye - hckel theory @xcite , the multivalent ions require an altogether different description that should account for such large correlation effects . the situation is , therefore , analogous to the one found in the case of asymmetric coulomb fluids confined between _ charged _ surfaces , where the dominant factor determining the behavior of multivalent counterions is again their electrostatic coupling to the local fields that are generated by the boundaries . the difference is however that in the the case of a charged surface the coupling is determined by the so - called electrostatic coupling parameter @xcite , depending on the charge density of the surface boundaries , enabling the so - called _ strong - coupling approximation _ for multivalent counterions at charged surfaces ( see , e.g. , refs . @xcite and references therein ) , or more generally , the _ dressed multivalent - ion theory _ @xcite , which provides a very good approximation for the study of asymmetric coulomb fluids over a wide range of parameters as verified by explicit - ion and implicit - ion simulations . the dressed multivalent - ion theory reproduces the mean - field debye - hckel theory and the standard strong - coupling theory for counterion - only systems @xcite as two limiting theories in the regime of large and small debye screening lengths and , therefore , bridges the gap between these two limits @xcite . the key step in this latter approach is to integrate out the degrees of freedom associated with monovalent ions by means of a linearization approximation , justified only for highly asymmetric coulomb fluids @xmath17 , and yielding an effective debye - hckel ( dh ) interaction between the remaining multivalent ions and the surface charges ( if any ) @xcite . then , one expands the partition function of the system in terms of the fugacity ( or bulk concentration , @xmath21 ) of multivalent ions or in terms of the inverse electrostatic coupling parameter for counterion - only systems @xcite . the leading order of the virial expansion can be cast into a simple analytic theory because of its single - particle structure , which can successfully describe various features of strongly charged systems containing multivalent ions ( see , e.g. refs . @xcite and references therein ) . the regime of applicability of this theory has been discussed extensively in recent literature @xcite , which we therefore do not reiterate here . in the case of strictly neutral ( charge - free ) surfaces , this strategy was shown to be effective as well @xcite , since , as is often the case in experimental systems containing asymmetric ionic mixtures @xcite , multivalent counterions of high valency ( for instance , cohex@xmath22 or polyamines such as spd@xmath22 , sp@xmath23 ) are present only in small bulk concentrations , of about a few mm , justifying fully the virial expansion scheme that underlies the dressed multivalent - ion theory . this is the approach of choice that we adopt also in the present context , where the surface boundaries carry random charges and are neutral only on the average . the only additional step here is that , due to the quenched disorder in the surface charge distribution , the free energy of the system follows by averaging over the whole ensemble of charge distributions , @xmath12 @xcite . without delving further into details available in previous publications @xcite , we proceed by giving the general expression for the free energy of the two - slab system considered in this work , which has the form @xcite @xmath24- \lambda_c k_{\mathrm{b}}t\int { \mathrm{d}}{{\bf r}}\ , \omega({{\bf r}})\ , \rme^{-\beta u({{\bf r}})}.\ ] ] here , @xmath25 and the green s function of the system is defined via @xmath26 where @xmath27 is the debye ( or salt ) screening parameter , which is non - zero only outside the region occupied by the dielectric slabs , i.e. , @xmath28 with @xmath29 being the total bulk concentration of monovalent ions . the first term in eq . ( [ eq : exact_gpot ] ) represents the free energy of the system in the absence of multivalent ions . it stems from _ self - interactions _ of disorder charges on the bounding surfaces of the slabs with their image charges ( note that the random charge distributions on the two slab are uncorrelated and , on the average , do not interact with each other ) . this term depends only on the disorder variance , @xmath30 , and is non - vanishing only in inhomogeneous systems with a finite dielectric discontinuity at the bounding surfaces and/or a spatially inhomogeneous distribution of salt ions . this contribution has been analyzed in the context of the fluctuation - induced forces between disordered surfaces in vacuum or in a weakly coupled coulomb fluid @xcite . the second term in eq . ( [ eq : exact_gpot ] ) represents the contribution from multivalent ions on the leading ( virial ) order , in which @xmath31 is the _ effective _ single - particle interaction energy @xcite @xmath32 ^ 2 . \label{eq : u2}\ ] ] the first term in the above expression represents the self - image interaction of individual multivalent ions in the slit and the second term represents the contribution of the surface charge disorder to the effective single - particle interaction energy . in the first term , @xmath33 gives the dielectric and/or salt polarization , or the image - charges effects ( corresponding to the generalized born energy ) , in which the free - space green s function , @xmath34 ( corresponding to the formation energy of individual ions in a homogeneous background ) , defined via @xmath35 , is subtracted from the total green s function . the second term in eq . ( [ eq : u2 ] ) is found to be proportional to the disorder variance and shows an explicit temperature dependence and a quadratic dependence on the green s function and the multivalent - ion charge valency , @xmath36 ( these latter features can be understood by noting that the disorder term indeed stems from the sample - to - sample fluctuations , or variance , of the sample - dependent single - particle interaction energy as discussed in ref . @xcite ) . in the strong - coupling limit or within the multivalent dressed - ion theory @xcite , the number density of multivalent counterions can be obtained in terms of the effective single - particle interaction energy as @xcite @xmath37 in the specific example of two planar slabs , we can take advantage of the translational invariance of the green s function with respect to transverse ( in - plane ) direction coordinates @xmath38 and @xmath39 , since @xmath40 is only a function of @xmath41 , @xmath3 and @xmath42 , and write the free energy in terms of its fourier - bessel transform @xmath43 defined through @xmath44 for two semi - infinite slabs , one has @xmath45.\end{aligned}\ ] ] where @xmath46 , and @xmath47 in the absence of salt screening ( @xmath48 ) , @xmath49 reduces to the bare dielectric discontinuity parameter @xmath50 which gives a measure of the magnitude of dielectric - image " charges ; while in a _ dielectrically homogeneous _ system , we have @xmath51 which gives a measure of the salt - induced image effects , or loosely speaking , salt - image " charges . both image charge effects lead to depletion of ions from the vicinity of dielectric surfaces in a medium of relatively large polarizability , but they exhibit some fundamental differences , which have been discussed in the context of charged surfaces in our previous works @xcite . ( a ) ( b ) 0.1 cm ( c ) in the two - slab geometry , we can then re - express the ( number ) density profile of multivalent ions in the slit region , @xmath11 , as @xmath52 where @xmath53.\\\end{aligned}\ ] ] the _ interaction _ free energy of the system ( per @xmath54 and per unit area , @xmath1 ) can be written as @xmath55 where we have subtracted additive terms that are independent of surface separation @xmath4 and defined @xmath56 let us first focus on the case of a dielectrically homogeneous system with @xmath57 . the density profiles of multivalent ions for this case are shown in fig . [ f : neutral_dist]a , where we have rescaled the density profiles with their bulk value @xmath9 and the position in the slit @xmath58 with the bjerrum length @xmath59 . the inter - surface distance and the screening parameter are rescaled in the same way . in the figure , we have fixed @xmath60 and @xmath61 , equivalent to mono- and multivalent salt concentrations of @xmath62 mm and @xmath63 mm when the bjerrum length is taken as @xmath64 nm ( appropriate for water at room temperature , i.e. , with @xmath19 k and @xmath18 ) and the multivalent ion valency is taken as @xmath65 ( note that throughout this paper we focus on the case of asymmetric coulomb fluids with tetravalent ions but the generalization of our results to other values of @xmath36 is straightforward ) . the surface charge disorder variance is shown in the figure in terms of the dimensionless _ disorder coupling ( or strength ) parameter _ @xcite @xmath66 which is varied in the figure as @xmath67 and 2 ( corresponding to strictly neutral surfaces with @xmath68 , and randomly charged surfaces with @xmath69 nm@xmath70 and 0.04 nm@xmath70 , respectively ) . as seen , for strictly neutral ( or disorder - free ) dielectrics , the density profile ( black solid curve ) shows a partial depletion of multivalent ions from the vicinity of the surface boundaries . this is caused by the salt - image repulsion ( corresponding to the first term in eq . ( [ eq : u2 ] ) ) , which , as noted in the previous section , are caused by the discontinuity in the distribution of salt ions across the dielectric interfaces , giving a non - vanishing @xmath49 according to eq . ( [ eq : delta_s_homogen ] ) . however , when the surfaces are randomly charged ( dashed curves ) , the situation turns out significantly different and the multivalent ions are attracted quite strongly towards the surfaces despite the soft " salt - image repulsions ( moreover , one can note from the shown profiles that a larger amount of multivalent ions are pulled into the slit from the bulk solution ) . in fact , the resulting disorder potential acting on individual multivalent ions ( corresponding to the second term in eq . ( [ eq : u2 ] ) ) exhibits a singular ( logarithmic ) behavior at the two surfaces and gives an algebraically diverging density as @xmath71 when @xmath72 . this kind of phenomena has been discussed in detail in the context of disordered charged surfaces bearing a net charge density @xcite , where we show that the singular behavior of the density profile is closely connected with the _ anti - fragile _ behavior of multivalent counterions : adding a quenched disorder component to an otherwise uniform distribution of surface charges leads to a lowered entropy ( or thermal disorder ) for the counterions . in the present context , with net - neutral surfaces , the disorder - induced effects are qualitatively similar as they are regulated only by the variance of the charge distribution , @xmath30 , rather than the net charge density of the surface . finite dielectric discontinuity at the bounding surfaces , @xmath73 , creates a much stronger depletion effect than in the case of dielectrically homogeneous system ( see fig . [ f : neutral_dist]b ) . in fact , the density profile of multivalent ions vanishes in the immediate vicinity of the surfaces because of the strong dielectric - image repulsion , provided again by the first term in eq . ( [ eq : u2 ] ) . this term is singular itself and diverges on approach to the surface as @xmath74 when @xmath72 , thus dominating over the singular potential created by the charge randomness on the boundaries and , leading to the vanishing contact density of multivalent ions regardless of the disorder strength , @xmath75 , as seen in the figure ( even though a larger amount of multivalent ions are found in the slit at larger values of @xmath75 ) . the competition between these two mechanisms of repulsion and attraction leads to a change in the shape of the density profile from unimodal to bimodal beyond a threshold value of the disorder strength , @xmath76 . the data in fig . [ f : neutral_dist]c show @xmath76 as a function of the dielectric discontinuity parameter , @xmath77 , for fixed @xmath60 , @xmath63 mm and different values of @xmath78 and 60 mm ( corresponding to @xmath79 and 0.58 , respectively ) . clearly , for larger salt screening and/or dielectric discontinuity parameter , a larger degree of charge disorder is required in order to counteract the image - charge repulsions . these features of the density profiles are qualitatively similar to those found in the case of charged surfaces @xcite and , therefore , we shall not delve further into the details of the behavior of the density profile and proceed with the analysis of the effective interaction between net - neutral surfaces . effective interactions between neutral dielectric slabs are standardly described in terms of the vdw interactions as , for instance , formulated within the lifshitz theory @xcite . the vdw interaction pressure for two plane - parallel slabs can be expressed as @xcite @xmath80 where the first term comes from the thermal zero - frequency mode of the electromagnetic field - fluctuations and the second term comes from the higher - order matsubara frequencies , pertaining to quantum fluctuations . @xmath81 is then the quantum part of the hamaker coefficient , which , based on an upper bound estimate in the case of hydrocarbon slabs interacting across an aqueous medium , can be taken as @xmath82 @xcite . the vdw interaction is dominant mostly at the scale of a few nanometers for the inter - surface distance @xcite . in the present model , one needs to account for the electrostatic contribution to the inter - surface pressure , @xmath83 , as well . this contribution can be written as the difference between the slit pressure of multivalent and monovalent ions and the bulk pressure @xmath84 , where @xmath29 is the total bulk concentration of monovalent ions , i.e. , @xmath85 . the slit pressure can be calculated as @xmath86 , in which the first term is the contribution of multivalent ions that follows from the derivative of the free energy expression , eq . ( [ e : f_neutral ] ) , w.r.t . the inter - surface distance when all other parameters are kept constant , while the second term is the contribution of monovalent ions that has been expressed in terms of the total mid - plane density of monovalent ions @xcite . this latter quantity can be estimated here through the relation @xmath87\big|_{q=1}$ ] for @xmath88 , which has been shown by means of explicit monte - carlo simulations of a disorder - free system @xcite to give an accurate estimate of the distribution of monovalent ions in the slit . the electrostatic pressure can thus be decomposed into its different components as @xmath89 where the components are obtained explicitly as @xmath90 -c_0\\ \label{eq : p_mon } & & \beta p_{mon}=n(0)-n_b.\end{aligned}\ ] ] the contribution @xmath91 arises from self - interactions of random charges on the surfaces of the two slabs with their ( dielectric and/or salt ) image charges . this contribution stems from the first term of the interaction free energy ( [ e : f_neutral ] ) and is present irrespective of multivalent ions , being typically comparable in range and strength with the vdw pressure . it can , however , scale differently with the inter - surface distance and can be repulsive or attractive depending on the sign of dielectric discontinuity parameter , @xmath77 ( e.g. , it is repulsive for @xmath73 , applicable to the cases studied in this paper ) , resulting thus in a rather diverse behavior for the inter - surface interaction between two net - neutral slabs in the absence of multivalent ions @xcite . the two terms @xmath92 and @xmath93 correspond to the disjoining or osmotic pressure contributions of multivalent and monovalent ions , respectively . the former stems from the second term of the interaction free energy ( [ e : f_neutral ] ) , while , as noted above , the latter is included heuristically through eq . ( [ eq : p_mon ] ) . , @xmath63 mm and @xmath94 and 35 mm as indicated on the graph ( colored dashed curves ) . the corresponding vdw pressure for these parameter values closely overlap ( black dashed curves ) . panel ( b ) shows the total pressure ( @xmath95 ) along with its electrostatic ( @xmath83 ) and vdw ( @xmath96 ) components for @xmath63 mm and @xmath62 mm . the light - colored dashed curves show @xmath83 for @xmath97 and 35 mm ( from right to left ) . , title="fig:"](a ) 0.3 cm , @xmath63 mm and @xmath94 and 35 mm as indicated on the graph ( colored dashed curves ) . the corresponding vdw pressure for these parameter values closely overlap ( black dashed curves ) . panel ( b ) shows the total pressure ( @xmath95 ) along with its electrostatic ( @xmath83 ) and vdw ( @xmath96 ) components for @xmath63 mm and @xmath62 mm . the light - colored dashed curves show @xmath83 for @xmath97 and 35 mm ( from right to left ) . , title="fig:"](b ) \(a ) + -0.4 cm ( b ) \(c ) + -0.4 cm ( d ) \(e ) + -0.4 cm ( f ) the total pressure acting on the slabs is finally given by @xmath98 this quantity is plotted in fig . [ f : neut_kappa]a as a function of the rescaled inter - surface distance for fixed @xmath99 , @xmath63 mm and four different values of @xmath94 and 35 mm , which correspond to @xmath100 and 0.45 , respectively ( colored dashed curves ) . for these same parameter values , we also show the vdw pressure in fig . [ f : neut_kappa]a ( black dashed curves ) , but these latter curves closely overlap and are not discernible at the implied resolution . this clearly indicates that the dependence of the total pressure on the salt screening parameter enters mainly through the electrostatic contribution ( see below ) . in fact , as one can see from the figure , the total pressure , @xmath101 , is less attractive than the standard vdw pressure at large separations , but , as @xmath7 decreases , the total pressure becomes more strongly attractive as compared with the vdw contribution . we show the electrostatic and vdw components of the total pressure separately in fig . [ f : neut_kappa]b for @xmath63 mm and @xmath62 mm ( @xmath102 ) . the electrostatic contribution , @xmath83 , shows a non - monotonic behavior : it is repulsive and decays to zero at large separations , while it becomes strongly attractive and diverges at small separations , showing thus a weakly repulsive maximum at intermediate inter - surface separations . the light - colored dashed curves show the same quantity for larger values of @xmath97 and 35 mm ( from right to left ) . clearly , the extent of the repulsion and the non - monotonic behavior of the electrostatic pressure become more pronounced as @xmath7 is increased . non - monotonic interaction pressures due to image - induced ion depletion have also been found in the absence of surface charge disorder but at relatively large bulk salt concentrations ( e.g. , above 250 mm ) @xcite . before proceeding with a more detailed analysis of the electrostatic contribution , we emphasize here that @xmath83 and @xmath96 show clearly different qualitative behaviors as one can see from fig . [ f : neut_kappa]b . the standard dlvo description of colloidal interactions @xcite in terms of the vdw interaction of neutral surfaces is therefore insufficient when the surfaces are neutral only _ on the average _ and otherwise carry random positive and negative charges , and/or when the system contains also an asymmetric coulomb fluid . our results show that only a small degree of charge randomness with , e.g. , a surface charge variance of @xmath103 nm@xmath70 ( corresponding to @xmath99 ) , will be enough to generate a sizable deviation from the standard vdw prediction . in order to gain further insight into the intriguing role of disorder - induced effects in the presence of multivalent ions , we examine the behavior of the effective electrostatic pressure , @xmath83 , and its components in more detail . first , we consider two strictly neutral ( disorder - free ) surfaces , in which case the disorder effects vanish and the electrostatic inter - surface pressure , @xmath83 , is given only by the osmotic pressures of multivalent and monovalent ions . the results are shown by black solid curves in figs . [ f : neutral_pressure](a , b ) and figs . [ f : neutral_pressure](c , d ) for a dielectrically homogeneous ( @xmath57 ) and a dielectrically inhomogeneous system ( @xmath104 ) , respectively . for the latter case , the contributing components of the electrostatic pressure , i.e. , @xmath105 and @xmath106 , are plotted in figs . [ f : neutral_pressure](e , f ) for comparison . the vdw pressure is shown by black dotted curves in panels a to d. in all cases , we have taken @xmath63 mm and @xmath62 mm . it turns out that the electrostatic pressure in a disorder - free system ( @xmath107 ) , which is given by @xmath108 , is negative ( attractive ) for all inter - surface separations and tends to zero at large separations because of salt screening effects . this means that the negative bulk pressure is stronger in magnitude than the slit pressure for both cases of @xmath57 and 0.95 . this is because of the image - induced depletion of ions generated by salt and/or dielectric - image charges that are of the same sign when the surfaces are immersed in a medium of larger dielectric constant as assumed here . in the limit @xmath109 , @xmath83 in an inhomogeneous system reduces to the bulk pressure , @xmath84 , due to complete depletion of ions from the slit under the strong repulsive force of dielectric - image charges , while in a dielectrically homogeneous system , complete depletion is not achieved ; hence , @xmath83 is less attractive in this latter case . on the other hand , for the given parameter values , the vdw pressure is more ( less ) attractive than the electrostatic pressure , @xmath83 , for @xmath104 ( @xmath57 ) as one can see from the figures . next , we discuss how the introduction of charge disorder on the inner surfaces of the net - neutral dielectrics affects the effective electrostatic interaction between them . to this end , we vary the disorder strength parameter from @xmath107 up to @xmath110 ( corresponding to @xmath111 nm@xmath70 ) and divide the results into two categories of `` weakly '' and `` strongly '' disordered cases . in the _ weak disorder regime _ ( typically @xmath112 ) , the electrostatic interaction pressure , @xmath83 , becomes gradullay more repulsive or positive ( especially at small separations ) as compared to that of strictly neutral surfaces when @xmath75 is increased up @xmath113 ( see colored dashed curves in figs . [ f : neutral_pressure]a and c for @xmath57 and @xmath104 , respectively ) . the repulsive interaction pressure is stronger for larger values of the dielectric discontinuity parameter , @xmath77 ( compare panels a and c ) . the most remarkable feature of our results is that @xmath83 in this regime adopts a non - monotonic behavior : for very small @xmath75 , it develops a point of zero electrostatic pressure followed by a shallow attractive minimum at intermediate separations ( see , e.g. , @xmath114 , red dashed curve , in panel a , or @xmath114 and 0.5 , red and blue dashed curves in panel c ) . the depth of this minimum decreases and the interaction profile instead develops a repulsive ( positive ) hump at large values of @xmath75 ( e.g. , @xmath113 ) . these behaviors are in clear contrast to what is observed in the case of strictly neutral ( disorder - free ) surfaces ( black solid curves ) . in order to elucidate the origin of this difference , we compare different components of @xmath83 in the case of an inhomogeneous system with @xmath104 and @xmath115 in fig . [ f : neutral_pressure]e . for comparison , the light - blue solid curve and the black solid curve ( which is seen more clearly in the inset ) show the osmotic contribution of monovalent and multivalent ions , @xmath106 and @xmath105 , in the case of strictly neutral surfaces ( @xmath107 ) . both contributions are attractive ( negative ) , while the former is the dominant one as expected since we have assumed that multivalent ions enter in small bulk concentrations ( here , @xmath63 mm ) . by introducing a weak charge randomness ( @xmath115 for the relevant curves in fig . [ f : neutral_pressure]e ) , the negative pressure of both multivalent ( @xmath92 , red dashed curves , inset ) and monovalent ions ( @xmath93 , blue dashed curve , main set ) decrease in magnitude and , thus , become less attractive as compared with the disorder - free case . it is important to note that the distribution of both monovalent and multivalent ions and , thus , their respective osmotic pressure , @xmath93 and @xmath92 , are affected by the surface charge disorder through the disorder - induced , single - ion potential , which is the second term in eqs . ( [ eq : u2 ] ) or ( [ eq : resu_neutral ] ) . the decrease in the magnitude of attractive pressure components @xmath93 and @xmath92 in the weak disorder regime is because more mono- and multivalent ions are attracted into the slit from the bulk solution in the presence of surface charge disorder ( see fig . [ f : neutral_dist ] and fig . [ f : sum_multi ] below ) . nevertheless , at very small separations , all ions are again totally depleted due to the repulsive forces of image charges and @xmath93 and @xmath92 reduce to the same bulk values as @xmath106 and @xmath105 do when @xmath116 , i.e. , @xmath117 and @xmath118 , respectively . the crucial role , however , is played by the repulsive disorder self - interaction component , @xmath91 ( green dashed curve ) , whose effect is amplified by increasing the charge disorder strength and/or the interfacial dielectric mismatch as follows from eq . . it is the interplay between this repulsive disorder - induced pressure ( which dominates at small to intermediate separations ) and the attractive osmotic pressure of ions ( which dominate at larger separations ) that gives rise to the non - monotonic behavior mentioned above . for sufficiently small @xmath75 , the pressure of multivalent ions is typically too small ( as compared to the other two components ) to change this qualitative behavior . this picture changes drastically in the _ strong disorder regime _ ( typically @xmath119 ) , where the pressure due to multivalent ions becomes a key factor . in fig . [ f : neutral_pressure]b and d , we plot @xmath83 for larger disorder strength parameters , @xmath120 up to 5 for @xmath57 and @xmath104 , respectively . in this regime , we find a reverse trend caused by the surface charge disorder : the repulsive hump at intermediate separations now diminishes and eventually disappears when @xmath75 is increased to larger values ; one thus finds a highly attractive ( negative ) interaction pressure with a range that can be much larger than that of the vdw interaction pressure ( black dotted curves ) , or the mere image - induced , ion - depletion pressure ( black solid curves with @xmath107 ) , at sufficiently large disorder strengths . the different components contributing to the electrostatic pressure for this case are shown in fig . [ f : neutral_pressure]f for @xmath99 with other parameters being the same as figs . [ f : neutral_pressure]c and d. as seen , while the disorder self - interaction contribution , @xmath91 , has become only slightly more repulsive ( green dashed curve ) , the other two components , @xmath92 and @xmath93 , show significant changes as compared to their counterparts in the weak disorder regime in fig . [ f : neutral_pressure]e . specifically , the attractive contribution due to the osmotic pressure of multivalent ions ( red dashed curve ) becomes much stronger than what we find in the weakly disordered or disorder - free cases , while the contribution from monovalent ions now becomes repulsive ( positive ) in contrast to its attractive ( negative ) behavior in these latter cases ( compare panels e and f ) . the monovalent contribution becomes positive because of an increase in the number of monovalent ions in the slit that creates a larger osmotic pressure . the average number of multivalent ions also increases in the slit as @xmath75 increases ; however , these ions correlate strongly with the disorder charges on the bounding surfaces of the slabs ( through the second term in eq . ( [ eq : resu_neutral ] ) which contributes directly to @xmath92 , eq . ( [ eq : p_c ] ) ) and , as a result , give rise to an effective attraction between the net - neutral slabs through the pressure component @xmath92 . this effect is in clear contrast with the image - induced , ion - depletion mechanism , in which a _ decrease _ in the number of mono- and multivalent ions in the slit gives rise to attractive osmotic pressures on the slabs similarly from both types of ions . this behavior originates from the ( singular ) attractive , single - ion potential , which is created by surface charge disorder . it is thus one of the most fundamental aspects of the coupling between surface charge disorder and mobile multivalent ions that follows from our results . , per rescaled surface area , @xmath121 , is plotted versus the rescaled inter - surface separation , @xmath122 , for a system similar to those considered in figs . [ f : neutral_pressure]c and d. ] , @xmath99 , @xmath123 mm and @xmath63 , 5 and 10 mm as indicated on the graph ( from top to bottom ) . the corresponding electrostatic and vdw components of the total pressure , i.e. , @xmath83 and @xmath96 , are shown in rescaled units as well ( @xmath83 is shown by the light - colored dashed curves and @xmath96 by the black one , which closely overlap ) . panel ( b ) shows @xmath95 along with @xmath96 , @xmath83 and the three components contributing to the latter , i.e. , @xmath91 , @xmath92 and @xmath93 for fixed @xmath124 mm and other parameter values as in ( a ) . , title="fig:"](a ) 0.3 cm , @xmath99 , @xmath123 mm and @xmath63 , 5 and 10 mm as indicated on the graph ( from top to bottom ) . the corresponding electrostatic and vdw components of the total pressure , i.e. , @xmath83 and @xmath96 , are shown in rescaled units as well ( @xmath83 is shown by the light - colored dashed curves and @xmath96 by the black one , which closely overlap ) . panel ( b ) shows @xmath95 along with @xmath96 , @xmath83 and the three components contributing to the latter , i.e. , @xmath91 , @xmath92 and @xmath93 for fixed @xmath124 mm and other parameter values as in ( a ) . , title="fig:"](b ) the remarks in the preceding discussions on the attraction ( depletion ) of multivalent ions to ( from ) the slit region can be corroborated by considering the average total number of ions in the slit . for multivalent ions , this quantity can be calculated from @xmath125 . it shows a very different behavior as a function of the inter - surface separation in the weak disorder and strong disorder regimes as one can see in fig . [ f : sum_multi ] ( we have used the same parameters as in figs . [ f : neutral_pressure]c and d ) . although , at fixed inter - surface distance , always more multivalent ions are attracted to the slit by increasing the disorder strength , the response of multivalent ions to the decrease in slab separation is quite different for different disorder regimes : for disorder - free and weakly disordered surfaces , multivalent counterions are quickly depleted from the slit by decreasing the slab separations , while , for sufficiently large disorder strengths , these ions ( as well as monovalent ions that are not shown here ) are more strongly attracted to the slit from the bulk solution due to stronger attraction experienced from the randomly charged , inner surfaces of the slabs . in the intermediate regime of disorder strengths , one find a non - monotonic behavior upon deceasing the inter - surface separation , with the average total number of multivalent ions first decreasing and then increasing at very small separations . one should , however , note that in dielectrically inhomogeneous systems such as the one considered in fig . [ f : sum_multi ] with @xmath104 , dielectric - image repulsions eventually win and , thus , even the curves for highly disordered systems eventually turn downward and we find @xmath126 for @xmath116 or , equivalently , @xmath127 for all cases in figs . [ f : neutral_pressure]c and d ( these limiting behaviors occur at very small separations that are not physically meaningful , e.g. , below @xmath128 , and , for the sake of presentation , they have not been shown in the plots ) . the crossover from weak to strong disorder regimes can be effected also by decreasing the salt screening parameter through a decrease in the bulk concentration of monovalent ions , @xmath7 ( and/or increasing the bulk concentration of multivalent ions , @xmath9 ) . for instance , at sufficiently large @xmath7 such as @xmath123 mm , the electrostatic interaction pressure , @xmath83 , at the disorder strength value of @xmath99 and multivalent salt concentration of @xmath63 mm shows a weak - disorder behavior ( see fig . [ f : kappa_c0]a ) , in contrast to what we found for @xmath99 at lower salt concentration of @xmath62 mm in fig . [ f : neutral_pressure]d . even though increasing @xmath9 up to @xmath124 mm enhances the attractive pressure as shown in the figure , the behavior of the effective interaction still remains in the weak - disorder regime as can be verified from the pressure components in fig . [ f : kappa_c0]b ( e.g. , compare @xmath93 and @xmath92 with those in fig . [ f : neutral_pressure]e ) . this is clearly because of strong salt screening effects at large @xmath7 , in which case the strong - disorder behavior can be achieved only by taking even larger values of @xmath75 . in this work , we have studied the effective interactions between _ net - neutral _ dielectric slabs that carry _ quenched _ ( fixed ) random charge distributions on their apposing surfaces while they are immersed in an asymmetric aqueous electrolyte ( coulomb fluid ) containing mobile monovalent and multivalent ions . the effective interaction between quenched , random charge distributions have been considered in a series of recent works @xcite , and , while the role of debye screening due to a weakly coupled ( monovalent ) salt solution on these interactions have been analyzed , the role of multivalent ions that couple strongly with surface charges has received much less attention @xcite . the interaction between ( electro- ) neutral dielectric slabs is traditionally described in terms of vdw or casimir - type forces @xcite that are always attractive between identical bodies in vacuum or in a polarizable medium such as water or an aqueous salt solution . in the dlvo context @xcite , the vdw attraction is counteracted by the mean - field electrostatic repulsion between like - charged surfaces , providing thus a mechanism for the stability of colloidal dispersions . in the case of strictly neutral surfaces , however , the electrostatic inter - surface repulsion is obviously absent but , in addition to the vdw interaction , there are image - induced , ion - depletion forces that are attractive as well and arise because the mobile solution ions are depleted from the vicinity of the dielectric interfaces as a consequence of the repulsion from their image charges ( these image charges are of the same sign when the surfaces are immersed in a medium of larger dielectric constant as compared with that of the slabs ) . this effect is intrinsically non - mean - field and occurs because of the discrete nature of ions , neglected in the collective mean - field description based on the standard poisson - boltzmann theory @xcite . image - induced , ion - depletion effects for strictly neutral dielectric surfaces have been studied extensively in recent years ( see , e.g. , refs . @xcite and references therein ) and the role of multivalent ions , in particular , has been considered in refs . @xcite . in the present work , we add a new feature to the current understanding of the effective interaction between neutral surfaces and consider the situation , in which the surfaces are neutral only _ on the average _ but otherwise carry a quenched random distribution of positive and negative charges . indeed , heterogeneously or randomly charged surfaces are commonplace in soft matter and have attracted a lot of attention in recent years ( see , e.g. , refs . we have thus shown that the presence of _ quenched surface charge disorder _ in conjunction with _ mobile multivalent ions _ in the ionic solution leads to remarkable and significant qualitative changes in the standard picture commonly accepted for the interaction between neutral bodies . the charge disorder has several manifestations in the present context : first , the self - interactions between the quenched random charges on the inner surfaces of the dielectric slabs and their image charges lead to a repulsive , short - ranged interaction pressure , @xmath91 , that tends to counteract the attractive vdw interaction pressure , @xmath96 as discussed thoroughly elsewhere @xcite . the surface charge disorder , on the other hand , modifies the single - ion interaction potentials in a way that creates a singular attractive potential acting between ions and the dielectric boundaries @xcite . this effect works against the ion - image repulsions that tend to deplete ions from the slit region between the slabs . thus , as a result of this attractive disorder - induced potential , more mono- and multivalent ions are accumulated in the slit when the confining dielectric boundaries are randomly charged . however , we find qualitative differences in the ways this excess attraction affects the osmotic pressure of ions . the osmotic pressure due to monovalent ions , @xmath93 , can become _ repulsive _ which , in fact , agrees with the standard mean - field picture since monovalent ions are only weakly coupled to the bounding surfaces and create a larger entropic pressure upon further accumulation in the slit . on the other hand , the osmotic pressure due to multivalent ions , @xmath92 , can become even more _ attractive _ ! this is in clear contrast with the image - induced , ion - depletion mechanism , in which a _ decrease _ in the number of mono- and multivalent ions in the slit gives rise to ( stronger ) attractive osmotic pressures on the slabs , similarly for both types of ions . this rich behavior stems from combined effects of surface charge disorder and the presence of mobile multivalent ions , with the latter creating strong inter - surface attractions upon further accumulation in the slit . as a result , the effective total interaction pressure between randomly charged , net - neutral dielectric surfaces can differ qualitatively from what one expects based on the standard vdw and/or image - induced , ion - depletion interactions for strictly neutral surfaces , where they predict purely attractive interactions ( unless at relatively large bulk salt concentrations , e.g. , above 250 mm , as noted in ref . @xcite ) . in particular , the net effect from the competing electrostatic components due to disorder , mono- and multivalent ions results in a distinct , disorder - induced _ non - monotonic _ behavior . in the _ weak disorder regime _ , the net electrostatic pressure becomes gradually more repulsive ( positive ) at intermediate to large separations as the disorder coupling parameter ( or equivalently the disorder variance , @xmath30 ) on the bounding surfaces of the slabs is increased from zero ( strictly neutral or disorder - free system ) , exhibiting first a shallow minimum and then a pronounced repulsive hump at intermediate separations . as the disorder coupling parameter is further increased into the _ strong disorder regime _ ( typically beyond @xmath129 ) , the behavior of the net electrostatic pressure is reversed ; the repulsive hump is diminished and one finds a strongly attractive ( negative ) interaction pressure with a range and magnitude larger than that of the vdw or the image - induced , ion - depletion interaction pressures between strictly neutral surfaces . the crossover from weak to strong disorder regimes can be effected also by decreasing the salt screening parameter ( through decreasing the bulk concentration of monovalent ions ) and/or increasing the bulk concentration of multivalent ions . the two different regimes of the disorder effects , the strong and the weak disorder regimes , bear also some resemblance to the strong - weak coupling dichotomy that exists for the net - charged surfaces , dependent on the electrostatic coupling parameter @xcite . it is no small feat to be able to partition the behavior of this complicated system by two dimensionless coupling parameters that effectively govern its salient characteristics . finally , we note the qualitative differences between the ion - mediated interactions we find between net - neutral surfaces in this work and those we reported between randomly charged surfaces carrying a finite mean surface charge density , @xmath130 , in refs . one of the key differences is that the repulsion between mean surface charges on the inner surfaces of the slabs , @xmath131 , in the latter case contributes a dominant repulsive pressure to the net electrostatic pressure , which is stronger , both in magnitude and range , than @xmath91 and @xmath96 . on the other hand , when the surfaces carry a finite mean surface charge density , the osmotic pressure of monovalent ions , @xmath93 , turns out to be always repulsive and comparable in magnitude and range to @xmath131 ; these repulsive pressure components completely mask the short - range interaction pressures , @xmath91 and @xmath96 , whose roles in competition with the attractive pressure due to multivalent ions , @xmath92 , are brought up only in the case of net - neutral surfaces . notably , we find that @xmath93 in the case of surfaces carrying a finite mean surface charge density hardly responds to changes in the disorder strength , while , in the case of net - neutral surfaces , it can change from a purely attractive component ( due to image - induced , ion - depletion effects ) for disorder - free and weakly disordered surfaces to a repulsive one ( due to disorder - driven ion accumulation in the slit ) for strongly disordered surfaces . another difference between the two cases is that multivalent ( counter- ) ions in the case of surfaces carrying a finite mean surface charge density correlate strongly also with the mean charge densities of the confining boundaries of the slit , in line with the strong - coupling paradigm for electrostatics of charged surfaces known to generate strong like - charged surface attractions ( see , e.g. , refs . such strong - coupling effects are clearly absent in the present case with net - neutral surfaces . in summary , our analysis provides new insight into the intricate role of surface charge disorder in the context of net - neutral surfaces and its fingerprints on the effective ion - mediated interactions between them . these interactions are predicted to occur with a range and magnitude comparable to or , for strongly disordered systems , much larger than the vdw and/or image - induced , ion - depletion effects considered previously only in the case of strictly neutral surfaces . our predictions will thus be amenable to numerical and/or experimental verification . a.n . acknowledges partial support from the royal society , the royal academy of engineering , and the british academy ( uk ) . acknowledges support from the school of physics , institute for research in fundamental sciences ( ipm ) , where she stayed as a short - term visiting researcher during the completion of this work . r.p . acknowledges support from the slovene agency of research and development ( arrs ) through grant nos . p1 - 0055 and n10019 as well as the travel grant from the simons foundation and the aspen center for physics , supported by national science foundation grant phy-1066293 . this constraint on multivalent counterions can be achieved by enclosing the outer regions ` behind ' the slabs in semi - permeable membranes , although , in the regime of parameters considered in this paper ( e.g. , with slab thicknesses much larger than the debye screening length ) , this constraint has no sizable impact on our results .

four - dimensional conformal field theories have attracted much attention in the last decade , mainly because of their relevance in the context of the ads / cft correspondence and its developments as well as its applications in completely different fields of which condensed matter is an example . the conformal symmetry imposes stringent constraints on the theory @xcite . in the case of two dimensions , for example , where scale invariance implies conformal invariance , the symmetry is enough to solve the theory @xcite . in four and higher dimensions , this is in general not true @xcite , since examples are known of theories with scale but not conformal symmetry @xcite , but meaningful statements can be made as well ( e.g. @xcite for the four - dimensional case ) . moreover , in the presence of supersymmetry , superconformal invariance puts restrictions on the operator scaling dimensions , independently of the space - time dimensionality @xcite . for these theories , a great deal of information can be obtained from the conformal group , that in four dimensions is @xmath3 and that acts non - linearly on the coordinates , due to the presence of operations such as the inversion and the special conformal transformations . however , it has been long known that it is possible to formulate the theory in such a way that the conformal group will act linearly on the coordinates , in the same fashion as the angular momentum does . in addition , correlation functions automatically exhibit manifest conformal symmetry and any results written in terms of the embedding coordinates are valid not just in minkowski space but in any conformally flat space - time . this approach goes back to @xcite and had applications to m / string theory branes @xcite as well as to more complicated conformal field theories in @xcite ( in particular @xcite generalizes the construction from scalars to more generic tensor fields ) . moreover , embeddings of chiral conformal superspaces were considered in @xcite , in terms of off - shell twistors @xcite , while @xcite and @xcite review the general ( chiral and non - chiral ) case . in particular , using the twistor notation , @xcite gives the two - point correlators between chiral and anti - chiral superfields for arbitrary @xmath2 . in some cases , @xmath4-point correlators have been calculated for various values of @xmath4 ( see for example @xcite where @xmath5 and the primaries are not chiral superfields ) . the supertwistor approach was also used in @xcite to study the superconformal structure of 4d @xmath6 compactified harmonic / projective superspace . the results presented here ( as well as in @xcite ) are equivalent to those obtained with twistor calculations . in the embedding method , one introduces two additional coordinates and embeds the four - dimensional space - time into a larger six - dimensional space with signature @xmath7 . in this space the conformal group generators act as angular momenta . the four - dimensional space is recovered by constraining the six - dimensional coordinates on the projective light - cone . the group @xmath3 is isomorphic to @xmath8 . as a consequence , one can re - express everything in spinor indices and introduce gamma matrices for the basis change from @xmath3 to @xmath8 . in @xcite , the authors generalize the embedding methods to @xmath0 superconformal field theories . the superconformal group is @xmath9 and the six - dimensional complex superspace is constructed out of the coordinates @xmath10 , which contain the standard space - time @xmath11 , one fermionic direction @xmath12 and an additional bosonic coordinate @xmath13 . @xmath14 transforms linearly under a superconformal transformation . upon projecting on the projective light - cone , the four - dimensional superspace @xmath15 is recovered . moreover , scalar and holomorphic fields are considered and it is shown that they correspond to four - dimensional @xmath0 chiral superfields whose @xmath16 component is an @xmath0 chiral primary operator @xcite . in this paper we generalize the embedding method to extended supersymmetry where the superconformal group is @xmath17 . most of the arguments are similar to the ones in @xcite and analogue results are obtained by considering more fermionic coordinates @xmath18 , with @xmath19 . automatically , more bosonic variables @xmath20 also appear in the six - dimensional superspace , which will then be removed once the light - cone constraint is imposed . we define such a set - up in section [ section : conformal and superconformal group ] , where the generators as well as the necessary representations are introduced . in section [ section : superspace ] we construct the @xmath2-extended four - dimensional superspace . this is done as a coset construction , achieved by applying translations and supersymmetry transformations to a reference origin . the resulting space is invariant under the superconformal and lorentz groups . in section [ section : the chiral sector ] we consider representations of the superconformal algebra in terms of superfields . in particular , we generalize the arguments of @xcite to @xmath2-extended chiral superfields , find their transformation rules under the @xmath21 symmetry subgroup of the superconformal algebra as well as under generic superconformal transformations . we also study in detail the case of chiral superfields in @xmath1 extended supersymmetry . as an aside result of our calculations , we have discovered an interesting connection between the number of component fields of a given kind in an @xmath2-extended supermultiplet and the so - called pascal s pyramid at layer @xmath2 . this aspect of number theory seems not to be mentioned explicitly in the literature . one can use it as a check that superfield expansions are correct . throughout this paper we use the conventions of @xcite and @xcite . in particular , the various definitions for the spinor indices and the gamma matrices are the same as in @xcite . in the appendix we give some details of the calculations as well as our notation . in particular , in appendix [ explicit su(2,2 ) transformations ] we compute explicitly the @xmath8 transformations of the coordinates @xmath11 . in appendix [ appendix useful identities ] we give our notation for the spinorial indices and show some identities that will be useful in our computations . in appendix [ 4d superspace barred ] we provide the necessary for computing the coordinates of the 4d barred superspace . in appendix [ appendix pascal pyramid ] we define the pascal pyramid , with some of its properties and symmetries , construct the @xmath5 chiral multiplet and relate it to the pyramid at layer @xmath22 . the superconformal group in four dimensions has fifteen generators that satisfy the @xmath3 algebra . it includes the poincar generators ( @xmath23 for translations and @xmath24 for lorentz transformations ) as a subalgebra . in addition there are the generators @xmath25 for special conformal transformations as well as the dilatation generator @xmath26 . special conformal transformations can be thought of as an inversion , followed by a translation and then by another inversion and act non - linearly on the coordinates @xmath27 . schematically : @xmath28 with real parameters @xmath29 , @xmath30 , @xmath31 and @xmath32 . here indices are raised and lowered by the four - dimensional metric @xmath33 . the conformal group @xmath3 is identical to the lorentz group in a space with six dimensions and signature @xmath7 . one can use this observation to make the conformal transformations act linearly on the coordinates . define new variables @xmath34 , with @xmath35 . under @xmath3 , @xmath34 transforms linearly : @xmath36 infinitesimally , @xmath37 , the conformal transformation @xmath38 is generated by the @xmath3 differential operators @xmath39 in order to recover the four - dimensional space , one has to demand that the coordinates @xmath34 are constrained on the projective light - cone : @xmath40 since the conformal group @xmath3 is equivalent to @xmath8 , we can transform everything into spinor notation . @xmath8 is the group of four by four special matrices that are unitary with respect to a fixed invariant matrix of signature @xmath41 . the connection is realized by using gamma matrices to transform the vector index @xmath42 into an anti - symmetric pair of spinor indices ( @xmath43 ) . a spinor @xmath44 of @xmath8 transforms as @xmath45 the @xmath8 index @xmath46 has an undotted component transforming in the fundamental representation of @xmath47 and a dotted component transforming in the complex conjugate representation , @xmath48 , with @xmath49 and @xmath50 ( e.g. @xcite ) . the vector @xmath34 becomes an anti - symmetric tensor @xmath51 , with each index transforming as in ( [ su(2,2 ) spinor transformation ] ) : @xmath52 with @xmath53 the light - cone constraint is written as @xmath54 with @xmath55 . similarly , one can express the @xmath3 generators @xmath56 in spinor notation . the result is @xmath57 , @xmath58 , with @xmath59 being the @xmath8 generators . the @xmath2-extended superconformal group in four dimensions is @xmath17 consisting of matrices of the form @xmath60 this is a block matrix , with the indices running in the range @xmath61 and @xmath62 . the diagonal blocks are commuting , while the off - diagonal ones are anti - commuting . an @xmath17 vector in the fundamental representation is written as @xmath63 where the @xmath44 component is fermionic and the @xmath2 @xmath64 fields are bosonic . indices are raised and lowered with the invariant matrix @xmath65 where @xmath66 is the @xmath8 invariant metric . in order to be an element of @xmath17 , the matrix @xmath67 must have unit superdeterminant : @xmath68 it must also satisfy @xmath69 where @xmath70 . one can introduce the @xmath17 generators @xmath71 as @xmath72 hence : @xmath73 with @xmath74 , @xmath75 , @xmath76 and @xmath77 . from the superdeterminant , it follows that @xmath71 have vanishing supertrace : @xmath78 where @xmath79 is the trace of @xmath80 , @xmath81 . from ( [ definition of su(2,2|n ) ] ) , it follows that @xmath82 hence , explicitly * for @xmath83 : @xmath84 , which implies that @xmath85 is a @xmath86 generator * for mixed indices : @xmath87 ( equivalently @xmath88 ) * for @xmath89 : @xmath90 ( equivalently @xmath91 ) , which states that @xmath80 is hermitian and hence @xmath92 is unitary , @xmath93 , as it should be since @xmath80 is the @xmath94-symmetry generator . in order to make @xmath85 an @xmath8 generator ( and hence @xmath95 ) , one can subtract its trace and replace @xmath71 by @xmath96 in the fundamental representation . by defining supercharges according to @xmath97 v_a\,,\ ] ] one has @xmath98 similarly , the @xmath94-symmetry generator is a non - abelian matrix @xmath99 which contains the @xmath100 block of the extended supersymmetry . many concepts here are similar to those presented in @xcite . if @xmath101 transforms in the fundamental representation , @xmath102 and @xmath103 in the anti - fundamental , @xmath104 then the product @xmath105 transforms as a tensor @xmath106 the singlet representation is given by its supertrace , @xmath107 where @xmath108 which , due to the property of the supertrace @xmath109 , is invariant under @xmath17 transformations . the adjoint representation is given by the part of @xmath110 with vanishing supertrace , @xmath111 tensors @xmath14 and @xmath112 transform ] ] as the products @xmath113 and @xmath114 : @xmath115 this implies that the product @xmath116 transforms as @xmath110 and hence the scalar product @xmath117 is @xmath17 invariant . the tensor @xmath14 contains the superspace components of six - dimensional conformal group . explicitly : @xmath118 where @xmath11 is anti - symmetric , @xmath119 are fermionic variables and @xmath120 is a symmetric matrix of bosonic coordinates . in order to use the invariant scalar product ( [ invariant scalar product ] ) , we need the conjugate of @xmath121 , @xmath112 , with components @xmath122 defining @xmath123 and @xmath124 , and using the invariant matrix ( [ invariant matrix ] ) , the components are @xmath125 , @xmath126 , @xmath127 . the full superspace is then described by the supercoordinates @xmath10 , endowed with the scalar product ( [ invariant scalar product ] ) . in components : @xmath128 the superconformal transformations of the coordinates can be computed by using ( [ superconformal transformations ] ) . for @xmath14 , they are @xmath129 , or in components : [ superconformal transformations for superspace ] @xmath130 for @xmath112 , they are @xmath131 , or in components : [ superconformal transformations for superspace bar ] @xmath132 in this section we construct the four - dimensional superspace , described by the projective coordinates @xmath133 , @xmath134 , on the six - dimensional light - cone @xmath135 we start from the origin defined as @xmath136 one can check that the origin satisfies the light - cone constraint ] ] . an arbitrary point in superspace is reached by applying all possible superconformal transformations ( [ superconformal transformations for superspace ] ) and ( [ superconformal transformations for superspace bar ] ) . under superconformal transformations , the origin transforms as [ transf of the origin ] @xmath137 explicitly , from table [ transformations of the origin ] in appendix [ explicit su(2,2 ) transformations ] , the origin is invariant under special conformal and lorentz transformations , and projectively invariant under dilatations . it is also invariant under supersymmetry transformations generated by a spinor of the form @xmath138 because in this case @xmath139 in ( [ transf of the origin ] ) has only the down ( dotted ) component . hence a generic point in superspace is reached from the origin by applying translations with parameter @xmath140 , and supersymmetry transformations with spinorial parameter @xmath141 then the four - dimensional superspace is a coset space where points are labelled by @xmath142 and the symmetries above have been modded out . we denote the whole set of theta coordinates by curly brackets : @xmath143 the explicit transformation that brings the origin to the point @xmath142 in superspace is a product of two commuting transformations , one in ordinary space - time and the other one in the theta directions : @xmath144 the space - time part is as in @xcite : @xmath145 where @xmath146 is the @xmath147 identity matrix . the theta part is a generalization of the expression used in @xcite ( see appendix [ appendix useful identities ] ) : @xmath148 this matrix is also equal to the product of analogous matrices , but with only one theta coordinate different from zero : @xmath149 where in the @xmath150-th factor @xmath151 if @xmath152 one has : @xmath153 ] . one can explicitly check that each of the single factors commutes with each other as well as with the space - time transformation ( [ space - time factor in the product ] ) and that , consequently , the full theta matrix ( [ theta factor in the product ] ) commutes with ( [ space - time factor in the product ] ) too . the complete transformation from the origin to any point in the superspace is explicitly @xmath154 where @xmath155 is the coordinate of the @xmath2-extended chiral superspace , or e.g. @xcite explicitly for @xmath1 . ] . we use eq . ( [ explicit full transformation in 4d superspace ] ) to deduce the coordinates of the 4d superspace . starting from the origin ( [ origin in spinor notation ] ) and using ( [ superconformal transformations ] ) , or equivalently ( [ superconformal transformations for superspace])-([superconformal transformations for superspace bar ] ) , one has . ] : [ coordinates of 4d superspace ] @xmath156 where @xmath157 and we use the epsilon tensor to lower spinorial indices , @xmath158 , so that @xmath159 . similarly , for the barred coordinates one finds ( more details in appendix [ 4d superspace barred ] ) : [ barred 4d superspace ] @xmath160 where @xmath161 @xmath162 and we have lowered the theta index , @xmath163 . under superconformal transformations , @xmath164 transform as the coordinates of the @xmath2-extended chiral superspace . in fact , using ( [ superconformal transformations for superspace ] ) with @xmath165 as the only non - vanishing parameter , we get the following transformations : [ 4d space - time superconformal transformations ] @xmath166 including also the barred half superspace given by @xmath112 , one gets the full superspace spanned by the coordinates @xmath167 . @xmath0 conformal chiral superfields , @xmath168 , were considered in @xcite . defining a chiral superfield on the light - cone @xmath169 and expanding in powers of theta @xmath170 with @xmath171 , it was checked that the component fields transform as a chiral multiplet under the @xmath0 super poincar group and have non - trivial rules under the special conformal transformations . in particular , the highest component @xmath172 is a chiral primary operator , in the sense that it is annihilated by the special superconformal generators @xmath173 and @xmath174 , which are related to the supercharges by @xmath175 moreover , under the @xmath176 @xmath94-symmetry , one recovers the fact that the @xmath94-charge is proportional to the scaling dimension @xmath177 at a superconformal fixed point @xcite , according to @xmath178 actually , this result continues to hold in the case of extended supersymmetry . under @xmath21 the indices @xmath179 transform with a unitary matrix : @xmath180 on the light - cone we define @xmath181 where the components are functions of ( @xmath182 ) . hence , as @xmath183 is a scalar , i.e. @xmath184 we find @xmath185 under @xmath186 . this result is the same as in the @xmath0 case , as we wanted to show . in the above equation we have used that @xmath187 and hence @xmath188 , @xmath189 , as well as the projective property @xmath190 with @xmath191 . under a generic superconformal transformation generated by the spinor ( [ superconformal parameter ] ) , using the scaling property of the superfield @xmath192 as well as its scalar nature , we have @xmath193 where @xmath194 and the variations @xmath195 and @xmath196 are given by ( [ 4d space - time superconformal transformations ] ) . the transformations for the component fields can be read off directly from this one by equating same powers of theta on both sides . special superconformal transformations are obtained by setting @xmath197 in these equations . @xmath2-extended superfields can be represented in terms of @xmath0 superfields by expanding all the theta coordinates but one . in the case of @xmath2-extended _ chiral _ superfields , the component fields are @xmath0 _ chiral _ superfields and one can repeat the above argument . the four - dimensional expansion strongly depends on the value of @xmath2 , since the number of components grows exponentially with @xmath2 . in the following subsection we will consider @xmath1 . as an example , consider an @xmath1 chiral superfield , that we denote by @xmath198 . the solution to the the chiral constraints @xmath199 and @xmath200 implies that @xmath201 does not depend explicitly on @xmath202 and @xmath203 , but implicitly through the combination @xmath204 ( cf . ( [ chiral coordinate ] and @xcite ) ) : @xmath205 expanding in powers of , say , @xmath206 , one finds ( e.g. @xcite ) @xmath207 where @xmath208 and @xmath209 are @xmath0 scalar superfields and @xmath210 is an @xmath0 spinor superfield . however , for our purposes , we will consider the full expansion in @xmath211 and @xmath206 : @xmath212 here all the components are functions of @xmath213 and the dots represent higher - order terms ( cubic and quartic in @xmath214 and @xmath215 as well as products thereof ) that will vanish on the light - cone . the fields @xmath216 , @xmath217 , @xmath218 , @xmath219 and @xmath26 are anti - symmetric in @xmath220 ( as well as in @xmath221 for @xmath26 ) . all the components transform as follows under a scaling transformation of the coordinates : @xmath222 , @xmath223 ( same for @xmath224 ) , @xmath225 ( same for @xmath226 and @xmath227 ) , etc . the four - dimensional @xmath1 chiral superfield is defined on the light - cone as @xmath228 in the following , we will be using such a notation for the components of this @xmath1 off - shell multiplet . there are a total of @xmath229 fields , of which : * 1 spin-0 scalar field @xmath230 * 2 spin-@xmath231 fields @xmath232 and @xmath233 * 1 spin-1 tensor field @xmath234 * 2 spin-1 fields @xmath235 and @xmath236 * 2 spin-@xmath237 fields @xmath218 and @xmath219 * 1 spin-2 field @xmath26 . this list agrees with the ones in e.g. @xcite . these numbers make up the pascal pyramid we give all the necessary material about the pascal pyramid . for the interested reader , more technical information can be found in @xcite . ] at layer 2 : @xmath238 in appendix [ appendix pascal pyramid ] we consider the @xmath5 case : there , the triangle analogue to ( [ pascal pyramid for n=2 chiral superfield ] ) is larger , contains 15 entries and corresponds to the pyramid at layer 4 . more generically , the component fields of various kind in an @xmath2-extended supermultiplets are equal in number to the elements of the pascal pyramid at layer @xmath2 . the connection with the trinomial with power @xmath2 , which is at the origin of the pyramid , arises from the fact that in the product @xmath239 the powers @xmath240 can only have three values , @xmath241 , with @xmath242 . using standard techniques , we can now identify the component fields . we find : [ n=2 components of chiral superfield ] @xmath243\\ \chi^a(y)&= & ( x^+)^{\delta+1 } \left[\chi^a(x_{\alpha\beta},0 ) -iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}\chi_{\dot{a}}(x_{\alpha\beta},0)\right]\\ \label{n=2 components of chiral superfield : z - term } z^{ab}(y)&= & i(x^+)^{\delta+1}\epsilon^{ab } \left[-4\frac{\partial a}{\partial\varphi_{12}}(x_{\alpha\beta},0 ) + x_{\alpha\beta}z_{\rm aver}^{\alpha\beta}(x_{\alpha\beta},0)\right]\\ b(y)&= & i(x^+)^{\delta+1 } \left[2\frac{\partial a}{\partial\varphi_{11}}(x_{\alpha\beta},0 ) -x_{\alpha\beta}v^{\alpha\beta}(x_{\alpha\beta},0)\right]\\ c(y)&= & i(x^+)^{\delta+1 } \left[2\frac{\partial a}{\partial\varphi_{22}}(x_{\alpha\beta},0 ) -x_{\alpha\beta}y^{\alpha\beta}(x_{\alpha\beta},0)\right]\\ e^a(y)&= & i(x^+)^{\delta+2 } \left[2\sqrt{2}\frac{\partial\lambda^a}{\partial\varphi_{12}}(x_{\alpha\beta},0 ) + 2\sqrt{2}\frac{\partial\chi^a}{\partial\varphi_{11}}(x_{\alpha\beta},0)\right.\nonumber\\ & & \phantom{i(x^+)^{\delta+2 } } -2\sqrt{2}iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}\frac{\partial\lambda_{\dot{a}}}{\partial\varphi_{12}}(x_{\alpha\beta},0 ) -2\sqrt{2}iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}\frac{\partial\chi_{\dot{a}}}{\partial\varphi_{11}}(x_{\alpha\beta},0)\nonumber\\ & & \phantom{i(x^+)^{\delta+2 } } \left .- x_{\alpha\beta}e^{\alpha\beta a}(x_{\alpha\beta},0 ) -x_{\alpha\beta}iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}e^{\alpha\beta}_{\phantom{\alpha\beta}\dot{a}}(x_{\alpha\beta},0)\right]\\ f^a(y)&= & i(x^+)^{\delta+2 } \left[2\sqrt{2}\frac{\partial\chi^a}{\partial\varphi_{12}}(x_{\alpha\beta},0 ) + 2\sqrt{2}\frac{\partial\lambda^a}{\partial\varphi_{22}}(x_{\alpha\beta},0)\right.\nonumber\\ & & \phantom{i(x^+)^{\delta+2}}-2\sqrt{2}iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}\frac{\partial\chi_{\dot{a}}}{\partial\varphi_{12}}(x_{\alpha\beta},0 ) -2\sqrt{2}iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}\frac{\partial\lambda_{\dot{a}}}{\partial\varphi_{11}}(x_{\alpha\beta},0)\nonumber\\ & & \phantom{i(x^+)^{\delta+2}}\left .- x_{\alpha\beta}f^{\alpha\beta a}(x_{\alpha\beta},0 ) -x_{\alpha\beta}iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}f^{\alpha\beta}_{\phantom{\alpha\beta}\dot{a}}(x_{\alpha\beta},0)\right]\\ \label{n=2 components of chiral superfield : d - term } d(y)&= & ( x^+)^{\delta+2 } \left[2\frac{\partial^2 a}{\partial\varphi_{12}^2}(x_{\alpha\beta},0 ) -2x_{\alpha\beta}\frac{\partial z_{\rm aver}^{\alpha\beta}}{\partial\varphi_{12}}(x_{\alpha\beta},0 ) -x_{\alpha\beta}x_{\gamma\delta}d^{\alpha\beta\gamma\delta}(x_{\alpha\beta},0)\right.\nonumber\\ & & \phantom{(x^+)^{\delta+2}}\left.+2x_{\alpha\beta}\frac{\partial v^{\alpha\beta}}{\partial\varphi_{22}}(x_{\alpha\beta},0 ) + 2x_{\alpha\beta}\frac{\partial y^{\alpha\beta}}{\partial\varphi_{11}}(x_{\alpha\beta},0)\right]\,.\end{aligned}\ ] ] in order to derive the components above , we have made use of the identity @xmath244 which is valid on the light - cone for each @xmath150 . note that a similar identity does not hold for @xmath245 if @xmath246 . for this reason , and in order to simplify the formulas above , we have introduced an auxiliary , or average , variable @xmath247 , defined as : @xmath248 the quantity @xmath247 makes the scaling properties of the component explicit . it appears in ( [ n=2 components of chiral superfield : z - term ] ) and ( [ n=2 components of chiral superfield : d - term ] ) and should be viewed as a shortcut of more lengthy expressions corresponding to the expansion : @xmath249=\nonumber\\ & = & ( x^+)^2\theta_{1a}\theta_{2b}\left[z^{ab}(x,\varphi)+iy^\mu(\bar{\sigma}_\mu)^{\dot{b}b}z^{a}_{\phantom{a}\dot{b}}(x,\varphi)+ iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}z_{\dot{a}}^{\phantom{\dot{a}}b}(x,\varphi)\right.\nonumber\\ & & \left.+\frac{1}{2}\left(y^2\epsilon^{\dot{a}\dot{b}}\epsilon^{ab}- 4y_\mu y_\nu ( \epsilon\sigma^{\lambda\mu})^{ab}(\bar{\sigma}^{\lambda\nu}\epsilon)^{\dot{a}\dot{b}}\right ) z_{\dot{a}\dot{b}}(x,\varphi)\right]\,.\end{aligned}\ ] ] in going from the first equality to the second we have exploited the fact that the product @xmath250 is symmetric in @xmath251 and hence we can use the symmetric version of the product of two ( barred ) sigma matrices ( see appendix [ appendix useful identities ] ) . the r.h.s . of ( [ zaverage ] ) is in components : @xmath252\ ] ] and hence one can derive the components of @xmath253 , using ( [ coordinates of 4d superspace ] ) for the field @xmath11 . expanding now both sides in @xmath20 , we are left with ordinary fields depending on @xmath11 only . we obtain : @xmath254\\ \theta_{1\alpha}\theta_{2\beta}4ix^+\theta_1\cdot\theta_2\frac{\partial z^{\alpha\beta}}{\partial\varphi_{12}}(x,0)&= & ( x^+)^2\theta_{1a}\theta_{2b}\big[4ix^+\theta_1\cdot\theta_2\left ( \frac{\partial z^{ab}}{\partial\varphi_{12}}(x,0)\right.\nonumber\\ & & + iy^\mu(\bar{\sigma}_\mu)^{\dot{b}b}\frac{\partial z^{a}_{\phantom{a}\dot{b}}}{\partial\varphi_{12}}(x,0)+ iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}\frac{z_{\dot{a}}^{\phantom{\dot{a}}b}}{\partial\varphi_{12}}(x,0)\nonumber\\ & & \left.+\frac{1}{2}\left(y^2\epsilon^{\dot{a}\dot{b}}\epsilon^{ab}- 4y_\mu y_\nu ( \epsilon\sigma^{\lambda\mu})^{ab}(\bar{\sigma}^{\lambda\nu}\epsilon)^{\dot{a}\dot{b}}\right ) \frac{\partial z_{\dot{a}\dot{b}}}{\partial\varphi_{12}}(x,0 ) \right ) \big]\,.\end{aligned}\ ] ] this modifies ( [ n=2 components of chiral superfield : z - term ] ) and ( [ n=2 components of chiral superfield : d - term ] ) , which explicitly read : @xmath255\right]\\ d(y)&= & ( x^+)^{\delta+2 } \left[2\frac{\partial^2 a}{\partial\varphi_{12}^2}(x,0 ) -x_{\alpha\beta}x_{\gamma\delta}d^{\alpha\beta\gamma\delta}(x,0 ) + 2x_{\alpha\beta}\frac{\partial v^{\alpha\beta}}{\partial\varphi_{22}}(x,0 ) + 2x_{\alpha\beta}\frac{\partial y^{\alpha\beta}}{\partial\varphi_{11}}(x,0 ) \right.\nonumber\\ & & \phantom{(x^+)^{\delta+2 } } \left . -i\epsilon_{ab}x^+\big [ \frac{\partial z^{ab}}{\partial\varphi_{12}}(x,0 ) + iy^\mu(\bar{\sigma}_\mu)^{\dot{b}b}\frac{\partial z^{a}_{\phantom{a}\dot{b}}}{\partial\varphi_{12}}(x,0)+ iy^\mu(\bar{\sigma}_\mu)^{\dot{a}a}\frac{z_{\dot{a}}^{\phantom{\dot{a}}b}}{\partial\varphi_{12}}(x,0 ) \right.\nonumber\\ & & \phantom{(x^+)^{\delta+2 } } \phantom{-i\epsilon_{ab}(x^+ ) } \left.+\frac{1}{2}\left(y^2\epsilon^{\dot{a}\dot{b}}\epsilon^{ab}- 4y_\mu y_\nu ( \epsilon\sigma^{\lambda\mu})^{ab}(\bar{\sigma}^{\lambda\nu}\epsilon)^{\dot{a}\dot{b}}\right ) \frac{\partial z_{\dot{a}\dot{b}}}{\partial\varphi_{12}}(x,0 ) \big ] \right]\,.\end{aligned}\ ] ] using ( [ component transformations from superfield ] ) we can determine the superconformal transformations for the components of the @xmath1 superfield . we obtain : @xmath256 special conformal transformations correspond to @xmath257 , as it was written in @xcite . by setting @xmath257 , we see that at the origin the @xmath258 component is invariant , namely @xmath259 , and hence @xmath172 is annihilated by the special superconformal generators @xmath260 , defined by @xmath261 this can be checked by using the appropriate generalizations of those operators from the @xmath0 case of @xcite , which tells us that @xmath192 is a chiral field generated by a chiral primary operator @xmath230 . this is a generic property of the chiral sector . in fact , using the transformation rules ( [ 4d space - time superconformal transformations ] ) , for special conformal transformations , @xmath262 ( the vertical bar denoting the @xmath263 component of the r.h.s . , which corresponds to taking the spinorial @xmath264-component of @xmath183 ) will always be proportional to @xmath265 and hence will vanish at the origin . in this paper we generalized the construction of superembedding methods for 4d @xmath0 in @xcite to embeddings of @xmath2-extended superconformal field theories . in this way , the superconformal group acts linearly on the coordinates of the ambient space , the conformal symmetry is manifest and moreover the method is valid for any conformally flat space , not just minkowski . we have considered explicitly the case of @xmath1 chiral superfields in four dimensions and concluded that any conformal chiral superfield of the @xmath2-extended supersymmetry is generated by a chiral primary operator sitting in its highest component . we have also noted a correspondence between the number of component fields of a certain type in any chiral multiplet of the @xmath2-extended supersymmetry and the entries of the pascal pyramid at layer @xmath2 . this correspondence has been explicitly showed in the cases of @xmath1 and @xmath5 supersymmetry . we have not considered correlation functions , that for @xmath0 were presented in @xcite , neither non - holomorphic operators and higher - rank tensors , which among other things are relevant in ads / cft and its applications . these points are left as open questions . we would like to acknowledge beatriz gato rivera for carefully reading the manuscript , the instituto de fisica teorica in madrid , ift - csic / uam , for hospitality during the completion of this work , warren siegel , andreas stergiou and dimitri sorokin for correspondence , and the npb referee for useful comments . this research has been supported by funding of the project consolider - ingenio 2010 , program cpan ( csd2007 - 00042 ) . for the six - dimensional vector @xmath266 the transformations under the conformal group @xmath3 are @xmath267 , with @xmath268 anti - symmetric . the four - dimensional coordinates @xmath27 are related to the six - dimensional ones by solving the light - cone constraint : @xmath269 using the conventions of @xcite , since and @xmath270 is @xmath271 ] @xmath272 in @xmath8 notation , these transformations are : @xmath273 in the next subsections , we consider the single conformal transformations explicitly . we will also apply these transformations to the origin ( [ origin in spinor notation ] ) and will find the summarizing table [ transformations of the origin ] , that is given here for convenience . .summary of @xmath8 transformations of the origin [ cols="^,^ " , ] [ transformations of the origin ] the only non - zero parameter is @xmath274 . hence @xmath275 the second line gives the four - dimensional translation @xmath276 and consistently the third line gives @xmath277 . using ( [ generic su(2,2 ) transformation ] ) one has : @xmath278 when evaluated at the origin , this becomes @xmath279 the only non - zero parameter is @xmath280 . this implies : @xmath281 the second line implies the four - dimensional lorentz transformation @xmath282 and consistently the third line gives @xmath283 . in @xmath8 notation , one has @xmath284 which vanishes at the origin , @xmath285 here @xmath286 is non - zero and @xmath287 the first two lines combined give the four - dimensional special conformal transformation @xmath288 with @xmath289 . in @xmath8 notation , one has @xmath290 which vanishes at the origin , @xmath285 the non - vanishing parameter is @xmath291 and @xmath292 the first two lines combined give the four - dimensional scale transformation @xmath293 while the third line gives consistently @xmath294 . in spinor notation , @xmath295 at the origin this becomes @xmath296 in this appendix we write down some spinor identities that are used in the calculations of the main text . our conventions for the spinors are the same as in @xcite , in particular : @xmath297 where @xmath298 the gamma matrices are given by : @xmath299 and @xmath300 the @xmath3 coordinate @xmath34 and the anti - symmetric @xmath8 tensor @xmath11 are related by @xmath301 and @xmath302 as it is standard in supersymmetry , for each @xmath150 , the product @xmath303 must be proportional to @xmath304 , i.e. @xmath305 , where the proportionality constant @xmath306 is fixed by multiplying both sides by @xmath307 and summing over @xmath308 and @xmath309 ( which gives @xmath310 ) . after anti - commuting the theta s , one finds @xmath311 and hence , for each @xmath150 : @xmath312 this expression was used in ( [ theta factor in the product ] ) . one can make similar manipulations in extended supersymmetry and write ( repeated indices are not summed ) : @xmath313 which were used , for example , in ( [ n=2 components of chiral superfield ] ) with @xmath314 . our conventions for the sigma matrices are as in @xcite and @xcite . in particular , @xmath315 and @xmath316 . these are related by @xmath317 this implies that @xmath318 the former equation was used in ( [ coordinates of 4d superspace ] ) . there , we also used the following relation : @xmath319 this comes from the more generic relation regarding the product of two sigma matrices , which can be expressed in terms of the generators @xmath320 and @xmath321 of the lorentz group in the spinor representation , which are anti - symmetric in the indices @xmath42 and @xmath4 ( see below ) . however , the symmetric part in the space - time indices and the anti - symmetric part in the spinorial indices is fully specified by the metric tensor and the epsilon tensor as : @xmath322}= -\frac{1}{2}\eta_{\mu\nu}\epsilon^{\dot{a}\dot{b}}\epsilon^{cd}\,.\ ] ] a similar formula holds for the anti - symmetric combination of the dotted indices as well as for the product of two ( unbarred ) sigma matrices . using the relations above , it is straightforward to check that : @xmath323 these identities were used in ( [ 4d space - time superconformal transformations ] ) . if we only look at the symmetric part in the space - time indices , instead , we have to consider the more general expression @xmath324\,,\ ] ] as it appears in @xcite , which was used in ( [ n=2 components of chiral superfield ] ) . in the translations given in ( [ superconformal transformations ] ) we need to know the inverse of ( [ explicit full transformation in 4d superspace ] ) . because of ( [ full transformation in 4d superspace ] ) , the inverse is a product of the two commuting transformations @xmath325 the space - time part is simply obtained by replacing @xmath326 in ( [ space - time factor in the product ] ) : @xmath327 to compute the theta part , it is convenient to use the following expression for the inversion formula of block matrices : @xmath328 with * @xmath329 * @xmath330 * @xmath331 * @xmath332 * @xmath333 hence : @xmath334 the complete inverse transformation is then the product of ( [ space - time factor in the product , inverse ] ) and ( [ theta factor in the product , inverse ] ) : @xmath335 where @xmath336 and we have again replaced , for each @xmath150 , @xmath337 . this is enough to derive the set of equations ( [ barred 4d superspace ] ) . in number theory , in analogy to the binomial expansion , whose coefficients can be organized into a triangle , one can consider the trinomial : @xmath338 where the integer numbers appearing in the pascal pyramid ( tetrahedron ) are just the coefficients @xmath339 of the expansion ( see figure [ pyramid ] ) of the trinomial . the power @xmath4 indicates the layer of the pyramid . the pyramid has a huge number of symmetries and interesting properties . for instance , by looking at specific examples or by using some combinatorics , one can check that there are @xmath340 elements at each layer . also , the sum of the entries of each layer is @xmath341 , as can be seen by taking @xmath342 in ( [ trinomial expansion ] ) . moreover , by superimposing two consecutive layers , it is possible to prove that each entry at layer @xmath4 is given by the sum of all the entries at layer @xmath343 that surround it . this is simply a property of the trinomial coefficients . in number theory , one generalizes binomials and trinomials to multinomials and , in a similar way , we speak of an @xmath42-simplex , which generalizes triangles and tetrahedrons . a multinomial is by definition @xmath344 where the coefficients are given by @xmath345 in ( [ pascal pyramid for n=2 chiral superfield ] ) we have written down the pascal pyramid at layer two and show that its elements are related to the number of certain kind of fields arising as components of an @xmath1 chiral supermultiplet . here we will do the same for @xmath5 chiral supermultiplets and will find the pyramid at layer four . as we have done in section [ section : the chiral sector ] , a six - dimensional @xmath5 chiral superfield can be expanded in the theta variables as @xmath346 here , @xmath347 can be raised / lowered with the flat ( identity ) metric , all the component fields are functions of @xmath348 and the dots in the end represent quantities that will vanish on the light - cone . the four - dimensional superfield as well as its components are defined on the light - cone and will be functions of @xmath349 only . moreover , the @xmath20-dependence is removed by replacing @xmath350 and expanding everything in @xmath351 . the result is : @xmath352 the fractional coefficients are such that each field appears only once in the sum . the four - dimensional components are expressed in terms of the six - dimensional ones appearing in ( [ 6d chiral n=4 ] ) . for example , for the first few components we have : @xmath353\nonumber\\ f^{ij , ab}(y)&= & ( x^+)^{\delta+1}\big[ix_{\alpha\beta}\epsilon^{ab}e^{ij,\alpha\beta}_{\rm aver.}(x,0 ) -4i\epsilon^{ab}\frac{\partial a_{\rm aver.}}{\partial \varphi_{ij}}(x,0)\big]\nonumber\\ f^{i}(y)&= & ( x^+)^{\delta+1}\big[-ix_{\alpha\beta}e^{i,\alpha\beta}(x,0)+2i\frac{\partial a}{\partial\varphi_{ii}}(x,0)\big]\nonumber\\ \upsilon^{i , abc}(y)&= & ( x^+)^{\delta+2}\big[\epsilon^{ab}ix_{\alpha\beta}\chi_{\rm aver.}^{i,\alpha\beta c}(x,0 ) + \epsilon^{ab}x^{\alpha\beta}y_\mu(\bar{\sigma}^\mu)^{\dot{a}c}\epsilon_{\dot{a}\dot{c}}\chi_{\rm aver.}^{i,\alpha\beta\dot{c}}(x,0)\nonumber\\ & & \phantom{(x^+)^{\delta+2 } } -24i\epsilon^{bc}\epsilon_{ijkl}\frac{\partial \lambda^{ja}}{\partial\varphi_{kl}}(x,0 ) -24\epsilon_{ijkl}\epsilon^{bc}y_\mu(\bar{\sigma}^\mu)^{\dot{a}a}\frac{\partial\lambda^j_{\dot{a}}}{\partial\varphi_{kl}}(x,0)\big]\nonumber\\ \upsilon^{i , a}(y)&= & ( x^+)^{\delta+2}\big[-ix_{\alpha\beta}\chi^{ij,\alpha\beta a}(x,0 ) -x_{\alpha\beta}y_\mu(\bar{\sigma}^\mu)^{\dot{c}a}\epsilon_{\dot{c}\dot{d}}\chi^{ij,\dot{d}}(x,0 ) + 2i\frac{\partial\lambda^{ia}}{\partial\varphi_{ij}}(x,0)\nonumber\\ & & \phantom{(x^+)^{\delta+2 } } + y_\mu(\bar{\sigma}^\mu)^{\dot{a}a}\frac{\partial\lambda^i_{\dot{a}}}{\partial\varphi_{jj}}(x,0 ) -i\frac{\partial\lambda^{ja}}{\partial\varphi_{ij}}(x,0 ) -y_\mu(\bar{\sigma}^\mu)^{\dot{a}a}\frac{\partial\lambda^j_{\dot{a}}}{\partial\varphi_{ij}}(x,0)\big]\nonumber\\ \dots&&\dots\qquad\qquad\dots\qquad\dots\qquad\dots\end{aligned}\ ] ] etc . here the averaged quantities are defined whenever @xmath246 as done in the main text , e.g. @xmath354 and should be regarded as shorter labels of longer expressions . * one scalar field @xmath355 * four spinors @xmath356 in the representation ( 1/2,0 ) * @xmath357 rank-2 tensors @xmath358 , @xmath359 in the representation ( 1,0 ) * four scalars @xmath360 * four spinors @xmath361 in the representation ( 3/2,0 ) * @xmath362 spinors @xmath363 , @xmath246 , in the representation ( 1/2,0 ) * one rank-4 tensor @xmath364 in the representation ( 2,0 ) * @xmath362 rank-2 tensors @xmath365 , @xmath366 and @xmath367 , in the representation ( 1,0 ) * @xmath368 scalars @xmath369 , @xmath359 and symmetric in @xmath370 * four spinors @xmath371 in the representation ( 3/2,0 ) * @xmath372 spinors @xmath373 , @xmath367 and symmetric in @xmath374 , in the representation ( 1/2,0 ) * six rank-2 tensors @xmath375 , @xmath359 and symmetric in @xmath374 , in the representation ( 1,0 ) * @xmath376 scalars @xmath377 , symmetric in @xmath378 * four spinors @xmath379 , in the representation ( 1/2,0 ) * one scalar @xmath380 . this field content can be summarized by using the pascal pyramid at layer @xmath5 : @xmath381 the total number of fields is given by the sum of the elements of the pyramid and amounts , as it should be , to @xmath382 . a. m. polyakov , `` non - 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emergent phenomena is one of the most profound topics in modern science addressing the ways that collectivities and complex patterns appear from multiplicity of components and simple interactions . ensembles of random hamiltonians allow one to explore the emergent phenomena in a statistical way , and thus to establish generic relations and rules . to study the many - body physics of interest we adopt a shell model approach with a two - body interaction hamiltonian . the sets of the two - body interaction strengths are selected at random resulting in the two - body random ensemble ( tbre ) . symmetries , such as rotational , isospin , and parity , entangled with complex many - body dynamics result in surprising regularities discovered recently in the low - lying spectrum . patterns exhibited by the random ensembles are remarkably similar to those observed in real nuclei . the high probability for the ground state spin to be zero is the most astounding feature of the tbre discovered in ref . signs of almost every collective feature seen in nuclei , namely , pairing superconductivity , deformation , and vibration , have been observed in random ensembles @xcite . while the systematics of the ground state quantum numbers is almost not sensitive to the short - range pairing matrix elements , the probability to find a coherent paired structure in the wave - functions of low - lying states is enhanced @xcite . the presence of rotational features in the spectra is another unexpected result seen in the tbre @xcite . the goal of this work is to study the emergence of collective mean - field dynamics in ensembles with random interactions . the discussion is organized as follows : in sec . [ sec : collectivity ] we briefly define the tbre , introduce signatures of collective motion , and discuss ways to detect them . in sec . [ sec : single_level ] we present our study of collectivities in single@xmath0 level models . more complex models are explored in secs . [ sec : two_levels ] and [ sec : realistic - model - space ] . we summarize our results in sec [ sec : summary ] with a discussion of the quadrupole - quadrupole hamiltonian which appears to be responsible for most of the observed phenomena . in the spirit of the traditional shell model approach , we define a model configuration as @xmath1 where @xmath2 nucleons occupy a set of single particle levels labeled by their angular momentum @xmath0 . in this work we assume that the single particle energies are degenerate . we examined other models for which this was not the case and the results are similar . the hamiltonians in the tbre are defined with a set of two - body matrix elements which are selected at random . the distribution of the matrix elements is gaussian so that , within a given symmetry class , the ensemble of hamiltonian matrices for two particles coincides with gaussian orthogonal ensemble . the presence of rotational symmetry and , where relevant , of parity and isospin symmetries is assumed . the typical number of random realizations was between @xmath3 and @xmath4 for all ensembles presented in this work . in the tbre the number of realizations where the ground state spin @xmath5 is disproportionally large . aiming at collective phenomena we select realizations with @xmath5 . with the exception of the ground state , labeled as @xmath6 , we denote the low - lying states by the value of their spin with an identifying subscript . the subscript is given in bold font if it refers to the absolute position of a given state in the spectrum . throughout the paper we give probabilities of finding realizations with certain features , these probabilities are always quoted in percent relative to the size of the ensemble ; however , all probability distribution plots are normalized to unit area . in order to identify and to analyze manifestations of collective phenomena in the spectra we use a set of observables . the goal is to choose a finite number of spectral observables that are likely to convey the most information about possible collective structures in a scale - independent way and with minimal model dependence . these quantities and the logic behind their selection are discussed in what follows . the geometry of the nuclear mean field is described by the multipole density operators @xmath7 with multipolarity @xmath8 and magnetic component @xmath9 . the structure of the multipole operators depends on the valence space , for each model it is addressed separately . the reduced transition rate from an initial state @xmath10 to a final state @xmath11 @xmath12 is one of the observables . here @xmath10 denotes a many - body state with angular momentum @xmath13 and magnetic projection @xmath14 the total transition strength from a state @xmath13 is given by the sum rule @xmath15 which provides a convenient normalization to assess the _ fractional collectivity _ of the transition @xmath16 the shape of a state is described by its multipole moments specified by the expectation value @xmath17 for a non - spherical system this moment describes the shape of a deformed nucleus measured in the lab frame . the intrinsic shape is characterized by the body - fixed ( intrinsic ) multipole moments @xmath18 a rotational spectrum ( band ) emerges for every fixed intrinsic shape . in a rigid rotor these intrinsic moments are the same for all states in the band and they determine the lab - frame observables in eqs . ( [ eq : be ] ) and ( [ eq : qlab ] ) . for the ground state band of interest , the intrinsic moments determine the total transition strength @xmath19 . in the axially symmetric case the quantum number @xmath20 , a projection of the angular momentum onto the body - fixed symmetry axis , is conserved . then for each rotational @xmath20-band the relations between the observables in the lab frame and in the intrinsic frame are expressed via clebsch - gordan coefficients @xmath21 and @xmath22 this limit of an axially symmetric rotor provides a convenient normalization to examine the multipole moments . in this work instead of @xmath23 we quote a normalized intrinsic moment @xmath24 which is computed as if the state is a member of the @xmath25 rotational ground state band . in this paper we only briefly touch the subject of collectivities other than quadrupole , see sec . [ sub : higher - multipole - collectivity ] ; thus for convenience the subscript @xmath8 is omitted for @xmath26 . the relation between the lab - frame moment of the @xmath27 state and its intrinsic moment is @xmath28 for the axially symmetric rotor the quadrupole transition sum rule for the @xmath6 is saturated by a single transition @xmath29 the quadrupole moment is @xmath30 for prolate or @xmath31 for oblate _ _ shapes . we normalize the total transition strength @xmath32 to its maximum possible value for a given valence space . taking the @xmath26 case as an example , we define the quadrupole - quadrupole ( qq ) hamiltonian as @xmath33 the eigenstate energy of the qq hamiltonian coincides with the total transition strength for that state : @xmath34 . thus , the absolute value of the ground state energy of the qq hamiltonian @xmath35 is the maximum possible value of the total transition strength @xmath36 for a given model space and for a given structure of the quadrupole operator . we therefore define a _ relative transition strength _ as@xmath37 to summarize , in our study we use the dimensionless variables defined in eqs . ( [ eq : b ] ) , ( [ eq : q ] ) , and ( [ eq : s ] ) . to shorten notations we define @xmath38 and @xmath39 for collective models of pairing , rotations , and vibrations @xmath40 we refer to a realization with @xmath41 as _ collective _ and with @xmath42 as _ non - collective_. the quadrupole moment @xmath43 allows one to separate different collective modes : @xmath44 for rotations and @xmath45 for vibrations and for paired states . in what follows we allude to collective realizations with @xmath46 as _ prolate _ and those with @xmath47 as _ oblate . _ for rotations the relative transition strength @xmath48 is proportional to the square of the intrinsic moment , and thus it is associated with the hill - wheeler deformation parameter @xmath49 . within elliot s su(3 ) model @xcite the relative transition strength @xmath48 can be thought to represent the expectation value of the casimir operator which identifies the irreducible representation . in cases where @xmath50 the ground state band structure is close to that of the qq hamiltonian . the collective structure is further analyzed using the following @xmath51 state . the types of collective modes can be classified by the ratio of the excitation energies measured relative to the energy of the @xmath6 state @xmath52 this ratio is close to @xmath53 for pairing , 2 for vibration , and 10/3 for rotation . the ratio of deexcitation rates@xmath54 is another measure . it is nearly 0 for pairing , 2 for vibrational mode , and 10/7 for rotational motion . typically , for models with the qq hamiltonian @xmath55 and @xmath56 are close to the rotational values , see summary in tab . [ tab : qq ] . a comprehensive review of different collective models , their analytic predictions , and comparisons with rotational spectra observed in real nuclei can be found in the textbooks @xcite . we begin our presentation with single @xmath0 level models . starting from the original paper @xcite the single @xmath0 level with identical nucleons has been at the center of numerous investigations ; a good summary may be found in the following reviews @xcite . with many issues understood and with still unanswered questions , the single @xmath0 model remains an important exploratory benchmark . the model , while simple , has a number of particularly attractive features which can be of both advantage and disadvantage @xcite : the hamiltonian is defined with a small number of parameters ; apart from an overall normalization constant , the multipole operators are uniquely defined ; a special role is played by the quasispin su(2 ) group ; and the particle - hole symmetry is exact . in fig . [ fig : jl19n6 ] the system with 6 nucleons in a single @xmath57 level is examined , we refer to this system as @xmath58 . here we select 10.4% of random realizations where the @xmath6 state is followed by the @xmath59 state . the distribution of the fractional collectivity @xmath60 in fig . [ fig : jl19n6](a ) points to highly collective nature of the quadrupole transition @xmath61 most realizations with @xmath6 and @xmath59 are collective @xmath62 , their fraction is 7.8% of the total number of samples . these realizations are shaded in red . this collectivity is not a statistical coincidence . the system @xmath58 has 1242 spin - states , among them there are 10 states with @xmath63 and 23 states with @xmath64 . thus , statistically the chance for the @xmath65 spin sequence to occur among all other possible outcomes is only 0.015% . the large fractional collectivity for the transition between these two states is even more unlikely , given that the transition strength is shared among 23 @xmath64 states , the chances for @xmath66 are of the order of 1 in @xmath4 . there are two peaks in the distribution of the quadrupole moment in fig . [ fig : jl19n6](b ) , they reflect prolate and oblate deformations . for most of the collective realizations , which are shaded in fig . [ fig : jl19n6 ] , the magnitude of the quadrupole moment is consistent with the value for the axially deformed rigid rotor @xmath67 . the ground state is most likely to be oblate , but in about one out of four collective cases a prolate mean field emerges . the collective realizations are further analyzed in fig . [ fig : jl19n6_col ] where the distribution of the relative transition strength @xmath48 is shown . in fig . [ fig : jl19n6_col ] the oblate @xmath68 and prolate @xmath69 cases are shaded with different patterns . the relative transition rate @xmath48 for the oblate samples is close to the maximum possible @xmath70 thus , for these realizations the ground state band structure is similar to that of the qq hamiltonian . the data on the qq hamiltonian for our models is summarized in sec.[sec : summary ] . for prolate systems the distribution of the relative transition strength peaks around @xmath71 . . ( a ) the distribution of the fractional collectivity @xmath72 . ( b ) the distribution the intrinsic quadrupole moment @xmath43 . only realizations with the @xmath65 spin sequence are included in both panels . there are 10.4% of such realizations . the 7.8% of collective realizations @xmath62 are shaded . , width=326 ] in fig . [ fig : jl19n6_col024 ] we focus on distributions of the deexcitation ratio @xmath56 and the energy ratio @xmath73 defined in eqs . [ eq : dx ] and [ eq : rr ] . we use the same shading for prolate and oblate realizations as in fig . [ fig : jl19n6_col ] , but slightly modify our selection of samples . we chose the collective realizations that have states @xmath27 and @xmath74 with @xmath27 being not higher than the forth excited state and @xmath51 state being above it . the collective oblate realizations comprise a peak in the distribution of @xmath56 in fig . [ fig : jl19n6_col024](a ) around the rotational limit of @xmath75 . for prolate realizations the distribution peaks near @xmath76 and has an extended shoulder . it is likely that the rotational structure is fragmented in instances with a weak prolate deformation . here the lower value of @xmath48 seen in fig . [ fig : jl19n6_col ] is used to suggest a weak deformation . therefore , the @xmath51 state is not purely rotational . the distribution of the ratio of the excitation energies @xmath55 in fig . [ fig : jl19n6_col024](b ) seems to contradict the rotational limit . for most of the collective realizations the values of the ratio fall between the pairing limit of @xmath77 and the vibrational limit of @xmath78 , while in the rotational limit the ratio of @xmath79 is expected . this discrepancy has been reconciled in ref . @xcite with the observation that the rotational ordering emerges for the ensemble - averaged excitation energies . the same conclusion is expected from the geometrical chaoticity arguments @xcite . excitation energies are sensitive to non - collective features , this leads to large fluctuations of @xmath55 . the experimental observations of realistic nuclei also show that when the quadrupole transition rates follow the rotational systematics , the excitation energy spectrum can deviate from rotational ; on occasions , the spectrum is closer to the vibrational limit @xcite . the coexistence of both prolate and oblate configurations in this @xmath58 system could be another reason for the distortion in the energy spectrum . within elliot s su(3 ) model analogous mixing of group representations was investigated in ref . @xcite . . the distribution of the relative transition strength @xmath48 for the collective realizations ( shaded area in fig . [ fig : jl19n6 ] ) . the quadrupole moments shown in the inset are separated into prolate ( @xmath46 ) and oblate ( @xmath47 ) shapes . the resulting distributions are shaded with a pattern and a uniform color , respectively . the fraction of oblate cases is 5.2% and the fraction of prolate cases is 1.3% relative to the total number of random realizations . , width=326 ] . ( a ) the distribution of the deexcitation ratio @xmath56 defined in eq . ( [ eq : dx ] ) . ( b ) the distribution of the excitation energy ratio @xmath55 defined in eq . ( [ eq : rr ] ) . the distributions are comprised of 13.6% of realizations that have the @xmath80 sequence with @xmath41 , the @xmath27 state is not higher than the fourth excited state , and @xmath81 . the prolate cases and oblate cases , that appear in the ensemble with probabilities 3.3% and 7.1% respectively , are shaded with the same patterns as in fig . [ fig : jl19n6_col ] . the values of @xmath56 and @xmath55 for the qq hamiltonian listed in tab . [ tab : qq ] are marked with the vertical grid lines.,width=326 ] the triaxiality is marked by the presence of low - lying levels @xmath82 . the excitation energies are subject to equalities @xmath83 and @xmath84 . it is remarkable that these relations appear to be well satisfied by the spectrum of the qq hamiltonian in the @xmath58 configuration , for which @xmath85 and @xmath86 here @xmath87 denotes the ratio of excitation energies . the rigid rotor hamiltonian , defined by three moments of inertia , is responsible for these correlations in the spectrum . in this work we examine two low - lying @xmath27 and @xmath88 states . these are the only states with spin 2 in the triaxial rotor model , they are mixed configurations of @xmath25 and @xmath89 . we use angle @xmath90 to express the level of the @xmath20-mixing . this angle is determined by the three reciprocal moments of inertia @xmath91 , @xmath92 in the rotor hamiltonian@xmath93 the ratio of the excitation energies of the @xmath27 and @xmath88 states is another parameter of the rotor hamiltonian . it is convenient to express this ratio @xmath94 in terms of the angle @xmath95 defined using the davydov - filippov model of irrotational flow @xcite as@xmath96 in our example that follows , the triaxiality is small and @xmath97 . thus , the rotor hamiltonian , given by the three moments of inertia , can be equivalently described by an overall energy scale , the @xmath20-mixing angle @xmath90 , and the angle @xmath95 . the quadrupole shape is parametrized by the hill - wheeler parameters @xmath98 and @xmath99 which define the quadrupole operator @xmath100 . the relation between the parameters of the rotor hamiltonian and the intrinsic shape is model - dependent . the irrotational - flow moments of inertia discussed in ref . @xcite result in @xmath101 and @xmath102-\gamma\right\ } /2 $ ] ; the latter implies @xmath103 for small triaxiality . a rather different result follows from the rigid - body moments of inertia . we determine @xmath104 @xmath105 and @xmath99 independently from the spectroscopic observables . the parameter @xmath95 is obtained from the energy spectrum , eq . ( [ eq : trixgammadf ] ) . following ref . @xcite one can view the sum rules @xmath106 and @xmath107 for the @xmath64 two - state model as the pythagorean theorem for amplitudes . the angles in the corresponding right - angled triangles are @xmath108 and @xmath109 therefore @xmath110 @xmath111 these equations allow one to determine the triaxiality @xmath99 and the @xmath20-mixing angle @xmath90 . all three angles @xmath112 @xmath104 and @xmath95 are small in our models with the qq hamiltonian , see discussion in sec . [ sec : summary ] . correspondingly , in the tbre the effects of triaxiality are weak but detectable . for our studies of triaxiality presented in fig . [ fig : jl19n6_trix ] we use the @xmath58 model . we recall that in the triaxial rotor model there is a second @xmath88 state with @xmath113 thus , we select collective realizations with @xmath6 and @xmath27 , and in addition to that we require that in the entire spectrum there is a @xmath88 state for which the equality @xmath114 holds within 20% of accuracy . in collective realizations of rotational type the magnitude of the @xmath115 is large as compared to the quadrupole moments of other many - body states . this simplifies the identification of the @xmath88 state . we find that practically for all collective realizations this second @xmath88 state exists . indeed , from the total number of random realizations a large fraction , 18.3% , satisfy all of the mentioned triaxiality conditions . in figs . [ fig : jl19n6_trix](a ) , [ fig : jl19n6_trix](b ) , and [ fig : jl19n6_trix](c ) we show the distributions of the triaxiality angle @xmath99 , @xmath20-mixing angle @xmath104 and @xmath95 , respectively . we use the same shading as in fig . [ fig : jl19n6_col024 ] to separate prolate and oblate shapes . . ( a ) the distribution of the triaxiality angle @xmath99 . ( b ) the distribution of the @xmath20-mixing angle @xmath90 . ( c ) the distribution of the triaxiality angle @xmath95 from the davydov - filippov model . the angles are obtained from eqs . ( [ eq : trixgamma ] ) , ( [ eq : trixgammaq ] ) and ( [ eq : trixgammadf ] ) . we select realizations with two states of spin 2 in the spectrum and require @xmath41 and @xmath116 ; 18.3% of realizations satisfy this set of restrictions . the realizations with prolate and oblate shapes are shaded with the same patterns as in figs . [ fig : jl19n6_col ] and [ fig : jl19n6_col024 ] . vertical grid lines indicate the triaxiality parameters calculated from the qq hamiltonian , which are : @xmath117 @xmath118 and @xmath119,width=288 ] in the @xmath58 model one often finds collective realizations with oblate intrinsic deformation and @xmath50 , these realizations are triaxial with @xmath120 , fig . [ fig : jl19n6_trix](a ) . this result , as well as @xmath121 in fig . [ fig : jl19n6_trix](b ) , is consistent with that of the qq hamiltonian . the less frequent prolate cases are nearly axially symmetric . in the tbre the angle @xmath95 , fig . [ fig : jl19n6_trix](c ) , appears on average to be higher than the corresponding angle in the qq hamiltonian . the peak in the @xmath95 distribution is also higher than the peak in the @xmath99 distribution , compare figs . [ fig : jl19n6_trix](a ) and [ fig : jl19n6_trix](c ) . ( we remind that @xmath101 in the irrotational flow model . ) nevertheless , no conclusions can be made from these two discrepancies . we believe that the excitation energies could be influenced significantly by non - collective features . the situation may be similar to the one in fig . [ fig : qqj19n8](b ) , where @xmath51 state is lower than expected for the rotor . similarly , if the @xmath88 state is lowered the resulting @xmath95 is larger . in both cases the lowering is relative to the excitation energy of the @xmath27 state . it is known that in the tbre the probability to find a @xmath6 state followed by either one of the states @xmath122 @xmath123 @xmath124 or @xmath125 is disproportionally large as compared to what is statistically expected . for the @xmath58 model the corresponding probabilities are 10.4% , 17.3% , 11.9% and 1.8% . in an attempt to understand this , we repeat the previous study but target the collective realizations of multipolarity @xmath126 , and 8 . for realizations with the @xmath127 state and with the first excited state of spin @xmath8 in fig . [ fig : multipole19n6 ] , we consider the fractional collectivity @xmath128 and the multipole moment @xmath129 . fig . [ fig : multipole19n6 ] shows evidences for intrinsic shapes with deformations of higher multipolarities . in particular , for @xmath130 and 8 there is a sizable number of collective realizations where @xmath131 . these realizations are shaded ( in red ) . the corresponding distributions of the multipole moments in figs . [ fig : multipole19n6](d ) and [ fig : multipole19n6](f ) have peaks which are centered at non - zero values of @xmath132 . the @xmath133 shape collectivity is nearly absent in the @xmath58 system : the realizations are mostly non - collective , @xmath134 ( shaded in blue ) , and the corresponding moment has a peak centered near zero . investigations of other single @xmath0 systems show presence of multipole collectivities with @xmath135 . generally , the collectivities corresponding to the intrinsic quadrupole shape are the most pronounced ones , however there are signatures of realizations with shapes of higher multipole deformation . the existence of such geometric structures may be related to the symmetries discussed in ref . @xcite . the distributions of the fractional collectivity @xmath136 are shown in panels ( a ) , ( b ) , and ( c ) . the distributions of the intrinsic multipole moments @xmath132 are shown in panels ( d ) , ( e ) , and ( f ) . the plots are organized in three rows corresponding to multipolarities with @xmath137 . here we include realizations where , in addition to the @xmath6 state , the first excited state is either @xmath123 or @xmath124 or @xmath138 the shaded areas correspond to collective and non - collective modes with @xmath131 and @xmath139 respectively . we use the same patterns as in fig . [ fig : jl19n6].,width=326 ] in this subsection we discuss the multipole structure of the two - body hamiltonian in the single @xmath0 level model . for this purpose we use a larger system of @xmath140 nucleons in the same @xmath57 model space , i.e. the @xmath141 model . the distributions of the fractional collectivity , the quadrupole moment , and the relative transition strength shown in figs . [ fig : qqj19n8 ] and [ fig : colqqj19n8 ] are similar to the distributions observed in the @xmath58 model in figs . [ fig : jl19n6 ] and [ fig : jl19n6_col ] . the main difference between the models is that , in contrast to fig . [ fig : jl19n6](b ) , only oblate ground state configurations are present in fig . [ fig : qqj19n8](b ) . . the same figure as fig . [ fig : jl19n6 ] but for the 8-particle system . ( a ) the distribution of the fractional collectivity @xmath72 . ( b ) the distribution of the intrinsic quadrupole moment @xmath43 . the histogram is comprised of 7.5% of random spectra with @xmath6 and @xmath59 states . shaded areas correspond to 4.6% of collective realizations ( @xmath41 ) and 1.9% of non - collective realizations ( @xmath142 ) . this figure is analogous to figs . [ fig : jl19n6 ] and [ fig : multipole19n6 ] , and the same shading is used in these figures.,width=326 ] . the same figure as fig . [ fig : jl19n6_col ] but for the 8-particle system . the distribution of the relative transition strength rule @xmath48 for the collective realizations . this figure is analogous to fig . [ fig : jl19n6_col ] , and the same shading is used as in figs . [ fig : jl19n6_col ] and [ fig : jl19n6_col024 ] , however only oblate shapes @xmath68 are seen . , width=326 ] the collectivities observed in the single @xmath0 studies are deeply rooted in the underlying geometric structure of the hamiltonian . to focus on this relation we express the two - body hamiltonian in the particle - hole channel in terms of the multipole operators @xmath143 the interaction parameters @xmath144 in the particle - hole channel are determined from those in the particle - particle channel @xmath145 via pandya transformation @xmath146 the transformation coefficients @xmath147 are given by the six-@xmath0 recoupling coefficients . on a single @xmath0 level only even values of the particle - pair angular momenta @xmath148 are allowed by the fermi statistics . thus , there are @xmath149 interaction parameters @xmath145 in eq . ( [ eq : pp2ph ] ) . in studies of the tbre a set of these parameters can be viewed as a random vector in the @xmath149 dimensional space . there is no such a restriction on the particle - hole momentum @xmath150 . thus , the inverse transformation @xmath151 may produce some unphysical @xmath145 with odd values of @xmath152 such pauli - forbidden terms in the hamiltonian do not generate any dynamics . therefore the @xmath153 parameters @xmath144 contain passive components which can be removed making @xmath144 linearly dependent @xcite . the interaction terms that correspond to the multipoles with momentum @xmath154 and @xmath155 are constants of motion @xcite . the @xmath154 term in eq . ( [ eq : h_int_ph ] ) , describes the nucleon - nucleon interaction that is the same for all angular momentum channels , @xmath156 , as follows from eq . ( [ eq : vlk ] ) . the resulting monopole hamiltonian is proportional to the number of particle - pairs in a system . this hamiltonian has no dynamical effect . thus , there is no change in results if one constrains the tbre by projecting out the monopole @xmath154 component as follows @xmath157 this effectively reduces the number of independent parameters @xmath145 . in a single @xmath0 model space the @xmath155 multipoles are proportional to the angular momentum operators @xmath158 . therefore the @xmath155 interaction leads to a rotational @xmath159 spectrum with @xmath160 determining the moment of inertia . in the particle - particle channel , the @xmath161 operator is obtained with @xmath162 . consistently , it was argued in refs . @xcite that those interactions that lead to the positive moment of inertia are likely to result in the @xmath5 . the exact @xmath161 operator component in the interaction can be removed by orthogonalization to @xmath163 following the procedure in eq . ( [ eq : projection ] ) . the changes in dynamics are no longer trivial when the quadrupole @xmath164 component in the interaction is modified . the role of different multipoles in the tbre is studied in fig . [ fig : j19n8gsspin ] and [ fig : multi_j19n8_jj ] where we remove different @xmath150 components from the interaction hamiltonian in eq . ( [ eq : h_int_ph ] ) using the graham - schmidt projection procedure . in the particle - particle channel the projection of pairing interaction @xmath165 has been extensively discussed in ref . the removal of pairing does not lead to any significant qualitative change , we thus forgo this topic in what follows . the probability to observe a certain ground state spin in the @xmath141 system is shown in fig . [ fig : j19n8gsspin ] . three cases of random ensembles are reviewed : ( a ) the traditional tbre where all @xmath149 interaction parameters @xmath145 are random gaussian variables , ( b ) the case where @xmath155 term is removed , and ( c ) the ensemble where _ @xmath155 _ and _ @xmath164 _ multipole components are removed from the hamiltonian . while the wave functions in ensembles ( a ) and ( b ) are identical , the ground state spin distributions are different . the role of the @xmath161 moment - of - inertial - like term has been discussed before in refs . @xcite ; it appears to be fully responsible for the cases with maximum possible spin . as seen in fig . [ fig : j19n8gsspin ] , the states with the maximum spin almost never occur as ground states in ensembles ( b ) and ( c ) where the @xmath161 interaction term ( @xmath155 ) is removed . . probabilities to observe a certain ground state spin @xmath166 for three random ensembles : ( a ) the tbre , ( b ) the tbre without a @xmath161 term ( the @xmath155 term in eq . ( [ eq : h_int_ph ] ) is removed ) , and ( c ) the tbre without both , @xmath161 and qq terms ( the @xmath155 and @xmath164 terms in eq . ( [ eq : h_int_ph ] ) are removed).,width=326 ] : the distribution of the fractional collectivity @xmath72 for the same three random ensembles as in fig . [ fig : j19n8gsspin ] . namely : ( a ) the traditional tbre , ( b ) the tbre without a @xmath161 term , and ( c ) the tbre without both , @xmath161 and qq terms . we select realizations with the @xmath6 state followed by the first excited state @xmath59 . the fraction of such cases for ensembles ( a ) , ( b ) , and ( c ) is 7.6% , 8.2% , and 4.7% respectively . , width=326 ] the ensembles ( b ) and ( c ) shown in fig . [ fig : j19n8gsspin ] appears to have similar ground state spin distributions but the behavior of the fractional collectivity is different . in fig . [ fig : multi_j19n8_jj ] for all three ensembles we show the distribution of the fractional collectivity of the transition between the @xmath6 and @xmath59 states . it is evident that the quadrupole collectivity disappears once the quadrupole component in the interaction is removed . thus , we conclude that the quadrupole - quadrupole component in the interaction generates the corresponding deformation and is responsible for the rotational behavior observed . in this section we expand the scope of our models and consider systems with two single - particle levels . the richer geometry allows one to study the effects of particle - hole conjugation , different structures of the multipole operators , and the role of parity . the distributions of the fractional collectivity and of the quadrupole moment are shown in figs . [ fig : ppvj13n6 ] for the @xmath167 system . here the model space is comprised of two levels with @xmath168 . both single - particle levels have the same positive parity , so that the effective spherical hartree - fock mean - field hamiltonian can contain terms of a mixed structure such as @xmath169 these terms are scalars for @xmath170 there is some arbitrariness in the choice of the single - particle matrix elements of the multipole operator @xmath100 which depend on the radial overlap of the operator @xmath171 . we choose the radial overlap to be diagonal @xmath172 and @xmath173 ; other possibility with @xmath174 has been explored and led to no substantial difference . a structurally different @xmath175 model is examined in fig . [ fig : vj13n6 ] where two levels of equal spin and different parity are considered . in this case the matrix elements of the hamiltonian are restricted by parity . the same structure of the quadrupole operator is used . the model space of this kind has been explored in ref . @xcite because it is the simplest model space that allows for quadrupole and octupole modes . the prevalence of the positive parity ground states is remarkable in this model . the ground state is most likely to have spin - parity @xmath176 , @xmath177 or @xmath178 with 35% , 19% , and 14% probability , respectively . in contrast , @xmath179 , the most probable negative parity ground state , happens only in 3% of realizations . for the @xmath175 model the number of many - body states is the same for both parities , 8,212 each . for both @xmath167 and @xmath175 models , the results related to the quadrupole collectivity are almost identical , see figs . [ fig : ppvj13n6 ] and [ fig : vj13n6 ] . moreover , these results are similar to those for the single @xmath0 level models , compare to figs . [ fig : jl19n6 ] and [ fig : qqj19n8 ] . the major features in the distributions of @xmath72 and @xmath43 persist despite a bigger number of random parameters defining the hamiltonians , more complex geometry of the two - level models , and a more chaotic resulting dynamics . there is a peak in the distribution of the fractional collectivity @xmath72 near 1 indicating a sizable number of collective cases . the distribution of the quadrupole moment for the collective realizations ( shaded in red ) has a well - defined peak on the oblate side . the non - collective realizations appear to have quadrupole moment distribution centered at zero ( shaded in blue ) . . ( a ) the distribution of the fractional collectivity @xmath72 . ( b ) the distribution of the intrinsic quadrupole moment @xmath43 . the @xmath180 of samples have the @xmath65 sequence . shaded areas correspond to 1.2% of collective realization and to 1.8% of non - collective realizations . this figure is analogous to figs . [ fig : jl19n6 ] , [ fig : multipole19n6 ] , and [ fig : qqj19n8 ] , and the same shading is used in these figures . , width=326 ] . ( a ) the distribution of the fractional collectivity @xmath72 . ( b ) the distribution of the intrinsic quadrupole moment @xmath43 . the @xmath181 of samples have the @xmath6 state and the first state @xmath27 , both of positive parity . the @xmath182 of collective and @xmath183 of non - collective realizations are shaded with patterns . this figure is analogous to figs . [ fig : jl19n6 ] , [ fig : multipole19n6 ] , [ fig : qqj19n8 ] , and [ fig : ppvj13n6 ] , and the same shading is used in these figures . , width=326 ] for systems with exact particle - hole symmetry the quadrupole moment for particles is equal in magnitude and opposite in sign to that of holes . moreover , properties such as excitation energies , spins of states , and transition rates , are exactly equal for particle - hole conjugated systems . the particle - to - hole transformation for any two - body hamiltonian amounts to the same hamiltonian for holes but with an additional one - body term . thus , the symmetry is not exact in a two - level model space . nevertheless in the tbre , where two - body matrix elements are selected symmetrically about zero , the one - body term averages to zero . therefore the results in figs . [ fig : ppvj13n22 ] and [ fig : ppvj13n6 ] for particle - hole conjugates systems @xmath184 and @xmath167 are nearly symmetric . the main difference is that the hamiltonian for holes contains random single - particle energies which leads to a different fraction of collective realizations in the ensembles . . ( a ) the distribution of the fractional collectivity @xmath72 . ( b ) the distribution of the intrinsic quadrupole moment @xmath43 . the system is particle - hole conjugated to that in fig . [ fig : ppvj13n6 ] . the percentage of samples with the @xmath65 sequence is @xmath185 which includes @xmath186 of collective and @xmath187 of non - collective . this figure is analogous to figs . [ fig : jl19n6 ] , [ fig : multipole19n6 ] , [ fig : qqj19n8 ] , [ fig : ppvj13n6 ] , and [ fig : vj13n6 ] , and the same shading is used in these figures . , width=326 ] the schematic models discussed in the previous sections all seem to possess the rotational low - lying spectrum which is an evidence of the intrinsic deformation . however , to what extend they reflect the dynamics of realistic nuclei remains a question . the oblate intrinsic deformation observed in our models seems to be inconsistent with the prolate dominance in real nuclei ( see discussion in sec . [ sec : summary ] ) , moreover , the semi - magic nuclei with one type of valence nucleons are generally not deformed . to attend to these issues we examine a realistic valence space consisting of the @xmath188 single - particle levels , allowing for both protons and neutrons . the matrix element of the quadrupole operator for this model are constructed using the harmonic oscillator single - particle wave functions , we use the same effective charge for both types of nucleons . the multipole operator in this form facilitates comparison with the su(3 ) group . in fig . [ fig:0f71p3 ] we present our results for the @xmath189 system with 8 nucleons : 4 protons and 4 neutrons . this corresponds to the configuration space of @xmath190 nucleus . in fig . [ fig:0f71p3](a ) , where the fractional collectivity @xmath72 is shown , a noticeable peak that corresponds to collective realizations is observed . the distribution of the quadrupole moment in fig . [ fig:0f71p3](b ) shows prolate and oblate peaks . the peaks are especially clear for the collective realizations ( shaded in uniform red ) . the non - collective cases in fig . [ fig:0f71p3](b ) are distributed around @xmath191 ( shaded in blue pattern ) . in agreement with the results in ref . @xcite , in this tbre the prolate intrinsic shape is more probable , as evident from a bigger prolate peak . . ( a ) the distribution of the fractional collectivity @xmath72 . ( b ) the distribution of the intrinsic quadrupole moment @xmath43 . the solid black line outlines the probability distribution for 31% of realizations with the @xmath192 state followed by the @xmath59 first excited state , both states with isospin @xmath193 the 8.8% of realizations are collective and the 12.8% are non - collective . this figure is analogous to figs . [ fig : jl19n6 ] , [ fig : multipole19n6 ] , [ fig : qqj19n8 ] , [ fig : ppvj13n6]-[fig : ppvj13n22 ] , and the same shading is used in these figures . , width=326 ] in fig . [ colfig:0f71p3 ] we focus on the 8.8% of realizations that are collective . the quadrupole moments in fig . [ fig:0f71p3](b ) are further separated into prolate @xmath46 and oblate @xmath47 cases as shown in the inset of fig . [ colfig:0f71p3 ] . the same shading is used in the main figure showing the distribution of the relative transition strength @xmath48 . the maximum possible value @xmath194 is reached when the ground state wave function of the randomly selected hamiltonian coincides with that of the qq hamiltonian . from the summary in tab . [ tab : qq ] one finds that the @xmath195 ground state of the qq hamiltonian , for which @xmath194 , is prolate in this valence space . indeed , the distribution of prolate realizations is peaked at around @xmath196 while the oblate shapes have @xmath48 near @xmath197 the distributions of @xmath56 and @xmath55 for collective realizations are shown in fig . [ fig:0f7p3_col024 ] . this figure can be compared to fig . [ fig : jl19n6_col024 ] . in both figures we use the same shading to separate the prolate and oblate collective cases . in contrast to fig . [ fig : jl19n6_col024](a ) , both prolate and oblate realizations in fig . [ fig:0f7p3_col024](a ) have a band structure with the deexcitation ratio @xmath56 that is consistent with the rotational value 10/7 . this ensemble , based on the more realistic model space , appears to have an energy spectrum that is closer to the rotational spectrum . the distribution of @xmath55 in fig . [ fig:0f7p3_col024](b ) is broad , but it has a peak around the rotor value of 10/3 . . the distribution of the relative transition strength @xmath48 for the collective realizations ( shaded with uniform red in fig . [ fig:0f71p3 ] ) . the 3.6% of prolate cases and 1.0% of oblate are identified with shades of color and pattern ( see the inset ) . this figure is analogous to fig . [ fig : jl19n6_col ] , and the same shading is used as in figs . [ fig : jl19n6_col]-[fig : jl19n6_trix].,width=326 ] . ( a ) the distribution of the deexcitation ratio @xmath56 defined in eq . ( [ eq : dx ] ) . ( b ) the distribution of the excitation energy ratio @xmath55 defined in eq . ( [ eq : rr ] ) . collective realization discussed in fig . [ fig:0f71p3 ] are selected and , in addition , we require that the second excited state has spin @xmath198 . the fraction of such cases is 4.2% , with 2.4% being prolate and 0.6% being oblate , they are shaded separately with the same patterns as in fig . [ colfig:0f71p3 ] . the values for @xmath56 and @xmath55 from the qq hamiltonian listed in tab.[tab : qq ] are shown with the vertical grid lines . this figure is analogous to fig . [ fig : jl19n6_col024 ] , and the same shading is used as in figs . [ fig : jl19n6_col]-[fig : jl19n6_trix ] and [ colfig:0f71p3].,width=326 ] as concluded in ref . @xcite , realizations with rotational features appear in random ensembles due to correlated interaction matrix elements . similarly to the single @xmath0 level model , it is natural to attribute this collectivity to the qq component in the hamiltonian . the overlap @xmath199 between the ground state wave functions of the two - body random ensemble @xmath200 and the fixed ground state wave function of the qq hamiltonian is defined as@xmath201 fig . [ fig : overlap ] shows the distribution of the overlap @xmath199 in the @xmath189 model . a similar approach has been used in investigations of pairing coherence in random ensembles , see review in ref . we select 56.3% of realizations where the ground state quantum numbers are @xmath5 and @xmath202 , the ground state of the qq hamiltonian has the same spin and isospin . the distribution of @xmath199 shown in fig . [ fig : overlap ] is compared with the porter - thomas @xmath203 distribution . the latter emerges for uncorrelated wave functions in the 126-dimensional space spanned by the @xmath204 wave functions . as shown in fig . [ fig : overlap ] the porter - thomas distribution drops abruptly , thus predicting that cases with large @xmath199 are extremely unlikely . according to the porter - thomas distribution the probability to find @xmath205 is only @xmath206 whereas in the tbre @xmath205 in @xmath207 of random realizations . to emphasize the relation between the collective structure and the large qq component of the wave function we show in fig . [ fig : overlap ] the histogram for collective realization ( with states @xmath6 and @xmath208 and @xmath41 ) . it is clear that the collective transitions and rotational structure emerge when the component of the wave functions that corresponds to the eigenstate of the qq hamiltonian is large . . the distribution of the overlap @xmath199 defined in eq . ( [ eq : x ] ) . the results for all @xmath209 states are unshaded ; the fraction of such realizations is 56.3% . collective realization that in addition to the @xmath6 state have the @xmath210 , @xmath27 first excited state and @xmath41 are shaded ( their fraction is 8.8% ) . solid line shows the porter - thomas distribution , which is expected for the overlap between uncorrelated states . , width=326 ] our studies show that a collective behavior that resembles realistic is quite likely to be present in the ensemble with two - body random interactions . this behavior appears to emerge due to the quadrupole - quadrupole interaction component in the hamiltonian . this component , as well as some higher multipoles can establish some noticeable coherence despite the overall many - body randomness and complexity . similarly to the moment - of - inertia - like @xmath161 term ( that is responsible for the ground state configurations with the maximum possible spin ) the qq component , while not a constant of motion , is dynamically prevailing . let us list the supporting arguments : * the fraction of random realizations that are quadrupole - collective is extremely large as compared to the statistically expected number . * in the two - body random ensemble , the quadrupole collectivity displayed by the transition rates disappears when the qq component in the interaction is removed , see fig . [ fig : multi_j19n8_jj ] . * from investigations in fig . [ fig : overlap ] , as well as indirectly form figs . [ fig : jl19n6_col ] , [ fig : colqqj19n8 ] , and [ colfig:0f71p3 ] , it follows that the collective states in the tbre have structure similar to that of the qq hamiltonian eigenstates . * in order to examine the shape and other quantitative characteristics of the deformed mean - field we turn to the qq hamiltonian , for which the geometry of the configuration space is the only parameter . the values of the quadrupole moments , transition rates , and level spacings for the models discussed in this paper are summarized in tab . [ tab : qq ] . in all cases the qq hamiltonian has a low - lying rotational spectrum . the type of the quadrupole deformation and most of the other quantitative measures in tab . [ tab : qq ] are consistent with those observed in the tbre . this again suggests that the collective features seen in the tbre arise from the coherent qq component . .characteristics of the qq hamiltonian . listed in the table are the values of the fractional collectivity @xmath72 , quadrupole moment @xmath43 , ratios of the transition rates @xmath56 and the ratios of excitation energies @xmath55 . the models are the same as those considered in our study of the tbre . [ cols="<,^,^,^,^",options="header " , ] practically all deformed nuclei in nature are known to have a prolate ground state shape . this prolate dominance has been widely discussed in the literature @xcite . an effort to pinpoint the origin of the phenomenon using the shell model approach with random interactions is presented in ref . @xcite . while in this work we do not explicitly pursue the question of prolate dominance , we are compelled to comment on the issue from the standpoint of our findings . our studies fully confirm the results in ref . however , conclusions supporting the prolate dominance are difficult to draw , instead we offer several observations . first , the quadrupole collectivity seen in the tbre is due to the qq component in the hamiltonian . this interaction and the geometry of the valence space determine the deformation type . thus , some questions of the shape systematics can be addressed by considering the qq hamiltonian and without invoking random interactions . second , the shape is determined by the valence configuration and by the positions of the single particle levels . the role of the single - particle level structure discussed by hamamoto in ref . @xcite is possible to pinpoint using the tbre as well as using analytic models , e.g. the seniority model and elliot s su(3 ) model @xcite . third , due to particle - hole symmetry , which does not need to be exact , the number of prolate and oblate configurations is approximately the same within a given valence space . the deviations from this symmetry affect only a few mid - shell systems where the two shapes compete . the effect of the particle - hole symmetry is seen in our results in figs . [ fig : ppvj13n6 ] and [ fig : ppvj13n22 ] . to conclude , in this work we examined the quadrupole collectivity that emerges in systems with two - body random interactions . a low - lying spectrum , characteristic of a rigid rotor , is commonly observed . the transition @xmath211 the quadrupole moment of the first @xmath27 state , and the deexcitation ratio @xmath212 are all consistent with that of the deformed rotor . a weak triaxiality is also identified . the coherent dynamical role of the quadrupole - quadrupole interaction component is established as a source of this behavior . the authors are thankful to v. zelevinsky and j.m . allmond for motivating discussions . support from the u. s. department of energy , grant de - fg02 - 92er40750 is acknowledged . the computing resources were provided by the florida state university shared high - performance computing facility .

attractors play a key role in the study of non - conservative dynamics . the description of attractors and the properties of their basins help predict the future behaviour of the orbits of a system . in this work we deal with physical measures i.e. an ergodic measure @xmath0 is physical if its basin of attraction has positive volume ( see section [ sec : preliminaries ] for precise definitions ) . we will think these measures as the attractors of our systems . in many cases , basins are ( essentially ) open sets and it is clear that if a point belongs to certain regions its trajectory goes , almost surely , to an attractor that is well determined . for instance , uniformly hyperbolic diffeomorphisms exhibit a finite number of physical measures and the union of their basins cover lebesgue almost every point the ambient manifold . moreover , each one of their basins is an open set ( modulo a set of null volume ) and then , we can clearly distinguish one attractor from the others . outside the uniformly hyperbolic world , this kind of behaviour of the basins of attractors is no longer true . open sets of diffeomorphisms of manifolds with boundary may have attractors with intermingled basins . more specifically , two or more basins are dense in the same open set . it was i. kan @xcite ( see also @xcite for a description of the example in terms of the partial hyperbolicty and lyapunov exponents ) who showed for the first time the existence of examples of partially hyperbolic endomorphisms defined on a surface and exhibiting two hyperbolic physical measures whose basins are intermingled . moreover , he showed that such phenomenon is robust among the maps preserving the boundary . we refer the reader to @xcite for a rigorous proof of kan example and @xcite for a generalization of the kan example and its relation with the sign of the schwarzian derivative . in @xcite the authors shown that the set of points that are not attracted by either of the components in the kan s example has hausdorff dimension less than the dimension of the phase space itself . following the same type of arguments , it is possible to construct a partially hyperbolic diffeomorphism defined on a 3-manifold with boundary exhibiting two intermingled physical measures , and such phenomenon still can be made robust . furthermore , it is well known that it is possible to extend such example to the 3-torus , but in this case it is no longer robust . we describe these examples in section [ sec : examples ] . the existence of these examples rise the question of how robust are the intermingled basins phenomenon for diffeomorphisms defined on boundaryless manifolds . in this work we show that partially hyperbolic diffeomorphisms on the 3-torus having hyperbolic physical measures with intermingled basins are not robust . in a recent work , okunev @xcite , studied attractors in the sense of milnor in the most restrictive case of @xmath1 partially hyperbolic skew products on @xmath2 with an anosov dffeomorphisms acting on the base @xmath3 . the author obtains results with the same flavour as ours without any explicit hypotheses about lyapunov exponent in the central direction . we are interested in diffeomorphisms defined on a 3-dimensional manifold @xmath4 , in particular we put our focus on @xmath5 . we give some basic definitions necessary to formulate the results , but the reader can find the precise definitions , properties and more detailed information in section [ sec : preliminaries ] and the references therein . a diffeomorphism @xmath6 is _ partially hyperbolic _ if the tangent bundle splits into three non trivial sub - bundles @xmath7 such that the strong stable sub - bundle @xmath8 is uniformly contracted , the strong unstable sub - bundle @xmath9 is uniformly expanded and the center sub - bundle @xmath10 may contract or expand , but this contractions or expansions are weaker than the strong expansions and contractions of the corresponding strong sub - bundles . it is known that there are unique foliations @xmath11 and @xmath12 tangent to @xmath9 and @xmath8 respectively @xcite but in general , @xmath10 , @xmath13 , and @xmath14 do not integrate to foliations ( see @xcite ) . the system is said to be _ dynamically coherent _ if there exist invariant foliations @xmath15 and @xmath16 tangent to @xmath17 and @xmath18 respectively . of course , if this is the case , there exists an invariant foliation tangent to @xmath10 obtained just by intersecting @xmath15 and @xmath16 . we will study dynamically coherent diffeomorphism with compact center leaves . as we mentioned above these diffeomorphisms are not always dynamically coherent although there are some results providing this property . just to mention one result , brin , burago , and ivanov have shown that every absolute partially hyperbolic system ( see subsection [ ssec : ph ] for the definition ) on the 3-torus is dynamically coherent @xcite . a set @xmath19 is _ @xmath20-saturated _ if it is the union of complete strong unstable leaves . the diffeomorphism @xmath21 is _ accessible _ if every pair of points @xmath22 can be joined by an arc consisting of finitely many segments contained in the leaves of the strong stable and strong unstable foliations . assuming that the center bundle is one - dimensional , k. burns , f. r. hertz , j. r. hertz , a. talitskaya and r. ures @xcite proved that the accessibility property is open and dense among the @xmath1-partially hyperbolic diffeomorphisms ( see also @xcite ) . our main theorem is the following . [ mteo : a ] let @xmath23 , @xmath24 , be partially hyperbolic , dynamically coherent with compact center leaves . let @xmath0 be a physical measure with negative center lyapunov exponent . assume that @xmath25 is a compact , @xmath21-invariant and @xmath20-saturated subset such that @xmath26 . then , @xmath27 contains a finite union of periodic 2-dimensional @xmath28-tori , tangent to @xmath29 . in particular @xmath21 is not accessible . we say that two physical measures @xmath0 and @xmath30 with disjoint supports have _ intermingled basins _ @xcite if for an open set @xmath31 we have @xmath32 and @xmath33 for any open set @xmath34 . [ mcor : b ] the set of dynamically coherent partially hyperbolic @xmath1-diffeomorphisms defined on @xmath2 , @xmath24 , exhibiting intermingled hyperbolic physical measures has empty interior . moreover , if @xmath35 is isotopic to a hyperbolic automorphism , there do not exist hyperbolic physical measures with intermingled basins . closely related , hammerlindl and potrie @xcite showed that partially hyperbolic diffeomorphisms on @xmath36-nilmanifold admit a unique @xmath20-saturated minimal subset . then , @xmath21 has a unique hyperbolic physical measure ( see section [ ssec : metric ] for more details ) and thus , it is not possible to have the intermingled basins phenomenon . we have as corollary of their work : [ mcor : c ] if @xmath4 is a @xmath36-nilmanifold , then there does not exist hyperbolic physical measures with intermingled basins . this paper is organized as follows . section [ sec : preliminaries ] is devoted to introduce the main tools in the proof : partial hyperbolic diffeomorphisms , physical measures , @xmath20-measures and lyapunov exponents . a toy example as well as kan - like examples are revisited in section [ sec : examples ] . proofs of theorem [ mteo : a ] and corollary [ mcor : b ] are developed in section [ sec : proofs ] . throughout this paper we shall work with a _ partially hyperbolic diffeomorphism _ @xmath21 , that is , a diffeomorphism admitting a nontrivial @xmath37-invariant splitting of the tangent bundle @xmath38 , such that all unit vectors @xmath39 ( @xmath40 ) with @xmath41 satisfy : @xmath42 for some suitable riemannian metric . @xmath21 also must satisfy that @xmath43 and @xmath44 . we also want to introduce a stronger type of partial hyperbolicity . we will say that @xmath21 is _ absolutely partially hyperbolic _ if it is partially hyperbolic and @xmath45 for all @xmath46 . for partially hyperbolic diffeomorphisms , it is a well - known fact that there are foliations @xmath47 tangent to the distributions @xmath48 for @xmath49 . the leaf of @xmath50 containing @xmath51 will be called @xmath52 , for @xmath49 . in general it is not true that there is a foliation tangent to @xmath10 . sometimes there is no foliation tangent to @xmath10 . indeed , there may be no foliation tangent to @xmath10 even if @xmath53 ( see @xcite ) . we shall say that @xmath21 is _ dynamically coherent _ if there exist invariant foliations @xmath54 tangent to @xmath55 for @xmath49 . note that by taking the intersection of these foliations we obtain an invariant foliation @xmath56 tangent to @xmath10 that subfoliates @xmath54 for @xmath57 . in this paper all partially hyperbolic diffeomorphisms will be dynamically coherent . we shall say that a set @xmath58 is _ @xmath59-saturated _ if it is a union of leaves of the strong foliations @xmath60 for @xmath61 or @xmath62 . we also say that @xmath58 is @xmath63-saturated if it is both @xmath64- and @xmath20-saturated . the accessibility class of the point @xmath41 is the minimal @xmath63-saturated set containing @xmath51 . in case there is some @xmath41 whose accessibility class is @xmath4 , then the diffeomorphism @xmath21 is said to have the _ accessibility property_. this is equivalent to say that any two points of @xmath4 can be joined by a path which is piecewise tangent to @xmath8 or to @xmath9 . in this section we consider @xmath65 be a diffeomorphism , not necessarily partially hyperbolic , defined on the riemannian manifold @xmath4 . we denote by @xmath66 the normalized volume form on @xmath4 . a point @xmath67 is _ birkhoff regular _ if the birkhoff averages @xmath68 @xmath69 are defined and @xmath70 for every @xmath71 continuous . we denote by @xmath72 the set of birkhoff regular points of @xmath21 . birkhoff ergodic theorem @xcite , implies that the set @xmath72 has full measure with respect to any @xmath21-invariant measure @xmath73 . when @xmath73 is an ergodic measure , @xmath74 for every @xmath75 in a @xmath73-full measure set @xmath76 . if @xmath73 is an @xmath21-invariant measure , the _ basin _ of @xmath73 is the set @xmath77 if @xmath73 is an @xmath21-invariant ergodic measure , then @xmath78,and so @xmath79 has full @xmath73-measure . an @xmath21-invariant probability measure @xmath0 is _ physical _ if its basin @xmath80 has positive lebesgue measure on @xmath4 @xcite . a physical measure is said to be _ hyperbolic _ if all its lyapunov exponents are nonzero @xcite . in the setting of partially hyperbolic diffeomorphims defined on a 3-dimensional manifold , a physical measure is hyperbolic if @xmath81 a point @xmath41 is _ lyapunov regular _ if there exist an integer @xmath82 , numbers @xmath83 and a decomposition @xmath84 into subspaces @xmath85 such that @xmath86 , and for every @xmath87 @xmath88 denote by @xmath89 the set of lyapunov regular points . the numbers @xmath90 are called the _ lyapunov exponents _ of @xmath51 . the splitting is called _ oseledets decomposition _ and the subspaces @xmath85 are called _ oseledets subespaces _ at @xmath51 . s theorem @xcite guarantee that the set @xmath89 has full measure with respect any invariant measure . in general the functions @xmath91 , @xmath92 , @xmath93 and @xmath94 are measurable . nevertheless , if @xmath73 is an ergodic invariant measure for @xmath21 , there is a subset @xmath95 , such that @xmath96 and there exist an integer @xmath97 , subspaces @xmath98 , numbers @xmath99 such that for every @xmath100 , we have * @xmath101 ; * @xmath102 , for every @xmath103 ; * @xmath104 , for every @xmath103 ; an ergodic measure @xmath73 is _ hyperbolic _ if @xmath105 , @xmath103 . in such case , for each @xmath100 we set @xmath106 @xmath107 we have @xmath108 , @xmath109 are constant and @xmath110 . the function @xmath111 and @xmath112 are measurables . if @xmath21 is @xmath1 , @xmath113 , pesin s theory @xcite guarantee the existence of invariant sub - manifolds @xmath114 , @xmath115 tangent to @xmath116 and @xmath117 respectively . more precisely , for every @xmath100 there is a @xmath1 embedded disk @xmath118 through @xmath51 such that * @xmath118 is tangent to @xmath116 at @xmath51 , * @xmath119 , * the stable set @xmath120 * there exist constant @xmath121 , @xmath122 such that , for every @xmath123 @xmath124 the @xmath1 disk @xmath118 is called _ pesin stable manifold_. similarly , every @xmath100 has an _ pesin unstable manifold _ @xmath125 satisfying the corresponding properties with @xmath126 in place of @xmath21 . the pesin manifolds above may be arbitrarily small , and they vary measurably on @xmath51 . for any integer @xmath127 , we may find _ hyperbolic blocks _ @xmath128 such that * @xmath129 , * @xmath130 , as @xmath131 . * the the size of the embedded disk @xmath118 is uniformly bounded from zero for each @xmath132 . moreover , for every @xmath132 , @xmath133 and @xmath134 in . analogous properties are satisfied by the unstable pesin s manifold @xmath125 . * the disk @xmath118 and @xmath125 vary continuously with @xmath135 . most important , the holonomy maps associated to the pesin stable lamination @xmath136 are absolutely continuous . more precisely , fix an integer @xmath127 , a hyperbolic block @xmath137 and a point @xmath132 . for @xmath138 , @xmath139 close to @xmath51 , let @xmath140 and @xmath141 be small smooth discs transverse to @xmath118 at @xmath138 and @xmath142 respectively . the holonomy map @xmath143 defined on the points @xmath144 by associate @xmath145 , the unique point in @xmath146 . if @xmath21 is @xmath1 , @xmath113 , then every holonomy map @xmath147 as before is absolutely continuous @xcite . of course , a dual statement holds for the unstable lamination . in our setting , @xmath21 is a @xmath1-partially hyperbolic diffeomorphism , @xmath113 , with splitting @xmath148 , where @xmath149 , @xmath150 . let @xmath73 be an ergodic measure and we consider any point @xmath100 . then @xmath151 and @xmath152 , @xmath153 and @xmath154 . moreover @xmath155 , @xmath156 and @xmath157 is called the _ center lyapunov exponent _ at @xmath51 . if we take @xmath158 , since @xmath159 we obtain that @xmath160 if we assume @xmath161 , then @xmath162 and @xmath163 for every @xmath100 . the local strong stable manifold @xmath164 is an embedded curve inside the pesin stable manifold @xmath118 which is a surface . on the other hand , the pesin unstable manifold @xmath125 coincides with the strong unstable manifold @xmath165 , for every @xmath100 . of course , analogous statement holds if we assume @xmath166 . assume now that @xmath21 is partially hyperbolic and @xmath167 . an @xmath21-invariant probability measure @xmath0 is a _ @xmath20-measure _ if the conditional measures of @xmath0 with respect to the partition into local strong - unstable manifolds are absolutely continuous with respect to the lebesgue measure along the corresponding local strong - unstable manifold . if @xmath21 is a @xmath1 partially hyperbolic diffemorphism , @xmath168 , then there exist @xmath20-measures for @xmath21 @xcite . several properties of the @xmath20-measures are well know ( see for instance @xcite , section 11.2.3 and the references therein , for a detailed presentation of such properties ) . for instance , the support of any @xmath20-measure is a @xmath20-saturated , @xmath21-invariant , compact set . if @xmath0 is a @xmath20-measure , then its ergodic components are @xmath20-measures as well . furthermore , the set of @xmath20-measures for @xmath21 is a compact , convex subset of the invariant measures . moreover , every physical measure for @xmath21 must be a @xmath20-measure . it is well know that if @xmath0 is an ergodic @xmath20-measure with negative center lyapunov exponent , then , @xmath0 is a physical measure @xcite . conversely , if @xmath0 is a physical measure with negative center lyapunov exponent , then @xmath0 is an ergodic @xmath20-measure . in this section we show some examples that motivated this paper . in the first example ( anosov times morse - smale ) there are no intermingled basins but there is a @xmath20-saturated set in the boundary of one of them . of course , we know a priori that this set consists of tori and it is not difficult to show that this situation is not robust . this example jointly with kan s was a source of inspiration to obtain theorem [ mteo : a ] . this is the easiest case where the theorem works . observe that there is only one physical measure . in the second case ( kan - like example ) the basins are intermingled . in the 3-torus @xmath169 , we consider the @xmath1-diffeomorphism , @xmath24 , @xmath170 defined by @xmath171 where @xmath172 is a linear anosov diffeomorphism with eigenvalues @xmath173 , and @xmath174 is a morse - smale diffeomorphisms with having exactly two hyperbolic fixed points , a source @xmath175 and a sink @xmath176 , satisfying @xmath177 and @xmath178 . we assume that @xmath179 satisfies : @xmath180 that means , @xmath179 is a partially hyperbolic diffeomorphism exhibiting a center foliation by compact leaves ( circles ) . furthermore , @xmath179 has a foliation by smooth 2-tori tangent to the @xmath181-sub - bundle . in particular , one of such leaves , the torus @xmath182 , is the only attractor of @xmath179 . the dynamics restricted to @xmath182 is hyperbolic , in fact , is given by @xmath183 . then , it supports the unique hyperbolic @xmath20-measure @xmath184 for @xmath179 ( actually the lebesgue measure on @xmath182 ) having negative center lyapunov exponent and so , it is physical . if @xmath185 denotes the basin of @xmath186 in the 2-torus @xmath182 under the anosov dynamics given by @xmath183 then , the basin of @xmath184 in @xmath2 is @xmath187 which is an open set modulus a set of zero lebesgue measure in @xmath2 . the boundary of @xmath188 contains the invariant 2-torus @xmath189 which is the only hyperbolic repeller of @xmath179 . this invariant torus is also a @xmath20-saturated set , tangent to @xmath181 . the dynamics restricted to @xmath189 is again hyperbolic and then , it supports a @xmath20-measure @xmath190 for @xmath179 ( actually lebesgue measure on @xmath3 ) but it is not physical . theorem a prevents the existence of such a @xmath20-saturated set from being robust . after a typical @xmath191-perturbation , the new map @xmath192 is partially hyperbolic and dynamically coherent . in fact , @xmath192 has a center foliation by compact leaves by classical results of normally hyperbolic foliations @xcite . typically @xmath192 does not preserve the invariant foliation by 2-tori tangent to @xmath181 , which exists for @xmath179 . nevertheless @xmath192 has two invariant compact subset @xmath193 and @xmath194 , the respective continuations of the hyperbolic basic sets @xmath189 and @xmath182 . of course , the dynamics of @xmath195 and @xmath196 are @xmath197-conjugated , so @xmath193 is ( homeomorphic to ) a continuous torus , and the dynamics of @xmath192 in @xmath193 is uniformly hyperbolic . the set @xmath193 remains to be a hyperbolic repeller and so @xmath64-saturated , but in general it is not @xmath20-saturated . similar conclusions hold for @xmath194 , the hyperbolic attractor of @xmath192 . it is a topological 2-torus , @xmath20-saturated , and it supports the unique physical measure of @xmath192 . note that the topological torus @xmath193 is contained in the boundary of the basin @xmath198 , but , in general , @xmath193 is no longer a @xmath20-saturated set . in @xcite kan provided the first examples of partially hyperbolic maps with intermingled basin . in this section we present the kan s examples with some variations , following @xcite , section 11.1.1 . the kan s example corresponds to a partially hyperbolic endomorphism defined on a surface with boundary exhibiting two intermingled hyperbolic physical measures . consider the cylinder @xmath199 $ ] , and @xmath200 the map defined by @xmath201 where @xmath202 is some integer , @xmath203 are two different fixed points of @xmath204 and @xmath205 $ ] is @xmath1 , @xmath24 , satisfying the following conditions : 1 . for every @xmath206 we have @xmath207 and @xmath208 . the map @xmath209\to[0,1]$ ] has exactly two fixed points , a hyperbolic source at @xmath210 and a hyperbolic sink in @xmath211 . analogously , the map @xmath212\to[0,1]$ ] has exactly two fixed points , a hyperbolic sink at @xmath210 and a hyperbolic source in @xmath211 . 3 . for every @xmath213 , @xmath214 , and 4 . @xmath215 and @xmath216 the dynamics along the @xmath217-direction is given by @xmath218 , so it is uniformly expanding . from [ k3 ] we conclude that the map @xmath27 is partially hyperbolic : the derivative in the @xmath219-direction is dominated by the derivative in the @xmath217-direction . condition [ k1 ] means @xmath27 preserves the boundary . then , each one of the boundary circles @xmath220 and @xmath221 supports an absolutely continuous invariant probability measure @xmath222 and @xmath186 , respectively . condition [ k4 ] implies that @xmath222 and @xmath186 have negative lyapunov exponent in the @xmath219-direction . so they are physical measures . moreover , their basin are intermingled . magic comes from condition [ k2 ] : take any curve @xmath223 inside the open cylinder and transverse to the @xmath219-direction . we can assume , up to taking some forward iterates , that @xmath223 crosses ( transversally ) the segments @xmath224 and @xmath225 $ ] . this is possible since @xmath21 is uniformly expanding along the @xmath217 direction and the angle between @xmath223 and the @xmath219-direction goes to @xmath226 due to the domination . then , there is a forward iterate of @xmath223 that intersects the basin of @xmath222 , in a set of positive lebesgue measure ( in @xmath223 ) , because @xmath223 intersects transversally @xmath227 . since @xmath223 also intersects transversally @xmath228 $ ] , then @xmath223 intersects the basin of @xmath186 in a set of positive lebesgue measure ( see figure [ fig : kan ] ) . fubini s theorem completes the argument . $ ] ] this example is robust among the maps defined on the cylinder preserving the boundaries . indeed , for @xmath229 , any map @xmath230 , @xmath1 close to @xmath27 and preserving the boundaries can be written as @xmath231 where @xmath232 is expanding along the @xmath217-direction and @xmath233 $ ] preservs the boundaries , that means @xmath234 satisfies [ k1 ] . moreover , if @xmath234 is chosen @xmath1 close enough of @xmath235 , then also their derivatives @xmath236 and @xmath237 are close for every @xmath213 and so @xmath234 satisfies [ k3 ] and [ k4 ] above . the two different fixed points of @xmath204 , @xmath203 , have continuations @xmath238 and the map @xmath239\to[0,1]$ ] has exactly two fixed points , a hyperbolic source at @xmath210 and a hyperbolic sink in @xmath211 . analogously , the map @xmath240\to[0,1]$ ] has exactly two fixed points , a hyperbolic sink at @xmath210 and a hyperbolic source in @xmath211 . then , @xmath234 satisfies [ k2 ] . arguing as before , we conclude that @xmath241 exhibits two intermingled hyperbolic physical measures supported on the boundary . the next example , corresponds to a partially hyperbolic diffeomorphism defined on a 3-manifold with boundary exhibiting two intermingled physical measures . the idea is to adapt the previous example , replacing @xmath242 with the torus @xmath3 and the expanding map @xmath204 with a hyperbolic automorphism of the 2-torus having at least two fixed points . more precisely , we can consider @xmath243 $ ] and diffeomorphisms @xmath244 where @xmath245 is a hyperbolic automorphism , and @xmath246 $ ] is @xmath1 , @xmath24 , satisfying the following conditions : 1 . for every @xmath247 we have @xmath248 and @xmath249 . 2 . for @xmath250 , fixed points of @xmath183 , we assume that the map @xmath251\to[0,1]$ ] has exactly two fixed points , a source at @xmath210 and a sink in @xmath211 . analogously , the map @xmath252\to[0,1]$ ] has exactly two fixed points , a sink at @xmath210 and a source in @xmath211 . 3 . for every @xmath253 , @xmath254 , and 4 . @xmath255 and @xmath256 as before , the dynamics along the @xmath75-direction of @xmath257 is uniformly hyperbolic . from [ kd3 ] we conclude that the map @xmath257 is partially hyperbolic : the derivative in the @xmath219-direction is dominated by the derivative in the unstable direction of @xmath183 and the stable direction of @xmath183 is dominated by the derivative in the @xmath219-direction . condition [ kd1 ] means @xmath257 preserves each boundary torus . then both boundary torus @xmath258 and @xmath259 support the measures @xmath222 and @xmath186 corresponding to the lebesgue measure in the torus . condition [ kd4 ] implies that @xmath222 and @xmath186 have negative lyapunov exponent in the center direction . so they are physical measures . as before , their basins are intermingled . the argument is the same : take any curve @xmath223 in the interior of @xmath260 and transverse to the @xmath18 distribution . up to some forward iterates , @xmath223 crosses ( transversally ) the surfaces @xmath261 and @xmath262 $ ] . this is possible since @xmath21 is uniformly expanding along the unstable direction and the domination improves the angle between @xmath223 and the center - stable direction . then , there is a forward iterate of @xmath223 that intersects the basin of @xmath222 in a set of positive lebesgue measure ( in @xmath223 ) , because @xmath223 intersects transversally the stable manifold @xmath263 . since @xmath223 also intersects transversally the stable manifold @xmath264 , then @xmath223 intersects the basin of @xmath186 in a set of positive lebesgue measure . fubini s theorem complete the argument . as before , this example is robust among the diffeomorphisms defined on @xmath260 preserving the boundary tori . the same construction can be done if @xmath260 is replaced with @xmath265 ( or even the mapping torus of a hyperbolic diffeomorphism ) and @xmath266 $ ] is replaced with @xmath267 . then , the four conditions are : 1 . for every @xmath247 we have @xmath268 and @xmath269 . 2 . for @xmath250 , fixed point of @xmath183 , we assume that the map @xmath270 has exactly two fixed points , a source at @xmath271 and a sink in @xmath211 . analogously , the map @xmath272 has exactly two fixed points , a sink at @xmath271 and a source in @xmath211 . 3 . for every @xmath253 , @xmath273 , and 4 . @xmath274 and @xmath275 . exactly the same proof gives that the basins of the lebesgue measures of the boundary tori are intermingled . the difference is that this phenomenon is no longer robust . in fact , there exists a unique physical measure after most perturbations ( see , for instance , @xcite ) . recently , bonatti and potrie announced that they are able to construct diffeomorphisms on the torus @xmath2 with exactly @xmath276 hyperbolic physical measures @xmath277 whose basins are all intermingled ( and dense on the whole torus ) , in fact , for every open set @xmath278 and every @xmath279 @xmath280 their example is partially hyperbolic in the following broad sense : the tangent space has an invariant splitting @xmath281 where @xmath282 dominates @xmath18 but the sub - bundle @xmath18 is indecomposable into dominated sub - bundles . we remark that partially hyperbolic diffeomorphisms on surfaces do not admit intermingled hyperbolic physical measures @xcite . the situation is different in the absence of domination as showed by fayad @xcite . inspired in the fayad example , melbourne and windsor @xcite give a family of @xmath283-diffeomorphisms on @xmath284 with arbitrary number of physical measures with intermingled basins . motivated by the latter situation , we say that a partially hyperbolic diffeomorphism @xmath21 is a _ kan - like differmorphisms _ if there exist , at least , two hyperbolic physical measures with intermingled basins . let @xmath285 , @xmath24 , be partially hyperbolic and dynamically coherent with compact center leaves . let @xmath0 be a hyperbolic physical measure for @xmath21 with @xmath286 . for further use let @xmath287 where @xmath288 are pesin blocks and @xmath289 . we assume that @xmath290 is invariant and its points are regular both in the sense of pesin s theory as in the sense of birkhoff s theorem . moreover , we will assume that every @xmath291 is a lebesgue density point of @xmath292 . suppose @xmath298 . fix @xmath299 . then , it is not difficult to see there is a sequence @xmath300 such that the distance between @xmath301 and @xmath302 converges to @xmath303 . indeed , if there is @xmath304 such that the distance of @xmath305 to @xmath302 is greater than @xmath303 you can construct a continuous which that takes the value @xmath306 for every point in @xmath302 and @xmath303 if the distance to @xmath302 is greater or equal to @xmath307 . since @xmath302 has positive @xmath0-measure this contradicts the fact that @xmath298 . denote by @xmath311 the space of center curves , that is , the quotient space obtained by the relation of equivalence @xmath312 if they are in the same center manifold . we denote by @xmath58 the space of compact subsets of @xmath4 . given a @xmath20-saturated closed subset @xmath19 , we define the function @xmath313 by @xmath314 . observe that this intersection is nonempty for every @xmath315 . on the other hand , since @xmath27 is saturated by strong unstable leaves and the unstable holonomy is continuous , the set of continuity points of @xmath316 is also saturated by strong unstable leaves . more precisely , if @xmath317 is a point of continuity of @xmath316 , then for every @xmath318 we have that @xmath319 is also a point of continuity of @xmath316 . let @xmath323 . taking iterates for the future , and recalling that almost every point returns infinitely many times to a positive measure set , we can assume that @xmath324 with @xmath325 where @xmath326 is the uniform size of the pesin stable manifolds of the points of the block @xmath327 . close to @xmath328 we take @xmath329 , with @xmath330 , and such that @xmath331 is a continuity point of @xmath332 . in particular , there is a @xmath333 such that , if @xmath334 then , there exists @xmath335 with @xmath336 . let @xmath337 and @xmath338 . the absolute continuity of the partition by pesin stable manifolds implies that @xmath339 . then , the ergodicity of the measure implies that there are infinitely many iterates of @xmath328 that belong to @xmath192 . in particular , there is an @xmath340 such that @xmath341 and @xmath342 . thus , we obtain that @xmath343 . the fact that @xmath344 implies that there is @xmath345 , such that @xmath346 . since @xmath347 we have that corresponding center arc @xmath348_c$ ] is completely contained in a pesin stable manifold . we take @xmath349 and this gives the conclusion of the lemma for the points of @xmath350 . in what follows we consider @xmath25 satisfying the hypotheses in theorem [ mteo : a ] . that is , @xmath27 is a compact , @xmath21-invariant and @xmath20-saturated subset such that @xmath26 . our strategy to prove theorem [ mteo : a ] will be to study the intersections of the set @xmath27 with the center manifolds of @xmath21 . let s begin with the proof . suppose on the contrary that for every @xmath354 there are @xmath356 and three points @xmath352 with @xmath357 for every pair of points @xmath358 . take the topological surfaces @xmath359 , @xmath360 and @xmath361 . without loose of generality we can assume that @xmath362 is in the center arc that joins @xmath51 and @xmath75 and has length less than @xmath354 . take @xmath363 such that @xmath364 and suppose that @xmath365 . since @xmath366 , lemma [ l1 ] implies that it can be approximated by a point @xmath367 belonging to @xmath368 . by lemma [ l2 ] we have that @xmath367 can be joined to @xmath369 by a center arc completely contained in @xmath370 . observe that @xmath367 is very close to @xmath371 and then , the length of this center arc is greater than , say , @xmath372 . still much larger than @xmath354 . this implies that the center arc joining @xmath367 and @xmath369 must intersect either @xmath359 or @xmath361 ( see figure [ fig:1 ] ) . [ finito ] let @xmath0 be an ergodic @xmath20-measure with negative center exponent and @xmath27 an invariant @xmath20-saturated set such that @xmath374 . then , the intersection of @xmath27 with each center manifold consists of finitely many points . let @xmath317 be a point of continuity of @xmath379 . continuity at @xmath317 implies that @xmath380 if @xmath381 is close enough to @xmath317 . the @xmath20-minimality , again , implies the inequality for every @xmath382 . suppose that the function @xmath378 is not constant . then , there is a dense set @xmath383 such that for @xmath384 we have that @xmath385 . continuity at @xmath317 implies that there are a point @xmath386 , a sequence @xmath387 and for each integer @xmath127 , a pair of points @xmath388 , @xmath389 so that both sequences @xmath390 , @xmath391 , converge to @xmath51 . then , taking @xmath260 large enough we can choose a center curve with two points @xmath392 and @xmath393 a very small @xmath353-distance . we will argue in a similar way to the arguments of the proof of lemma [ lemah ] . we want to obtain three points that are very close to each other in the same center manifold and surfaces through them that are not in @xmath370 , to arrive to a contradiction with lemma [ l2 ] . since @xmath375 is @xmath20-minimal we can find @xmath394 very close to @xmath395 . continuity of the holonomy gives that there are center manifolds converging to the center manifold of @xmath75 and pairs of points @xmath396 of @xmath375 in each of these center manifolds converging to @xmath75 . finally , fix an integer @xmath397 large enough ( in such a way that the @xmath353 distance between @xmath398 and @xmath399 is much smaller than the one between @xmath395 and @xmath400 ) and call @xmath401 and @xmath402 . denote @xmath403 the center leaf that contains @xmath404 . because of the choices we have made , @xmath405 intersects @xmath403 in a point @xmath406 that is close to @xmath407 and @xmath408 but at a greater distance than @xmath409 . that means that one of the two points @xmath410 lies in between the other two ( see figure [ fig:3 ] ) . let @xmath376 be @xmath20-minimal and closed . lemma [ constant ] shows that @xmath375 is locally the graph of a continuous function and then , it is a closed topological surface topologically transverse to the center foliation . since it is foliated by unstable leaves , that are lines , we have that @xmath375 is a torus . moreover , proposition [ finito ] implies that the torus @xmath375 is periodic . thus , all that remains is to prove that the strong stable manifolds of the points of @xmath375 are completely contained in @xmath375 . as @xmath376 is periodic , we can take an iterate @xmath127 such that @xmath411 . by simplicity we assume that @xmath412 . suppose that there is a point @xmath413 such that its strong stable manifold @xmath414 has a point @xmath362 that does not belong to @xmath375 . since @xmath375 is closed , there exists an open neighbourhood @xmath415 of @xmath362 such that @xmath416 . by the continuity of the strong stable foliation , reducing @xmath417 if necessary , we can find an open neighbourhood @xmath31 of @xmath51 with the property that the strong stable manifold of every point in @xmath417 has a point in @xmath418 , in particular , in @xmath375 . we know that @xmath419 , then @xmath420 . hence , there is @xmath421 and if we take @xmath422 , then @xmath423 ( see figure [ fig:2 ] ) . in particular , @xmath424 and its omega limit is contained in @xmath369 . since @xmath375 is @xmath21-invariant , then @xmath425 which contradicts the hypotheses @xmath426 . this finishes the proof of the theorem [ mteo : a ] . first of all , observe that neither @xmath0 nor @xmath30 can have positive center lyapunov exponent . this is a consequence of the well - known fact that under our hypotheses the basin of attraction of such a measure would be essentially open ( see for instance @xcite where the conservative case is discussed with details and recently @xcite for a discussion about the non conservative case . ) . suppose that the center exponents are negative . if their basins are intermingled then @xmath427 . indeed , it is not difficult to see that the definition of intermingled basins implies that there is a point of the stable manifold ( in the sense of pesin ) of a regular point of @xmath30 that is accumulated by points of the basin of @xmath0 . since @xmath30 is ergodic the orbit of a regular point is dense in its support . by forward iteration we obtain the desired inclusion . then , as consequence of theorem [ mteo : a ] applied to @xmath428 , @xmath21 is not accessible . as mentioned above accessibility is an open an dense property , and then we obtain the first assertion . for the second statement , the works of a. hammerlindl @xcite and r. potrie @xcite proved that the center foliation of every dynamically coherent partially hyperbolic diffeomorphism on the 3-torus is homeomorphic to the corresponding foliation of a linear toral automorphism . as a consequence , there are two possibilities : either the center foliation is by circles or the diffeomorphism is homotopic to a hyperbolic automorphism , it is always dynamically coherent and the center foliation is by lines . we have already studied the first case . in the second case , potrie @xcite ( see also @xcite ) proved that if @xmath21 is isotopic to a hyperbolic automorphism , there is a unique minimal @xmath20-saturated set . this implies that @xmath21 has at most one physical measure with negative center exponent . a. bonifant and j. milnor . schwarzian derivatives and cylinder maps . in _ holomorphic dynamics and renormalization _ , volume 53 of _ fields inst . _ , pages 121 . soc . , providence , ri , 2008 . k. burns , d. dolgopyat , and y. pesin . partial hyperbolicity , lyapunov exponents and stable ergodicity . , 108(5 - 6):927942 , 2002 . dedicated to david ruelle and yasha sinai on the occasion of their 65th birthdays . a. fathi , m. herman , and j. yoccoz . a proof of pesin s stable manifold theorem . in _ geometric dynamics ( rio de janeiro , 1981 ) _ , volume 1007 of _ lecture notes in math . _ , pages 177215 . springer , berlin , 1983 .

atom interferometry has opened new frontiers in precision metrology . highly sensitive gravimeters , gravity gradiometers , and gyroscopes have been constructed , and promising work has been done to integrate these sensors into a robust apparatus that can operate outside the laboratory with applications in inertial navigation and geodesy @xcite . moreover , atom interferometers have been used to make competitive measurements of the fine structure constant @xcite . since atom interferometric measurements of the fine structure constant do not assume the validity of quantum electrodynamics ( qed ) , while determinations of the fine structure constant based on measurements of the electron magnetic moment do make this assumption , comparison between the results of these two methods provides a stringent test of qed @xcite . in addition , an experiment to test einstein s equivalence principle with unprecedented precision is underway @xcite , and atom interferometric gravitational wave detectors offer the possibility to study gravitational radiation in frequency ranges complementary to ligo and lisa @xcite . atom interferometers have traditionally relied on matter gratings or light pulses to act as beam splitters and mirrors for matter waves , with atomic wave packets traveling freely between these interaction zones . light - pulse schemes using either raman pulses ( where the internal state of the atom is changed ) or bragg pulses ( where the internal state of the atom remains unchanged ) have been implemented , such as those described in @xcite . for a number of applications of light - pulse atom interferometers , such as measurements of gravity and rotation , the sensitivity is proportional to the separation in momentum that can be attained between the two arms @xcite . in measurements of the fine structure constant , the sensitivity scales as the square of this separation @xcite . therefore , significant efforts have been devoted to the development of large momentum transfer ( lmt ) beam splitters . lmt beam splitters achieving momentum splittings of @xmath0 using multi - photon bragg pulses have recently been demonstrated @xcite . however , the required laser intensities to make significant improvements on this result may prove to be prohibitive @xcite . in contrast , lmt beam splitters that use several two - photon bragg pulses or a multi - photon bragg pulse of relatively small order to separate the two arms of the interferometer in momentum space , followed by the acceleration of one of the arms with an optical lattice , could potentially provide multiple order of magnitude increases in attainable momentum separations with relatively modest laser intensity requirements . an atom interferometer that uses this method has been successfully operated in a proof of principle experiment ( with a maximum demonstrated momentum splitting of @xmath1 ) @xcite . in a separate experiment , an atom interferometer with @xmath2 lmt beam splitters has been realized using a similar technique @xcite . alternatively , both arms of the interferometer could be simultaneously accelerated in opposite directions by two different optical lattices after the initial splitting . using this second scheme , an interferometer with @xmath0 lmt beam splitters that achieves 15@xmath3 contrast and an individual beam splitter that provides an @xmath4 momentum separation have been demonstrated @xcite . the utility of atom interferometry hinges upon the ability to precisely calculate the phase accumulated along the different arms of an interferometer @xcite , of which the phase acquired during interactions of the atoms with light is an important component . indeed , the phase obtained by an atom during a raman or bragg pulse is well - understood @xcite . analogously , in order to take full advantage of the potential of lattice beam splitters , we must have a detailed understanding of the phase evolution of an atom in an optical lattice . in this paper , we provide a rigorous analytical treatment of this problem . to our knowledge , such a treatment has not been previously presented in the literature . based on this analysis , we propose atom interferometer geometries in which optical lattices are used to continuously guide the atoms , so that the atomic trajectories are precisely controlled for the duration of the interferometer sequence , with a different lattice guiding each arm of the interferometer ( as illustrated in fig . [ fig : guidedinterf ] ) . we point out here a distinction in terminology between a lattice waveguide and a lattice beam splitter . here , a lattice waveguide is the use of a lattice to continuously control the trajectory of an arm of an atom interferometer . we note that two separate lattice waveguides can independently control the two arms of an interferometer , or a single lattice waveguide can simultaneously control both arms . in contrast , a lattice beam splitter is an interaction of relatively short time ( in comparison to a waveguide ) with the primary purpose of splitting the arms of the interferometer in momentum space rather than providing continuous trajectory control . the underlying physics behind lattice waveguides and lattice beam splitters is the same , and they can be treated with a common formalism . a single lattice waveguide that simultaneously transfers @xmath5 of momentum to the two arms of a ramsey - bord interferometer has been previously achieved in @xcite . however , to our knowledge , our idea of using optical lattice waveguides to create a fully confined atom interferometer has not been previously considered . our analysis indicates that these lattice interferometers will offer unprecedented sensitivities for a wide variety of applications and that they will be able to operate effectively over distance scales previously considered too small to be studied by precision atom interferometry . for example , one particularly interesting configuration involves using two optical lattice waveguides to continuously pull the two arms of the interferometer apart , subsequently holding the two arms a fixed distance from each other in a single lattice waveguide that is common to the two arms , and then using two lattice waveguides to recombine the arms . such a configuration could be used , for instance , as a gravimeter . the sensitivity of lattice interferometers is illustrated by the fact that , given the experimental parameters stated in @xcite ( @xmath6 atoms / shot and @xmath7 shots / s ) , a shot noise limited lattice gravimeter whose arms are separated by 1 m for an interrogation time of 10 s has a sensitivity of @xmath8 @xmath9/hz@xmath10 . we perform phase shift calculations for these lattice interferometers using the theoretical groundwork formulated in this paper , and we discuss how lattice interferometers can both exceed the performance of conventional atom interferometers in many standard applications and expand the types of measurements that can effectively be carried out using atom interferometry . the paper is organized as follows . ii . describes the hamiltonian for an atom in an optical lattice in the different frames we use in the paper . iii . discusses the phase evolution of an atom in an optical lattice under the adiabatic approximation . iv . introduces the formalism of perturbative adiabatic expansion to calculate corrections to the adiabatic approximation , and sec . v. applies this formalism to calculate phase corrections to a lattice beam splitter . vi . proposes a number of interferometer geometries that make use of lattice manipulations of the atoms . the main results of the paper are eqs . ( [ eqn : phasecorrection ] ) and ( [ eqn : phasecorrectionref ] ) , which show how to obtain analytical corrections to the lowest order phase shift estimates . these corrections are surprisingly large , and understanding them is vital to realizing the full accuracy of the sensor geometries proposed in sec . vi . , as well as other geometries utilizing optical lattice manipulations of the atoms . for example , the gravitational wave detector proposed in @xcite will likely make use of lattice beam splitters and/or waveguides . previously , the phase evolution induced by lattice manipulations was not sufficiently well - understood to allow for a detailed design of the atom optics system or an estimation of the corresponding systematic effects . an optical lattice is a periodic potential formed by the superposition of two counter - propagating laser beams . atoms can be loaded into the ground state of the lattice by ramping up the lattice depth adiabatically , and the lattice can then be used to impart momentum to the atoms and/or to control the atoms trajectories . optical lattices are thus a useful tool for atom optics . we begin our discussion of the lattice - atom interaction by finding a useful form for the hamiltonian . as is typical for many applications of atom interferometry , to minimize decoherence we assume that we work with atomic gases dilute enough so that the effects of atom - atom interactions are negligible . we first consider the hamiltonian in the lab frame , where for now we assume a vertical configuration with constant gravitational acceleration @xmath9 so that we have a gravitational potential given by @xmath11 . we expose the atom to a superposition of an upward propagating beam with phase @xmath12 and a downward propagating beam with phase @xmath13 , which couples an internal ground state @xmath14 to an internal excited state @xmath15 . the two - photon rabi frequency is @xmath16 , where we let @xmath17 denote the single - photon rabi frequency of the upward propagating beam , @xmath18 denote the single - photon rabi frequency of the downward propagating beam , and @xmath19 denote the detuning from the excited state . we depict the physical setup in fig . [ fig : setup ] . making the rotating wave approximation and adiabatically eliminating the excited state as is standard procedure @xcite , we obtain the following hamiltonian where the periodic term in the potential arises from a spatially varying ac stark shift and where @xmath20 is the magnitude of the wave vector of the laser beams @xcite : @xmath21+mg\hat{x}\ ] ] note that where the difference between the frequency of the upward propagating beam and the frequency of the downward propagating beam is denoted by @xmath22 , we will have the relation @xmath23 . for a given @xmath24 , the lattice standing wave will be translated by @xmath25 in the @xmath26 direction from the origin . thus , the velocity of the lattice in the lab frame is : @xmath27 and we rewrite the lab frame hamiltonian as : @xmath28+mg\hat{x}\ ] ] in order to most readily describe the dynamics of an atom in an accelerating optical lattice , it is useful to work in momentum space . the @xmath29 term that appears in the lab frame hamiltonian makes such an approach difficult , especially when considering non - adiabatic corrections to the phase shift . however , we can change frames by performing a unitary transformation in order to obtain a hamiltonian that is easier to handle analytically . in the end , we will see that approaching the problem from the point of view of dressed states provides a convenient hamiltonian for our purposes . we consider the transformation procedure from the lab frame to the dressed state frame in appendix a , where we also introduce an intermediate frame that freely falls with gravity ( which we call the freely falling frame ) . we note that the general form of the unitary transformations considered in appendix a as well as the specific transformations to the different frames we consider can also be found in the appendix of @xcite . and the lasers are detuned from the transition between the atom s internal ground state and excited state so that the atom s external momentum states are coupled through two - photon transitions , creating an effective lattice potential.,width=672 ] it is convenient to absorb the initial velocity @xmath31 of the atom in the lab frame into the dressed state frame , so that velocity @xmath31 in the lab frame corresponds to velocity zero in the dressed state frame . the hamiltonian in the dressed state frame is , as derived in appendix a : @xmath32 now , we will show how working in momentum space allows us to represent @xmath33 as an infinite dimensional , discrete matrix . this matrix is discrete because the optical lattice potential term , @xmath34 , only couples a momentum eigenstate @xmath35 to the eigenstates @xmath36 and @xmath37 @xcite . for the moment , we will examine the evolution of individual eigenstates of the dressed state hamiltonian @xmath33 . these eigenstates reduce to single momentum eigenstates @xmath35 when @xmath38 . the knowledge of how each of these eigenstates evolves under @xmath33 will allow us to describe the dynamics of an entire wavepacket . for the moment , we will only consider momentum eigenstates corresponding to an integer multiple of @xmath39 , since we have boosted away the initial velocity @xmath31 of the atom in the lab frame . we note that it is always possible to transform to a particular dressed state frame in which a given momentum eigenstate in the lab frame corresponds to zero momentum in that dressed state frame . the results we derive here can thus be readily generalized to arbitrary momentum eigenstates in a wavepacket , as we discuss in greater detail in appendix b. we consider a discrete hilbert space spanned by the momentum eigenstates @xmath40 for integers @xmath41 , so that we can express any vector in this hilbert space as : @xmath42 since this hilbert space is discrete , it is natural to adopt the normalization convention that @xmath43 . when considered as an operator acting on this discrete hilbert space , @xmath33 can be written as @xcite : @xmath44 where we drop the common light shift . now , it is convenient to introduce the recoil frequency @xmath45 and the recoil velocity @xmath46 . in order to make our notation as compact as possible , we will be interested in the quantity @xmath47 , which is the velocity of the lattice in the dressed state frame in units of @xmath48 . we can express the second term of @xmath49 in a useful way by noting that @xmath50 . furthermore , we define @xmath51 . we can now write the discrete hamiltonian in a simplified form : @xmath52 \\\ ] ] the matrix elements of this hamiltonian are : @xmath53\ ] ] in matrix notation , the schrodinger equation for the discrete hilbert space takes the form @xmath54 , where we let @xmath55 be the matrix whose element in the @xmath56th row and @xmath41th column is given by @xmath57 and we let @xmath58 be the column vector whose @xmath41th entry is @xmath59 . now that we have determined the hamiltonian matrix @xmath55 , we have the appropriate machinery in place to describe the phase evolution of an atom in an optical lattice . we consider the process in which momentum is transferred to the atom through bloch oscillations . reference @xcite provides a thorough and insightful description of bloch oscillations in a number of different pictures . given our choice of hamiltonian , we work in the dressed state picture , which is discussed in sec . iv.b . of @xcite , making extensive use of bloch s theorem , the concept of brillouin zones , and the band structure of the lattice @xcite . as in the previous discussion , we consider the evolution of single eigenstates of the dressed state hamiltonian , noting that we can easily generalize our results to the case of a wave packet of finite width , as we address in appendix b. initially , we consider the system to be in a momentum eigenstate . first , we adiabatically ramp up the lattice depth by increasing the laser power so that we load the system into an eigenstate of the hamiltonian . for the purposes considered here , we want the system to enter the ground eigenstate ( corresponding to the zeroth band of the lattice ) . in order for this to be achieved , a resonance condition must be met , which states that the velocity of the lattice must match the velocity of the atom to within @xmath48 . the loading will be adiabatic if the adiabatic condition @xmath60 is satisfied , where @xmath61 and @xmath62 respectively denote the ground state and the first excited state of the hamiltonian @xcite . this condition will be easier to meet near the center of the band ( where the velocity of the lattice is identical to the velocity of the atom ) , because the energy gap @xmath63 between the zeroth band ( which corresponds to ground state ) and the first band ( which corresponds to the first excited state ) becomes smaller as the velocity difference between the lattice and the atom becomes larger ( which corresponds to moving toward the border of the first brillouin zone ) . the resonance condition is discussed further in appendix c. in the lab frame , the atom accelerates under gravity , increasing the deviation between its velocity and the lattice velocity during the loading process . in the freely falling and dressed state frames , in which gravity is boosted away , this corresponds to the lattice accelerating upward while the atom remains at rest . this effect can negatively impact the loading efficiency if the loading sequence is sufficiently long so that the accrued velocity difference becomes a significant fraction of @xmath48 . in such a scenario , the effect can be ameliorated by accelerating the lattice in the lab frame to fall with the atom , which corresponds to the lattice velocity remaining constant in the freely falling and dressed state frames . furthermore , we note that in the case of a lattice lmt beam splitter , the lattice should only be resonant with one arm of the interferometer , so that negligible population from the other arm is affected . otherwise , the signal could be distorted by multi - path interference , causing a systematic error in the estimation of the interferometer phase shift . conditions for when the negative effects of off - resonant lattices can be avoided can be estimated using the hamiltonian matrix for an off - resonant lattice given in eq . ( [ eqn : offresmatrix ] ) . we discuss off - resonant lattices quantitatively and in more detail in sec . vi . after the adiabatic loading of the atom into the ground state of the hamiltonian , the frequency difference between the laser beams is swept to accelerate the lattice , periodically imparting momentum to the atom in units of @xmath39 . in the dressed state frame , this phenomenon can be understood in terms of avoided line crossings , which occur because the coupling between the atom and the laser beams lifts the degeneracy at the crossing points . we refer the reader to fig . 10 of @xcite for a clear illustration of these avoided crossings . as the frequency difference is swept so that the system passes through the avoided crossings , the system remains in the ground state of the dressed state hamiltonian as long as the process is adiabatic . consequently , at each of the avoided crossings , the momentum of the atom increases by @xmath39 , which corresponds to a bloch oscillation . finally , after the acceleration of the lattice , the frequency difference is held constant while the lattice depth is adiabatically ramped down , delivering the system into the momentum eigenstate @xmath64 , where @xmath65 is the momentum before the bloch oscillations and @xmath66 is the number of bloch oscillations . [ fig : latticeaccel ] depicts the lattice depth and velocity as functions of time for the process described above and shows a numerical simulation of a particular instance of this process : the adiabatic loading of the lattice from an initial state @xmath67 , the transfer of @xmath2 of momentum through 5 bloch oscillations , and the ramping down of the lattice to deliver the system into the final state @xmath68 . since under the adiabatic approximation we assume that the atom always stays in the ground state of the hamiltonian , the phase @xmath69 of the atom evolves as follows : @xmath70 where @xmath71 is the instantaneous ground state eigenvalue of the hamiltonian and @xmath72 is the initial phase of the atom . in addition to eq . ( [ eqn : adiabaticphase ] ) , there is also a berry s phase term @xcite . however , this term is zero for a linear external potential . therefore , there is no contribution from the berry s phase under the semiclassical approximation , as long as the external potential is treated as linear we discuss the validity and ramifications of this approximation at the end of appendix a. we note that any such contribution would arise from the residual external potential terms of the dressed state hamiltonian that are non - linear in @xmath26 ( we collectively denote these terms as @xmath73 in appendix a and explain why they can often be neglected ) . of momentum . the lattice depth and velocity are shown as a function of time in the right panel , where in this particular case the relevant parameters are @xmath74 , @xmath75 , @xmath76 , and @xmath77 ( corresponding to a final lattice velocity of 10 @xmath48 ) . first , we adiabatically ramp up the lattice to a depth of @xmath78 so that the lattice is loaded into the ground state of the dressed state hamiltonian . subsequently , we accelerate and ramp down the lattice , leaving the atom in a single momentum eigenstate.,width=720 ] we now consider how the eigenvectors and eigenvalues of @xmath55 change with @xmath79 , which we recall is the dimensionless velocity of the atom in the dressed state frame . say that the eigenvalues of @xmath80 are given by @xmath81 with corresponding eigenvectors @xmath82 , where the index @xmath41 runs from @xmath83 to @xmath84 . we choose to index the eigenvalues so that @xmath81 denotes the the @xmath41th eigenvalue labeled in order of increasing value . moreover , we let @xmath85 be the @xmath86th element of the column vector @xmath82 , so that @xmath87 . the transformation properties of the eigenvectors and eigenvalues under changes in @xmath79 can be deduced from bloch s theorem . it can be shown that when the lattice velocity is increased by @xmath88 , which corresponds to @xmath79 being increased by two , while @xmath89 is kept fixed , the new eigenvectors can be obtained through the following relation @xcite : @xmath90 this simply represents a shift of the wavefunction in momentum space by @xmath39 , which is exactly what we expect , since increasing the lattice velocity by @xmath91 corresponds to undergoing a single bloch oscillation . the dependence of the eigenvalues on the lattice velocity can be expressed as follows : @xmath92 where @xmath93 is periodic in @xmath79 such that @xmath94 for integer @xmath56 holds for all @xmath79 and @xmath93 vanishes when the condition @xmath95 holds . this periodicity follows directly from bloch s theorem , since increasing @xmath79 by @xmath96 corresponds to increasing the lattice velocity by @xmath97 , meaning that @xmath79 and @xmath98 are at the same point in the first brillouin zone . note that the dependence of the eigenvalue on @xmath89 can be calculated using the truncated matrix approximation discussed in sec . the relevance of this dependence to the phase shift of an interferometer and how this dependence varies with momentum are discussed in sec . v. and appendix b. the first term in eq . ( [ eqn : evalue ] ) has a simple physical interpretation . where @xmath99 is the velocity of the lattice in the dressed state frame ( so that @xmath100 ) , note that : @xmath101 which is simply the kinetic energy of an atom traveling along a classical trajectory defined by the motion of the lattice . the @xmath102 and @xmath93 terms represent the band structure of the lattice , with the @xmath93 term accounting for the bands deviating from being flat . for interferometer geometries in which lattices act as waveguides for the atoms , the net contributions to the phase shift from the @xmath103 and @xmath104 terms in the ground state energy and from corrections to the adiabatic approximation are often negligible , as explained in the following sections . in this case , the phase difference between the two arms of the interferometer is given by ( assuming that the two arms arrive at the same endpoint ) : @xmath105 \\ \nonumber & = & \frac{1}{\hbar}\left [ \int_{0}^{t } \frac{1}{2}mv_{\text{lattice}}^{\text{arm1}}(t)^2d t-\int_{0}^{t } \frac{1}{2}mv_{\text{lattice}}^{\text{arm2}}(t)^2 d t \right ] \\\end{aligned}\ ] ] where @xmath106 is the time elapsed during the interferometer sequence . observe that this expression for the phase difference can be obtained by assuming that the lattice potential acts as a constraint that forces the atoms in each arm to traverse the classical path traveled by the lattice guiding that arm . in this case the phase shift is just the difference of the respective action integrals over the two classical paths , as we would expect from the feynman path integral formulation of quantum mechanics @xcite . since the lattice is the only potential in the freely falling and dressed state frames , the action integrals yield eq . ( [ eqn : feynman ] ) ( for an insightful treatment of the applications of path integrals in atom interferometry , we refer the reader to @xcite ) . the terms that we have neglected in eq . ( [ eqn : feynman ] ) embody corrections to the simple picture of the lattice as a force of constraint arising from the quantum nature of the motion ( e.g. , a small portion of the population leaving the ground state of the lattice ) , which can sometimes be important . however , our simple picture provides physical intuition into the lattice phase shift and is often sufficient to derive quantitative results . to summarize , the eigenvalues of the lattice consist of a kinetic energy term , a term that depends only on the lattice depth , and a term that is periodic in @xmath79 , and the eigenvectors transform under a simple shift operation when @xmath79 is changed by @xmath107 for integer @xmath56 . these properties are a direct result of bloch s theorem . the symmetries that we have discussed allow us to conclude that if , for a given @xmath89 , we know the eigenvalues and eigenvectors of the hamiltonian for all @xmath79 within any range @xmath108 $ ] , we can subsequently determine the eigenvalues and eigenvectors for arbitrary @xmath79 . this result will prove to be useful from a computational standpoint , since the dynamics of the system are completely described by the solution within a finite range of @xmath79 . we now present the method of perturbative adiabatic expansion @xcite to determine corrections of arbitrary order to the adiabatic approximation . we note that the particular adiabatic approximation that we correct here refers to the adiabatic evolution of the ground state of the dressed state hamiltonian , rather than the adiabatic elimination of the excited state during the raman process , which is treated in @xcite . the corrections we consider will always be present to some extent , since lattice depth and velocity ramps occurring over a finite time can never be perfectly adiabatic . in addition , non - adiabatic corrections can be caused by perturbations arising from laser frequency noise and amplitude noise . although our analytical and numerical computations indicate that the contribution of non - adiabatic corrections to the overall phase shift will be highly suppressed for many interferometer geometries , it is important to have a generalized framework with which to treat these corrections in order to determine when they are important and to precisely calculate them when necessary . during the course of this derivation , it is convenient to parameterize the hamiltonian , eigenvalues , and eigenvectors using the single variable @xmath109 rather than the two variables @xmath110 and @xmath111 . note that much of our discussion will follow a similar outline as the proof of the adiabatic theorem in @xcite . a more detailed version of the derivation presented here can be found in @xcite . for all times @xmath109 , we can express any state vector @xmath58 in hilbert space as a linear combination of the instantaneous eigenvectors @xmath112 of the dressed state hamiltonian , where in general the coefficients of each eigenvector can vary in time . the instantaneous eigenvectors satisfy the relation @xmath113 . choosing coefficients with a phase @xmath114 factored out , we can write : @xmath115 to simplify matters further , we choose the phase of @xmath112 so that each element of @xmath112 is real for all @xmath109 and varies continuously with @xmath109 , which we can do because the particular hamiltonian matrix @xmath116 we consider is a real - valued , hermitian matrix . when applied to eq . ( [ eqn : stateexpansion ] ) , the schrodinger equation gives us the relation : @xmath117}\ ] ] we note that the inner products @xmath118 are zero for the case of a linear external potential , as verified numerically , and we will thus drop them from the sum . these inner products are closely related to the berry s phase , because integrating @xmath118 with respect to time gives the berry s phase for the @xmath86th eigenvector . ( [ eqn : coefderiv ] ) provides us with a relation which we could directly use to perform adiabatic expansion . but to see more clearly how the adiabatic expansion series relates to the rate at which the hamiltonian changes in time , we will express @xmath119 in a more transparent form . as long as @xmath120 , we can express @xmath121 in terms of a matrix element of @xmath122 and an energy difference @xcite : @xmath123 we can therefore write eq . ( [ eqn : coefderiv ] ) as : @xmath124 } \right ) - \left(\sum_{n \in s_{nd}(t ) } b_n(t ) \frac{\vec{\psi}^{\dag}_j(t ) \dot{h}(t ) \vec{\psi}_n(t)}{\varepsilon_n(t)-\varepsilon_j(t ) } e^{i \left[\varphi_n(t ) - \varphi_j(t ) \right ] } \right)\ ] ] where @xmath125 is the set of all @xmath41 such that @xmath126 and @xmath127 and @xmath128 is the set of all @xmath41 such that @xmath129 . ( [ eqn : non - adiabaticexpansion ] ) illuminates the rationale behind the adiabatic approximation . under the adiabatic approximation , we assume that @xmath116 and hence also its eigenvectors vary slowly enough in time so that the conditions @xmath130 ( for @xmath129 ) and @xmath131 ( for @xmath127 and where @xmath132 is the time scale of the approximation ) hold . the righthand side of eq . ( [ eqn : non - adiabaticexpansion ] ) can therefore be approximated as zero . then , all the coefficients @xmath133 are constant in time . to compute higher order corrections , we employ the method of adiabatic expansion , which mathematically follows in the spirit of the born approximation . to zeroth order , we take the coefficients @xmath133 to be constant as dictated by the adiabatic approximation . to obtain the first order corrections to these coefficients , we substitute the constant zeroth order coefficients @xmath134 into the righthand side of eq . ( [ eqn : coefderiv ] ) with @xmath126 and integrate to find the first order coefficients @xmath135 . we can repeat this process recursively , substituting the coefficients @xmath135 into eq . ( [ eqn : coefderiv ] ) to calculate the coefficients @xmath136 and continuing until we know the coefficients to the necessary precision . this method provides a way to construct a series expansion for each @xmath133 . we adopt a matrix notation for the terms in this series to facilitate the discussion . the first order correction to @xmath133 consists of contributions from each nonzero @xmath137 where @xmath126 , and we depict the contribution from @xmath137 as @xmath138 . so to first order , we can write : @xmath139 where : @xmath140 } dt^{\prime}\ ] ] the second order solution for @xmath133 will then be : @xmath141 } dt^{\prime } = b^{(0)}_j+ \sum_{n \neq j } c_{n \rightarrow j } + \sum_{n \neq j } \sum_{m \neq n } c_{m \rightarrow n \rightarrow j}\ ] ] where : @xmath142 } dt^{\prime}\ ] ] under the implicit assumption that the time variable associated with @xmath143 is appropriate for the context in which it appears . the calculation of corrections of higher order is discussed in appendix d. to find the eigenvectors and eigenvalues that we need to calculate the terms that make a non - negligible contribution to the expansion , we must approximate the infinite dimensional hamiltonian matrix as a finite dimensional truncated matrix . at the end of sec . , we concluded that the problem of determining the eigenvalues and eigenvectors for all @xmath79 reduces to finding the eigenvalues and eigenvectors for a range @xmath144 $ ] for arbitrary @xmath145 . in addition , it suffices to calculate the inner products @xmath121 just in this range of @xmath79 , which follows from the symmetry @xmath146 for integer @xmath56 @xcite . to make the calculation less cumbersome , we can look at the range @xmath147 $ ] . for @xmath89 not too large and @xmath79 in this range , the eigenvectors with lower energies are populated almost entirely by momentum eigenstates @xmath148 with relatively small @xmath149 . this is the case because for @xmath79 in the range , the diagonal elements of the hamiltonian will be smallest for values of @xmath56 close to zero . we note that for @xmath150 , the diagonal elements are the eigenvalues . in the limit of @xmath151 , each eigenvector will consist of only a single momentum eigenstate , where in general eigenvectors corresponding to momentum eigenstates with @xmath56 closer to zero will have lower eigenvalues . increasing @xmath89 will allow the lower eigenvectors to spread out in momentum space to a certain extent , but this will not change the fact that the lower eigenvectors will be linear combinations of momentum eigenstates corresponding to smaller values of @xmath149 . as discussed previously , the eigenvectors we care about for calculational purposes will be those with eigenvalues closer to the ground state eigenvalue . we can thus consider a truncated @xmath152 hamiltonian matrix centered around @xmath153 , where we choose @xmath41 to be large enough so that for the particular dynamics being described , a sufficient number of eigenvectors and eigenvalues can be calculated . we illustrate the above method by calculating phase corrections to a lattice beam splitter . in this example , we consider the case where two optical lattices of the same depth but different accelerations are used to separate the two arms of the interferometer ( after an initial momentum space splitting is achieved through bragg diffraction ) . we note that the analysis here is equally applicable to the situation where two separate optical lattice waveguides are used to address the two arms of the interferometer , an example of which is illustrated in fig . [ fig : conj ] . to calculate the phase shift for applications in precision measurement , we need to determine the non - adiabatic correction to the phase difference between the two arms that accrues during the beam splitter . in practice , we do this by first calculating corrections to the ground state coefficient @xmath154 and then evaluating how these corrections affect the phase difference between the arms . we note that the dominant contribution to the phase difference will come from the zeroth order term as given in eq . ( [ eqn : feynman ] ) . for this example , we consider the situation shown in fig . [ fig : latticeaccel ] , where the interaction of the atoms with the lattice is divided into three distinct parts . from @xmath155 to @xmath156 we ramp up the lattice , from @xmath156 to @xmath157 we accelerate the lattice , and from @xmath157 to @xmath158 we ramp down the lattice . for the sake of simplicity , the ramps are chosen to be symmetric so that the lattice depth decrease ramp is the time reversed lattice depth increase ramp . we assume that initially all of the population is in the ground state , so that @xmath159 , and we make the lattice depth and velocity ramps adiabatic enough so that almost all of the population remains in the ground state . in order to find the non - adiabatic correction to the phase shift , we determine the non - adiabatic correction to the phase of the ground state for each arm . since to lowest order only the ground state is populated , the leading corrections to @xmath154 will come at second order . using the formalism in eq . ( [ eqn : secondorder ] ) , the second order correction will be : @xmath160 the largest contribution comes from @xmath161 , and it is this term on which we focus . because the ground state is non - degenerate , eqs . ( [ eqn : non - adiabaticexpansion ] ) and ( [ eqn : factor2 ] ) give us : @xmath162 } \\ \nonumber & = & -\int_{0}^{t } dt_1 \left(-\int_{0}^{t_1 } dt_2 \frac{m_{10}(t_2)}{\delta \varepsilon_{10}(t_2 ) } e^{-\frac{i}{\hbar } \int_{0}^{t_2 } \delta \varepsilon_{10}(t_3 ) dt_3 } \right ) \frac{m_{10}(t_1)}{- \delta \varepsilon_{10}(t_1 ) } e^{\frac{i}{\hbar } \int_{0}^{t_1 } \delta \varepsilon_{10}(t_3 ) dt_3 } \\\end{aligned}\ ] ] where we define @xmath163 ( note that the two matrix elements are equal because we choose the eigenvectors to be real ) and @xmath164 . we examine the ultimate contribution of @xmath165 to @xmath166 . since during the ramp up and ramp down stages @xmath167 depends only on @xmath168 but not on @xmath169 , some portions of @xmath165 will be common to both arms of the interferometer , because we assume that the lattice interaction processes for the two arms differ only in the magnitude of the lattice acceleration . we denote these common terms as @xmath170 . the remaining terms depend on the lattice acceleration and will thus differ between the arms . there will be a term @xmath171 that depends both on @xmath168 and @xmath169 . however , it can be shown that under the assumption that the lattice depth decrease ramp is the time reversed lattice depth increase ramp , this term is zero @xcite . finally , there will be a term @xmath172 that depends quadratically on @xmath169 and is not explicitly dependent on @xmath168 . we note that @xmath172 implicitly depends on the maximum lattice depth @xmath173 , which can be seen by the fact that @xmath173 affects what value the energy gap @xmath174 takes on during the acceleration stage ( a deeper lattice leads to a larger energy gap ) . we can thus write : @xmath175 in calculating @xmath172 , it is useful to note that @xmath167 takes on a convenient form during the acceleration stage . recalling the form of the hamiltonian matrix from eq . ( [ eqn : hamiltonianelements ] ) , we observe that @xmath122 will be a diagonal matrix with matrix elements @xmath176 . we can thus write @xmath177 , where @xmath178 is a weighted dot product . in order to more clearly illuminate the general points we are illustrating with this example , we make the simplifying assumption that @xmath179 is constant throughout the acceleration stage . moreover , we assume that the lattice is deep enough so that @xmath180 and @xmath181 are also constant during the acceleration stage , which is an accurate approximation for typical experimental situations . during the acceleration stage , we respectively denote these constant quantities as @xmath169 , @xmath182 , and @xmath183 . note that these assumptions , along with the assumption of mirror symmetry between the ramp up and ramp down stages , are certainly not necessary to carry out the calculation . they only serve to make the final result take a particularly simple form that provides physical insight into the process . in the absence of these assumptions , the calculation will be only slightly more complicated and can easily be performed . we note in particular that the mirror symmetry assumption is not stringent , for even when this symmetry is largely violated , the @xmath171 term is typically an order of magnitude or more smaller than the @xmath172 term , as we verify by estimating the relevant integrals in eq . ( [ eqn : correctionintegrals ] ) . if needed , @xmath171 can be calculated by evaluating these integrals . in addition , we note that the treatment given in this example can readily be generalized to the case where the lattice depth and velocity are changed simultaneously . performing the necessary integrals , we find that : @xmath184\ ] ] we note that for @xmath185 , we can solve the problem by employing adiabatic expansion over a single time interval . however , there may be times when we must divide the problem into multiple parts , as discussed in appendix d. for typical experimental parameters , the term proportional to @xmath186 in @xmath172 will dominate both the second term in @xmath172 and the @xmath170 term . we have verified that the @xmath170 term ( which embodies the non - adiabatic loading of the lattice ) is typically much smaller than the second term in @xmath172 by estimating the integrals in eq . ( [ eqn : correctionintegrals ] ) that correspond to @xmath170 and by checking these estimates numerically . thus , we can express the condition @xmath185 as : @xmath187 ) , as calculated using a simplified adiabatic expansion method in which we keep only leading terms versus numerical simulations of the schrodinger equation . the curve represents the prediction made by the adiabatic expansion method , while the red dots represent the numerical results . the lattice depth ramps and acceleration time are identical to those of the acceleration sequence shown in fig . [ fig : latticeaccel ] , with an acceleration time of @xmath188 and with @xmath76 . in addition to the leading second order term , we keep the leading fourth order correction term in @xmath189 , @xmath190 , calculated in @xcite , which becomes significant for larger accelerations ( e.g. , for an acceleration of @xmath191 , this term is smaller than the leading second order correction by a factor of @xmath192 ) . we neglect corrections arising from the term @xmath170 , for these corrections are common to both arms of the interferometer to lowest order . even the simple approximation used to obtain the curve agrees remarkably well with the simulations ( with an rms deviation of @xmath193 radians ) , and we note that we could easily improve this approximation by including more terms in the adiabatic expansion series . as expected , we observe that the correction scales quadratically with acceleration.,width=672 ] this condition will often hold , since in many experimentally relevant cases the acceleration time or acceleration will be sufficiently small . we now show how to determine the correction to the phase shift between the two arms arising from the non - adiabatic correction @xmath194 in the case where this correction is small . as we recall from eq . ( [ eqn : stateexpansion ] ) , the coefficient of the ground state eigenvector at time @xmath195 is @xmath196 . now , note that the argument of any complex number @xmath197 can be written as @xmath198 $ ] . the non - adiabatic correction to the phase of the coefficient of the ground state eigenvector for an arm with acceleration @xmath169 is thus : @xmath199 \\ \nonumber \\ \nonumber & \approx & \left|g_{\text{ramp}}\right| \sin \left(\arg\left[g_{\text{ramp}}\right]\right ) + \left|g_{\text{accel}}\right| \sin \left(\arg\left[g_{\text{accel}}\right]\right ) \\\end{aligned}\ ] ] where we have taylor expanded to first order in small quantities . to calculate the correction to the phase shift between two arms , we take the difference @xmath200 . the @xmath201\right ) $ ] term is common to both arms . the non - adiabatic correction to the phase difference between an arm with acceleration @xmath169 and an unaccelerated arm is : @xmath202\right)\ ] ] eqs . ( [ eqn : phasecorrection ] ) and ( [ eqn : phasecorrectionref ] ) provide us with a means to calculate the leading correction to the phase shift , and comparison with numerical results for a variety of experimentally conceivable lattice depth ramps shows excellent agreement . a comparison of the adiabatic expansion method with numerical calculations is illustrated in fig . [ fig : correction ] . in particular , fig . [ fig : correction ] shows @xmath203 as a function of @xmath169 for an acceleration time of @xmath188 with @xmath76 . the numerical values shown in the figure come from a numerical simulation of the schrodinger equation . the non - adiabatic correction to the phase shift between the two arms of the interferometer for arbitrary arm acceleration differences is @xmath204 . note that the leading order results depicted in fig . [ fig : correction ] will often be sufficient , but it may sometimes be necessary to calculate higher order corrections , examples of which can be found in @xcite . in an experimental implementation , the two arms of the beam splitter are addressed by two different lattices . therefore , any imbalance in the depths of the two lattices will lead to a phase error between the arms . where the intensities of the two beams forming a given lattice are @xmath205 and @xmath206 , we note that the lattice depth is proportional to the product @xmath207 , meaning that intensity imbalances lead to lattice depth imbalances . we calculate that once a lattice is ramped up , the @xmath103 term in the expression for the lowest eigenvalue of the dressed state hamiltonian from eq . ( [ eqn : evalue ] ) is much larger than the @xmath104 term . thus , for an interaction lasting from time @xmath208 to time @xmath209 , the dominant contribution to the phase error , which we denote as @xmath210 , arising from the imbalance in the lattice depths is : @xmath211 d t\ ] ] where we note that @xmath103 can be calculated using the truncated matrix approximation discussed in sec . iv . in order to put eq . ( [ eqn : balance ] ) in a more convenient form for making order of magnitude estimates , we use the fact that for @xmath212 , @xmath213 . this result is verified by direct comparison with the values of @xmath103 obtained with the truncated matrix approximation . recalling that @xmath214 , it is convenient to rephrase this statement in terms of @xmath215 as follows : for @xmath216 , @xmath217 . this range of @xmath218 is of considerable experimental interest and is amenable to simple approximation . however , if needed , smaller lattice depths can be treated using the general relation given in eq . ( [ eqn : balance ] ) and the truncated matrix approximation . substituting the above result into eq . ( [ eqn : balance ] ) , we obtain : @xmath219 where @xmath220 and @xmath221 respectively denote the average values of @xmath222 and @xmath223 between time @xmath208 and time @xmath209 . note that fluctuations in the difference between @xmath222 and @xmath224 that occur at frequencies that are large with respect to the beam splitter time @xmath225 will largely average out , suppressing their net effect on the phase shift . furthermore , in many geometries , arm 1 will be addressed by one pair of laser beams , that we call lattice a , during the splitting of the arms and then will be addressed by a second pair of laser beams , that we call lattice b , during the recombination of the arms . conversely , arm 2 will be addressed by lattice b during the splitting stage and by lattice a during the recombination stage . in such a geometry , the effect on the phase shift of a constant offset in depth between lattice a and lattice b will cancel , and the effect on the phase shift of fluctuations in the depth difference between lattice a and lattice b that occur at low frequencies with respect to the time scale of the interferometer sequence will be highly suppressed . also , as we discuss in the following section , we will often be interested in the difference in the phase shifts of two interferometers in a differential configuration . if both of the interferometers remain well within the rayleigh ranges of the laser beams so that beam divergence is a small effect , any lattice depth imbalance will be largely common to the two interferometers , suppressing its effect on the phase shift difference by orders of magnitude . light - pulse atom interferometer geometries have had tremendous success in performing many types of high - precision measurements . however , in many cases , we would like to be able to push the capabilities of atom interferometry by making more precise measurements using spatially compact interferometers . atom interferometers that use optical lattices as waveguides for the atoms offer the potential to make such measurements attainable . in such a scheme , we can use an initial beam splitter composed of multiple bragg pulses , a multi - photon bragg pulse , or a hybrid bragg pulse / lattice acceleration scheme as described in @xcite to split the arms of the interferometer in momentum space . we can then control each arm independently with an optical lattice . we will once again use bragg pulses during the @xmath226-pulse and final @xmath227-pulse stages of the interferometer sequence , with lattices acting as waveguides between these stages . the preceding analysis has developed the theoretical machinery for calculating phase shifts for these lattice interferometers . we now examine several of the most promising applications of lattice interferometers . lattice interferometers can be used to make extremely precise measurements of the local gravitational acceleration @xmath9 . we proceed to calculate the phase shift for a lattice gravimeter . it is essential to note that whenever the two arms are addressed by different lattices they will be in different dressed state frames ( where we recall that a dressed state frame is defined by the velocity of the corresponding lattice and by the distance that the lattice has traveled since the beginning of the interferometer sequence ) . let the velocities in the lab frame of the two lattices be denoted as @xmath228 and @xmath229 , respectively . the lattice velocities for the two arms in their respective dressed state frames will thus be @xmath230 and @xmath231 . we note that @xmath31 is the velocity of the atom before the initial bragg diffraction that splits the arms in momentum space , so that the two arms have different momenta after the bragg diffraction . during the lattice loading period , the two lattices must be resonant ( as described in sec . iii and in appendix c ) with the respective portions of the atomic wavefunction that they are addressing . also , let @xmath232 be the velocity difference between the two arms and @xmath233 be the distance between the two arms . where the interferometer sequence lasts for a time @xmath106 , we can derive the phase shift for a lattice gravimeter using eq . ( [ eqn : feynman ] ) . we note that an additional contribution to the phase shift will arise if the two arms of the interferometer end up in slightly different dressed state frames . the arms begin in the same dressed state frame , and if @xmath234 they will end up in the same dressed state frame , in which case eq . ( [ eqn : feynman ] ) provides the final say on the phase shift . if @xmath235 differs slightly from zero , the two arms will end up in slightly different dressed state frames . since we must compare phases in a common frame , it is convenient to boost both arms into the freely falling frame using the transformation given in eq . ( [ eqn : dstransform ] ) and the mathematical framework discussed in appendix b. in addition to the phase difference from eq . ( [ eqn : feynman ] ) , we will then have an additional term in the phase shift equal to @xmath236 , where @xmath237 is the lab frame momentum at time @xmath106 of a particular momentum eigenstate in the atomic wavepacket . after averaging , it follows from the discussion in @xcite that @xmath237 will take on the value of the center of the momentum space wavepacket . we can combine the contribution from eq . ( [ eqn : feynman ] ) and the contribution from boosting the two arms to a common frame to calculate the phase shift , where we also include a term @xmath238 to embody the net contribution to the phase shift arising from the bragg pulses : @xmath239 -\frac{1}{\hbar } m(v_f - v_0 + gt ) \delta d(t ) + \phi_{\text{bragg}}\\ \nonumber \\ \nonumber & = & \frac{1}{\hbar}\left [ \int_{0}^{t } \frac{1}{2}m(v_{\text{lab}}^{\text{arm1}}(t)^2-v_{\text{lab}}^{\text{arm2}}(t)^2)d t+\int_{0}^{t}mg \delta v(t ) t dt - \int_{0}^{t}m v_0 \delta v(t ) dt \right ] -\frac{1}{\hbar } m(v_f - v_0 + gt ) \delta d(t ) + \phi_{\text{bragg}}\\\end{aligned}\ ] ] integrating the second term by parts and simplifying yields : @xmath240 when expressed in terms of lab frame quantities , it is apparent that @xmath241 contains terms corresponding to the propagation phase and the separation phase that typically appear in standard atom interferometer phase shift calculations . we note that in the dressed state frame , ideal bragg pulses simply yield contributions in units of @xmath242 to the overall phase shift between the arms , and these contributions can easily be made to cancel so that @xmath243 . however , we note that in some cases , corrections to the simplified picture of an ideal bragg pulse due to such factors as gravity gradients , finite pulse and detuning effects ( which can sometimes lead to a non - negligible propagation phase during the bragg pulse ) , phase noise , or population loss may need to be considered . to avoid unnecessarily complicating our presentation , we will not present these corrections here . instead , we emphasize that they are well - understood effects and refer the reader to other sources for further discussion @xcite . moreover , we note that these effects gain additional suppression for interferometers in a differential configuration so that they will often be below the mrad level @xcite , as in the case of the gravity gradiometer discussed below . in the symmetric case where the velocities of the two arms in the lab frame are either opposite to each other or equal so that @xmath244 . since we need the two arms of the interferometer to overlap at time @xmath106 , it is convenient for us to choose @xmath234 . thus , the first term in eq . ( [ eqn : phaseshift ] ) will constitute the only contribution to @xmath241 . however , in an experiment , the parameters in eq . ( [ eqn : phaseshift ] ) will undergo small fluctuations around their desired values from shot to shot , so that the other terms in eq . ( [ eqn : phaseshift ] ) act as a source of noise . in order to cancel the effects of this noise , we can adopt a gradiometer setup in which an array of two or more gravimeters interacts with the same lattice beams . although fluctuations in @xmath245 will still affect phase differences between gravimeters , which take the form @xmath246 , modern phase lock techniques will typically allow us to control the phase differences between the lattice beams well enough so that these effects are smaller than shot noise @xcite . when measuring a gravity gradient , the value of @xmath9 will vary due to the gradient over the range of a single gravimeter . for linear gradients , we can calculate the phase shift in the presence of a gravity gradient by assuming that the value of @xmath9 corresponding to the gravimeter is equal to its value at the center of mass position of the atom ( see @xcite for a rigorous justification of this procedure ) . when higher order derivatives of the gravitational field become sizable in comparison to the first derivative , this simple prescription may not suffice , and we can treat the problem perturbatively . if we want to measure an acceleration as well as a gravitational gradient in a noisy environment in which the fringes of the individual gravimeters are washed out , we can use dissimilar conjugate interferometers whose phase noise is strongly correlated as suggested in @xcite . appropriate statistical methods can then be used to extract the desired signal @xcite . the effects on the phase shift of non - adiabatic corrections , lattice depth imbalances , and the finite spread of the atomic wavefunction in momentum space are considered in sec . iv . , sec . v. , and appendix b. based on the analysis in these sections , we conclude that a wide range of experimentally feasible gravimeter geometries exist that contain sufficiently adiabatic lattice depth and velocity ramps and that make use of symmetry in such a way that the net contribution of these corrections to the phase shift will be below the mrad level in a gradiometer configuration . we note that this paper has developed the mathematical machinery to calculate any such corrections to arbitrary precision if necessary . when two lattices are used to manipulate the arms of an atom interferometer , one lattice will be on resonance with a given arm , while the other will be highly detuned . as long as we keep this detuning large enough and/or employ geometries with sufficient symmetry between the arms , the net effect of the off - resonant lattices on the final phase shift can often be made to be smaller than the mrad level in a differential configuration ( e.g. , a gradiometer ) . for arm 1 , the detuned lattice will manifest as an additional term in the discrete hamiltonian , given by : @xmath247 where @xmath248 4 \omega_rdt^{\prime}=-2 k \delta d(t)$ ] and where @xmath249 is the lattice depth parameter corresponding to the detuned lattice that addresses arm 2 @xcite . to obtain the correction to the hamiltonian for arm 2 , which comes from the detuned lattice addressing arm 1 , we replace @xmath250 with @xmath251 and @xmath249 with @xmath252 . for large detunings , @xmath250 will vary rapidly with time so that the contribution from the detuned hamiltonian will be small due to the rotating wave approximation . corrections arising from the detuned hamiltonian can be solved for perturbatively using methods such as adiabatic perturbation theory . but we emphasize again that we can often avoid situations where this will be necessary . for example , we find from perturbation theory that to avoid population loss due to the off - resonant lattice as described in sec . , we should choose the off - resonant lattice to have a velocity that differs from that of the particular arm of the interferometer under consideration by an amount @xmath253 ( where @xmath89 is the depth parameter of the off - resonant lattice ) . we have verified this result with numerical simulations . however , in this regime , the off - resonant lattice can still sometimes cause a non - negligible energy shift , which we estimate with perturbation theory . the energy shift for arm 1 is @xmath254 . analagously , the energy shift for arm 2 is @xmath255 . the relevant quantity in determining the correction to the phase difference between the arms is the difference in energy shifts @xmath256 , which is determined by the lattice depth imbalance between the arms . where we let @xmath257 and @xmath258 : @xmath259 this result agrees with numerical simulations . [ fig : shift ] illustrates the dependence of @xmath260 on @xmath261 . for the same reasoning as in the discussion in the previous section , the net effect of lattice depth imbalances on the correction term treated here will often be further suppressed by orders of magnitude for interferometers in a differential configuration . thus , as stated previously , in many cases the net effect of phase corrections due to the off - resonant lattices will be below the mrad level in a differential configuration . on @xmath261 as given in eq . [ eqn : offresshift ] , which quantifies the effect of the off - resonant lattices . this plot takes the lattice depth for arm 1 to be @xmath257 and the lattice depth for arm 2 to be @xmath258 , where @xmath262 and @xmath263 . the recoil frequency is taken to be @xmath264.,width=720 ] a lattice gravimeter can provide extraordinary levels of sensitivity . this sensitivity can be achieved over small distance scales by implementing a hold sequence in which the two arms are separated , manipulated into the same momentum eigenstate , held in place by a single lattice , and then recombined . in achieving compactness , we note that the fact that lattice interferometers are confined and can thus keep the atoms from falling under gravity during the separation and recombination stages of the interferometer as well as during the hold sequence is essential . otherwise , for many configurations , the desired arm separation could not be reached without the atoms falling too great a distance , which would ruin the compactness of the interferometer . hold times will be limited by spontaneous emission , which decreases contrast . modern laser technology will allow us to use detunings of hundreds or even thousands of ghz , making hold times on the order of 10 s within reach @xcite . gravimeter sensitivities using the hold method greatly exceed the sensitivities of light - pulse gravimeters while simultaneously allowing for a significantly smaller interrogation region . for example , for @xmath265 atoms / shot and @xmath266 shots / s , a shot noise limited conventional light - pulse interferometer with a 10 m interrogation region can achieve a sensitivity of @xmath267 @xmath9/hz@xmath10 . with similar experimental parameters , a shot noise limited lattice interferometer with a 10 s hold time and an interrogation region of 1 cm will have a sensitivity of @xmath268 @xmath9/hz@xmath10 . if we expand the interrogation region to 1 m , we obtain a sensitivity of @xmath269 @xmath9/hz@xmath10 . this remarkable sensitivity has a plethora of potential applications . extremely precise gravimeters and gravity gradiometers can be constructed to perform tests of general relativity , make measurements relevant to geophysical studies , and build highly compact inertial sensors . moreover , the fact that lattice interferometers can operate with such high sensitivities over small distance scales makes them prime candidates for exploring short distance gravity . one could set up an array of lattice gravimeters to precisely map out gravitational fields over small spatial regions , as shown in fig . [ fig : array ] . the knowledge obtained about the local gravitational field could be useful in searching for extra dimensions @xcite as well as in studying the composition and structure of materials . , @xmath208 , @xmath209 , and @xmath270 . such an array could be used to study general relativistic effects , search for extra dimensions , examine local mass distributions , or measure newton s constant . in addition , the ability of lattice interferometers with hold sequences to provide extremely precise measurements with a small interrogation region makes them ideal candidates for compact , mobile sensors.,width=288 ] ultra - high precision gravitational measurements are certainly among the most promising applications of lattice interferometers , but the usefulness of lattice interferometers is certainly not limited to the study of gravity . by exposing the two arms of a lattice interferometer to different electrostatic potentials , tests of atom charge neutrality with unprecedented accuracy could be achieved @xcite . the main advantage of a lattice interferometer in such a measurement is that the interrogation time can be significantly increased in comparison to the interrogation time achievable in a light - pulse geometry through the use of a hold sequence . where @xmath271 is the duration of the hold , @xmath272 is the electrostatic potential difference between the two arms of the interferometer , @xmath273 is the electron charge , and @xmath274 is the ratio of the atomic charge to the electron charge , the phase shift is given by @xmath275 @xcite . based on the results of sec . iv . , sec . v. , and appendix b , and assuming an identical configuration to that described in @xcite except for the inclusion of a hold sequence , we estimate that the phase error induced by the undesirable effects we consider will be below the proposed shot noise limit for this experiment ( 1 mrad ) . any systematic phase error can be characterized by the methods we have developed . for a hold time of 10 s and for integration over @xmath276 shots , atom charges can be probed down to the region of @xmath277 . the ratio @xmath278 is of particular interest because of its direct relation to the fine structure constant . atom interferometry has previously been used to provide exquisite measurements of @xmath278 . the most precise atom interferometric measurement to date was performed by cadoret _ _ , who performed an elegant experiment combining bloch oscillations and a ramsey - bord interferometer to measure the fine structure constant to within a relative uncertainty of @xmath279 @xcite . ( [ eqn : phaseshift ] ) indicates that if we apply different accelerations to the two arms of the interferometer , we will see a phase shift proportional to @xmath280 that depends on the kinetic energy difference between the arms , which can be made extremely large . a differential configuration using conjugate interferometers ( shown in fig . [ fig : conj ] ) could reduce the net contribution of such unwanted effects as laser phase noise and cancel the gravitational phase shift up to gradients @xcite . such a geometry could provide an extremely precise measurement of @xmath278 , as illustrated by the fact that we can achieve a phase shift of @xmath281 radians for a 5 m interrogation region and a 0.6 s interrogation time , corresponding to a shot noise limited sensitivity of @xmath282/hz@xmath10 for the experimental parameters stated above . in this situation , the dominant unwanted effect would arise from non - adiabatic corrections , and the methods for calculating these corrections that are presented in the previous sections would need to be applied ( we estimate a phase error @xmath283 10 rad ) . we note that even if the shot noise limit is not reached , the technique we have proposed could still improve over current @xmath278 measurements . we emphasize again that to take full advantage of the sensitivity offered by lattice manipulations in atom interferometry , the methods we develop in this paper for calculating non - adiabatic corrections are absolutely essential . . the trajectories shown hold in the lab frame . bragg pulses are used at times @xmath284 , @xmath208 , and @xmath209 ( at @xmath284 , a sequence of multiple bragg pulses is necessary to split the system into the two arms of the two conjugate interferometers ) . the phase shift of the lower interferometer is subtracted from the phase shift of the upper interferometer , which suppresses the effects of laser phase noise and eliminates the gravitational phase shift up to gradients @xcite . with such a scheme , a measurement of @xmath278 to a part in @xmath285 may be possible , which could lead to the most accurate determination of the fine structure constant to date.,width=288 ] the fact that all terms in eq . ( [ eqn : phaseshift ] ) are proportional to @xmath56 ( except for the @xmath238 term , which as we have explained , will often be negligible ) can be exploited to provide high - precision measurements of isotope mass ratios by using an interferometer geometry in which the two isotopes follow identical trajectories . isotope mass ratios could be relevant to studies of advanced models of the structure of the nucleus @xcite . conversely , if we know the mass ratio of two isotopes sufficiently well , we can use such a geometry to precisely measure accelerations , where the two isotopes provide two dissimilar conjugate interferometers . the phase noise of these two interferometers will be extremely well correlated because they are topologically identical . this also eliminates the need for the additional lattice beams required to form the second , topologically distinct interferometer that would be needed if we only had a single isotope . note that this scheme to construct dissimilar , topologically identical conjugate accelerometers would not be possible for a light - pulse geometry , for the leading order phase shift of light - pulse accelerometers is independent of isotope mass . lattice interferometers can also be used to build compact and highly sensitive gyroscopes . there are multiple possible schemes in which optical lattices can enhance gyroscope sensitivity . one such scheme is to modify a typical atom - based gyroscope by replacing the raman pulses with lmt lattice beam splitters , increasing the enclosed area of the interferometer and hence its sensitivity to rotations . the phase shift can be written in sagnac form as @xmath286 , where @xmath287 is the rotation rate vector and @xmath288 is the normal vector corresponding to the enclosed area of the interferometer @xcite . the gyroscope described in @xcite achieves a sensitivity of @xmath289/hz@xmath10 . replacing the raman pulses in this experiment with @xmath290 lattice beam splitters would increase the sensitivity by a factor of 100 . in such a configuration , we estimate that each beam splitter could introduce a non - adiabatic phase error of @xmath283 1 rad if the arms of the interferometer are not split symmetrically . however , beam splitter configurations that exploit symmetry between the arms can reduce this effect by orders of magnitude . another option is to use optical lattices along multiple axes to provide complete control of the motion of the atoms in two or three dimensions ( this control is only achieved in the region in which the lattices overlap , necessitating the use of wide beams ) . analagous to a fiber - optic gyroscope , the atoms could be guided in repeated loop patterns , with the two arms rotating in opposite directions . geometries in which atomic motion is controlled in multiple dimensions could also expand the possibilities for other applications of lattice interferometry ( such as measurements of gravity ) by allowing for the measurement of potential energy differences between arbitrary paths . for instance , a compact array of three orthogonal lattice gravity gradiometers could be used to measure the nonzero divergence of the gravitational field in free space predicted by general relativity @xcite . we have presented a detailed analytical description of the interaction between an atom and an optical lattice , using the adiabatic approximation as a starting point and then proceeding to rigorously develop a method to calculate arbitrarily small corrections to this approximation using perturbative adiabatic expansion . we have applied this theoretical framework to calculate the phase accumulated during a lattice acceleration in an lmt beam splitter . and we have proposed atom interferometer geometries that use optical lattices as waveguides and discussed applications of such geometries , using our theoretical methods to add rigor to this discussion . we are working toward the experimental implementation of lattice interferometers and lmt lattice beam splitters , and we hope to explore the applications we have discussed . in this experimental work , we realize that we will have to contend with a number of unwanted systematic effects , such as spatially varying magnetic fields , imperfections in the lattice beam wavefronts , and inhomogeneity of the lattice depth across the atomic cloud . we have studied these effects using both analytical and numerical methods , and we are optimistic that they can be significantly mitigated for a wide range of experimental parameters a conclusion that we hope to verify experimentally . many of the unwanted systematic effects that are relevant to lattice interferometers are also shared by light - pulse interferometers and can therefore be dealt with using similar methods . therefore , we believe that it will likely be possible to realize lattice interferometers in existing apparatuses originally constructed with light - pulse geometries in mind . we would like to thank philippe bouyer , sheng - wey chiow , gerald gabrielse , and holger mueller for valuable discussions . tk acknowledges support from a fannie and john hertz foundation fellowship and an nsf fellowship . tk and dj acknowledge support from a stanford graduate fellowship . for the purposes of this paper , we consider unitary transformations that consist of a translation in position space , a boost in momentum space , and a time - dependent change of phase . such a transformation has the general form : @xmath291 we note we are free to choose the boost parameters @xmath292 and @xmath293 arbitrarily . that is , we do not have to choose them so that @xmath293 is the rate of change of @xmath292 . the operators @xmath294 and @xmath295 transform under @xmath296 as follows : @xmath297 we now consider the hamiltonian in a frame that is freely falling with gravity , which takes the form : @xmath298\ ] ] we can transform from the freely falling frame to the lab frame by applying the appropriate galilean transformation @xmath299 , which corresponds to specifying @xmath300 , @xmath301 , and @xmath302 , so that @xmath303 . since the only position dependence in @xmath304 comes from the lattice potential , we could readily approach the problem of finding the dynamics of the system by working in the freely falling frame . however , it will prove to be useful from a calculational standpoint to transform to a third frame , with a hamiltonian resembling that describing the atom - light interaction from the point of view of dressed states . we note that although we could have performed a boost directly from the lab frame to the dressed state frame , it is useful to introduce the freely falling frame for pedagogical reasons , since it is the frame in which calculations for atom interferometry are typically performed . in appendix b , we use the transformation between the freely falling frame and the dressed state frame to highlight the parallels between a lattice beam splitter and a typical light pulse beam splitter . we absorb the initial velocity @xmath31 of the atom in the lab frame ( and hence also in the freely falling frame ) into the dressed state frame , so that velocity @xmath31 in the lab frame corresponds to velocity zero in the dressed state frame . the hamiltonian in the dressed state frame is : @xmath32 the unitary transformation @xmath305 that transforms from the dressed state frame to the freely falling frame , so that @xmath306 , corresponds to boost parameters @xmath307 and @xmath308 with a time - dependent phase factor : @xmath309 the ability to boost to a frame in which the hamiltonian contains no position dependent terms outside of the lattice potential is contingent upon the assumption that the external potential in the lab frame ( not including the lattice potential ) is linear in @xmath26 . however , real - world potentials such as the potential corresponding to earth s gravitational field will deviate somewhat from this assumption . any such deviations would manifest as residual position dependent terms in the dressed state hamiltonian , which we collectively refer to as @xmath73 . under the semiclassical approximation , we neglect the effects of @xmath73 on the time evolution of the atomic wavepacket . this approximation is valid when the energy scale of @xmath73 over the spread of the atom s wavefunction ( which is on the order of magnitude of the expectation value of @xmath73 in the atomic wavepacket ) is much smaller than the energy scale of the lattice potential and is small relative to the time scale of the experiment , which is the case for a wide class of experimental parameters . for example , in the case of a rubidium atom wavepacket with a spatial spread of @xmath310 m in the gravitational field at the earth s surface ( which has a gradient of @xmath311 ) , the energy scale of @xmath73 will be @xmath312 @xmath313 ( @xmath314 hz ) . this energy scale is smaller than that of a lattice of typical experimental depth ( @xmath283 @xmath315 , where @xmath316 is the recoil energy @xmath317 by a factor of @xmath318 and is small on a time scale of @xmath192 s. the effects of linear gradients and of more general potentials can be accounted for through a straightforward generalization of the results presented in this paper , as discussed in greater depth in @xcite . in sec . ii . , we discretized the hamiltonian using the basis of momentum states @xmath40 for integer @xmath41 . we now generalize our results to the case of a finite wavepacket . throughout our analysis , we have worked mainly in the dressed state frame , since this frame is particularly convenient for describing phase evolution in a lattice . for interferometers where we use lattices as waveguides for the atoms , we will want to perform the entire phase shift calculation in this frame . however , phase shift calculations for light - pulse atom interferometers are often performed in the freely falling frame . thus , for applications where lattice manipulations are used for beam splitters and mirrors in a light - pulse geometry , it is useful to convert the phase evolution we calculate in the dressed state frame to the freely falling frame . we thus present the general results derived in this appendix in the freely falling frame . in sec . , we considered a particular dressed frame in which the initial velocity of the atom is boosted to zero . we now introduce an unboosted dressed state frame that is related to the freely falling frame by a translation in position space with no boost in momentum space , so that the unitary transformation @xmath319 transforms from the unboosted dressed state frame to the freely falling frame . the hamiltonian in this frame is : @xmath320 now , say that before the lattice acceleration , the state that we are accelerating is described by @xmath321 in the freely falling frame . we can then transform this state vector to the unboosted dressed state frame , describe its evolution to the final time @xmath322 in this frame , and transform back to the freely falling frame . where @xmath323 is the time evolution operator the takes us from time @xmath109 to time @xmath324 in the unboosted dressed state frame , we can write : @xmath325 denoting the initial momentum space wavefunction in the freely falling frame as @xmath326 , we can express eq . ( [ eqn : evolution ] ) as : @xmath327 where we have used the linearity of the operators to bring them inside the integral . we now consider how each momentum eigenstate @xmath35 evolves in the unboosted dressed state frame . that is , we must calculate @xmath328 for each @xmath329 . in order to do so , we introduce a class of boosted dressed state frames @xmath330 parameterized by @xmath329 , so that momentum @xmath329 in the unboosted dressed state frame ( which is just the frame @xmath331 ) corresponds to momentum zero in the frame @xmath330 . in essence , where @xmath332 , frame @xmath330 travels with velocity @xmath333 with respect to the unboosted dressed state frame . in frame @xmath330 , the hamiltonian takes the form : @xmath334 where we note that the unitary transformation that transforms from frame @xmath330 to the unboosted dressed state frame is : @xmath335}\ ] ] where @xmath336 is the time evolution operator in frame @xmath330 , we can write : @xmath337 } \hat{u}_p(t_f ) \hat{t}_{\text{ds}_p}(t_f , t_0 ) \left | 0 \right > \\\end{aligned}\ ] ] the problem of determining the evolution of the momentum eigenstate @xmath35 in the unboosted dressed state frame thus reduces to evolving the momentum eigenstate @xmath61 in the frame @xmath330 , which we know how to do from the preceding sections . we assume that the momentum space wavefunction is narrow enough so that the momentum eigenstates we consider are resonant with the lattice ( that is , as we recall from sec . iii . , the magnitudes of their velocities with respect to the lattice are less than @xmath48 ) . where the lattice acceleration is chosen so as to transfer a momentum of @xmath338 to the atom , the result from sec . iii . tells us that the time evolution in eq . ( [ eqn : boostedds ] ) yields : @xmath339 for phase : @xmath340 where @xmath341 is the lattice velocity in frame @xmath330 . for the sake of pedagogy , at the moment we neglect the phase arising from the lattice depth , the phase arising from the small periodic variations in the lowest eigenvalue , and the phase arising from non - adiabatic corrections , where we note that we could calculate these contributions if needed . evaluating the integral in eq . ( [ eqn : velocityphase ] ) , we obtain : @xmath342v+ \frac{m}{2 \hbar}v^2(t_f - t_0)\ ] ] substituting this result into eq . ( [ eqn : boostedds ] ) , applying the transformation @xmath343 , and canceling terms in the exponentials , we find that : @xmath344 we can now evaluate eq . ( [ eqn : freelyfallingevolution ] ) , which gives us the final result : @xmath345 where : @xmath346 for @xmath347 . observe that the second term in eq . ( [ eqn : pulsephase ] ) is what would typically be called the laser phase in a light - pulse atom interferometer for an @xmath41-photon beam splitter . we note that as long as the resonance condition is met for the momentum eigenstates we are considering so that they are accelerated , the phase evolved during a lattice acceleration in the @xmath348 dressed state frame , is independent of @xmath329 , as shown in eq . ( [ eqn : ds0phase ] ) . ( this is true up to non - adiabatic corrections and the small periodic variations in the ground state eigenvalue ) . this makes dressed state frames particularly convenient for performing calculations involving wavepackets . in contrast , in the freely falling frame , the accumulated phase @xmath349 is dependent on @xmath329 , but this dependence cancels in the final expression for the phase shift between the two arms of an interferometer as long as the distance travelled by the atom while locked into a lattice is the same for both arms . note that momentum dependent contributions to the total phase shift must arise when we treat the problem purely in terms of dressed states , since the total phase shift is an observable quantity and must therefore be independent of the frame in which it is calculated . the key point to realize is that if the total distance traveled in the lattice is not the same for both arms , then the two arms will end up in two different dressed state frames . we now consider the effect on phase evolution of the periodic term @xmath104 in the expression for the ground state eigenvalue of the dressed state hamiltonian given in eq . ( [ eqn : evalue ] ) . this term leads to a momentum dependent correction @xmath350 to the evolved phase @xmath351 described in eq . ( [ eqn : velocityphase ] ) , where : @xmath352 we note that @xmath353 can be calculated using the truncated matrix approximation described in sec . since @xmath353 is periodic in @xmath354 with period @xmath355 , the contribution @xmath356 of this correction term to the phase difference between the two arms of an interferometer will be highly suppressed if both arms load the atom into the lattice near the center of the zeroth band ( as discussed in sec . iii . ) and undergo a nearly integral number of bloch oscillations so that the effects of this periodic variation in the ground state eigenvalue will be largely common to the two arms , as verified by estimation of the integral in eq . ( [ eqn : periodicphase ] ) and by numerically solving the schrodinger equation . here , we derive the resonance condition stated in sec . iii . , which says that for a sufficiently slow lattice depth ramp , the atom will be loaded into the ground state of the lattice if the initial velocity of the lattice is within @xmath48 of the initial velocity of the atom . first , we examine the eigenvalues of @xmath55 when @xmath357 . for @xmath357 , @xmath55 is diagonal matrix . since @xmath55 is a matrix defined in terms of the basis of momentum eigenstates @xmath40 for integer @xmath41 , when @xmath55 is diagonal these momentum eigenstates are also the eigenstates of @xmath55 , with corresponding eigenvalues @xmath358 ( which are just the diagonal elements of @xmath55 ) . we note that the effect of the @xmath359 term in the hamiltonian is to encapsulate the dependence of the eigenvalues and eigenvectors of the hamiltonian on the lattice velocity for any given lattice depth , which can be best conceptually understood from the point of view of the dressed state picture explained in @xcite . the resonance condition states that if the initial state is @xmath360 , then the initial velocity of the lattice must be within @xmath48 of @xmath361 . thus , the initial value @xmath145 of the dimensionless velocity @xmath79 must be in the range @xmath362 . we will now show why this condition on @xmath145 implies that the ground state of @xmath55 with @xmath357 is @xmath360 . where we write @xmath363 and let @xmath364 for integer @xmath365 , the eigenvalues of @xmath55 with the lattice depth set to zero are : @xmath366= 4 e_r \left [ -n_0 ^ 2 + ( \delta n)^2 -n_0 \delta- \delta n \delta \right]\ ] ] for @xmath367 , @xmath368 will be most negative when @xmath369 , so @xmath360 will indeed be the ground state of the hamiltonian when @xmath357 . therefore , as long as we ramp up the lattice slowly enough and keep the lattice velocity constant while doing so , the adiabatic theorem tells us that the atom will remain in the lattice ground state throughout the ramp up process provided that the ground state never passes through a point of degeneracy , which is indeed the case for fixed @xmath79 and @xmath367 . the maximum rate at which we can increase the lattice depth while still maintaining the validity of the adiabatic approximation depends on @xmath370 . this statement follows from the adiabatic condition @xmath60 , where @xmath61 and @xmath62 respectively denote the ground state and the first excited state of the hamiltonian . when the resonance condition holds , note that the first excited state will be @xmath371 . as @xmath370 increases toward 1 , the gap between @xmath372 and @xmath373 ( i.e. the energy gap between the zeroth and first bands ) decreases . to see why this is true , we observe that : @xmath374 therefore , for the adiabatic approximation to hold , the maximum rate at which we can ramp up the lattice decreases as we increase @xmath370 . this result is identical to the statement in sec . iii . that adiabatic loading is more difficult to achieve near the border of the first brillouin zone . after ramping up the lattice , we accelerate it until its velocity is within @xmath48 of @xmath375 where @xmath376 is the target momentum eigenstate . the ground state of the hamiltonian when the lattice has finished ramping down will therefore be @xmath376 , and as long as the entire process is adiabatic so that the atom remains in the ground state , the atom will indeed end up in the target state . by analogy to the previous discussion , the maximum rate at which we can adiabatically ramp down the lattice increases as the difference between the final lattice velocity and @xmath375 goes to zero . in order to calculate non - adiabatic corrections at arbitrary order , we define @xmath377 recursively in the natural way based on the notation of sec . iv : where the last term includes @xmath20 sums and @xmath380 . in order for the expansion to be practical from a calculational standpoint , the series must converge quickly enough so that we do not have to find an unreasonable amount of terms . there are two types of terms that we can often neglect . first , where we assume that the system is initially in the ground state , it is often possible to neglect many terms of the form @xmath381 so that we only need to examine a relatively small number of eigenvectors . terms of this form indeed decrease rapidly as @xmath41 increases because the overlap of the @xmath41th eigenstate with the zeroth eigenstate is essentially nonexistent for large enough @xmath41 . second , we can often neglect all terms of order greater than some cutoff value . we will be able to do so as long as the time scale over which we are solving the problem is small enough so that integrating a @xmath20th order correction term against a factor of the form @xmath121 yields a @xmath382th order correction term that is much smaller than the correction term of order @xmath20 . note an important nuance in how we have phrased the above condition we have mentioned nothing about a hamiltonian that varies slowly in time . the convergence of the perturbative series depends on the time interval of the solution . as long as the hamiltonian does not vary at an infinitely fast rate , we can always work on a small enough time scale so that the series converges rapidly . to solve the problem on time scales for which the series does not converge quickly , we can simply break the problem into multiple parts . this method provides with us with a means to describe the system for a hamiltonian that changes arbitrarily fast in time . having a slowly varying hamiltonian just serves to allow us to solve the problem without dividing it into as many parts ( much of the time we will not have to divide the problem at all , and in sec . v. we derive conditions for when this will be the case ) , thus making the calculation significantly easier . j. m. hogan , d. m. s. johnson , and m. a. kasevich , in _ proceedings of the international school of physics `` enrico fermi '' on atom optics and space physics _ , edited by e. arimondo , w. ertmer , and w. p. schleich ( ios press , amsterdam , 2009 ) , pp . 411 - 447 ( arxiv:0806.3261v1 ) .

compact binary ( neutron star - neutron star , ns@xmath1 , or black hole - neutron star , bh - ns ) mergers are prime sources of gravitational radiation . the gw detectors ligo @xcite , virgo @xcite and geo600 @xcite are designed to optimally detect merger signals . these detectors have been operational intermittently during the last few years reaching their nominal design sensitivity @xcite with detection horizons of a few dozen mpc for ns@xmath1 and almost a hundred mpc for bh - ns mergers ( the ligo - virgo collaboration adopts an optimal canonical distance of 33/70mpc ; @xcite ) . both ligo and virgo are being upgraded now and by the end of 2015 are expected to be operational at sensitivities @xmath2 times greater than the initial ligo @xcite , reaching a few hundred mpc detection horizon for ns@xmath1 mergers and a gpc for bh - ns mergers ( 445/927 mpc are adopted by the ligo - virgo collaboration as canonical values ; @xcite ) . understanding the observable em signature of compact binary mergers has several observational implications . first , once the detectors are operational it is likely that the first detection of a gw signal will be around or even below threshold . detection of an accompanying em signal will confirm the discovery , thereby increasing significantly the sensitivity of gw detectors @xcite . second , the physics that can be learned from observations of a merger event through different glasses is much greater than what we can learn through em or gw observations alone . finally , even before the detectors are operational , detection of em signature will enable us to determine the expected rates , a question of outmost importance for the design and the operation policy of the advanced detectors . the current constraints on the rates are rather loose . the last ligo and virgo runs provided only weak upper limits on the merger rates : 8700 myr@xmath3(@xmath4 corresponding to @xmath5 yr@xmath3 gpc@xmath6 for ns@xmath1 and @xmath7 myr@xmath3 ( @xmath4 ( @xmath8 ) for bh - ns @xcite . estimates based on the observed binary pulsars in the galaxy are highly uncertain , with values ranging from @xmath9 @xcite . it has been suggested @xcite that short gamma - ray bursts ( grbs ) arise from neutron star merger events . the estimated rate of short grbs are indeed comparable to binary pulsar estimates @xcite . however , while appealing , the association is not proven yet @xcite . if correct , the observed rate of short grbs , @xmath10 , provides a lower limit to the merger rate . the true rate depends on a poorly constraint beaming angle , resulting in an uncertainty of almost two orders of magnitude . there are no direct estimates of bh - ns mergers , as no such system has ever been observed , and here one has to relay only on a rather model dependent population synthesis ( e.g. , * ? ? ? * ; * ? ? ? possible em signals from coalescence events were discussed by several authors . there are several suggestions @xcite of a prompt ( coinciding with the gw signal ) short lived em signals , mostly in low radio frequencies , whose amplitudes are highly uncertain . @xcite suggested that the radioactive decay of ejected debris from the merger will drive a short lived supernova like event . @xcite calculated the radioactive heating during this process self - consistently . they find that if @xmath11 is ejected then the optical emission from a merger at @xmath12 mpc peaks after @xmath13 day at @xmath14 . if the mass ejection is lower then the optical emission will be even fainter . finding , and especially identifying the origin of , such rare and faint events in the crowded variable optical sky is an extremely challenging task , even for current and future optical searches like ptf @xcite , panstarr , and lsst @xcite . an intriguing possibility is that mergers produce short grbs @xcite . however , short grbs are expected to be beamed , and only rarely this em signal will point towards us . a beamed grb that is observed off - axis will inevitably produce a long lasting radio orphan " afterglow @xcite . a key point in estimating the detectability of grb orphan afterglows is that the well constrained observables are the _ isotropic _ equivalent energy of the flow and the rate of bursts that point towards earth . however , the detectability of the orphan afterglows depends only on the _ total _ energy and _ true _ rate , namely on the poorly constraint jet beaming angle . @xcite have shown that while large beaming increases the true rate it reduces the total energy , and altogether reduces the detectability of radio orphan afterglows . this counterintuitive result makes the detectability of late emission from a decelerating jet , which produced a grb when it was still ultra - relativistic , less promising . however , regardless of amount of ulrtra - relativistic outflow that is launched by compact binary mergers , and of whether they produce short grbs or not , mergers are most likely do launch an energetic sub - relativistic and mildly - relativistic outflows . the interaction of these outflows with the surrounding matter will inevitably produce blast waves and possibly stronger radio counterparts than that of ultra - relativistic outflows . below we first discuss ( in 2 ) the current estimates of mass and energy ejection from compact binary mergers . in 3 we calculate the radio emission resulting from the interaction of this ejecta ( sub - relativistic , mildly relativistic and off - axis relativistic ) with the surrounding interstellar matter ( ism ) . the calculations follow to a large extent models of radio ib / c supernovae @xcite and long grb radio afterglows @xcite . the success of radio supernova ( sn ) modeling , where the observations are superb , indicates that the microphysics is well constrained ant that equipartition parameters describe well the physical conditions . hence the main uncertainty in the predicted radio signal is in the amount of matter ejected from the merger event and its velocity . luckily the estimates of this important quantity can be significantly improved even using existing numerical models . we discuss the observational implications for detectability of merger remnants in 4 . we estimate the expected rates of detection of different outflows in 4.1 . we devote in 4.2 a special attention to short grb orphan afterglows that are a special case of our model , in which the outflow is launched relativistically , but the radio emission peaks only during its mildly relativistic phase . the estimates of orphan afterglows detectability is independent of whether they are the products of binary mergers or not . in 4.3 we examine possible other radio transients that may hinder the identification of merger remnants . finally in 4.4 we examine blind transient searched done in the past and we identify rt 19870422 as a possible and even likely merger remnant . we conclude in 5 . numerical simulations of compact binary mergers have been carried out by various groups with two different approaches . some ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) use newtonian dynamics ( modified to allow for gravitational radiation back - reaction ) with detailed microphysics . others ( e.g. , * ? ? ? * ; * ? ? ? * ) use full general relativistic dynamics with different levels of approximate microphysics , with or without mhd . in almost all ns@xmath1 simulations one finds an accretion disk surrounding a rapidly rotating massive object that eventually collapses to a black hole . an exception is the recent general relativistic simulations of @xcite who find no disks in some configurations . the system lifetime is at least a few dozen milliseconds and possibly longer . the fate of an accretion disk in a bh - ns merger is expected to depend on the mass ratio , the bh spin and the ns compactness . in some cases the disk is very small while in others it is substantial ( e.g. , * ? ? ? * ; * ? ? ? all simulations find some form of relativistic or sub - relativistic mass ejection . first , matter is ejected as tidal tails during the first stages of the merger . in bh - ns mergers the ejected energy can be very high and its velocity is mildly relativistic . for example @xcite find @xmath15 c ejecta with @xmath16 erg , where c is the light speed . in a ns@xmath1 mergers a lower , but yet significant , amount of energy can be ejected ( e.g. , @xcite find @xmath17 erg ) at a lower velocities of @xmath18 c. this mass ejection is expected also if no significant disk is formed . disk formation leads to several additional outflow sources . first , neutrino heating drives a wind from the disk surface ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? the energy in this wind is substantial with predictions ranging between @xmath19 erg and @xmath20 erg for 0.01 - 0.1 m@xmath21 disk . the outflow velocity is @xmath22 c from the outskirts of the disk and it is increasing , possibly up to relativistic velocities , for wind that is ejected from close to the central object . the mass ejection becomes even stronger when neutrino heating shuts - off and the wind is driven by viscous heating and by he - synthesis @xcite , leading to an ejection of @xmath23 of the initial disk mass at @xmath18 c. additional energy source is neutrino - antinuetrino annihilation above the disk , which can deposit up to @xmath24 erg , into an amount of mass that is not well constrained , leading possibly to a relativistic outflow . finally , more speculative , but yet very plausible , source of outflow are em processes that tap the rotational energy of the central object , such as the blandford - znajek mechanism @xcite . these are likely to produce relativistic outflows with an energy that can be as high as @xmath25 erg , and are the most probable engines of short grbs , if those are produced by compact binary mergers . the conclusion is that a significant mass and energy ejection is a prediction of almost all compact binary merger modelings . in ns@xmath1 mergers an ejection of @xmath26 erg at @xmath18 c is a fairly robust prediction . faster ejecta ( relativistic or mildly relativistic ) with energy @xmath27 is also quite likely from inner parts of the 0.01 - 0.1 m@xmath21 disk that is typically found in simulations . the outflow from bh - ns mergers was explored only by a few authors , but it is also seems to be significant and potentially even more energetic and at faster velocities than the outflow from ns@xmath1 mergers . consider a spherical outflow with an energy @xmath28 and an initial lorentz factor @xmath29 , with a corresponding velocity @xmath30 , that propagates into a constant density , @xmath31 , medium . if the outflow is not ultra relativistic , i.e. , @xmath32 it propagates at a constant velocity until , at @xmath33 , it reaches radius @xmath34 , where it collects a comparable mass to its own : @xmath35 and @xmath36 where we approximate @xmath37 and ignore relativistic effects . here and in the following , unless stated otherwise , @xmath38 denotes the value of @xmath39 in c.g.s . units . at a radius @xmath40 the flow decelerates assuming the sedov - taylor self - similar solution , so the outflow velocity can be approximated as : @xmath41 if the outflow is collimated , highly relativistic and points away from a generic observer , as will typically happen if the mergers produce short grbs , the emission during the relativistic phase will be suppressed by relativistic beaming . observable emission is produced only once the external shock decelerates to mildly relativistic velocities and the blast - wave becomes quasi spherical . this takes place when @xmath42 namely at @xmath43 . from this radius the hydrodynamics and the radiation become comparable to that of a spherical outflow with an initial lorentz factor @xmath44 . this behavior is the source of the late radio grb orphan afterglows @xcite . our calculations are therefore applicable for the detectability of mildly and non - relativistic outflows as well as for radio orphan grb afterglows . emission from newtonian and mildly relativistic shocks is observed in radio sne and late phases of grb afterglows . these observations are well explained by a theoretical model involving synchrotron emission of shock accelerated electrons in an amplified magnetic field . the success of this model in explaining the detailed observations of radio ib / c sne ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) allows us to employ the same microphysics here . energy considerations show that both the electrons and the magnetic field carry significant fractions of the total internal energy of the shocked gas , @xmath45 . these values are consistent with those inferred from late radio afterglows of long grbs ( e.g. , * ? ? ? * ; * ? ? ? the observed spectra indicate that the distribution of the accelerated electrons lorentz factor , @xmath46 , is a power - law @xmath47 at @xmath48 where @xmath49 in mildly relativistic shocks ( e.g. , the radio emission from grb associated sne and late grb afterglows ) and @xmath50 in newtonian shocks ( as seen in typical radio sne ; ( * ? ? ? * and references therein ) ) . the value of @xmath51 is not observed directly but it can be calculated based on the total energy of the accelerated electrons , @xmath52 the radio spectrum generated by the shock is determined by two characteristic frequencies . one is @xmath53 the typical synchrotron frequency of electrons with the typical ( also minimal ) lorentz factor @xmath51 . the other is @xmath54 , the synchrotron self - absorption frequency . we show below that since we are interested at the maximal flux at a given observed frequency , @xmath54 may play a role only if it is larger than @xmath55 . its value in that case is @xmath56 , and frequencies , @xmath54 and @xmath55 . the arrows show the temporal evolution of the characteristic values . the temporal dependence before @xmath33 is noted below / to the left of the arrows while the temporal dependence after @xmath33 is noted above / to the right of the arrows . note that the evolution of @xmath55 and @xmath57 , marked only in the left spectrum , is relevant for both spectra . the evolution of @xmath54 , marked only in the right spectrum , is correct only when @xmath58 and is therefore relevant only in that spectrum.,title="fig:",width=491 ] + figure [ fig : spec ] illustrates the two possible spectra , depending on the order of @xmath54 and @xmath55 . the flux at any frequency can be found using these spectra and the unabsorbed synchrotron flux at @xmath55 : @xmath59 where @xmath60 is the distance to the source ( we neglect any cosmological effects ) . note that this is the real flux at @xmath55 only if @xmath61 ( see figure [ fig : spec ] ) . as long as the shock is moving with a constant velocity i.e. , at @xmath62 , the flux across the whole spectrum increases ( see equations [ eq num]-[eq fm ] ) . the flux evolution at later times depends on the spectrum at @xmath33 , namely on @xmath63 and if @xmath64 then possibly on @xmath65 the flux at that time can be found using the unabsorbed synchrotron flux at @xmath66 : @xmath67 consider now a given observed frequency @xmath68 . we are interested at the light curve near the peak flux at this frequency . there are three possible types of light curves near the peak corresponding to : ( i ) @xmath69 , ( ii ) @xmath70 and ( iii ) @xmath71 . where we define @xmath72 as the frequency at which this equality will never take place . in that case @xmath73 is the frequency at which this equality would have happened if @xmath29 would have been large enough ( see figure [ fig : cases ] ) ] @xmath74 . in figure [ fig : cases ] we show a schematic sketch of the time evolution of @xmath54 and @xmath55 and the corresponding ranges of @xmath68 in which each of the cases is observed . and @xmath55 in two cases , @xmath75 ( top ) and @xmath64 ( bottom ) . also marked is the value of @xmath73 . the vertical dashed line marks @xmath76 . the ranges of @xmath68 at which each of the cases is observed is separated by horizontal dashed lines and marked on the right . note that in the bottom panel @xmath55 and @xmath54 are not crossing each other at @xmath77 and only two types of light curves , cases ( i ) and ( iii ) , can be observed.,title="fig:",width=453 ] + to estimate the time and value of the peak flux we recall , that at all frequencies the flux increases until @xmath33 . in case ( i ) , @xmath69 , the deceleration time , @xmath76 , is also the time of the peak . the reason is that while @xmath57 increases , @xmath55 decreases fast enough so that @xmath78 decreases after @xmath33 . note that in that case @xmath54 plays no role since it decreases after deceleration . overall , in this case the flux peaks at @xmath33 and @xmath79 . in the two other cases , ( ii ) and ( iii ) , @xmath80 and/or @xmath81 and the flux keeps rising at @xmath77 until @xmath82 or @xmath83 , whichever comes last . to find out which one of the two frequencies is it , we compare @xmath68 with @xmath73 . at @xmath84 , @xmath55 decreases faster than @xmath54 . therefore in case ( ii ) where @xmath85 , the last frequency to cross @xmath68 is @xmath55 and the peak flux is observed when @xmath86 . in case ( iii ) where @xmath87 , and the last frequency to cross @xmath68 is @xmath54 and the peak flux is observed when @xmath88 . now , it is straight forward to calculate the peak flux , @xmath89 and the time that it is observed , @xmath90 , for different frequencies . it is also straight forward to calculate the flux temporal evolution prior and after @xmath33 using equations [ eq num]-[eq fm ] and the relation @xmath91 which holds at @xmath62 and @xmath92 at @xmath84 . the peak fluxes , the times of the peak and the temporal evolution of the three different cases are summarized in table [ table1 ] . the overall different light curves are depicted in fig . [ fig : lightcurves ] .[table1 ] the observed flux before and after @xmath90 in the three different regimes . @xmath93 the temporal evolution only during the last power - law segment before @xmath90 . at earlier times the temporal evolution may be different . [ cols="<,^,^,^,^ " , ] the most sensitive radio facilities are at frequencies of @xmath94 ghz and higher . equations [ eq numdec ] and [ eq nuadec ] imply that in this frequency range , for most realistic scenarios , it is a case ( i ) light curve , i.e. , @xmath95 . therefore , a newtonian and mildly relativistic outflows as well as relativistic grb orphan afterglows peak at @xmath33 with : @xmath96 the regime of @xmath97 at lower radio frequencies ( @xmath98 ghz ) depends on the various parameters . if the outflow is newtonian or the density is low or the energy is low then @xmath99 mhz and equation [ eq fpeak reg1 ] is applicable . otherwise low radio frequencies are in regime ( iii ) , i.e. , @xmath100 . the flux peaks in this case at @xmath101 with @xmath102 in the last two equations we used @xmath103 ( other @xmath104 values in the range 2.1 - 3 yield slightly different numerical factors and power laws ) . to date , the best observed signal from a mildly relativistic blast waves is the radio emission that follows grb associated sne . the main difference is that in these cases the circum burst medium is typically a wind ( i.e. , @xmath105 ) and therefore the density at early times is much larger then in the ism and self absorption plays the main role in determining the light curve . a good example for comparison of equation [ eq fpeak reg1 ] with observations is the light curve of sn 1998bw . this light curve is observed at several frequencies at many epochs , enabling a detailed modeling that results in tight constraints of the blast wave and microphysical parameters . @xcite find that at the time of the peak at @xmath94 ghz , about 40 days after the sn , taking @xmath106 , the energy in the blast wave is @xmath107 erg , its lorentz factor is @xmath108 and the external density at the shock radius is @xmath109 . the peak is observed when @xmath110 and it depends only on these parameters ( it is only weakly sensitive to the density profile ) . therefore , equation [ eq fpeak reg1 ] is applicable in that case . indeed , plugging these numbers into equation [ eq fpeak reg1 ] we obtain a flux of 20 mjy at the distance of sn 1998bw ( 40 mpc ) , compared to the observed flux of 30 mjy . this is not surprising given that the model we use is based on that of radio sne . in the discussion above we considered an outflow with a characteristic energy and velocity . however a compact binary merger may produce an outflow with an energy dependent velocity , e.g. , @xmath111 . if we consider a case ( i ) signal ( @xmath69 ; e.g. , above @xmath112 ghz ) then we find @xmath113 . therefore if @xmath114 the flux is dominated by the mildly relativistic ejecta , assuming that the relativistic part of the outflow is not pointing towards the observer . otherwise the flux is dominated by the slowest ejecta . for canonical parameters the strongest signal is expected around @xmath94 ghz , conveniently where the sensitivity of radio telescopes is high . if the signal peaks at lower frequencies it decreases from the peak as @xmath115 . since 1.4 ghz receivers are ten times more sensitive than lower frequency ones they are still more likely to detect a signal even in this case . therefore , @xmath94 ghz is the optimal frequency to look for radio remnants of compact binary mergers and we therefore consider here the delectability at a @xmath94 ghz survey . the number of events in a single whole sky snapshot is @xmath116 , where @xmath117 is the detectable volume at the survey flux limit @xmath118 , @xmath119 is the time that the flux is above the detection limit and @xmath120 is the event rate . since at @xmath94 ghz the relevant light curve is case ( i ) ( see [ sec theory ] ) we use equations [ eq : tdec ] and [ eq fpeak reg1 ] , and the approximation @xmath121 , to find that the number of radio coalescence remnants in a single @xmath94 ghz whole sky snapshot is @xmath122 where @xmath123 mjy and @xmath124 is the merger rate in units of @xmath125 . since the microphysical parameters are reasonably constrained by radio sne , the main uncertainty in the signal detectability is the blast - wave properties and the circum - merger density . the latter is expected to vary significantly between different merger events , dropping down to @xmath126 for mergers that take place outside of their host galaxies . however , the observed galactic double - ns population reveals that a significant fraction of compact binary mergers ( at least ns@xmath1 ) take place in the disks of milky - way - like galaxies where @xmath127 . as discussed in 2 there is a good chance that ns@xmath1 mergers eject mildly relativistic outflows with @xmath128 erg . in that case the em counterparts of these gw sources are detectable by current facilities . a deep radio survey that covers a significant fraction of the sky will most likely detect their radio remnants to a distance of @xmath129 mpc . these will appear as transients , varying on a several weeks time scale , with an optically thin spectrum at @xmath130 ghz at all time . the radio remnants will be identified in random places within their host galaxies , which should be easily detectable at that distance . it will not be accompanied by any optical counterpart with similar variability time scales . given a gw trigger with localization of @xmath131 , a deep search of the error box will detect a remnant in cases that the surrounding density is not very low . ns@xmath1 mergers are also expected to eject energetic , @xmath132 erg , outflow at lower velocities , @xmath133 . the detectability is very sensitive to the velocity , @xmath134 for @xmath103 , and to the energy , @xmath135 . therefore , even if there is no mildly relativistic component to the outflow , the number of detectable remnants that are dominated by slow moving ejecta is expected to be significant . for example , assuming that an energy of @xmath136 erg is ejected at @xmath137 we expect @xmath138 . the variability time scale of these transients is expected to be @xmath139 yr . if the velocity is instead @xmath140c the number of events drops by an order of magnitude and the variability timescale increases to @xmath141 yr . these time scales increase the difficulty in the identification of the remnants as transients , and require a long term survey . the spectrum of these transients will be optically thin also at low frequencies ( @xmath142 mhz ) . other characteristics of these transients ( e.g. , location within the host ) are expected to be similar to that of a mildly relativistic outflow . a gw triggered search increases of course the probability to find a remnant and a @xmath143 erg outflow can be detected up to @xmath12 mpc even if its velocity is @xmath144c . the detectability of bh - ns mergers is much harder to predict . first due to their virtually unconstrained rate and second since the properties of the outflow are less certain . the latter can be significantly improved by current and future merger simulations that put focus on the ejected mass . in any case the potential of these mergers to throw out a considerable amount of energy , @xmath17 erg , at mildly relativistic velocities can make them detectable to their gw detection horizon , which is much further than the ns@xmath1 horizon . finally , if compact binary mergers launch also collimated ultra - relativistic outflows , and produce short grb , then orphan short grb afterglows are also part of the post merger em signal . the detectability of radio orphan afterglows can be estimated based on observations of short grbs and is independent of whether they are produced by mergers or not . we discuss their detectability in the next subsection . all together , the range of the current predictions is rather large , but with most parameters we expect detectable radio signals . some of the new generation radio telescopes have large fields of view ( e.g. , askap with @xmath145 and apertif with @xmath146 ; @xcite ) and improved sensitivities , making them ideal for large scale sub - mjy blind survey . the evla , which has a smaller field of view but a remarkable sensitivity , is the best facility for gw triggered search . it is also currently the fastest radio - survey instrument and it can carry - out a sub - mjy blind survey . all these observatories have a very good chance to detect compact binary merger remnants in dedicated blind searches . in fact with very reasonable parameters ( e.g. , @xmath147 erg of mildly relativistic ejecta ) a sub - mjy whole sky survey can detect thousands of binary - merger radio remnants . short grb outflow begins highly relativistic and probably highly beamed . eventually it slows down ( see [ sec theory ] ) and become detectable from all directions . therefore , the rate estimate equation [ eq rate ] is also applicable for radio orphan afterglows when @xmath148 . however some of the parameters in equation [ eq rate ] are not directly observable . the observed quantities are isotropic equivalent @xmath46-ray energy , @xmath149 , and the rate of bursts that point to the observer @xmath150 , while equation [ eq rate ] depends on @xmath151 and @xmath152 , where @xmath153 is the fraction of the @xmath154 steradian covered by the jet and @xmath155 is the isotropic equivalent energy in the afterglow blast wave . x - ray observations indicate that @xmath46-ray emission in short grbs is very efficient and that in general @xmath156 @xcite . we assume that this is the case in the following discussion . @xmath149 of short grbs ranges at least over four orders of magnitude ( @xmath157 erg ) . the rate of observed short grbs is dominated by @xmath19 erg bursts , and the luminosity function can be well approximated by a power - law , at least in the range @xmath158 erg , such that @xmath159 where @xmath160 @xcite . plugging these into equation [ eq rate ] we obtain @xmath161 this equation is similar to equation 9 of @xcite , with the observed luminosity function already folded in . narrower beamed bursts ( with lower @xmath162 ) are more numerous and they produce less total energy per burst . the positive dependence of equation [ eq orphan_rate ] on @xmath162 implies that overall the lower energy is winning " over the increased rate , and the detectability of narrower bursts is lower . using , equation [ eq orphan_rate ] we can put a robust upper - limit on the orphans rate since all the parameters are rather well constraint by observations , with the exception of @xmath162 which is @xmath163 by definition . therefore , assuming that short grbs are beamed , the detection of the common @xmath24 erg bursts in a blind survey , even with next generation radio facilities , is unlikely @xcite . however , brighter events should be detectable . if the beaming is energy independent , detectability increases with the burst energy . the luminosity function possibly breaks around @xmath20 erg , in which case the orphans number is dominated by @xmath20 erg bursts . for @xmath164 we expect , from these bursts @xmath141 orphan afterglows at a 0.1 mjy in a single @xmath94 ghz whole sky snapshot . so far we discussed detectability in a blind survey . a followup dedicated search would be , of course , more sensitive . if compact binary mergers produce short grbs than the energy of most gw detected bursts will be faint with @xmath165 erg . the chance to detect their orphan afterglows again depended on their total energy and thus on @xmath162 . equation [ eq fpeak reg1 ] shows that if @xmath164 then detection should be difficult but possible in a dedicated search mode . note that since the energy of the burst is low , the radio emission will evolve quickly , reaching a peak and decaying on a week time scale , so a prompt and rather deep search will be needed . @xcite carried out a 5 ghz survey looking for transients on timescales of a week to a year . the survey sensitivity for transients with variability scale @xmath166 day is 0.37 mjy with an effective are of @xmath167 . events with variability time scale of two months where surveyed at sensitivity of 0.2 mjy with an effective area of @xmath168 . @xcite report the detection of 10 transients . the most interesting of those , in our context , is rt 19870422 , which has a variability time scale of two months . it is found within a star forming galaxy at a distance of 1.05 gpc , but at a significant offset from the host nucleus . its luminosity and time scale are those expected from a @xmath169 erg mildly relativistic outflow that propagates in the ism . it is , therefore , a prime candidate for a compact binary merger radio remnant . based on this single event @xcite infer a best estimate rate of @xmath170 for rt 19870422-like events . taking a @xmath171 poisson error , the best estimate translates to a range of @xmath172 , fully consistent with the estimates of compact binary mergers . if this is indeed a merger remnant then , since for optically thin spectrum the fluxes at 1.4 ghz and 5 ghz are not very different , a sub - mjy 1.4 ghz whole sky survey would detect hundreds to thousands of radio remnants . @xcite suggested that this transient is a radio sne similar to sn 1998bw , but brighter . this is certainly a viable possibility , however , if true then this radio sn is brighter by an order of magnitude than the brightest radio sn ever observed before . unfortunately , lacking optical search for a sn or a multi - wavelength measurement that determines the transient spectrum , it is impossible to rule out any of the two possibilities . an additional interesting candidate is rt 19840613 . it is variable on less than 7 days and it has a host galaxy at a distance of 140 mpc . even assuming a variability time scale of 7 days it is marginal as a merger remnant candidate . assuming 7 days variability @xcite find a rate of @xmath173 , which is again only marginally consistent with current estimates of compact binary merger rates . therefore , while this may be a merger remnant it is not a very promising candidate . @xcite suggest that this is also a sn 1998bw - like event . while this possibility can not be ruled out , the inferred rate is at least an order of magnitude larger than that of 98bw - like events ( note that radio sne as bright as sn 1998bw are very rare compared to typical radio sne ) . the additional 8 events detected by @xcite have no clear host galaxies and are therefore probably not merger remnants . a key issue with the detection of compact binary merger remnants in blind surveys is their identification . @xcite present a census of the transient radio sky . luckily the transient radio sky at @xmath94 ghz are relatively quite . the main contamination source are radio active galactic nuclei ( agns ) , however their persistent emission is typically detectable in other wavelength and/or deeper radio observations . moreover , the signal from a compact binary merger is expected to be located within its host galaxy ( otherwise the density is too low ) , but away from its center . the host and the burst location within it , should be easily detectable at the relevant distances . the only known , and guaranteed , transient 1.4 ghz source with similar properties are radio sne . among these typical radio sne are the most abundant . transient search over 1/17 of the sky with @xmath174 mjy at 1.4ghz @xcite finds one radio sn . this rate translates to @xmath175 sne in a whole sky @xmath176 mjy survey . these contaminators can be filtered in three ways . first , by detection of the sn optical light . however , the optical signal may be missed if it is heavily extincted , and given the large number of radio sne , misidentifying even a small fraction of them may render the survey useless for our purpose . the second filter is the optically thick spectrum at high radio frequency ( @xmath141 ghz ) at early times , which is a result of the blast wave propagation in a wind . thus , a multi - wavelength radio survey can identify radio sne . the last filter is the luminosity - time scale relation of typical radio sne that is induced by the outflow velocity ( e.g. , figure 2 in * ? ? ? type ii sn outflows are slow , @xmath177c , and therefore their radio emission is longer / fainter than that expected for merger remnants . the common type of ib / c radio sne is produced by @xmath178c blast waves but with much less energy than what we expect from a binary merger outflow , and therefore their radio emission is much fainter . the combination of any two of these filters will hopefully be enough to identify all the typical radio sne . slightly different contaminators are grb associated sne . their outflows is as fast and as energetic as those that we expect from a binary merger and therefore their radio signature is similar in time scales and luminosities . sn1998bw - like events are detectable by a 0.1 mjy survey at 1.4 ghz up to a distance of several hundred mpc for 40 days and their rate is @xmath179 @xcite , implying at least several sources at any whole sky snapshot . here only the first filter ( sn optical ligth ) and possibly the second ( optically thick spectrum ) can be applied . however , given the high optical luminosity of grb associated sne and their relatively low number this should be enough in order to filter them out . these contaminators highlight the importance of a multi - wave length strategy where an optical survey accompany the radio survey to best utilize both surveys detections . the results of @xcite implies that thousands of sources with properties similar to rt 19870422 are expected in a 5 ghz sub - mjy all sky survey . if these events are optically thick during their whole evolution than these are not merger remnants and they could be easily filtered out . moreover , in that case their rate in a 1.4 ghz survey should be lower by two orders of magnitude . if these events show a synchrotron optically thin spectrum and no optical counterparts , then they should be abundant also in a 1.4 ghz survey , and their origin is most likely compact binary mergers . finally , radio is the place to look for blast waves in tenuous mediums , regardless of their origin . any source of such explosion , being a binary merger , a grb or a sn , produces a radio signature . therefore , all the strong explosions may be detectable is a deep radio survey , this include for example long grb on - axis and off - axis afterglows and giant flares from extra galactic soft gamma - repeaters . the difference between the radio signatures of the different sources ( amplitude , spectrum and time evolution ) depends on the blast wave energy and velocity and on the external medium properties . we thus will be able to identify the characteristics of binary mergers outflows . if , however , there is a different source of @xmath169 erg of mildly relativistic outflow that explodes in the ism it will be indistinguishable from binary mergers ( at least not in the radio ) . currently we are not aware of any such source , with the exception of long grbs at the low end of the luminosity function , but these are too rare to contaminate a survey . any other source of such outflows , if exist , will probably be a part of the family of collapsing / coalesing compact objects . compact binaries are expected to eject sub - relativistic , mildly relativistic and possibly ultra - relativistic outflows as part of their merger process . we have shown that these outflows will inevitably produce a long lived radio remnant . these are the most robust predictions of an em counterpart of the merger gw signal . the radio remnant appears weeks to years after the merger and remains bright for a similar time . therefore , a trigger following a detection of gw signal can wait for a week after the event and no online triggering is needed . in addition the long lasting remnants enable a detection in a blind survey . for mildly relativistic outflows with @xmath19 erg that propagate in the ism we expect a few weeks radio transients with a 1.4 ghz flux of @xmath180 0.3 mjy from sources at @xmath129 mpc , the advanced ligo - virgo horizon for ns@xmath1 mergers . the bh - ns gw horizon is farther , but current numerical simulations suggest they involve higher energy outflow resulting in a comparable flux . follow up observations of gw candidate events , at a level of @xmath181jy are feasible and are very likely to show a radio transient for either ns@xmath1 merger of bh - ns merger . we find that the optimal frequency to carry out a search for merger remnants is @xmath94 ghz . assuming a mildly relativistic outflow with @xmath19 erg the canonical ns@xmath1 merger rate of 300 gpc@xmath6 yr@xmath3(and a range of 20 - @xmath182 gpc@xmath6 yr@xmath3 ) implies a detection of @xmath183 ( 1 - 1200 correspondingly ) radio ns@xmath1 remnants in a @xmath140 mjy all sky survey . this rate depends quadratically on the outflow energy , so a very plausible ejected energy of @xmath136 erg increases the rate by two orders of magnitude , making them detectable even in a survey that covers only a small fraction of the sky or that is at a mjy sensitivity . therefore carrying out a large field - of - view and sensitive ghz survey by currently available facilities has a great potential to constrain the rate of binary mergers , a piece of information that is of great importance for the design and operation of gw detectors . even if mergers do not launch a significant mildly relativistic ejecta they are still expected to produce an energetic ( @xmath184 erg ) sub - relativistic ( 0.1 - 0.2)c outflows . these outflows will also produce radio remnants . these remnants will be fainter , detectable only to a distance of @xmath142 mpc at 0.1 mjy , and will evolve more slowly , on time scales of 3 - 10 yr . these transients are also detectable at a rate of @xmath141 over the whole sky at 0.1 mjy , although identifying them as transients is harder and it requires a long term survey . we estimate the detectability of short grb orphan afterglows , which may also be produced by compact binary mergers if they are launching also ultra - relativistic outflows . these estimates are based on short grb observations and are therefore indifferent to whether short grbs are binary mergers or not . the main uncertainty in the rate estimates is the grb beaming factor . we find that assuming @xmath164 there are expected to be about 10 orphan afterglows at a 0.1 mjy in a single @xmath94 ghz whole sky snapshot . the duration of these afterglows is several weeks . if binary mergers are short grbs than a gw triggered event will most likely be of a low energy grb , @xmath165 erg , and a true energy , after beaming correction , that is even lower . the radio orphan afterglow will probably still be detectable in a deep search . however its variability time scale is short , about a week , so the search should be done promptly . remarkably , the observed 5 ghz transient rt 19870422 , detected by @xcite fits very well our estimates of the typical expected properties of a compact binary merger radio remnant . at a distance of 1 gpc and a duration of two months this transient is what expected from a mildly relativistic outflow with @xmath169 erg . the rate inferred from this single event is also fully consistent with that of ns@xmath1 mergers . this transient is an excellent candidate to be the first observed radio remnant of a merger . unfortunately , one can not rule out the possibility , suggested by @xcite , that this is an especially bright radio sn . note , however , that this interpretation requires a sn brighter by an order of magnitude than any radio sne previously observed . simultaneous optical observations or multi - wavelength radio observations could have easily distinguished between the two possibilities . the first could have determined if there was a sn or not . the second could have distinguished between an optically thick radio spectrum expected in radio sne vs the optically thin spectrum expected in merger remnants at these frequencies at all times . unfortunately , no such observations were available . however , the rate implied by this even is very high and similar events should be detected in a sub - mjy survey of even a small fraction of the sky . therefore , the nature of this type of events can be easily probed with current facilities . our results show the great potential of @xmath94 ghz radio transient observations at the sub - mjy level for the detection of ns@xmath1 mergers . on the observational side these predictions provide an excellent motivation for carrying out a whole sky sub - mjy survey using the evla or other upcoming radio telescopes . the main source of contamination in such surveys would be radio supernova and those could be distinguished from compact binary mergers by their optical signal , spectrum and other characteristic properties . while it is clear that compact binary mergers produce sub - relativistic to relativistic outflows , details of those outflows are not well determined at present . this is to large extent because of lack of interest rather than because of specific difficulties in analyzing their properties . our analysis elucidate the importance of a detailed quantitative estimates concerning these outflows , a task that is within the scope of current simulations . we thank dale frail , shri kulkarni , andrew macfadyen , eran ofek and stephan rosswog for helpful discussions . this research was supported by an erc advanced research grant , by the israeli center for excellent for high energy astrophysics , by the israel science foundation ( grant no . 174/08 ) and by an irg grant .

within the shubnikov phase of type ii superconductors the applied magnetic field enters the sample in the form of flux lines . the standard mean - field type calculation@xcite shows that in an isotropic material two straight vortices repel at all distance scales , with an interaction strength @xmath1 , @xmath2 being the basic energy scale in the vortex matter , @xmath3 is the zero - order modified bessel function , @xmath0 is the inter - vortex distance , and @xmath4 the magnetic penetration length ( @xmath5 denotes the flux quantum ) . however , it has recently been shown @xcite that in layered and strongly anisotropic superconductors the thermal fluctuations of the flux lines give rise to a long range attraction @xmath6 of the van der waals type between the vortices , where @xmath7 denotes the temperature and @xmath8 is the interlayer separation . the strongly fluctuating and layered high temperature superconductors are particularly well suited to exhibit this attractive component in the vortex - vortex interaction . alternatively , the attraction is induced through static vortex distortions due to an underlying pinning landscape , an effect recently studied by mukherji and nattermann@xcite and by volmer _ et al._@xcite following a suggestion of nelson@xcite , the statistical mechanics of vortices can be mapped to the imaginary time quantum mechanics of two - dimensional ( 2d ) bosons . the particular interaction between the flux lines renders the bosons charged ( with a charge screened on the scale of the london penetration depth @xmath4 ) . this type of long range interaction can be formulated in terms of a massive gauge field theory@xcite . within the resulting 2d massive electrodynamics , the vortex matter acts as a dielectric medium and we can define a casimir problem , see fig . 1 : under the vortex @xmath9 boson mapping two half - spaces of vortices separated by a gap of width @xmath0 act as two dielectric planar media which attract each other via a casimir force . = 8.5 cm in the present work , we determine the casimir force between two dielectric half planes of charged bosons . for dilute media , the macroscopic casimir force can be related to the microscopic van der waals force between the media s constituents via pairwise summation . here , we present a derivation of the van der waals force between vortex lines via this alternative route , calculating first the casimir force between two dielectric planar media in the boson picture and then reconstructing the van der waals force in the reductionist way . since the discovery of the casimir effect@xcite in 1948 , several hundred papers have dealt with this phenomenon , disseminating its fascination into many branches of physics @xcite . casimir forces between macroscopic bodies are a quantum effect caused by a shift in the zero - point energy of gauge field fluctuations such as the electromagnetic one . the casimir effect is bound to a number of system properties , such as topology and dielectric permittivity , and the reduction to an analogous van der waals interaction is not always possible . however , the interpretation in terms of a van der waals attraction is possible for the case of rarefied media and appropriate geometries , such as the parallel plate setup@xcite . in our derivation of the van der waals force from the casimir effect we will make use of these special conditions . in the following , we discuss the relationship between the casimir effect and the van der waals interaction within a path integral formulation ( sec . ii ) before deriving the appropriate action for the 2d charged bosons from the london functional describing the vortices ( sec . iii ) . in section iv , we briefly review the derivation of the van der waals force in the original vortex language and then proceed with the calculation of the 2d casimir effect in section v , the main section of the paper containing the new results . we consider two parallel material slabs made from fluctuating dipoles and separated by a vacuum gap . summing pairwise over all microscopic van der waals interactions between the dipoles provides the macroscopic casimir interaction between the slabs . on the other hand , we can determine the dielectric properties of the individual slabs as produced by the fluctuating dipoles . the specific boundary conditions due to the dielectric properties of the slabs influence the spectrum of the electromagnetic field confined in between . the change in the spectrum as a function of the separation of the slabs produces the casimir force . this analogy between the casimir and the van der waals force is transparently brought out within a path integral formulation , see fig . 2 : assume the system under consideration can be described by an action @xmath10 $ ] depending on the gauge field @xmath11 and a particle current @xmath12 . carrying out the partial integration in the partition function @xmath13{\cal d}[{\bf j}]\ , e^{-{\cal s } [ { \bf j},{\bf a}]/\hbar}$ ] over the matter field @xmath12 or over the gauge field @xmath11 , we obtain an effective action @xmath14 describing the conjugate field alone : from the effective action @xmath15 $ ] describing the gauge field we can derive the casimir effect , while the interaction of the particle currents as described by @xmath16 $ ] will give us the van der waals attraction . the casimir- and van der waals forces then can be related to one another via the pairwise summation of the interparticle forces@xcite : consider two @xmath8-dimensional homogeneous macroscopic dielectric bodies of density @xmath17 with parallel interfaces separated by a distance @xmath0 . the two bodies attract one another due to a microscopic particle - particle interaction @xmath18 , @xmath19 , of the van der waals type . the interaction energy can be written in the form @xmath20 where @xmath21 is a hypercube of size @xmath22 parallel to the interface and @xmath23 , with @xmath24 the @xmath25 dimensional in - plane coordinate , while @xmath26 is the coordinate along the direction perpendicular to the plane . we then find for the casimir force density@xcite @xmath27 the result @xmath28 } { \gamma(\alpha/2 ) } \frac{1}{\alpha - d } \frac{1}{r^{\alpha - d}}. \label{f_of_r}\end{aligned}\ ] ] the result ( [ f_of_r ] ) then allows to infer the parameters @xmath29 and @xmath30 , characterizing the van der waals interaction @xmath31 , from the macroscopic casimir force density @xmath32 . ( 8.5,5 ) ( 0.1,0.1)(8.3,4.7 ) ( 2.25,2)(0,-1)0.75 ( 6.25,2)(0,-1)0.75 ( 4.25,0.75)(-1,0)1 ( 4.25,0.75)(1,0)1 ( 3.5,4)(-1,-1)1 ( 5,4)(1,-1)1 ( 3.8,4.25)@xmath10 $ ] ( 1.8,2.4)@xmath33 $ ] ( 5.9,2.4)@xmath16 $ ] ( 0.75,0.7)casimir effect ( 0.75,0.35)in dielectrics ( 5.75,0.7)van der waals ( 5.75,0.35)attraction ( 3.4,0.35)summation ( 3.6,1)pairwise ( 0.9,3.75)integrate ( 0.9,3.4)over @xmath12 in @xmath34 ( 6,3.75)integrate ( 6,3.4)over @xmath11 in @xmath34 after mapping the thermally fluctuating vortex matter to a system of 2d quantum charged bosons , we will arrive at an action @xmath35 $ ] with two gauge fields @xmath11 and @xmath36 , see eq . ( [ edr ] ) . the casimir effect we are interested in here is the one produced by the fake gauge field @xmath11 : the integration over the matter field @xmath12 will produce the dielectric properties of the media , while the integration over the physical gauge field @xmath37 renders the fake field massive and hence will always exponentially confine the casimir force to finite distances . we start from the london free energy in an isotropic superconductor@xcite @xmath38 = \frac{1}{8\pi}\int { \rm d}^3r \ , [ { \bf b}^2 + \lambda^2(\nabla \times { \bf b})^2 ] , \label{london}\ ] ] with @xmath39 the magnetic field and @xmath4 denoting the london penetration depth . in order to acquaint for the vortices , we add the current term @xmath40 with @xmath41 where @xmath42 , @xmath43 , the coordinates @xmath44 denote the position , the vectors @xmath45 the direction of the vortex lines , and @xmath5 is the unit of flux . ignoring screening , the interaction between the vortex lines is long ranged and thus can conveniently be expressed through a mediating gauge field @xmath11 . as usual , we introduce the gauge field @xmath11 as an auxiliary field such that @xmath46 ) = \exp(-\beta { \cal f})$ ] , with @xmath47 the inverse temperature@xcite ( we set the boltzmann constant @xmath48 to unity and fix the gauge through the condition @xmath49 ) , @xmath50 & = & \int { \rm d}^3{x } \ , \left[i{\bf a}\cdot\left({\bf j}-\frac{1}{\phi_0}(\nabla\times{\bf a})\right ) \right . \label{fprime } \\ & & \quad + \left . \frac{1}{8\pi}(\nabla \times { \bf a})^2 + \frac{1}{2g^2}(\nabla \times { \bf a})^2 \right ] , \nonumber\end{aligned}\ ] ] where @xmath51 and the line energy @xmath52 is the basic energy scale in the problem . in ( [ fprime ] ) , we have accounted for screening by introducing back into the model the real gauge field @xmath37 . the anisotropy of uniaxial layered materials is most conveniently introduced through a rescaling of the scalar fake magnetic field @xmath53 @xcite , @xmath54 where the anisotropy factor @xmath55 is determined through the effective masses perpendicular ( @xmath56 ) and parallel ( @xmath57 ) to the @xmath58-plane ( the fake electric field @xmath59 remains unchanged ; the subscript ` @xmath60 ' identifies the planar component of a vector , @xmath61 ) . we map the statistical mechanics of the vortex system to the imaginary time quantum mechanics of 2d bosons through the replacements@xcite @xmath62 ( imaginary boson time ) , @xmath63 ( the boson s planck constant ) , @xmath64 ( the boson action ) , and @xmath65 ( the light velocity in the boson system ) . the boson partition function is given by @xmath66 { \cal d}[{\bf a}]{\cal d}[{\bf a } ] e^{-{\cal s}[\{{\bf r}_\mu\},{\bf a},{\bf a}]/\hbar^{\scriptscriptstyle b } } , \label{zzz}\ ] ] with @xmath67 and @xmath68 = \int \!\ ! { \rm d}\tau \sum_\mu \left[\frac{m}{2 } \left(\partial_\tau { \bf r}_\mu ( \tau)\right)^2 - \mu^{\scriptscriptstyle b } \right],\label{plm}\\ & & { \cal s}_{\rm int } [ \{{\bf r}_\mu\},{\bf a},{\bf a } ] = \int \!\ ! { \rm d } \tau { \rm d}^2 r \left[i{\bf a}\cdot \left({\bf j}-\frac{1}{\phi_0 } ( \nabla \times { \bf a})\right ) \right . \label{edr } \\ & & \quad + \left . \frac{1}{8\pi } ( \nabla \times { \bf a})^2 + \frac{1}{2g^2}\left((\nabla\times{\bf a})_{xy}^2 + \frac{1}{\varepsilon^2 } ( \nabla\times { \bf a})_\tau^2\right ) \right ] . \nonumber\end{aligned}\ ] ] here , @xmath69 is the chemical potential of the bosons , @xmath70 is the boson density , and @xmath71 is the boson mass ( we have introduced a term @xmath72 to account for the external magnetic field @xmath73 producing the vortices in the superconductor ; @xmath74 denotes the planar coherence length ) . in the thermodynamic limit , we have @xmath75 , corresponding to @xmath76 , i.e. , we are interested in the ground state physics of the boson system . note that the free boson action @xmath77 contains the bare mass @xmath78 due to the vortex core energy . the retarded self - interaction of the bosons via their gauge fields then produces the mass renormalization @xmath79 , where @xmath80 is the dispersive line tension@xcite of the vortex lines . to set the stage , we briefly review the derivation of the van der waals interaction as presented in ref . ; this will allow us to fix some flaws in the previous derivation and will provide us with a check on the results for the casimir force derived later . for a simple qualitative analysis we consider two vortices in a layered superconductor ( with layers separated by the distance @xmath8 ) . ignoring the coupling between the layers , the fluctuating pancake vortices interact via a logarithmic potential @xmath81 . second order perturbation theory then provides us with a van der waals interaction @xmath82 , the energy scale @xmath83 being set by the driving action of the thermal fluctuations . at long distances @xmath84 a finite interlayer coupling changes this result as the josephson interaction reduces the cutoff @xmath85 on the @xmath86 modes to the new value @xmath87 ; the van der waals interaction crosses over to @xmath88 . the two results have their analogue in the van der waals attraction between neutral atoms , where the interaction potential exhibits a crossover from a @xmath89 at short to a @xmath90 behavior at large distances@xcite . we proceed with the derivation of the long range van der waals interaction between vortex lines / bosons . following the scheme in fig . 2 , we integrate over the gauge fields @xmath37 and @xmath11 in the partition function ( [ zzz ] ) and obtain the effective current - current interaction @xmath91 = \frac{\varepsilon_0}{2}\int { \rm d}^3r \,{\rm d}^3 r'\ , { j}_\alpha({\bf r})v_{\alpha\beta}^{\rm int}({\bf r}-{\bf r } ' ) { j}_\beta({\bf r } ) \label{aa1}\ ] ] ( we return to the more natural statistical mechanics notation in this section , @xmath92 ) . inserting the expression ( [ current ] ) for the currents , we can cast eq . ( [ aa1 ] ) into the standard form@xcite @xmath93 = \frac{\varepsilon_0}{2 } \sum_{\mu,\nu } \int { \rm d}r_{\mu\alpha } { \rm d}r'_{\nu\beta } v_{\alpha\beta}^{\rm int } ( { \bf r}_\nu - { \bf r}'_\mu ) , \label{aa2}\ ] ] with the vortex positions @xmath94 and the interaction potential @xmath95 , conveniently expressed within a fourier representation , @xmath96 with @xmath97 . \label{vab}\end{aligned}\ ] ] here @xmath98 , @xmath99 and @xmath100 , @xmath101 . the free energy ( [ aa2 ] ) can be split into the self energy part @xmath102 ( @xmath103 ) and the interaction part @xmath104 ( @xmath105 ) . restricting ourselves to two vortices a distance @xmath0 apart , we obtain @xmath106}e^{ik_z(z_1-z_2 ) } \nonumber \\ & \times & [ v_{zz}^{\rm int}({\bf k } ) + t_{1\alpha}(z_1)t_{2\beta}(z_2 ) v_{\alpha \beta}^{\rm int}({\bf k } ) ] , \end{aligned}\ ] ] where we have split the vortex positions @xmath107 into a mean field part @xmath108 and a fluctuating part @xmath109 . up to a constant , the free energy is given by @xmath110\rangle_0.\ ] ] the average @xmath111 has to be taken with respect to the self - energy @xmath102 of the free vortices . performing a cumulant expansion , we obtain the effective vortex - vortex interaction in the form @xmath112/2t$ ] , where @xmath113 denotes the sample thickness . splitting into longitudinal ( to the induction , @xmath114 ) and transverse ( @xmath115 ) parts , the longitudinal interaction produces the standard repulsive vortex - vortex interaction@xcite , to lowest order in @xmath116 , @xmath117 while higher orders in @xmath116 merely renormalize the prefactor . the transverse part produces the van der waals interaction@xcite @xmath118 using the decomposition @xmath119 we can reduce the average in ( [ vdw_def ] ) to the simpler form @xmath120 ^ 2 \langle t^2(k_z ) \rangle_0 ^ 2,\ ] ] with the partial fourier transform @xmath121 in strongly anisotropic and layered material the single vortex mean squared amplitude of fluctuations @xmath122 is limited by the electromagnetic interaction through the dispersive elasticity @xmath80 , @xmath123 with @xmath124 the evaluation of the partial fourier transform ( [ vpartial ] ) is carried out in appendix a and making use of the results ( [ app_a_x ] ) and ( [ app_a_y ] ) in the limit @xmath125 , we find the van der waals interaction in the decoupled limit @xmath126 the interlayer distance @xmath8 provides the large @xmath86 cutoff on the thermal fluctuations of the individual lines . in the continuous anisotropic case the single vortex elasticity is still dominated by its electromagnetic contribution . at intermediate distances @xmath127 we recover again the result ( [ v4_vdw ] ) as the josephson coupling is not effective yet . at large distances @xmath128 , however , the interlayer coupling becomes important through cutting off the @xmath86 modes at @xmath87 and we find the result @xmath129 these results coincide with the ones found by blatter and geshkenbein@xcite up to the numerical prefactor as they missed a factor @xmath130 in eq . ( [ t^2 ] ) , as well as a term in the partial fourier transform which contributes to the result ( [ v5_vdw ] ) to order @xmath131 . the same mistakes show up in their results for the attraction of the vortex to the sample surface , eqs . ( 19 ) and ( 20 ) of ref . . we give the correct expressions here : @xmath132 for the decoupled case , and @xmath133 in the continuous anisotropic situation . in this section , we derive the casimir force between two parallel cubes of vortex matter and then show , via the method of pairwise summation , that the results coincide with those found by means of the van der waals approach . again , we begin with a simple analysis based on dimensional estimates , instructing us what to expect . consider two @xmath8-dimensional hypercubes of size @xmath134 separated by a distance @xmath135 , see fig . 1 . we wish to determine the casimir force @xmath32 per unit area acting between the interfaces . being due to the zero point fluctuations of the gauge field , the only relevant dimensional quantities entering the expression for the force are planck s constant @xmath136 , the velocity of light @xmath137 , and the distance @xmath0 between the slabs . the combination @xmath138 then provides us with a dimensionally correct expression for the force ; in two dimensions we then find @xmath139 for the retarded casimir effect . in the non - retarded case , @xmath140 and we have to replace the combination @xmath141 by the frequency scale @xmath142 where the matter becomes transparent ( for the vortex matter in a layered superconductor this ` frequency ' is given by the layer separation @xmath8 , @xmath143 ) . dimensional estimates then give us for the force per unit length in two dimensions @xmath144 . in order to derive the corresponding microscopic inter - particle potential from these estimates , we apply the method of pairwise summation and obtain the result @xmath145 in the retarded case and @xmath146 in the non - retarded limit . these simple estimates produce the correct power laws for the van der waals potential ; in order to find the correct sign and the complete prefactors , we have to go through the detailed analysis below . the casimir energy is given through the difference in the sum over all cavity modes minus the free field contribution@xcite . while the calculation is rather straightforward for the simple case of a metallic cavity , an appropriate formalism has to be set up to treat more general configurations involving dielectric media . we start from a quadratic lagrangian density @xmath147 in @xmath8 dimensions , with @xmath148 a vector boson field with @xmath149 components and @xmath150 the green function matrix . the partition function @xmath34 is expressed through the usual imaginary time path integral formalism @xmath151\ , \exp\left[-\frac{1}{\hbar } \int { \rm d}^d x \int_0^{\hbar / t } \!\!\!\!\!\!\!\!d\tau \ , { \cal l}[{\bf a}]\right ] = [ \det { \bf g}]^{-1/2},\ ] ] and the free energy @xmath152 is given by @xmath153 we consider the classic parallel - plate geometry with isotropic media separated by a gap of width @xmath0 along the @xmath154-direction . for each polarization @xmath155 we determine the eigenstates obeying the boundary conditions at the @xmath156-dimensional hypersurfaces placed at @xmath157 and @xmath0 . the individual modes are characterized by their polarization @xmath158 , the ( matsubara ) frequency @xmath159 , @xmath160 , the transverse wave vector @xmath161 , and the longitudinal wave vector @xmath162 , @xmath163 . given the set @xmath164 , the discrete wave vectors @xmath162 are obtained as the solutions of the boundary condition @xmath165 , where @xmath166 is a short hand for the other indices @xmath167 ( the function @xmath168 derives from the determinant associated with the set of boundary conditions ; for the classic casimir effect , @xmath169 with @xmath0 the distance between the metal plates ) . expressing the trace through all the indices we have @xmath170 where the prime on the sum indicates that we count the @xmath171 term with a weight @xmath172 . in ( [ f_ln ] ) we have made explicit the boundary condition at the hypersurfaces separating the media from the gap . = 5.5 cm we make use of cauchy s theorem to perform the sum over the longitudinal momenta @xmath162 : we first rewrite the sum in the form @xmath173 = \int { \rm d}y \sum_{d_{\{\alpha\}}(k)=0 } \frac{1}{g_{\{\alpha\}}^{-1 } + y}.\ ] ] next , we let the boundary condition @xmath165 produce the desired sum in a cauchy loop integral in the complex @xmath174 plane , see fig . 3 . with @xmath175 we have @xmath176 where it is understood that both @xmath177 and @xmath178 further depend on the other indices @xmath166 as well as on the parameter @xmath179 in the case of @xmath178 . we define the function @xmath180 as the zeroes of the expression @xmath181 + y = 0 $ ] . taking the derivative of the last equation with respect to @xmath179 , @xmath182 , and using the result in ( [ c_loop ] ) , we obtain the desired result @xmath183 , \label{sum_k}\ ] ] where we have used that the green function exhibits the proper asymptotics @xmath184 and the symmetry @xmath185 . inserting the sum on @xmath174 , eq . ( [ sum_k ] ) with @xmath186 , back into the free energy expression ( [ f_ln ] ) we obtain @xmath187 , \label{f_lnd}\ ] ] where @xmath188 \equiv l^{d-1}\sum_{\nu = 1}^r { \sum^{\infty}_{s=0 } } \!\!\phantom{|}^\prime \int \frac{{\rm d}^{d-1}q}{(2\pi)^{d-1 } } \ , [ \dots]\ ] ] denotes the sum / integral over all modes @xmath189 . the above formal manipulations require a regularization such that @xmath190 \rightarrow 1 $ ] . we carry out the appropriate renormalization by the subtraction of the free energy of the free field without boundary conditions , corresponding to replacing the partition function through the ratio @xmath191 , with @xmath192 the partition function of the free field . for the latter , we define the ` boundary condition ' @xmath193 in such a way as to produce the fraction @xmath194 of the free field energy in the volume @xmath134 via the expression ( [ f_lnd ] ) ; for the classic casimir problem , @xmath195\}^{r / l}$ ] . with @xmath196 we finally obtain the regularized free energy of the casimir problem@xcite @xmath197 . \label{fren}\ ] ] we illustrate the use of this formalism with a brief derivation of the classic casimir result , the attraction between two parallel metallic plates of size @xmath198 and a distance @xmath0 apart . the dispersion relation for the free electromagnetic field is @xmath199 , such that the function @xmath200 takes the form @xmath201 . the boundary condition requiring the fields to vanish at @xmath157 and @xmath0 can be cast into the form @xmath202 , producing the modes with a longitudinal @xmath174 vector @xmath203 . the ` boundary condition ' describing the free field takes the form @xmath204 , such that @xmath205 , where @xmath206 . combining the results for @xmath207 and @xmath208 , we have to carry out the mode summation over the logarithm of @xmath209 = \left[1-\exp\left(-2r \sqrt{\frac{\xi_s^2}{c^2}+q^2}\right)\right]^2,\ ] ] where taking the square accounts for the two polarization modes of the electromagnetic field ( the single polarization mode at @xmath210 has been properly taken into account here ) . the force density @xmath32 between the plates is given through the derivative of the energy ( [ fren ] ) with respect to @xmath0 , @xmath211 where we have replaced the matsubara sum through an integral over complex frequencies @xmath74 . the remaining integrals are easily evaluated by changing to polar coordinates @xmath212 , @xmath213 and we obtain the classic result due to casimir@xcite @xmath214 in the following , we apply the above formalism to the 2d charged bosons described by the action @xmath215 as given in eqs . ( [ plm ] ) and ( [ edr ] ) . following the scheme outlined in fig . 2 , we first have to integrate over the currents @xmath216 . we split the boson positions @xmath217 into the mean field part @xmath218 and a fluctuating part @xmath219 , @xmath220 . in carrying out the integration over the boson positions @xmath221 we can ignore the fluctuating part @xmath219 in the gauge field @xmath11 as the latter is smooth on the scale of the amplitude @xmath219 , @xmath222 \exp\bigg\{-\frac{1}{\hbar^{\scriptscriptstyle b } } \int{\rm d}\tau\sum_\mu \big[\frac{m}{2}(\partial_\tau { \bf u}_\mu)^2 \nonumber \\ & & \qquad\qquad\qquad\qquad\qquad + i(\partial_\tau { \bf u}_\mu,1)\cdot { \bf a}({\bf r}_\mu^0,\tau)\big]\bigg\ } \nonumber \\ & & \propto \exp\left[-\frac{1}{\hbar^{\scriptscriptstyle b } } \!\ ! \int \ ! { \rm d}^2 r{\rm d}\tau \left(\frac{1}{2 } \pi({\bf r } ) a^2_{xy}({\bf r } ) + i \rho({\bf r } ) a_\tau({\bf r } ) \right)\right ] , \nonumber\end{aligned}\ ] ] where @xmath223 denote the polarizability and the density of the 2d bosons , respectively . second , we have to integrate over the true gauge field @xmath37 ( we have fixed the gauge to @xmath49 ) . the resulting term @xmath224 in the action then renders the fake gauge field @xmath11 massive ( physically , this mass term expresses the finite range @xmath4 of the interaction between the charged bosons / vortices ) . third , we introduce the free green function matrix @xmath225 for the gauge field , in fourier representation , @xmath226 , \nonumber \\ \noalign{\vskip 5 pt } { \rm with } \quad ( { \bf g}_{xy})_{\alpha \beta}^{-1 } & = & k^2 \delta_{\alpha\beta } + ( c^2 - 1 ) k^2 p_{\alpha \beta}({\bf k } ) , \nonumber \\ g_\tau^{-1 } & = & k^2 , \nonumber\end{aligned}\ ] ] and the transverse projector @xmath227 . combining the results of the above three steps , we arrive at the desired effective action for the fake gauge field @xmath11 , @xmath228 & = & \int\!\!{\rm d}^2 r { \rm d}\tau \left [ { \bf a}_{xy } \frac{1}{2g^2}\left({\bf g}_{xy}^{-1 } + \frac{1}{\lambda^2 } + g^2\pi \right ) { \bf a}_{xy } \right . \nonumber \\ & & \left . + a_\tau \frac{1}{2g^2}\left(g_\tau^{-1 } + \frac{1}{\lambda^2}\right)a_\tau + i\rho a_\tau \right ] . \label{thing}\end{aligned}\ ] ] the euler - lagrange equations of this action determine the ( imaginary time ) field equations for the transverse and longitudinal parts of @xmath11 ( we remind that @xmath229 ) , @xmath230 and @xmath231 with @xmath232 . in the last equation of ( [ trans_eom ] ) we have made use of the gauge condition @xmath233 and have ignored the time dependence in the longitudinal field component @xmath234 , see below . the longitudinal field @xmath234 is generated by the source @xmath235 and has no dynamics on its own . placing two bosons at a distance @xmath0 , a simple integration gives the vortex - vortex interaction ( c.f . , eq . ( [ rep_eq ] ) ) @xmath236 here , we ignore the fluctuations of the vortex position in the calculation of @xmath234 as they merely produce a small renormalization of the prefactor in the repulsive potential . similarly , we can neglect such corrections in the calculation of the transverse modes below and set the r. h. s. of ( [ trans_eom ] ) equal to zero . the dynamical transverse field @xmath237 generates the casimir force . here , we have in mind a geometry as sketched in fig . 1 , with two parallel cubes of vortex matter of density @xmath17 , separated by a vortex free region of thickness @xmath0 . we choose the @xmath26-axis to lie perpendicular to the ` plates ' , the @xmath238-axis is directed along the vortices . within the boson language we deal with two parallel planes of 2d bosons lying in the @xmath60-plane , the direction along the vortices now transforming to the imaginary time coordinate ( below , we consider the limit @xmath239 , implying vortex lines of infinite length , @xmath240 ) . note that in the present 2d case we have to consider only one polarization mode for the transverse gauge field @xmath237 , a consequence of the gauge condition @xmath49 . going over to fourier space we can cast ( [ trans_eom ] ) into the form ( we remind the reader that @xmath241 ; @xmath242 and @xmath174 are the wave vectors along and perpendicular to the ` plates ' ) @xmath243 { \bf a}_{xy } = 0 , \label{trans_eom_k}\ ] ] from which we obtain the green function @xmath244 and the function @xmath208 , @xmath245 here , we have defined the ` dielectric ' constant @xmath246 where we go over to vortex parameters in the last equation ( @xmath247 ) and make use of the expression ( [ elasticity ] ) for the dispersive vortex elasticity @xmath248 . next , we have to formulate the boundary conditions in terms of the zeroes of the function @xmath249 . the translational invariance in the @xmath179-direction allows for a plane - wave ansatz @xmath250 , while the sequence of ` dielectric ' and ` vacuum ' regions along the @xmath26-axis , requires to match the plane waves @xmath251 , with ( see ( [ trans_eom_k ] ) ) @xmath252 the gauge condition @xmath253 is satisfied through the ansatz @xmath254 with piecewise constant amplitudes @xmath255 and @xmath256 . the six coefficients are determined by the boundary conditions at @xmath257 and @xmath258 , requiring the parallel electric field @xmath259 and the magnetic field @xmath260 to be continuous , see appendix b. two further conditions force the fields to vanish at @xmath261 , @xmath262 . requiring the determinant of the resulting @xmath263 matrix problem to vanish , we find @xmath264 & = & e^{-ik_l l } e^{-ik_0 r } e^{-ik_r l } \label{dvdw } \\ & & \left[1 - \frac { ( 1 + ( \lambda\xi)^{-2})k_l-(\epsilon_{\scriptscriptstyle v}+ ( \lambda\xi)^{-2})k } { ( 1 + ( \lambda\xi)^{-2})k_l+(\epsilon_{\scriptscriptstyle v}+ ( \lambda\xi)^{-2})k } \right . \nonumber \\ & & \times \left . \frac { ( 1 + ( \lambda\xi)^{-2})k_r-(\epsilon_{\scriptscriptstyle v}+ ( \lambda\xi)^{-2})k } { ( 1 + ( \lambda\xi)^{-2})k_r+(\epsilon_{\scriptscriptstyle v}+ ( \lambda\xi)^{-2})k } e^{2 i k r}\right ] . \nonumber\end{aligned}\ ] ] for the free field , we find @xmath265 and combining the above results for @xmath208 , @xmath177 , and @xmath193 , eqs . ( [ fvdw ] ) , ( [ dvdw ] ) , and ( [ dvdw_0 ] ) , we can construct the function @xmath266 $ ] and obtain the formal expression for the free energy ( [ fren ] ) of the vortex casimir problem . we note that the cut introduced through the definitions of @xmath267 and @xmath268 in eq . ( [ ki ] ) disappears from the boundary condition @xmath177 and hence does not contribute to the loop integral ( [ c_loop ] ) . as a result , the free energy expression ( [ fren ] ) for the casimir free energy remains valid . in the present two dimensional situation the expression for the casimir energy eq . ( [ fren ] ) per length reads @xmath269,\ ] ] and produces the casimir force per length @xmath270 @xmath271^{-1 } } . \nonumber\end{aligned}\ ] ] we introduce the new variables @xmath272 and arrive at the simplified expression @xmath273^{-1 } } . \nonumber\end{aligned}\ ] ] eq . ( [ force_gen ] ) gives the two dimensional version for massive photons of the casimir force between dielectric media , first derived by lifshitz@xcite for the conventional 3d situation . in the following we will be interested in the situation where the ` dielectric ' media are dilute ( dilute vortex matter at small magnetic induction @xmath274 ) . in this case , we expand ( [ force_gen ] ) in the small correction @xmath275 giving the deviation of the dielectric constant from its vacuum value . furthermore , if the integral is dominated by large frequencies @xmath74 we can ignore the mass term everywhere . under these conditions , we approximate @xmath276 and @xmath277 , such that @xmath278 , while @xmath279 , and obtain the simplified force expression ( the weak logarithmic disperion in @xmath280 is approximated through an appropriate constant ) @xmath281 in ( [ force_gen ] ) , ( [ force_dil ] ) we have to account for three relevant frequency- or length scales . the first is introduced by the discrete nature of our superconductor : the layered structure limits the @xmath74-integration through the frequency cutoff @xmath282 ( this corresponds to the medium becoming transparent at high frequencies @xmath283 ) . the second frequency scale is introduced through the dispersion in the dielectric constant : at low frequencies @xmath284 , the dielectric constant takes the form @xmath285 and the dielectric response changes over to a mass renormalization . the third frequency scale @xmath286 appears through the exponential @xmath287 in the integrand of ( [ force_gen ] ) ; as @xmath74 goes beyond @xmath288 the exponent cuts off the integrand in ( [ force_gen ] ) . thus , depending on the frequency @xmath74 , the media are either transparent , give a dielectric response , or produce a mass renormalization . and depending on the position of @xmath288 with respect to these response regimes , we will find a different behavior of the casimir force . below , we will analyze the various regimes and use the pairwise summation technique to show that the results are in agreement with those obtained for the van der waals interaction potential in sec . iv above . at intermediate distances @xmath127 , we have @xmath289 such that the cutoff on the @xmath74 integral is given by @xmath290 . as the integral is dominated by large values of @xmath74 we can ignore the mass terms @xmath291 everywhere . the exponential @xmath292 restricts the @xmath293-integral to values @xmath294 , admitting large values of @xmath293 such that we can drop the corrections to @xmath295 in ( [ force_dil ] ) , e.g. , @xmath296 . transforming @xmath293 to the new variable @xmath297 , we find the casimir force on intermediate scales @xmath298 using the result ( [ f_of_r ] ) of the pairwise summation and inserting the expression ( [ epsilon_v ] ) for the dielectric constant , we obtain the van der waals interaction describing the decoupled limit , @xmath299 in agreement with the result ( [ v4_vdw ] ) ( note that the density @xmath17 cancels in the final result for the van der waals interaction ) . at large distances , the @xmath74-integral is cutoff on the scale @xmath288 , i.e. , the casimir force is limited by the distance @xmath0 through the exponential @xmath292 rather than by the transparency of the material . we still can ignore the mass terms in ( [ force_gen ] ) as we assume that @xmath301 , providing us with a large regime for the @xmath74-integration where @xmath302 . transforming the energy variable @xmath74 to @xmath303 , we obtain the following expression for the casimir force in the low density limit , @xmath304 following the usual scheme , this result produces the retarded van der waals interaction @xmath305 in agreement with eq . ( [ v5_vdw ] ) . the situation at very large distances @xmath307 is most conveniently analyzed in the original formulation ( [ force_gen_0 ] ) . the mode summation is limited by the exponential @xmath308 , implying the following restrictions on the integration variables @xmath74 and @xmath242 : @xmath309 and @xmath310 . the integral @xmath311 $ ] then is given by the area @xmath312 times the @xmath313 limit of the integrand . in the limit @xmath314 the mass terms are relevant and the dispersion in the dielectric constant ( [ epsilon_v ] ) produces the additional mass renormalization @xmath315 the casimir force then decays exponentially following the law @xmath316 for small densities , we can expand @xmath317 and obtain for the screened van der waals interaction in the regime @xmath306 the expression ( note that the pairwise summation formula ( [ f_of_r ] ) has to be modified for the exponential factor in the interaction @xmath318 ) @xmath319 we have derived the casimir force between two bodies of vortex matter and have inferred from the result the strength of the van der waals interaction between vortex lines via the method of pairwise summation . while the calculation of the casimir force is carried out in the boson formulation , the van der waals attraction is determined in the vortex picture . the agreement between the results is once more an illustration of the equivalence of the two formalisms @xcite . the physical implications of these results lead to interesting modifications of the @xmath320-@xmath7 phase diagram of layered / anisotropic type ii superconductors@xcite : the attraction between the lines produces a generic vortex solid of density @xmath321 at low temperatures . an interesting problem appearing in this context is the accurate determination of the entropic repulsion between the vortex lines , the latter giving an important contribution to the gibbs free energy . various approaches to this problem have been discussed by blatter and geshkenbein@xcite , by volmer , mukherji , and nattermann@xcite , and by volmer and schwartz@xcite . the transition from the meissner state to the van der waals vortex solid takes place via a sharp first - order transition at the lower critical field @xmath322 . the concurrent phase separation then can be observed in a decoration experiment , though very clean samples are required . furthermore , the phase separation leads to the concentration of the vortex lines into dense regimes separated by vortex free regions . the interaction between the vortex domains then is given by the casimir force calculated here , thus providing a natural realization of the physics described in this paper . a further remark is in place concerning the physical character of the van der waals interaction discussed here . the usual van der waals interaction between neutral atoms arises from dipolar fluctuations in the charge distribution of the atoms , resulting in a force which is mediated through the scalar ( longitudinal ) potential , at least within the non - retarded regime at small distances . comparing to the van der waals force between the 2d charged bosons / vortex lines discussed here , we note that the elementary objects do not have an internal structure producing a fluctuating dipole . the van der waals interaction then arises from _ current _ fluctuations and thus involves the ( transverse ) vector field . in principle , such a ` transverse ' van der waals force is also present in an electron gas : while the longitudinal repulsive interaction is screened on the thomas - fermi length , the transverse attractive interaction of the van der waals type due to fluctuating currents survives at long distances ( and in principle induces a superconducting instability ) . however , the transverse van der waals force in an electronic system involves the small parameter @xmath323 ( @xmath324 is the fermi velocity ) , rendering the effect small . in the 2d boson system discussed here , the light velocity is given by the anisotropy of the superconductor , @xmath241 , and thus is a tunable parameter . we thank vadim geshkenbein , a. van otterlo , and christoph bruder for fruitful discussions . in this section we sketch , by way of example , the calculation of @xmath325 . we start from eq . ( [ vab ] ) and set @xmath326 ( here , we ignore the exponential cutoff function due to the vortex core ; the cases @xmath327 and @xmath328 , @xmath329 follow trivially ) @xmath330 performing a contour - integral over @xmath331 , we obtain @xmath332 with @xmath333 and @xmath334 . carrying out the integral@xcite over @xmath335 in @xmath336 we find @xmath337 the second integral @xmath338 can be calculated in the same manner , @xmath339 here , @xmath3 denotes the @xmath157-th order modified bessel functions of the second kind and @xmath340 . collecting results and carrying out the derivatives , we obtain the desired result . here , we quote the final expressions for the case @xmath341 : @xmath342 \right . \nonumber \\ & & \left . \qquad\qquad\qquad -\frac{a}{\lambda_c r } k_1\left(\frac{ar}{\lambda_c}\right ) \right\ } , \label{app_a_x } \\ v_{yy}^{\rm int}(r , k_z ) & = & \frac{1}{2 \pi a^2 } \left \{\frac{a^2}{\lambda_c^2}\left[k_0\left(\frac{ar}{\lambda_c}\right ) + \frac{\lambda_c}{ar}k_1\left(\frac{ar}{\lambda_c}\right ) \right ] \right . \nonumber \\ & & \left . \qquad\qquad\qquad -\frac{a}{\lambda r } k_1\left(\frac{ar}{\lambda}\right ) \right\}. \label{app_a_y}\end{aligned}\ ] ] the third component @xmath343 vanishes for @xmath344 . the important terms in ( [ app_a_x ] ) and ( [ app_a_y ] ) are those involving the large screening length @xmath345 in the argument of the bessel functions . the first term @xmath346 in ( [ app_a_y ] ) has been missed in ref . . we sketch the main features of the 2d real time massive electrodynamics underlying the present work ( we denote the mass by @xmath347 and restrict ourselves to the isotropic case ) . we introduce the metric @xmath348 and define the gauge field @xmath349 . in addition , we define the derivatives @xmath350 , where @xmath351 . the field tensor @xmath352 is given by @xmath353 and its dual takes the form @xmath354 ( @xmath355 is the antisymmetric tensor ) . the lagrangian of the massive field coupled to an external current @xmath356 is given by @xmath357 with @xmath358 the coupling constant . this lagrangian produces an imaginary time action @xmath359 = \int \!\ ! { \rm d } \tau { \rm d}^2 r \left [ \frac{1}{2g^2}(\nabla\times{\bf a})^2 + \frac{1}{2 g^2 \lambda^2}{\bf a}^2 + { \bf a } { \bf j } \right].\ ] ] the transformation from real time to imaginary time is obtained via the formal rules @xcite @xmath360 , @xmath361 , @xmath362 . the magnetic field remains unchanged , while the electric field transforms to @xmath363_{\perp } = i { \bf e}$ ] . the free field real time lagrangian @xmath364 then goes over into the simple form @xmath365 within the imaginary time formalism . the functional derivative of eq . ( [ l_aj ] ) provides us with the inhomogeneous maxwell equations ( the homogeneous one , @xmath366 , follows from the antisymmetric structure of @xmath352 ) , @xmath367 making use of gauss and stokes theorems for a loop encircling the boundary between two media , we obtain the boundary conditions @xmath368 continuous and @xmath369 continuous . the quantization of the theory can be developed via the gupta - bleuler formalism within the lorentz gauge @xmath370 . in the case of a massive theory this method leads to two polarization modes , while in a massless theory the gauge invariance reduces the number of polarization modes by one . within the present theory , the coupling of @xmath11 to the real @xmath37 field in the form @xmath371 produces a gauge invariant mass term @xmath372 . as the gauge invariance is conserved in our theory we end up with a single polarization mode .

in the minimal supersymmetric standard model ( mssm ) , the ( susy ) particles must be produced in pairs . the phase space is largely suppressed in pair production of susy particles due to the important masses of the superpartners . the r - parity violating ( ) extension of the mssm contains the following additional terms in the superpotential , which are trilinear in the quarks and leptons superfields , @xmath7 where @xmath8 are flavour indices . these couplings offer the opportunity to produce the scalar particles as resonances @xcite . although the coupling constants are severely constrained by the low - energy experimental bounds @xcite , the resonant superpartner production reaches high cross sections both at leptonic @xcite and hadronic @xcite colliders . the resonant production of susy particle has another interest : since its cross section is proportional to a power @xmath9 of the relevant coupling , this reaction would allow an easier determination of the couplings than the pair production provided the coupling is large enough . as a matter of fact in the pair production study , the sensitivity on the couplings is mainly provided by the displaced vertex analysis of the lightest supersymmetric particle ( lsp ) decay which is difficult experimentally , especially at hadronic colliders . neither the grand unified theories ( gut ) , the string theories nor the study of the discrete gauge symmetries give a strong theoretical argument in favor of the r - parity violating or r - parity conserving scenarios @xcite . hence , the resonant production of susy particle through couplings is an attractive possibility which must be considered in the phenomenology of supersymmetry . the hadronic colliders have an advantage in detecting new particles resonance . indeed , due to the wide energy distribution of the colliding partons , the resonance can be probed in a wide range of the new particle mass . this is in contrast with the leptonic colliders for which the center of mass energy must be fine - tuned in order to discover new narrow width resonances . at hadronic colliders , either a slepton or a squark can be produced at the resonance respectively through a @xmath10 or a @xmath11 coupling constant . in the hypothesis of a single dominant coupling constant , the resonant scalar particle can decay through the same coupling as in the production , leading to a two quark final state for the hard process @xcite . in the case where both @xmath10 and @xmath12 couplings are non - vanishing , the slepton produced via @xmath10 can decay through @xmath12 giving rise to the same final state as in drell - yan process , namely two leptons @xcite . however , for reasonable values of the coupling constants , the decays of the resonant scalar particle via gauge interactions are typically dominant if kinematically allowed @xcite . + the main decay of the resonant scalar particle through gauge interactions is the decay into its partner plus a gaugino . indeed , in the case where the resonant scalar particle is a squark , it is produced through @xmath11 interactions so that it must be a right squark @xmath13 and thus it can not decay into the @xmath14-boson , which is the only other possible decay channel via gauge interactions . besides , in the case where the resonant scalar particle is a slepton , it is a left slepton produced via a @xmath10 coupling but it can not generally decay as @xmath15 or as @xmath16 . the reason is that in most of the susy models , as for example the supergravity or the gauge mediated models , the mass difference between the left charged slepton and the left sneutrino is due to the d - terms so that it is fixed by the relation @xmath17 @xcite and thus it does not exceed the @xmath14-boson mass . nevertheless , we note that in the large @xmath18 scenario , a resonant scalar particle of the third generation can generally decay into the @xmath14-boson due to the large mixing in the third family sfermion sector . for instance , in the sugra model with a large @xmath18 a tau - sneutrino produced at the resonance can decay as @xmath19 , @xmath20 being the lightest stau . + the resonant scalar particle production at hadronic colliders leads thus mainly to the single gaugino production , in case where the decay of the relevant scalar particle into gaugino is kinematically allowed . in this paper , we study the single gaugino productions at tevatron run ii . the single gaugino productions at hadronic colliders were first studied in @xcite . later , studies on the single neutralino @xcite and single chargino @xcite productions at tevatron have been performed . the single neutralino @xcite and single chargino @xcite productions have also been considered in the context of physics at lhc . in the present article , we also study the single superpartner productions at tevatron run ii which occur via @xmath0 processes and do not receive contributions from resonant susy particle productions . the singly produced superpartner initiates a cascade decay ended typically by the decay of the lsp . in case of a single dominant @xmath11 coupling constant , the lsp decays into quarks so that this cascade decay leads to multijet final states having a large qcd background @xcite . nevertheless , if some leptonic decays , as for instance @xmath21 , @xmath22 being the chargino and @xmath23 the neutralino , enter the chain reaction , clearer leptonic signatures can be investigated @xcite . in contrast , in the hypothesis of a single dominant @xmath10 coupling constant , the lsp decay into charged leptons naturally favors leptonic signatures @xcite . we will thus study the single superpartner production reaction at tevatron run ii within the scenario of a single dominant @xmath2 coupling constant . in section [ theoretical ] , we define our theoretical framework . in section [ discussion ] , we present the values of the cross sections for the various single superpartner productions via @xmath2 at tevatron run ii and we discuss the interesting multileptonic signatures that these processes can generate . in section [ analysis1 ] , we analyse the three lepton signature induced by the single chargino production . in section [ analysis2 ] , we study the like sign dilepton final state generated by the single neutralino and chargino productions . our framework throughout this paper will be the so - called minimal model ( msugra ) which assumes the existence of a grand unified gauge theory and family universal boundary conditions on the supersymmetry breaking parameters . we choose the 5 following parameters : @xmath24 the universal scalars mass at the unification scale @xmath25 , @xmath26 the universal gauginos mass at @xmath25 , @xmath27 the trilinear yukawa coupling at @xmath25 , @xmath28 the sign of the @xmath29 parameter ( @xmath30 , @xmath31 denoting the running scale ) and @xmath32 where @xmath33 and @xmath34 denote the vacuum expectation values of the higgs fields . in this model , the higgsino mixing parameter @xmath35 is determined by the radiative electroweak symmetry breaking condition . note also that the parameters @xmath26 and @xmath36 ( @xmath37 wino mass ) are related by the solution of the one loop renormalization group equations @xmath38 with @xmath39 , where @xmath40 are the beta functions , @xmath41 is the at @xmath25 and @xmath42 $ ] , @xmath43 $ ] corresponding to the gauge group factors @xmath44 . we shall set the unification scale at @xmath45 and the running scale at the @xmath46-boson mass : @xmath47 . we also assume the infrared fixed point hypothesis for the top quark yukawa coupling @xcite that provides a natural explanation of a large top quark mass @xmath48 . in the infrared fixed point approach , @xmath18 is fixed up to the ambiguity associated with large or low @xmath18 solutions . the low solution of @xmath18 is fixed by the equation @xmath49 , where @xmath50 for @xmath51 . for instance , with a top quark mass of @xmath52 @xcite , the low solution is given by @xmath53 . the second important effect of the infrared fixed point hypothesis is that the dependence of the electroweak symmetry breaking constraint on the @xmath54 parameter becomes weak so that @xmath35 is a known function of the @xmath24 , @xmath26 and @xmath18 parameters @xcite . finally , we consider the extension of the msugra model characterised by a single dominant coupling constant of type @xmath2 . at hadronic colliders , either a sneutrino ( @xmath55 ) or a charged slepton ( @xmath56 ) can be produced at the resonance via the @xmath2 coupling . as explained in section [ intro ] , for most of the susy models , the slepton produced at the resonance has two possible gauge decays , namely a decay into either a chargino or a neutralino . therefore , in the scenario of a single dominant @xmath2 coupling and for most of the susy models , either a chargino or a neutralino is singly produced together with either a charged lepton or a neutrino , through the resonant superpartner production at hadronic colliders . there are thus four main possible types of single superpartner production reaction involving @xmath2 at hadronic colliders which receive a contribution from resonant susy particle production . the diagrams associated to these four reactions are drawn in fig.[graphes ] . as can be seen in this figure , these single superpartner productions receive also some contributions from both the @xmath57 and @xmath58 channels . note that all the single superpartner production processes drawn in fig.[graphes ] have charge conjugated processes . we have calculated the amplitudes of the processes shown in fig.[graphes ] and the results are given in appendix [ formulas ] . in this section , we discuss the dependence of the single gaugino production cross sections on the various parameters . we will not assume here the radiative electroweak symmetry breaking condition in order to study the variations of the cross sections with the higgsino mixing parameter @xmath59 . first , we study the cross section of the single chargino production @xmath60 which occurs through the @xmath2 coupling ( see fig.[graphes](a ) ) . the differences between the @xmath61 , @xmath62 and @xmath63 production ( occuring respectively through the @xmath64 , @xmath65 and @xmath66 couplings with identical @xmath67 and @xmath68 indices ) cross sections involve @xmath69 lepton mass terms ( see appendix [ formulas ] ) and are thus negligible . the @xmath60 reaction receives contributions from the @xmath70 channel sneutrino exchange and the @xmath57 and @xmath58 channels squark exchanges as shown in fig.[graphes ] . however , the @xmath57 and @xmath58 channels represent small contributions to the whole single chargino production cross section when the sneutrino exchanged in the @xmath70 channel is real , namely for @xmath71 . the @xmath57 and @xmath58 channels cross sections will be relevant only when the produced sneutrino is virtual since the @xmath70 channel contribution is small . in this situation the single chargino production rate is greatly reduced compared to the case where the exchanged sneutrino is produced as a resonance . hence , the @xmath57 and @xmath58 channels do not represent important contributions to the @xmath72 production rate . the dependence of the @xmath72 production rate on the @xmath54 coupling is weak . indeed , the rate depends on the @xmath54 parameter only through the masses of the third generation squarks eventually exchanged in the @xmath57 and @xmath58 channels ( see fig.[graphes ] ) . similarly , the dependences on the @xmath54 coupling of the rates of the other single gaugino productions shown in fig.[graphes ] are weak . therefore , in this article we present the results for @xmath73 . later , we will discuss the effects of large @xmath54 couplings on the cascade decays which are similar to the effects of large @xmath18 values . * @xmath18 dependence : * the dependence of the @xmath72 production rate on @xmath18 is also weak , except for @xmath74 . this can be seen in fig.[xstan ] where the cross section of the @xmath75 reaction occuring through the @xmath76 coupling is shown as a function of the @xmath18 parameter . the choice of the @xmath76 coupling is motivated by the fact that the analysis in sections [ analysis1 ] and [ analysis2 ] are explicitly made for this coupling . in fig.[xstan ] , we have taken the @xmath76 value equal to its low - energy experimental bound for @xmath77 which is @xmath78 @xcite . + at this stage , some remarks on the values of the cross sections presented in this section must be done . first , the single gaugino production rates must be multiplied by a factor 2 in order to take into account the charge conjugated process , which is for example in the present case @xmath79 . furthermore , the values of the cross sections for all the single gaugino productions are obtained using the cteq4l structure function @xcite . choosing other parametrizations does not change significantly the results since proton structure functions in our kinematical domain in bjorken @xmath80 are known and have been already measured . for instance , with the set of parameters @xmath81 , @xmath82 , @xmath83 , @xmath84 and @xmath85 , the @xmath86 production cross section is @xmath87 for the cteq4l structure function @xcite , @xmath87 for the bep structure function @xcite , @xmath88 for the mrs ( r2 ) structure function @xcite and @xmath89 for the grv lo structure function @xcite . * @xmath59 dependence : * in fig.[xsmu ] , we present the cross sections of the @xmath90 and @xmath91 productions as a function of the @xmath59 parameter . we observe in this figure the weak dependence of the cross section @xmath92 on @xmath59 for @xmath93 . the reason is the smooth dependence of the @xmath94 mass on @xmath59 in this domain . however , the rate strongly decreases in the region @xmath95 in which the @xmath94 chargino is mainly composed by the higgsino . nevertheless , the small @xmath96 domain ( @xmath35 smaller than @xmath97 for @xmath98 , @xmath99 , @xmath100 and @xmath101 ) is excluded by the present experimental limits derived from the lep data @xcite . + in contrast , the cross section @xmath102 increases in the domain @xmath103 . the explanation is that the @xmath104 mass is enhanced as @xmath35 increases . the region in which @xmath102 becomes important is at small values of @xmath35 , near the lep limits of @xcite . we also remark in fig.[xsmu ] that the single @xmath105 production rate values remain above the single @xmath106 production rate values in all the considered range of @xmath59 . in this figure , we also notice that the cross section is smaller when @xmath59 is negative . to be conservative , we will take @xmath107 in the following . * @xmath24 and @xmath108 dependences : * in fact , the cross section @xmath109 depends mainly on the @xmath24 and @xmath108 parameters . we present in fig.[xs02 ] the rate of the @xmath86 production as a function of the @xmath24 and @xmath108 parameters . the rate decreases at high values of @xmath24 since the sneutrino becomes heavier as @xmath24 increases and more energetic initial partons are required in order to produce the resonant sneutrino . the decrease of the rate at large values of @xmath108 is due to the increase of the chargino mass and thus the reduction of the phase space factor . in fig.[xscl ] , we show the variations of the @xmath110 cross sections with @xmath24 for fixed values of @xmath108 , @xmath59 and @xmath18 . the cross sections corresponding to the @xmath86 production through various couplings of type @xmath65 are presented . in this figure , we only consider the couplings giving the highest cross sections . the values of the considered @xmath65 couplings have been taken at their low - energy limit @xcite for a squark mass of @xmath111 . the rate of the @xmath112 production through @xmath76 is also shown in this figure . we already notice that the cross section is significant for many couplings and we will come back on this important statement in the following . + the @xmath113 rates decrease as @xmath24 increases for the same reason as in fig.[xs02 ] . a decrease of the rates also occurs at small values of @xmath24 . the reason is the following . when @xmath24 decreases , the @xmath55 mass is getting closer to the @xmath22 masses so that the phase space factor associated to the decay @xmath114 decreases . + we also observe that the single @xmath106 production rate is much smaller than the single @xmath105 production rate , as in fig.[xsmu ] . + the differences between the @xmath86 production rates occuring via the various @xmath65 couplings are explained by the different parton densities . indeed , as shown in fig.[graphes ] the hard process associated to the @xmath86 production occuring through the @xmath65 coupling constant has a partonic initial state @xmath115 . the @xmath86 production via the @xmath76 coupling has first generation quarks in the initial state which provide the maximum parton density . we now discuss the rate behaviours for the reactions @xmath116 , @xmath117 and @xmath118 which occur via @xmath76 , in the susy parameter space . the dependences of these rates on the @xmath54 , @xmath18 , @xmath59 and @xmath108 parameters are typically the same as for the @xmath62 production rate . the variations of the @xmath119 , @xmath120 and @xmath121 productions cross sections with the @xmath24 parameter are shown in fig.[allxs ] . the @xmath122 , @xmath123 and @xmath124 production rates are comparatively negligible and thus have not been represented . we observe in this figure that the cross sections decrease at large @xmath24 values like the @xmath62 production rate . however , while the single @xmath4 productions rates decrease at small @xmath24 values ( see fig.[xscl ] and fig.[allxs ] ) , this is not true for the single @xmath3 productions ( see fig.[allxs ] ) . the reason is that in msugra the @xmath3 and @xmath125 ( @xmath126 ) masses are never close enough to induce a significant decrease of the cross section associated to the reaction @xmath127 , where @xmath126 ( see fig.[graphes](c)(d ) ) , caused by a phase space factor reduction . therefore , the resonant slepton contribution to the single @xmath3 production is not reduced at small @xmath24 values like the resonant slepton contribution to the single @xmath4 production . for the same reason , the single @xmath3 productions have much higher cross sections than the single @xmath4 productions in most of the msugra parameter space , as illustrate fig.[xscl ] and fig.[allxs ] . we note that in the particular case of a single dominant @xmath66 coupling constant and of large @xmath18 values , the rate of the reaction @xmath128 ( see fig.[graphes](d ) ) , where @xmath129 is the lightest tau - slepton , can be reduced at low @xmath24 values since then @xmath130 can be closed to @xmath131 due to the large mixing occuring in the staus sector . by analysing fig.[xscl ] and fig.[allxs ] , we also remark that the @xmath132 ( @xmath133 ) production rate is larger than the @xmath62 ( @xmath134 ) one . the explanation is that in @xmath135 collisions the initial states of the resonant charged slepton production @xmath136 have higher partonic densities than the initial states of the resonant sneutrino production @xmath137 . this phenomenon also increases the difference between the rates of the @xmath138 and @xmath86 productions at tevatron . + although the single @xmath4 production cross sections are smaller than the @xmath3 ones , it is interesting to study both of them since they have quite high values . at hadronic colliders , the single productions of susy particle via @xmath2 can occur through some @xmath0 processes which do not receive contributions from any resonant superpartner production . these non - resonant superpartner productions are ( one must also add the charge conjugated processes ) : * the gluino production @xmath139 via the exchange of a @xmath140 ( @xmath141 ) squark in the @xmath57 ( @xmath58 ) channel . * the squark production @xmath142 via the exchange of a @xmath141 squark ( @xmath143 quark ) in the @xmath57 ( @xmath70 ) channel . * the squark production @xmath144 via the exchange of a @xmath141 squark ( @xmath145 quark ) in the @xmath57 ( @xmath70 ) channel . * the squark production @xmath146 via the exchange of a @xmath147 squark ( @xmath148 quark ) in the @xmath57 ( @xmath70 ) channel . * the squark production @xmath149 via the exchange of a @xmath140 squark ( @xmath148 quark ) in the @xmath57 ( @xmath70 ) channel . * the sneutrino production @xmath150 via the exchange of a @xmath148 or @xmath143 quark ( @xmath151 sneutrino ) in the @xmath57 ( @xmath70 ) channel . * the charged slepton production @xmath152 via the exchange of a @xmath148 or @xmath145 quark ( @xmath125 slepton ) in the @xmath57 ( @xmath70 ) channel . * the sneutrino production @xmath153 via the exchange of a @xmath143 quark ( @xmath125 sneutrino ) in the @xmath57 ( @xmath70 ) channel . * the charged slepton production @xmath154 via the exchange of a @xmath145 quark ( @xmath151 sneutrino ) in the @xmath57 ( @xmath70 ) channel . the single gluino production can not reach high cross sections due to the strong experimental limits on the squarks and gluinos masses which are typically about @xmath155 @xcite . indeed , the single gluino production occurs through the exchange of squarks in the @xmath57 and @xmath58 channels , as described above , so that the cross section of this production decreases as the squarks and gluinos masses increase . for the value @xmath156 which is close to the experimental limits , we find the single gluino production rate @xmath157 which is consistent with the results of @xcite . the cross sections given in this section are computed at a center of mass energy of @xmath158 using the version 33.18 of the comphep routine @xcite with the cteq4 m structure function and an coupling @xmath81 . similarly , the single squark production cross section can not be large : for @xmath159 , the rate @xmath160 is of order @xmath161 . the production of a slepton together with a massive gauge boson has a small phase space factor and does not involve strong interaction couplings . the cross section of this type of reaction is thus small . for instance , with a slepton mass of @xmath162 we find the cross section @xmath163 to be of order @xmath164 . as a conclusion , the non - resonant single superpartner productions have small rates and will not be considered here . nevertheless , some of these reactions are interesting as their cross section involves few susy parameters , namely only one scalar superpartner mass and one coupling constant . in this section , we study the three lepton signature at tevatron run ii generated by the single chargino production through @xmath2 , @xmath165 , followed by the cascade decay , @xmath166 , @xmath167 ( the indices @xmath8 correspond to the indices of @xmath2 ) . in fact , the whole final state is 3 charged leptons + 2 hard jets + missing energy ( @xmath168 ) . the two jets and the missing energy come respectively from the quarks and the neutrino produced in the cascade decay . in the msugra model , which predicts the @xmath3 as the lsp in most of the parameter space , the @xmath169 reaction is the only single gaugino production allowing the three lepton signature to be generated in a significant way . since the @xmath170 production rate is dominant compared to the @xmath171 production rate , as discussed in section [ cross1 ] , we only consider the contribution to the three lepton signature from the single lightest chargino production . for @xmath172 , the branching ratio @xmath173 is typically of order @xmath174 and is smaller than for the other possible decay @xmath175 because of the color factor . + since in our framework the @xmath3 is the lsp , it can only decay via @xmath2 , either as @xmath176 or as @xmath177 , with a branching ratio @xmath178 ranging between @xmath179 and @xmath180 . the three lepton signature is particularly attractive at hadronic colliders because of the possibility to reduce the associated background . in section [ back1 ] we describe this background and in section [ cut1 ] we show how it can be reduced . the first source of background for the three leptons final state is the top quark pair production @xmath181 or @xmath182 . since the top quark life time is smaller than its hadronisation time , the top decays and its main channel is the decay into a @xmath183 gauge boson and a bottom quark as @xmath184 . the @xmath185 production can thus give rise to a @xmath186 final state if the @xmath183 bosons and one of the b - quarks undergo leptonic decays simultaneously . the cross section , calculated at leading order with pythia @xcite using the cteq2l structure function , times the branching fraction is @xmath187 ( @xmath188 ) with @xmath189 at @xmath190 for a top quark mass of @xmath191 ( @xmath192 ) . the other major source of background is the @xmath193 production followed by the leptonic decays of the gauge bosons , namely @xmath194 and @xmath195 . the value for the cross section times the branching ratios is @xmath196 ( @xmath189 ) at leading order with a center of mass energy of @xmath190 . + the @xmath193 production gives also a small contribution to the 3 leptons background through the decays : @xmath197 and @xmath198 , @xmath194 and @xmath198 or @xmath197 and @xmath199 , if a lepton is produced in each of the b jets . similarly , the @xmath200 production followed by the decays @xmath201 ( @xmath202 ) , @xmath203 , where one of the @xmath204 decays into lepton while the other decays into jet , leads to three leptons in the final state . within the same framework as above , the cross section is of order @xmath205 . + the @xmath200 production can also contribute weakly to the 3 leptons background via the decays : @xmath199 and @xmath198 or @xmath198 and @xmath198 , since a lepton can be produced in a b jet . it has been pointed out recently that the @xmath206 ( throughout this paper a star indicates a virtual particle ) and the @xmath207 productions could represent important contributions to the trilepton background @xcite . the complete list of contributions to the 3 leptons final state from the @xmath208,@xmath207 and @xmath209 productions , including cases where either one or both of the gauge bosons can be virtual , has been calculated in @xcite . the authors of @xcite have found that the @xmath208 , @xmath207 and @xmath209 backgrounds ( including virtual boson(s ) ) at the upgraded tevatron have together a cross section of order @xmath210 after the following cuts have been implemented : @xmath211 , @xmath212 , @xmath213 ; @xmath214 ; @xmath215 ; @xmath216 ; @xmath217 ; @xmath218 ; @xmath219 . + we note that there is at most one hard jet in the 3 leptons backgrounds generated by the @xmath208 , @xmath220 and @xmath209 productions ( including virtual boson(s ) ) . since the number of hard jets is equal to 2 in our signal ( see section [ signal1 ] ) , a jet veto can thus reduce this background with respect to the signal . other small sources of background have been estimated in @xcite : the productions like @xmath221 , @xmath222 or @xmath223 . after applying cuts on the geometrical acceptance , the transverse momentum and the isolation , these backgrounds are expected to be at most of order @xmath224 in @xmath135 collisions with a center of mass energy of @xmath225 . we have checked that the @xmath221 production gives a negligible contribution to the 3 lepton signature . there are finally some non - physics sources of background . first , the 4 leptons signal , which can be generated by the @xmath200 and @xmath185 productions , appears as a 3 leptons signature if one of the leptons is missed . besides , the processes @xmath226 would mimic a trilepton signal if @xmath227 fakes a lepton . monte carlo simulations using simplified detector simulation , like for example shw @xcite as in the present study ( see section [ cut1 ] ) , can not give a reliable estimate of this background . a knowledge of the details of the detector response as well as the jet fragmentation is necessary in order to determinate the probability to fake a lepton . in @xcite , using standard cuts the background coming from @xmath226 has been estimated to be of order @xmath228 at tevatron with @xmath225 . the authors of @xcite have also estimated the background from the three - jet events faking trilepton signals to be around @xmath229 . hence for the study of the background associated to the 3 lepton signature at tevatron run ii , we consider the @xmath193 production and both the physics and non - physics contributions generated by the @xmath200 and @xmath185 productions . if an excess of events is observed in the three lepton channel at tevatron , one would wonder what is the origin of those anomalous events . one would thus have to consider all of the productions leading to the three lepton signature . in the present context of r - parity violation , multileptonic final states can be generated by the single chargino production involving couplings , but also by the particle pair production which involves only gauge couplings @xcite . in models , the superpartner pair production can even lead to the trilepton signature @xcite . as a matter of fact , both of the produced particles decay , either directly or through cascade decays , into the lsp which is the neutralino in our framework . in the hypothesis of a dominant @xmath10 coupling constant , each of the 2 produced neutralinos can decay into a charged lepton and two quarks : at least two charged leptons and four jets in the final state are produced . the third charged lepton can be generated in the cascade decays as for example at the level of the chargino decay @xmath230 . .cross section ( in @xmath231 ) of the sum of all the superpartners pair productions at tevatron run ii as a function of the @xmath24 and @xmath26 parameters for @xmath83 , @xmath232 and @xmath233 at a center of mass energy of @xmath225 . these rates have been calculated with herwig @xcite using the ehlq2 structure function . [ cols="^,^,^,^,^,^",options="header " , ] in table [ cutsusp ] , we give the number of like sign dilepton events generated by the susy background ( all superpartners pair productions ) at tevatron run ii as a function of the @xmath24 and @xmath26 parameters for cut 3 . this number of events decreases as @xmath24 and @xmath26 increase due to the behaviour of the summed superpartners pair production cross section in the susy parameter space ( see section [ susyback1 ] ) . we first present the reach in the msugra parameter space obtained from the analysis of the like sign dilepton final state at tevatron run ii produced by the single neutralino and chargino productions via @xmath76 : @xmath234 , @xmath235 and @xmath236 . the sensitivities that can be obtained on the @xmath65 ( @xmath67 and @xmath68 being not equal to @xmath237 simultaneously ) , @xmath64 and @xmath66 coupling constants will be discussed at the end of this section . in fig.[fig2d ] , we present the @xmath238 and @xmath239 discovery contours and the limits at @xmath240 confidence level in the plane @xmath24 versus @xmath26 , for @xmath232 , @xmath83 , @xmath233 and using a set of values for the luminosity . those discovery potentials were obtained by considering the @xmath241 , @xmath242 and @xmath243 productions and the background originating from the standard model . the signal and background were selected by using cut @xmath244 described in section [ cut2 ] . the reduction of the sensitivity on @xmath26 observed in fig.[fig2d ] as @xmath24 increases is due to the decrease of the @xmath241 , @xmath242 and @xmath243 productions cross sections with the @xmath24 increase observed in fig.[xscl ] and fig.[allxs ] . in fig.[fig2d ] , we also see that the sensitivity on @xmath26 is reduced in the domain @xmath245 . this reduction of the sensitivity is due to the fact that in msugra at low @xmath18 and for large values of @xmath26 and small values of @xmath24 , the lsp is the right slepton @xmath246 ( @xmath247 ) . therefore , in this msugra region the dominant decay channel of the lightest neutralino is @xmath248 ( @xmath247 ) so that the @xmath249 production , which is the main contribution to the like sign dilepton signature , leads to the @xmath250 final state only in a few cases . there are two reasons . first , in this msugra scenario the charged lepton produced in the main @xmath3 decay is not systematically a muon . secondly , if the lsp is the right slepton @xmath246 it can not decay in the case of a single dominant @xmath2 coupling constant and it is thus a stable particle . the sensitivities presented in the discovery reach of fig.[fig2d ] which are obtained from the like sign dilepton signature analysis are higher than the sensitivities shown in fig.[fig2 ] which correspond to the trilepton final state analysis . this is due to the 3 following points . first , the rate of the @xmath249 production ( recall that it represents the main contribution to the like sign dilepton final state ) is larger than the @xmath251 cross section in most of the msugra parameter space ( see section [ cross1 ] ) . secondly , the @xmath3 decay leading to the like sign dilepton final state in the case of the @xmath249 production has a larger branching ratio than the cascade decay initiated by the @xmath4 which generates the trilepton final state ( see sections [ signal1 ] and [ signal2 ] ) . finally , at tevatron run ii the background of the like sign dilepton signature is weaker than the trilepton background ( see tables [ cutsusy ] and [ cutsusp ] ) . it is clear from fig.[fig2d ] that at low values of the @xmath24 and @xmath26 parameters , high sensitivities can be obtained on the @xmath76 coupling constant . we have found that for instance at the msugra point defined as @xmath252 , @xmath253 , @xmath232 and @xmath83 , @xmath76 values of @xmath254 can be probed through the like sign dilepton analysis at tevatron run ii assuming a luminosity of @xmath255 . this result was obtained by applying cut @xmath244 described in section [ cut2 ] on the susy signal ( @xmath256 , @xmath242 and @xmath243 productions ) and the standard model background . we expect that , as in the three lepton signature analysis , interesting sensitivities could be obtained on other @xmath65 coupling constants . + the sensitivities obtained on the @xmath66 couplings from the like sign dilepton signature analysis should be weaker than the sensitivities on the @xmath65 couplings deduced from the same study . indeed , in the case of a single dominant @xmath66 coupling the same sign leptons generated by the @xmath257 production would be 2 tau leptons ( see fig.[graphes](d ) and section [ signal2 ] ) . therefore , the like sign dileptons ( @xmath258 or @xmath259 ) produced by the signal would be mainly generated in tau decays and would thus have higher probabilities to not pass the analysis cuts on the particle energy . moreover , the requirement of @xmath258 or @xmath259 events would decrease the efficiency after cuts of the signal due to the hadronic decay of the tau . finally , the selection of two same flavour like sign dileptons ( @xmath258 or @xmath259 ) would reduce the signal , since each of the 2 produced taus could decay either into an electron or a muon , and hence would not be an effective cut anymore . + the sensitivities obtained on the @xmath64 couplings from the like sign dilepton signature study are expected to be identical to the sensitivities on the @xmath65 couplings obtained from the same study . indeed , in the case of a single dominant @xmath64 coupling constant , the only difference in the like sign dilepton signature analysis would be that @xmath258 events should be selected instead of @xmath260 events ( see fig.[graphes](d ) and section [ signal2 ] ) . nevertheless , a smaller number of @xmath64 couplings is expected to be probed since the low - energy constraints on the @xmath64 couplings are generally stronger than the limits on the @xmath65 couplings @xcite . in the high @xmath18 case , the lightest stau @xmath261 can become the lsp instead of the lightest neutralino , due to a large mixing in the third generation of charged sleptons . in such a situation , the dominant decay channel of the lightest neutralino is @xmath262 . two scenarios must then be discussed : if the single dominant coupling is not of the type @xmath66 , the @xmath129-lsp is a stable particle so that the reaction @xmath263 , representing the main contribution to the like sign dilepton final state , does not often lead to the @xmath250 signature . if the single dominant coupling is of the type @xmath66 , the @xmath257 production can receive a contribution from the resonant @xmath264 production ( see fig.[graphes](d ) ) and the @xmath129-lsp decays via @xmath66 as @xmath265 so that the @xmath250 signature can still be generated in a significant way by the @xmath266 reaction . we end this section by some comments on the effect of the @xmath267 conserving background to the like sign dilepton signature . in order to illustrate this discussion , we consider the results on the @xmath76 coupling constant . + we see from table [ cutsusp ] that the susy background to the like sign dilepton final state can affect the sensitivity on the @xmath76 coupling constant obtained by considering only the background , which is shown in fig.[fig2d ] , only in the region of small superpartners masses , namely in the domain @xmath268 for @xmath83 , @xmath232 and assuming a luminosity of @xmath269 . + in contrast with the susy signal amplitude which is increased if @xmath76 is enhanced , the susy background amplitude is typically independent on the value of the @xmath76 coupling constant since the superpartner pair production does not involve couplings . therefore , even if we consider the susy background in addition to the one , it is still true that large values of the @xmath76 coupling can be probed over a wider domain of the susy parameter space than low values , as can be observed in fig.[fig2d ] for @xmath270 . note that in fig.[fig2d ] larger values of @xmath76 still respecting the indirect limit could have been considered . + finally , we mention that further cuts , as for instance some cuts based on the superpartners mass reconstructions ( see section [ reconsp ] ) , could allow to reduce the susy background to the like sign dilepton signature . the @xmath3 and @xmath6 mass reconstructions can be performed in a model independent way via the like sign dilepton analysis . we have simulated these mass reconstructions based on the like sign dimuon events generated in the scenario of a single dominant @xmath65 coupling constant . in this scenario , the main susy contribution to the like sign dilepton signature , namely the @xmath271 production , has the final state @xmath272 ( see section [ signal2 ] ) . indeed , the produced @xmath3 decays into @xmath273 through @xmath65 . the muon generated together with the @xmath3 can be identified as the leading muon for relatively large @xmath274 mass differences ( see section [ cut2 ] ) . note that for nearly degenerate values of @xmath275 and @xmath131 the @xmath271 production rate and thus the sensitivity on the susy parameters would be reduced ( see section [ cross1 ] ) . the muon created in the @xmath3 decay can thus be identified as the softer muon so that the @xmath3 can be reconstructed from the the softer muon and the 2 jets present in the @xmath271 production final state . the other contributions to the like sign dimuons events can lead to some missing energy and at most 4 jets in the final state ( see section [ signal2 ] ) . hence , we have chosen to reconstruct the @xmath3 from the 2 leading jets when the final state contains more than 2 jets . once the @xmath3 has been reconstructed , the @xmath276 has been reconstructed from the @xmath3 and the leading muon since the dominant contribution to the @xmath249 production is the reaction @xmath277 . these mass reconstructions are represented in fig.[rec2s ] . in this figure , we also represent the same mass reconstructions obtained by applying a cut in the upper left plot of fig.[dienmu ] excluding the peak associated to the @xmath278 and @xmath279 productions ( see section [ cut2 ] ) . the interest of this cut , as can be seen in fig.[rec2s ] , is to select the @xmath271 production and thus to improve the accuracy on the @xmath3 and @xmath276 reconstructions which are based on this production . we observe in fig.[rec2s ] that the @xmath3 reconstruction has less combinatorial background than the @xmath276 reconstruction . this comes from the fact that the selection of the softer muon and the 2 leading jets allows to reconstruct the @xmath3 even in the dimuon events generated by the @xmath278 and @xmath279 productions , while the selection of the 2 muons and the 2 leading jets does not allow to reconstruct the @xmath276 in the dimuon events generated by the @xmath278 and @xmath279 productions ( see section [ signal2 ] ) . we have represented on the plots of fig.[rec2s ] the fits of the invariant mass distributions . we see from these fits that the distributions are well peaked around the @xmath3 and @xmath276 generated masses . the average reconstructed masses are @xmath280 and @xmath281 . we note that the accuracy on the @xmath3 ( and thus on the @xmath276 ) mass reconstruction could be improved if the distributions in the upper plots of fig.[rec2s ] were recalculated by selecting the muon giving the @xmath3 mass the closer to the mean value of the peak obtained in the relevant upper plot of fig.[rec2s ] . in the hypothesis of a single dominant coupling constant of type @xmath64 or @xmath66 , exactly the same kind of @xmath3 and @xmath276 mass reconstructions can be performed by selecting the @xmath282 or @xmath283 events , respectively . as a conclusion , the @xmath3 and @xmath276 mass reconstructions based on the like sign dilepton signature generated by the @xmath241 , @xmath242 and @xmath284 productions at tevatron can easily give precise results , in contrast with the mass reconstructions performed in the superpartner pair production analysis at hadronic colliders which suffer an high combinatorial background @xcite . in our theoretical framework ( see section [ theoretical ] ) , the values of the @xmath35 and @xmath18 ( up to the ambiguity of low / high solution ) parameters are predicted . this has no important effects on the results presented in sections [ lp211p ] as the single gaugino production cross sections vary weakly with these parameters ( see section [ cross1 ] ) . however , since we have worked within the msugra model , the @xmath285 mass was typically larger than the @xmath3 mass . in a situation where @xmath286 would approach @xmath131 , the rate of the @xmath287 production , representing in msugra the main contribution to the like sign dilepton signature ( see section [ signal2 ] ) , would decrease . therefore , within a model allowing degenerate @xmath285 and @xmath3 masses or even a @xmath285 lighter than the @xmath3 , other single gaugino productions than the @xmath263 reaction could represent the major contribution to the like sign dilepton signature in some parts of the susy parameter space . besides , in a situation where the lsp would not be the @xmath3 , the branching ratios of the @xmath3 decays violating @xmath267 would be reduced with respect to the case where the lsp is the @xmath3 , as often occurs in msugra . however , in such a situation , the like sign dilepton signature could receive a significant contribution from a decay of the @xmath3 different from the channel . in those kinds of scenarios where the lsp is not the @xmath3 , the @xmath287 production would not represent systematically the main contribution to the like sign dilepton signature . in the several scenarios described above where the @xmath287 production is not the major contribution to the like sign dilepton signature , this signature could receive quite important contribution from the other single gaugino productions described in section [ resonant ] . the single gaugino productions at tevatron reach important cross sections thanks to the contributions of the resonant slepton productions . hence , the analysis of the 3 charged leptons and like sign dilepton signatures generated by the single gaugino productions at tevatron run ii would allow to obtain high sensitivities on many coupling constants , compared to the low - energy limits , in wide domains of the susy parameter space . this is also due to the fact that the standard model backgrounds associated to the 3 charged leptons and like sign dilepton final states at tevatron can be greatly suppressed . from the supersymmetry discovery point of view , superpartner masses well beyond the present experimental limits could be tested through the analysis of the the 3 charged leptons and like sign dilepton signatures generated by the single gaugino productions at tevatron run ii . if some of the coupling constants values were close to their low - energy bounds , the single gaugino productions study based on the 3 charged leptons and like sign dilepton signatures would even allow to extend the region in the @xmath24-@xmath26 plane probed by the superpartner pair production analyses in the 3 charged leptons and like sign dilepton channels at tevatron run ii . the reason is that the single superpartner production has a larger phase space factor than the superpartner pair production . + besides , the 3 charged leptons and like sign dilepton signatures generated by the single gaugino productions at tevatron run ii would allow to reconstruct in a model independent way the @xmath3 , @xmath4 , @xmath5 and @xmath6 masses with a smaller combinatorial background than in the superpartner pair production analysis . we end this summary by a comparison between the results obtained from the studies of the 3 charged lepton and like sign dilepton signatures generated by the single gaugino productions at tevatron run ii . in the msugra model , the like sign dilepton signature analysis would give rise to higher sensitivities on the susy parameters than the study of the 3 charged lepton final state . this comes notably from the fact that in msugra , the @xmath3 is lighter than the @xmath288 so that the cross section of the @xmath289 production , which is the main contribution to the like sign dilepton signature , reaches larger values than the cross section of the @xmath290 production , representing the main contribution to the 3 charged lepton final state . other interesting prospective studies concerning hadronic colliders are the analyses of the single gaugino productions occuring through resonant squark productions via @xmath291 coupling constants which we will perform in the next future . we would like to thank emmanuelle perez , robi peschanski and auguste besson for fruitful discussions and reading the manuscript . in this appendix , we give the amplitudes for all the single productions of particle at hadronic colliders , which can receive a contribution from a slepton resonant production . these single productions occur via the coupling @xmath2 and correspond to the four reactions , @xmath292 , @xmath293 , @xmath294 , @xmath295 . each of those four processes receives contributions from both the t and u channel ( see fig.[graphes ] ) and have charge conjugated diagrams . note also that the contributions coming from the exchange of a right squark in the u channel involve the higgsino components of the gauginos . these contributions , in the case of the single chargino production , do not interfere with the s channel slepton exchange since the initial or final states are different ( see fig.[graphes ] ) . in the following , we give the formulas for the probability amplitudes , squared and summed over the polarizations . our notations closely follow the notations of @xcite . in particular , the matrix elements @xmath296 are defined in the basis of the photino and the zino , as in @xcite . @xmath297 \cr \vert m_u ( u^k \bar d^j \to \tilde \chi_a^+ \nu_i ) \vert ^2 & = & { { \l'_{ijk}}^2 g^2 m_{d^k}^2 \vert u_{a2 } \vert^2 \over 24 m_w^2 \cos \beta ^2 ( u - m_{\tilde d^k_r}^2)^2 } ( m_{\tilde \chi_a^+}^2+m_{u^k}^2-u ) ( m_{d^j}^2-u ) \cr 2 re [ m_s m_t^ * ( \tilde \chi_a^+ \bar \nu_i ) ] & = & { { \l'_{ijk}}^2 g^2 \over 6 ( s - m_{\tilde l^i_l}^2 ) ( t - m_{\tilde d^j_l}^2 ) } \bigg [ { \vert u_{a1 } \vert ^2 \over 2 } [ ( m_{u^j}^2+m_{\tilde \chi_a^+}^2-t ) ( m_{d^k}^2-t ) \cr & + & ( m_{u^j}^2+m_{d^k}^2-s ) ( m_{\tilde \chi_a^+}^2-s ) -(m_{u^j}^2-u ) ( m_{\tilde \chi_a^+}^2+m_{d^k}^2-u ) ] \cr & - & ( m_{d^k}^2-t ) { re(u_{a1}v_{a2 } ) m_{\tilde \chi_a^+ } m_{u^j}^2 \over \sqrt{2 } m_w \sin \beta } \bigg ] , \label{fchanu}\end{aligned}\ ] ] @xmath301 \cr \vert m_u ( d_j \bar d_k \to \tilde \chi_a^0 \bar \nu_i ) \vert ^2 & = & { { \l'_{ijk}}^2 \over 6 ( u - m_{\tilde d^k_r}^2)^2 } ( m_{d^j}^2-u ) \bigg [ ( m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) \bigg ( { g^2 m_{d^k}^2 \vert n'_{a3 } \vert ^2 \over 4 m_w^2 \cos^2 \beta } + { e^2 \vert n'_{a1 } \vert ^2 \over 9 } \cr & + & { g^2 \sin^4 \theta_w \vert n'_{a2 } \vert ^2 \over 9 \cos^2 \theta_w } - { 2 e g re(n'_{a1}n'_{a2 } ) \sin^2 \theta_w \over 9 \cos \theta_w } \bigg ) \cr & - & { 2 m_{\tilde \chi_a^0 } m_{d^k}^2 g \over m_w \cos \beta } \bigg ( -{e re(n'_{a1}n'_{a3 } ) \over 3 } + { g \sin^2 \theta_w re(n'_{a2}n'_{a3 } ) \over3 \cos \theta_w } \bigg ) \bigg ] \cr 2 re [ m_s m_t^ * ( \tilde \chi_a^0 \bar \nu_i ) ] & = & - { { \l'_{ijk}}^2 g \over 12 \cos \theta_w ( s - m_{\tilde \nu^i_l}^2 ) ( t - m_{\tilde d^j_l}^2 ) } \bigg [ ( m_{d^k}^2-t ) { m_{\tilde \chi_a^0 } m_{d^j}^2 g re(n'_{a2}n'_{a3 } ) \over m_w \cos \beta } \cr & + & \bigg ( -{e re(n'_{a1}n^*_{a2 } ) \over 3 } + { g \vert n'_{a2 } \vert ^2 \over \cos \theta_w } ( { \sin^2 \theta_w \over 3 } -{1 \over 2 } ) \bigg ) [ ( m_{d^j}^2+m_{\tilde \chi_a^0}^2-t ) ( m_{d^k}^2-t ) \cr & + & ( m_{d^j}^2+m_{d^k}^2-s ) ( m_{\tilde \chi_a^0}^2-s ) -(m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) ( m_{d^j}^2-u ) ] \bigg ] \cr 2 re [ m_t m_u^ * ( \tilde \chi_a^0 \bar \nu_i ) ] & = & { { \l'_{ijk}}^2 \over 6 ( u - m_{\tilde d^k_r}^2 ) ( t - m_{\tilde d^j_l}^2 ) } \bigg [ ( m_{d^k}^2-t ) { g m_{\tilde \chi_a^0 } m_{d^j}^2 \over m_w \cos \beta } \bigg ( { g \sin^2 \theta_w re(n'_{a2}n'_{a3 } ) \over 3 \cos \theta_w}-{e re(n'_{a1}n'_{a3 } ) \over 3 } \bigg ) \cr & + & [ ( m_{d^j}^2-u ) ( m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) + ( m_{d^k}^2-t ) ( m_{d^j}^2+m_{\tilde \chi_a^0}^2-t ) -(m_{\tilde \chi_a^0}^2-s ) ( m_{d^j}^2+m_{d^k}^2-s ) ] \cr & & \bigg ( -{egre(n'_{a1}n'_{a2 } ) \over 3 \cos \theta_w } ( { 2 \sin^2 \theta_w \over 3 } -{1 \over 2 } ) + { e^2 \vert n'_{a1 } \vert ^2 \over 9 } + { g^2 \sin^2 \theta_w \vert n'_{a2 } \vert ^2 \over 3 \cos^2 \theta_w } ( { \sin^2 \theta_w \over 3}-{1 \over 2 } ) \bigg ) \cr & - & { m_{\tilde \chi_a^0 } m_{d^k}^2 g \over m_w \cos \beta } \bigg ( -{e re(n'_{a1}n'_{a3 } ) \over 3 } + { g re(n'_{a2}n'_{a3 } ) \over \cos \theta_w } ( { \sin^2 \theta_w \over 3}-{1 \over 2 } ) \bigg ) ( m_{d^j}^2-u ) \cr & + & { m_{d^j}^2 m_{d^k}^2 g^2 \vert n'_{a3 } \vert ^2 \over 2 m_w^2 \cos^2 \beta}(m_{\tilde \chi_a^0}^2-s ) \bigg ] \cr 2 re [ m_s m_u^ * ( \tilde \chi_a^0 \bar \nu_i ) ] & = & { { \l'_{ijk}}^2 g \over 12 \cos \theta_w ( s - m_{\tilde \nu^i_l}^2 ) ( u - m_{\tilde d^k_r}^2 ) } \bigg [ - { m_{\tilde \chi_a^0 } m_{d^k}^2 g re(n'_{a2}n'_{a3 } ) \over m_w \cos \beta } ( m_{d^j}^2-u ) \cr & + & \bigg ( -{e re(n^*_{a1}n'_{a2 } ) \over 3 } + { \vert n'_{a2 } \vert ^2 g \sin^2 \theta_w \over 3 \cos \theta_w } \bigg ) [ ( m_{d^j}^2+m_{d^k}^2-s ) ( m_{\tilde \chi_a^0}^2-s ) \cr & + & ( m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) ( m_{d^j}^2-u ) -(m_{d^j}^2+m_{\tilde \chi_a^0}^2-t ) ( m_{d^k}^2-t ) ] \bigg ] , \label{fnenu}\end{aligned}\ ] ] @xmath305 \cr \vert m_t ( u_j \bar d_k \to \tilde \chi_a^0 \bar l_i ) \vert ^2 & = & { { \l'_{ijk}}^2 \over 6 ( t - m_{\tilde u^j_l}^2)^2 } ( -t+m_{l^i}^2+m_{d^k}^2 ) \bigg [ \bigg ( { g^2 m_{u^j}^2 \vert n'_{a4 } \vert ^2 \over 4 m_w^2 \sin^2 \beta } + { 4 e^2 \vert n'_{a1 } \vert ^2 \over 9 } \cr & + & { g^2 \vert n'_{a2 } \vert ^2 \over \cos^2 \theta_w } ( { 1 \over 2}-{2 \sin^2 \theta_w \over 3 } ) ^2 + { 4 e g re(n'_{a1}n'_{a2 } ) \over 3 \cos \theta_w } ( { 1 \over 2}-{2 \sin^2 \theta_w \over 3 } ) \bigg ) ( -t+m_{u^j}^2+m_{\tilde \chi_a^0}^2 ) \cr & + & { 2 g m_{u^j}^2 m_{\tilde \chi_a^0 } \over m_w \sin \beta } \bigg ( { 2 e re(n'_{a1}n'_{a4 } ) \over 3 } + { g re(n'_{a2}n'_{a4 } ) \over \cos \theta_w } ( { 1 \over 2}-{2 \sin^2 \theta_w \over 3 } ) \bigg ) \bigg ] \cr \vert m_u ( u_j \bar d_k \to \tilde \chi_a^0 \bar l_i ) \vert ^2 & = & { { \l'_{ijk}}^2 \over 6 ( u - m_{\tilde d^k_r}^2)^2 } ( m_{u^j}^2+m_{l^i}^2-u ) \bigg [ \bigg ( { e^2 \vert n'_{a1 } \vert ^2 \over 9 } + { g^2 \sin^4 \theta_w \vert n'_{a2 } \vert ^2 \over 9 \cos^2 \theta_w } -{2 e g re(n'_{a1}n'_{a2 } ) \sin^2 \theta_w \over 9 \cos \theta_w } \cr & + & { g^2 m_{d^k}^2 \vert n'_{a3 } \vert ^2 \over 4 m_w^2 \cos^2 \beta } \bigg ) ( m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) - { 2 g m_{d^k}^2 m_{\tilde \chi_a^0 } \over m_w \cos \beta } \bigg ( -{e re(n'_{a1}n'_{a3 } ) \over 3 } \cr & + & { g \sin^2 \theta_w re(n'_{a2}n'_{a3 } ) \over 3 \cos \theta_w } \bigg ) \bigg ] \cr 2 re [ m_s m_t^ * ( \tilde \chi_a^0 \bar l_i ) ] & = & - { { \l'_{ijk}}^2 \over 6 ( s - m_{\tilde l^i_l}^2 ) ( t - m_{\tilde u^j_l}^2 ) } \bigg [ -{m_{l^i}^2 m_{u^j}^2 g^2 re(n'_{a3}n^*_{a4 } ) \over 2 m_w^2 \sin \beta \cos \beta } ( m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) \cr & + & \bigg ( { -2 e^2 \vert n'_{a1 } \vert ^2 \over 3 } + { e g re(n^*_{a1}n'_{a2 } ) \over 3 \cos \theta_w } ( 4 \sin^2 \theta_w-{5 \over 2 } ) \cr & + & { g^2 \vert n'_{a2 } \vert ^2 \over \cos^2 \theta_w } ( { 1 \over 2}-{2 \sin^2 \theta_w \over 3 } ) ( \sin^2 \theta_w-{1 \over 2 } ) \bigg ) \cr & & [ ( m_{u^j}^2+m_{d^k}^2-s ) ( m_{\tilde \chi_a^0}^2+m_{l^i}^2-s ) + ( m_{u^j}^2+m_{\tilde \chi_a^0}^2-t ) ( m_{l^i}^2+m_{d^k}^2-t ) \cr & - & ( m_{u^j}^2+m_{l^i}^2-u ) ( m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) ] + { g m_{u^j}^2 m_{\tilde \chi_a^0 } \over m_w \sin \beta } \bigg ( -e re(n'_{a1}n'_{a4})+{g re(n'_{a2}n'_{a4 } ) \over \cos \theta_w } \cr & & ( \sin^2 \theta_w-{1 \over 2 } ) \bigg ) ( m_{l^i}^2+m_{d^k}^2-t ) -(s - m_{u^j}^2-m_{d^k}^2 ) { g m_{l^i}^2 m_{\tilde \chi_a^0 } \over m_w \cos \beta } \bigg ( { 2 e re(n'_{a1}n'_{a3 } ) \over 3 } \cr & + & { g re(n'_{a2}n'_{a3 } ) \over \cos \theta_w } ( { 1 \over 2}-{2 \sin^2 \theta_w \over 3 } ) \bigg ) \bigg ] \cr 2 re [ m_t m_u^ * ( \tilde \chi_a^0 \bar l_i ) ] & = & { { \l'_{ijk}}^2 \over 6 ( u - m_{\tilde d^k_r}^2 ) ( t - m_{\tilde u^j_l}^2 ) } \bigg [ { g m_{u^j}^2 m_{\tilde \chi_a^0 } \over m_w \sin \beta } ( m_{l^i}^2+m_{d^k}^2-t ) \bigg ( -{e re(n'_{a1}n'_{a4 } ) \over 3 } \cr & + & { g \sin^2 \theta_w re(n'_{a2}n'_{a4 } ) \over 3 \cos \theta_w } \bigg ) - { m_{\tilde \chi_a^0 } g m_{d^k}^2 \over m_w \cos \beta } \bigg ( { 2 e re(n'_{a1}n'_{a3 } ) \over 3 } + { g re(n'_{a2}n'_{a3 } ) \over \cos \theta_w } ( { 1 \over 2}-{2 \sin^2 \theta_w \over 3 } ) \bigg ) \cr & & ( m_{l^i}^2+m_{u^j}^2-u ) - { g^2 re(n'_{a3}n^*_{a4 } ) m_{u^j}^2 m_{d^k}^2 \over 2 m_w^2 \cos \beta \sin \beta } ( s - m_{l^i}^2-m_{\tilde \chi_a^0}^2 ) + \bigg ( -{2 e^2 \vert n'_{a1 } \vert ^2 \over 9 } \cr & + & { e g re(n^*_{a1}n'_{a2 } ) \over 3 \cos \theta_w } ( -{1 \over 2}+ { 4 \sin^2 \theta_w \over 3 } ) + { g^2 \sin^2 \theta_w \vert n'_{a2 } \vert ^2 \over 3 \cos^2 \theta_w } \cr & & ( { 1 \over 2}-{2 \sin^2 \theta_w \over 3 } ) \bigg ) [ ( m_{l^i}^2+m_{u^j}^2-u ) ( m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) + ( m_{l^i}^2+m_{d^k}^2-t ) ( m_{\tilde \chi_a^0}^2+m_{u^j}^2-t ) \cr & - & ( m_{l^i}^2+m_{\tilde \chi_a^0}^2-s ) ( m_{d^k}^2+m_{u^j}^2-s ) ] \bigg ] \cr 2 re [ m_s m_u^ * ( \tilde \chi_a^0 \bar l_i ) ] & = & { { \l'_{ijk}}^2 \over 6 ( s - m_{\tilde l^i_l}^2 ) ( u - m_{\tilde d^k_r}^2 ) } \bigg [ -{g m_{l^i}^2 m_{\tilde \chi_a^0 } \over m_w \cos \beta } \bigg ( -{e re(n'_{a1}n'_{a3 } ) \over 3 } + { g \sin^2 \theta_w re(n'_{a2}n'_{a3 } ) \over 3 \cos \theta_w } \bigg ) \cr & & ( s - m_{d^k}^2-m_{u^j}^2 ) - { g m_{d^k}^2 m_{\tilde \chi_a^0 } \over m_w \cos \beta } \bigg ( -e re(n'_{a1}n'_{a3})+{g re(n'_{a2}n'_{a3 } ) \over \cos \theta_w } ( \sin \theta_w^2-{1 \over 2 } ) \bigg ) \cr & & ( m_{l^i}^2+m_{u^j}^2-u ) + { g^2 m_{l^i}^2 m_{d^k}^2 \vert n'_{a3 } \vert ^2 \over 2 m_w^2 \cos^2 \beta } ( m_{\tilde \chi_a^0}^2+m_{u^j}^2-t ) + \bigg ( { e^2 \vert n'_{a1 } \vert ^2 \over 3 } \cr & - & { e g re(n^*_{a1}n'_{a2 } ) \over 3 \cos \theta_w } ( 2 \sin \theta_w^2-{1 \over 2 } ) + { g^2 \vert n'_{a2 } \vert ^2 \sin^2 \theta_w \over 3 \cos^2 \theta_w } ( \sin^2 \theta_w-{1 \over 2 } ) \bigg ) \cr & & [ ( m_{l^i}^2+m_{u^j}^2-u ) ( m_{\tilde \chi_a^0}^2+m_{d^k}^2-u ) -(m_{l^i}^2+m_{d^k}^2-t ) ( m_{\tilde \chi_a^0}^2+m_{u^j}^2-t ) \cr & + & ( m_{l^i}^2+m_{\tilde \chi_a^0}^2-s ) ( m_{d^k}^2+m_{u^j}^2-s ) ] , \bigg ] \label{fnel}\end{aligned}\ ] ] @xmath308 \cr \vert m_t ( d_j \bar d_k \to \tilde \chi_a^- \bar l_i ) \vert ^2 & = & { g^2 { \l'_{ijk}}^2 \over 3 ( t - m_{\tilde u^j_l}^2)^2 } ( t - m_{d^k}^2-m_{l^i}^2 ) \bigg [ ( t - m_{\tilde \chi_a^+}^2-m_{d^j}^2 ) ( { \vert v_{a1 } \vert ^2 \over 4 } + { m_{d^j}^2 \vert u_{a2 } \vert^2 \over 8 m^2_w \cos^2 \beta } ) \cr & + & { re(v_{a1}u_{a2 } ) m_{\tilde \chi_a^+ } m^2_{d^j } \over \sqrt{2 } m_w \cos \beta } \bigg ] \cr \vert m_u ( \bar u_k u_j \to \tilde \chi_a^- \bar l_i ) \vert ^2 & = & { g^2 { \l'_{ijk}}^2 \over 24 ( u - m_{\tilde d^k_r}^2)^2 } ( m_{\tilde \chi_a^+}^2+m_{u^k}^2-u ) ( m_{l^i}^2+m_{u^j}^2-u ) { \vert u_{a2 } \vert^2 m_{d^k}^2 \over m_w^2 \cos^2 \beta } \cr 2 re [ m_s m_t^ * ( \tilde \chi_a^- \bar l_i ) ] & = & { g^2 { \l'_{ijk}}^2 \over 12 ( s - m_{\tilde \nu^i_l}^2 ) ( t - m_{\tilde u^j_l}^2 ) } \bigg [ \vert v_{a1 } \vert ^2 [ -(m_{l^i}^2+m_{d^j}^2-u ) ( m_{\tilde \chi_a^+}^2+m_{d^k}^2-u ) \cr & + & ( m_{l^i}^2+m_{d^k}^2-t ) ( m_{\tilde \chi_a^+}^2+m_{d^j}^2-t ) + ( m_{l^i}^2+m_{\tilde \chi_a^+}^2-s ) ( m_{d^k}^2+m_{d^j}^2-s ) ] \cr & + & { re(v_{a1}u_{a2 } ) m_{\tilde \chi_a^+ } \sqrt{2 } \over m_w \cos \beta } [ m^2_{l^i } ( s - m_{d^j}^2 - m_{d^k}^2 ) - m^2_{d^j } ( m_{l^i}^2+m_{d^k}^2-t ) ] \cr & - & { \vert u_{a2 } \vert^2 m_{l^i}^2 m^2_{d^j } \over m_w^2 \cos^2 \beta } ( m_{\tilde \chi_a^+}^2+m_{d^k}^2-u ) \bigg ] , \label{fchal}\end{aligned}\ ] ] s. dimopoulos and l.j . hall , phys . * b 207 * , 210 ( 1988 ) . h. dreiner and g. g. ross , nucl . b 365 * , 597 ( 1991 ) . h. dreiner , published in _ perspectives on supersymmetry _ , ed . kane , world scientific ( 1998 ) , hep - ph/9707435 . g. bhattacharyya , invited talk presented at ` beyond the desert ' , castle ringberg , tegernsee , germany , 8 - 14 june 1997 ; susy 96 , nucl . b ( proc . suppl . ) * 52a * 83 ( 1997 ) . r. barbier et al . , report of the group on the r - 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following classical protocols , quantum information protocols were initially formulated in terms of qubits that are manipulated unitarily in order to realize computational and communication tasks that may over perform their classical analogs @xcite . a different , but widespread approach to process quantum information involves using continuous variables ( cv ) @xcite . among the most important advances in the field of cv quantum information are the realization of quantum teleportation @xcite , as well as quantum cryptography protocols , which rely on states defined in a continuous variables representation @xcite . universality for manipulation of continuous variables quantum states was defined in ref . @xcite , and subsequently measurement based quantum computation was generalized from the discrete to the continuous realm @xcite . manipulating quantum information in continuous or discrete variables , on its usual circuit based formulation , involves the application of unitary gates . while such gates , in the discrete variable ( dv ) case can be expressed in terms of su(2 ) transformations , a general unitary in cv is composed by polynomials of conjugate operators with a continuous unbounded spectrum , such as position and momentum of a particle , or the electromagnetic field quadratures . even though , in the general case , a direct correspondence between universal operations in dv and cv has not been established , it was shown by gottesman , kitaev and preskill ( gkp ) in ref . @xcite , that such a correspondence exists for a family of states that , while being defined in cv , can be used to encode a qubit . moreover , the gkp encoding is also at the heart of the demonstration showing that fault - tolerant measurement - based quantum computation with cv cluster states is possible @xcite . a drawback of this encoding is that it relies on non - physical states . furthermore , physical states that are close to them are of extremely challenging experimental realization with optical field quadratures . in addition , the gkp encoding has the specific purpose of quantum computation applications . in the more general context of quantum mechanics , quantum computing presents the particular aspect of requiring measurements realized in the computational basis only . however , a number of important quantum mechanics or quantum information related tasks , such as bell inequalities violation @xcite , quantum state tomography @xcite , and fundamental tests of quantum mechanics @xcite , rather rely on the recovery of binary information through the measurement of different mutually unbiased bases . for these applications , one should build a formalism offering an analogy of pauli matrices in phase space not only from the operational point of view , as proposed in @xcite , but also from a measurable perspective . moreover , in order to build a complete toolbox to manipulate and measure discrete quantum information encoded in cv , one should also define how to perform rotations between different measurement bases . in the present article we create a framework to manipulate and measure binary quantum information encoded in continuous variables using the formalism of modular variables ( mv ) @xcite , which allows us to naturally identify discrete structures in continuous variable states . we further introduce adapted operations and observables which enable us to manipulate and readout the encoded discrete quantum information in terms of the corresponding cv logical states . our formulation shows that , if one is interested in recovering quantum information by measuring binary observables defined in cv , one can loosen the requirements imposed on the gkp states . our results have immediate experimental impact which we demonstrate by applying them to the transverse degrees of freedom of single photons . the structure of this paper is as follows . in the next section we give an introduction to the modular variables formalism , including the definition of the modular position and momentum operators and the resulting representation in terms of their common eigenstates . in sec . [ sec : qipframework ] we present our main results and show how to process discrete quantum information encoded in logical states expressed in the modular variables representation . further on , measurements of judiciously chosen modular variables are revealed to enable the readout the encoded discrete quantum information from the corresponding logical states . section [ sec : expproposal ] is devoted to the discussion of an experimental implementation of our ideas using the transverse degrees of freedom of single photons . finally , we conclude in sec . [ sec : conclusion ] . in the modular variables ( mv ) formalism , pairs of canonically conjugate observables are expressed in terms of modular and integer parts , respectively . in the case of the position and momentum operators this leads to @xcite : @xmath0 where @xmath1 , @xmath2 ( @xmath3 ) has integer eigenvalues , and @xmath4-\ell/4 $ ] ( @xmath5-\pi/\ell$ ] ) is the modular position ( momentum ) operator with eigenvalues in the interval @xmath6 ( @xmath7 ) ( see fig . [ fig_1 ] ( a ) and ( b ) ) . the center of the domains containing modular variables @xmath8 and @xmath9 is not of further importance and was chosen for future convenience . and @xmath10 , respectively , and of ( b ) the bounded spectra of the modular position and momentum operator . the red arrows indicate the displacements implementing the logical pauli operators @xmath11 , @xmath12 , and @xmath13 . ( c ) representation of the transverse distribution of a single photon created in a source s with a gaussian wave function which is transformed into a periodic diffraction pattern by passing through a grating with slit distance @xmath14 . in experiments , such a diffraction grating can be implemented easily using a spatial light modulator ( slm ) . ] the separation ( [ eq : defmodvar ] ) , in modular and integer parts , proved itself useful for the detection of entanglement in spatial interference patterns @xcite . furthermore , measurements of more general modular variables , namely periodic functions of position and momentum operators , have been used recently in proposals to test the clauser - horne - shimony - holt ( chsh ) @xcite , the leggett - garg @xcite and noncontextuality inequalities @xcite . later on , in sec . [ sec : readoutmodvar ] , we provide a general framework suitable to deal with measurements of such modular variables based on the above introduced modular representation . further on it can be shown that the modular position and momentum operators , @xmath15 and @xmath16 , commute @xcite : @xmath17=0 $ ] , which leads to the definition of the modular basis @xmath18 , consisting of the common eigenstates of the these two modular operators . consequently , in terms of the modular basis we can write @xmath19 , with a normalized wave function @xmath20 written in the modular representation . the modular representation is especially convenient when dealing with periodically symmetric states , _ e.g. _ the gkp states . as a matter of fact , using the definition of @xcite and the notation @xmath21 for the logical gkp qubits , one has simply that , in the modular basis : @xmath22 and @xmath23 , showing that the gkp states naturally emerge from the modular basis . a formal definition of the modular basis and related expressions , as well as an example of @xmath20 will be discussed in the next section . formally , the modular eigenstates @xmath24 can be defined as superposition of position or momentum eigenstates ( distinguished by subscripts @xmath25 and @xmath26 , respectively ) : @xmath27 fulfilling the completeness and the orthogonality relation : @xmath28 where @xmath29 and @xmath30 are dirac @xmath31 functions defined on the intervals @xmath32 and @xmath7 , respectively ( for brevity , we will omit in the following the superscripts @xmath33 and @xmath10 ) . inversely , we can define the position and momentum eigenstates in terms of the new modular eigenstates , as : @xmath34 hence , an arbitrary state in position representation @xmath35 transformed to the modular representation , reads : @xmath36 where @xmath37 is called modular wave function of @xmath38 . the same representation was introduced by j. zak in 1967 under the term @xmath39-representation @xcite . the modular variables representation is particularly well suited for wave functions that obey a certain periodicity in position or momentum space . for example , the state @xmath40 , representing a comb of @xmath31 functions with distance @xmath14 in position space , becomes in the modular representation @xmath41 , namely a single @xmath31 peak at the origin , if we set @xmath42 . this state , together with @xmath43 , are examples of the logical qubit state introduced in the gkp paper @xcite . instead , a more physical state can be obtained if we replace the @xmath31 comb by a comb of finitely squeezed gaussian spikes with width @xmath44 and a gaussian envelope with width @xmath45 ( see fig . [ fig_2](b ) ) . the wave function of such a state in position representation , reads : @xmath46 with a normalization factor @xmath47 . in the limit @xmath48 and @xmath49 , of a large envelope and sufficiently thin spikes , respectively , the latter can be approximated by @xmath50 . then , transforming eq . ( [ app : eq : approxgkp ] ) to the modular representation with the help of eq . ( [ app : eq : modvareigenstates ] ) , yields : @xmath51 where @xmath52 and @xmath42 . to obtain the above result we used that according to the poisson sum formula we have @xmath53 , and that in the limit of large gaussian envelopes we can approximate @xmath54 . a possible experimental platform allowing for the production of such periodic states is given by the transverse degrees of freedom of single photons , as illustrated in fig . [ fig_1](c ) . labeling quantum states using bounded continuous variables enables the definition of two disjoint sets of equal size for each one of the variables . such a splitting can be done in infinitely many ways , and in order to illustrate the principles of our ideas we discuss in detail the splitting of the domain of the variable @xmath8 into two subintervals @xmath55 and @xmath56 . as a consequence , we obtain a continuum of two - level systems spanned by the states @xmath57 in terms of which we can express a general state @xmath38 as : @xmath58 where @xmath59 with a complex function @xmath60 such that @xmath61 and two real functions , @xmath62 and @xmath63 , defined on @xmath64 . the mathematical expressions allowing to switch back and forth between the position and modular representation can be found in appendix [ app : logicalqubits ] . equation ( [ eq : generalstate ] ) can be seen as a weighted continuous superposition of pure qubit states @xmath65 for each subspace with fixed @xmath8 and @xmath9 . we stress that , so far , no approximation has been made , and state ( [ eq : generalstate ] ) is simply an alternative way of writing an arbitrary state expressed in a continuous basis . note that the choice of @xmath33 is also arbitrary , and modifying it for a given state modifies the definition of the functions appearing in ( [ eq : generalstate ] ) and ( [ eqn : continuousqubit ] ) . in the following , in order to encode discrete quantum information in cv states , we assume that @xmath66 and @xmath67 are constant functions such that eq . ( [ eq : generalstate ] ) becomes @xmath68 with logical qubit states , defined as : @xmath69 the logical qubit states ( [ eq : logical0 ] ) and ( [ eq : logical1 ] ) reflect a dichotomization of the hilbert space with respect to the modular position @xmath8 . the exact choice of @xmath60 is arbitrary as long as it emerges from a properly defined modular wave function @xmath20 ( see appendix [ app : logicalqubits ] ) . in fig . [ fig_2](a ) we plot the modulo square of these states in the case where @xmath60 is given by a two dimensional gaussian function centered at the origin with standard deviations @xmath44 and @xmath70 in the modular position and momentum variables , respectively . in position space the same states are represented by two shifted combs of gaussian spikes with width @xmath44 and gaussian envelope of width @xmath71 ( see fig . [ fig_2](b ) ) . this example corresponds to the well known non - perfect gkp states introduced in @xcite . we now discuss the manipulation of the introduced logical states through appropriate unitary operations and their analogy to ordinary pauli matrices . and @xmath72 , respectively , in the modular representation ( see eq . ( [ eq : generalstate ] ) ) with @xmath60 , @xmath62 and @xmath63 chosen as explained in the text and @xmath73 . the full widths at half maximum ( fwhm ) of the distributions are indicated by @xmath74 and @xmath75 . ( b ) plot of the wave function @xmath76 of the same two logical states in the position representation . in momentum space the same functions with switched roles of @xmath77 and @xmath78 represent the logical states @xmath79 . ] in this section , we introduce some single qubit logical operations acting on the qubit structure ( [ app : eq : contqubitrep2 ] ) that is naturally embedded in every state @xmath38 . we start by expressing the single mode phase space displacement operator @xmath80 in the modular representation . to do so , we calculate first its action on a modular eigenstate ( [ app : eq : modvareigenstates ] ) , yielding : @xmath81 over - lined expressions denote the corresponding modular parts in position or momentum , respectively . equation ( [ app : eq : dispop ] ) shows that a phase space displacement by @xmath82 leads to a displacement of the corresponding modular position and momentum accompanied by the generation of additional phase factors . the latter encode information about the change of the discrete position and momentum values , @xmath83 and @xmath84 , induced by the displacement . now , by setting the displacements in eq . ( [ app : eq : dispop ] ) equal @xmath85 , @xmath86 and @xmath87 , @xmath88 , respectively , and by splitting the @xmath8-integration we reveal the following two operators : @xmath89 where @xmath90 with @xmath91 . we thus see that the weyl - heisenberg operators act as ordinary pauli operators ( @xmath92 and @xmath93 ) on each of the qubit subspaces defined by @xmath94 . the analog of the third pauli operator @xmath95 can be obtained from the product of the former two @xmath96 , yielding : @xmath97 with @xmath98 on the other hand , using the commutation rules for the phase space displacement operator , we find that @xmath99 , yielding @xmath100 . thus , as illustrated in fig . [ fig_1](b ) , the three displacements implementing the logical pauli operations ( [ eq : sigmaz ] ) , ( [ eq : sigmax ] ) and ( [ eq : sigmay ] ) , form a triangular in phase space that encloses an area of @xmath101 . the latter is closely related to the fact that the anti - commutators between the displacements ( [ eq : sigmax ] ) , ( [ eq : sigmaz ] ) and ( [ eq : sigmay ] ) , vanish @xcite , yielding the anti - commutation relations of our logical pauli operators : @xmath102 as expected form the algebra of pauli matrices . however , despite the similarities , the logical operations defined above are not completely equivalent to a pauli algebra in the general case . this becomes apparent from their commutation relations which are found to be @xmath103=2i\hat z^\dagger$ ] , @xmath104=2i\hat y^\dagger$ ] and @xmath105=2i\hat x^\dagger$ ] . they resemble those of the pauli matrices , but since the operators @xmath11 , @xmath12 and @xmath13 are not hermitian , they deviate in the fact that the commutator between each of them yields the adjoint of the third one . we further note that , in the subspace spanned by the gpk states @xmath106 and @xmath107 , the operators @xmath11 , @xmath12 and @xmath108 are hermitian ( @xmath109 , @xmath110 and @xmath111 ) . hence , in this subspace , the above commutation relations become equal to those of a real pauli algebra . the fact that we are dealing with a nonperfect pauli algebra has some consequences . for instance , if we calculate the square of one of the logical pauli operators we get : @xmath112 with @xmath113 , which differs from an identity through the appearance of an @xmath8 dependent phase factor under the integral . similarly , such phase factors also appear when manipulating the states @xmath114 and @xmath115 with one of the logical operations ( [ eq : sigmaz ] ) , ( [ eq : sigmax ] ) or ( [ eq : sigmay ] ) . we will see later on that these phase factors become irrelevant if one considers only protocols involving a specific class of modular variables as readout observables . in light of the definition of these readout observables , which will be given in sec . [ sec : readoutmodvar ] , we will also introduce appropriate rotation operators allowing to perform measurements according to different mutually unbiased bases of the logical space ( see sec . [ sec : modularrotations ] ) . consequently , the logical states and pauli operations , together with the modular readout observables and the corresponding rotations ( see secs . [ sec : readoutmodvar ] and [ sec : modularrotations ] ) , establish a solid framework to handle cv quantum information from a quantum measurement point of view . a qubit phase gate @xmath116 can be realized using the shear operation @xmath117 with @xmath118 . it transforms the logical pauli operators , ( [ eq : sigmaz ] ) and ( [ eq : sigmax ] ) , as : @xmath119 and @xmath120 . in this case , the shear implements a rotation of @xmath11 around the @xmath121-axis of the bloch sphere . further on , the hadamard gate @xmath122 can be directly realised using a rescaled fourier transform @xmath123 , with @xmath124 chosen as above , which transforms the logical pauli operators as @xmath125 and @xmath126 . in combination with the above defined logical phase - gate , we can define the fourier transformed shear @xmath127 which then implements a @xmath101-rotation of @xmath13 around the @xmath25-axis , namely @xmath128 and @xmath129 . finally , the two - qubit clifford generator @xmath130 can be realized by the two - mode gaussian unitary @xmath131 which implements the operations @xmath132 and @xmath133 , with @xmath134 . note , that the logical controlled - phase gate @xmath135 follows from @xmath130 by an additional application of @xmath136 on the second mode . note that these logical operations implement the desired clifford group operation only when acting on perfect gkp logical states @xmath41 and @xmath43 . therefore , the finite squeezing of the logical states @xmath114 and @xmath72 leads to a faulty implementation of the above defined logical clifford operations which manifests itself by a washing out of the signal @xcite . in order to circumvent such errors one can apply gkp error correction to the encoded states @xcite , which keeps the squeezing on a tolerable level . this problem does not occur if we manipulate our logical qubits with the rotation operator defined in sec . [ sec : modularrotations ] . an experimental implementation of these rotations using the transverse degrees of freedom of single photons will be discussed later on in sec . [ sec : expproposal ] . we will move on now and show how measurements of suitably defined modular variables allow to retrieve binary quantum information that is encoded in terms of our modular logical states . we start by defining the observables @xmath137 , which are the analogous to the pauli matrices from the point of view of a measurement : @xmath138 with @xmath139 , @xmath140 a real and bounded function and @xmath141 defined as in eqs . ( [ eq : sigz ] ) , ( [ eq : sigx ] ) and ( [ eq : sigmaxpy ] ) . as we show in appendix [ app : gammaprop ] , the sum over @xmath142 of the modulo squares of the expectation value of these observables is bounded by @xmath143 . moreover , restricted to our logical space , defined by @xmath114 and @xmath72 , these expectation values can be expressed as @xmath144 where @xmath145 and @xmath146 . hence , we find that the expectation values of the observables ( [ eq : gammas ] ) are proportional to the bloch vector of the encoded qubit states , whereas the proportionality factors are determined by the overlap of @xmath147 with @xmath140 . we can also define spatial and temporal correlators among observables of the kind of eq . ( [ eq : gammas ] ) , which have already been proven to be useful in the context of testing quantum mechanical properties in cv in @xcite . all these works involve measurements of particularly chosen modular variables that can be expressed in the form of eq . ( [ eq : gammas ] ) . the form of the operators ( [ eq : gammas ] ) is chosen so as to be operationally analogous to the logical pauli operations , defined in eqs . ( [ eq : sigmaz ] ) , ( [ eq : sigmax ] ) and ( [ eq : sigmay ] ) . interestingly , unwanted phase factors , appearing when manipulating the states @xmath114 and @xmath115 with some logical operation , disappear . for instance , if we consider the operator @xmath148 ( see eq . ( [ eq : logicalzsquared ] ) ) and apply it to an arbitrary state of the logical space @xmath149 , we obtain @xmath150 , where @xmath151 differs from @xmath38 by a modular position ( momentum ) dependent phase factor , but we have @xmath152 , for all @xmath139 . therefore , for implementations of protocols involving measurements of the expectation values @xmath153 , the @xmath148 operator acts as the identity . similarly , phase factors that appear due to the application of the logical pauli operations ( [ eq : sigmaz ] ) , ( [ eq : sigmax ] ) and ( [ eq : sigmay ] ) to a logical state @xmath38 , are invisible to measurements of the expectation values of ( [ eq : gammas ] ) . consequently , this allows one to establish a solid framework for handling discrete quantum information encoded in the cv logical states @xmath114 and @xmath115 . in appendix [ app : conditions ] , we discuss the conditions a general phase space observable @xmath154 , where @xmath155 is a real - valued function , has to fulfill such that it can be written in the form ( [ eq : gammas ] ) . in general , we find that the observables ( [ eq : gammas ] ) are given by periodic phase space observables , meaning observables that can be expressed as : @xmath156 where @xmath157 are complex fourier coefficients obeying the normalization condition @xmath158 , and @xmath14 and @xmath159 denote to the periodicities in position and momentum of the corresponding phase space representation of @xmath154 . all three observables ( [ eq : gammas ] ) can be expressed as ( [ eq : doublefourier ] ) with different choices of @xmath157 , @xmath14 and @xmath159 ( see appendix [ app : conditions ] ) . examples of such observables are , for instance , @xmath160 , @xmath161 and @xmath162 , where @xmath163 , @xmath164 and @xmath165 . we note that the general definition of the observables ( [ eq : gammas ] ) leads only in the case @xmath166 , for all @xmath142 , to a real set of pauli operators . however , if one wants to favour the experimental accessibility of such observables via positive operator valued measurements ( see sec . [ sec : measurementpovm ] and ref . @xcite ) , it is desirable to keep the freedom of choice of the functions @xmath140 . this leads , in the general case where @xmath140 is given by a continuous function , to an operator with a continuous spectrum . once we have created the possibility of retrieving quantum information through measurement of binary observables defined in cv , one can complete the set of qubit like operations by defining arbitrary rotations on the encoded subspaces . however , since the operators @xmath11 , @xmath12 and @xmath13 are unitary but not hermitian , arbitrary rotations can not be generated by simply exponentiating them with the proper multiplicative factors , as is true for the pauli matrices @xcite . this can be done only if we consider the operators ( [ eq : gammas ] ) in the special case , @xmath166 , for all @xmath8 , @xmath9 and @xmath142 ( in the following denoted by @xmath167 ) , which then can be used to define @xmath168 where @xmath169 and @xmath170 indicates the axis of rotation . equation ( [ eq : rotations ] ) allows to perform rotations of the general observables ( [ eq : gammas ] ) without changing the function @xmath140 and thus to implement measurements in different mutually unbiased bases of the corresponding logical space . note that , in contrary to logical operations introduced in @xcite , the operators ( [ eq : rotations ] ) perform well not only on the subspace spanned by perfect gkp states but also on the space spanned by the more general logical states @xmath114 and @xmath72 . a proposal of an experimental implementation of these rotation operations using the spatial distribution of single photons is discussed in sec . [ app : implementationlogops ] . in the following , we assume that the coordinates @xmath171 and @xmath172 refer to the transverse position and momentum of a single photon . these variables are related to the object or source plane ( position plane ) and the fourier plane ( momentum plane ) of a single - photon field . if we remain in the paraxial approximation , @xmath173 ( see fig . [ fig_1](c ) ) , the wave function of this field can be seen as the wave function of a single point particle , here being the photon . we restrict ourselves to the one dimensional case because the hilbert space associated to the two dimensional spatial photon field is a tensor product of the hilbert spaces associated with the two orthogonal transverse directions of the photon @xcite . a general quantum state of the transverse momentum ( or position ) of the photon can be written in the modular basis , as shown in ( [ app : eq : contqubitrep1 ] ) . , or optionally @xmath174 if placing lenses ( l1,l2 ) before and after the slm . finally , the photons are measured with a spatially resolving detector ( d ) . , scaledwidth=50.0% ] one major advantage in using the transverse degrees of freedom of single photons is that we can very efficiently produce states with a periodic wave function , as those presented in fig . [ fig_2 ] . to do so , we simply pass the photons through a periodic diffraction grating , as indicated in fig . [ fig_3 ] . if the photon , which is impinging on the grating , has a gaussian transverse wave function @xmath175 , with width @xmath71 , and the transmission function of the grating is given by @xmath176 , where @xmath177}$ ] with slit width @xmath44 and distance @xmath14 , the resulting wave function of the diffracted photons has the form ( [ app : eq : approxgkp ] ) . hence , by by adjusting the slit widths and distances of the grating , we can produce the logical qubit states @xmath178 and @xmath179 . in fig . [ fig_3 ] , the photons are sent through a lens system before the grating in order to prepare them in approximate plane waves . this allows us additionally to adjust the width @xmath71 of the gaussian envelope of the wave function ( [ app : eq : approxgkp ] ) , which corresponds to the quality of the prepared plane waves . experimentally gratings are often realized using spatial light modulators ( slms ) ( see fig . [ fig_1](c ) ) . note that the free propagation of the diffracted photons will lead to a blurring in the photon s transverse wave function . this effect is mainly due to the finite envelope of the photons transverse wave function and can be minimized by improving the quality of the prepared plane waves before the grating . the dependency of the blurring on the number of irradiated slits was studied in @xcite by calculating the fidelity of the initially prepared wave function with its revivals after multiples of a specific propagation distance , referred to as talbot distance . it was shown that , for currently available diffraction gratings , a fidelity higher than @xmath180 can be maintained for a propagation distance of about 10 times the talbot distance . for larger propagation distances one could possibly improve the fidelity using the earlier mentioned gkp error correction procedure to the transverse field of the photons . once it is known how to prepare single photons in the corresponding logical states @xmath114 and @xmath72 , we can try to entangle a pair of photons and further use it for the realization of quantum information protocols . one possibility to do so is by producing polarization entangled states in a type-2 parametric down - conversion process and subsequently swapping the polarization entanglement to the spatial distributions of the two photons , as suggested in @xcite . therefore , the photons are sent through a mach - zehnder interferometer made up of polarization dependent beam - splitters , half- and quater - wave plates and diffraction gratings . the output states of the interferometers will yield to 50% the desired spatially entangled photons , while the other half of the emerging photons has to be discarded . details of the procedure can be found in @xcite . all our following discussion about experimental implementations of single qubit rotations or measurements of readout observables rely on an optical element , called spatial light modulator ( slm ) . a slm consists usually of some transparent or reflective screen that is divided into a certain number of pixels whose diffraction index can be adjusted individually . in this way , one can impose spatial phase or amplitude modulations on a light beam that is transmitted or reflected by the slm . in particular , it is possible to implement operations of the form @xmath181 , where @xmath182 is an arbitrary user - defined function . similarly , one can implement phase modulations in momentum space by combining a slm with the fourier transform @xmath123 , which itself is realized optically with lenses , where @xmath124 is related to the focal length @xmath183 of the lens and the wave length @xmath184 through the relation @xmath185 ( see also fig . [ fig_3 ] ) @xcite . the above discussion leads us to a linear optical implementation of the logical rotation operator @xmath186 where @xmath169 and @xmath170 defines the axis of rotation . we focus on rotation around the two main axes corresponding to the operators @xmath187 and @xmath188 , which by composition allow to implement any desired single qubit rotation . therefore , we remind the reader that @xmath189 with the step function @xmath190 that takes the value @xmath191 if @xmath192 and @xmath193 if @xmath194 . by means of the discussion in the appendix [ app : conditions ] we know that eq . ( [ app : eq : gamma1z ] ) reads in the position representation as follows : @xmath195 where @xmath196 is a @xmath33-periodic rectangular function taking the value @xmath191 if @xmath197 and @xmath193 if @xmath198 , with integers @xmath199 . hence , the rotation operator ( [ app : eq : rotations ] ) reads : @xmath200 which is a simple position phase gate that can be implemented through the slm operation @xmath201 with @xmath202 . similarly , we can write @xmath203 where @xmath204 takes the value @xmath205 if @xmath192 or @xmath205 if @xmath194 . and again , by following the arguments in appendix [ app : conditions ] , we find that eq . ( [ app : eq : gamma1x ] ) can be written in the momentum representation as @xmath206 where @xmath207 is a @xmath208-periodic rectangular function taking the value @xmath191 if @xmath209 and @xmath193 if @xmath210 , with integers @xmath211 . from eq . ( [ app : eq : gammaxmom ] ) then follows directly @xmath212 which is a position phase gate that can be implemented with a slm operation programmed with the function @xmath213 , sandwiched between two fourier transforms , as discussed previously . we first recall that the observables @xmath214 , with @xmath139 , correspond to phase space operators @xmath154 fulfilling certain periodicity constraints , as discussed in the appendix [ app : conditions ] . if we further consider only those readout observables that can be expressed as a function of a general quadrature @xmath215 , we can write them in the corresponding diagonal form @xmath216 where the subscript denotes the @xmath217-representation . examples , as mentioned previously , are @xmath218 , @xmath219 and @xmath220}$ ] , being functions of @xmath221 , @xmath222 and @xmath223 , where @xmath224 and @xmath225}$ ] . accordingly , the expectation value of the operator ( [ app : eq : fdiagquad ] ) reads : @xmath226 which is solely determined by the probability density @xmath227 . we can reproduce the same reasoning in a bipartite system where we get for a product of two readout observables @xmath228 : @xmath229 and the corresponding expectation value : @xmath230 with the joint - probability density @xmath231 . in an experimental setup with pairs of single photons we can determine the position or momentum probability densities @xmath232 or @xmath233 , by detecting the position of the photons in the near- or far - field with respect to the output plane of the source of the photons . position measurements of single photons can be performed either by scanning a single photon counter in the transverse plane of the photon or by using a single - photon sensitive camera @xcite . arbitrary quadratures @xmath217 can be assessed via fractional fourier transforms realized with lens systems @xcite , allowing to determine arbitrary distributions @xmath234 . finally , we can use eq . ( [ app : eq : expecfquad ] ) to calculate expectation values of the desired readout observables @xmath214 . the same measurement schemes can be applied to entangled pairs of photons ( see sec . [ sec : creationwavefkt ] ) , using respectively two single photon counters or two single photon sensitive cameras , in order to determine the joint - probability distributions @xmath235 . the expectation values of the observables @xmath214 , @xmath139 , can also be measured indirectly . figure [ fig_4 ] ( a ) shows the quantum circuit that allows for a indirect measurement of the expectation values of ( [ eq : gammas ] ) in their general form by coupling the cv system to an ancilla qubit and performing controlled unitary operations . in the following , we assume that the spectrum of the operators @xmath214 is bounded by one . let us define the following povm elements ( effects ) : @xmath236 which satisfy the relation @xmath237 . the probability to obtain the outcome @xmath238 or @xmath239 is thus given by @xmath240 or @xmath241 , respectively , and we can calculate @xmath242 . hence , the expectation value of every @xmath214 can always be measured in terms of a two - valued povm . more generally , if the spectrum of @xmath214 is bounded between @xmath243 and @xmath244 , one can simply rescale the spectrum of the corresponding povm to reproduce the same argument @xcite . ) . @xmath122 depict hadamard gates and a controlled unitary gate @xmath245 is applied if the ancilla is in the state @xmath246 . the expectation value of ( [ eq : gammas ] ) is given by @xmath247 , where @xmath248 ( @xmath249 ) are the probabilities of detecting the ancilla in the state @xmath246 ( @xmath250 ) . in the case of the specific example mentioned in the text we choose @xmath251 , @xmath12 , @xmath13 . ( b ) proposal of an experimental implementation of circuit ( a ) using the spatial field of single photons passing through a mach - zehnder interferometer . controlled unitaries are realized by linear optical transformations inserted in one arm of the interferometer . unitaries of the form @xmath252 or @xmath253 can be implemented using a slm and lenses ( l ) allowing us to switch from the position to the momentum space . ] consider the quantum circuit shown in fig . [ fig_4](a ) which implements the operation @xmath254 on the initial state @xmath255 . hence , by measuring the ancilla state in the basis @xmath256 , we project the system state onto @xmath257 with the probability @xmath258 , which is equivalent to measuring the povm @xmath259 with the corresponding effects @xmath260 . with a general unitary operator @xmath261 , where @xmath262 is a real function of position and momentum operator , we can also write @xmath263\}$ ] , leading to @xmath264\rangle}$ ] . now , in order to measure any of the observables @xmath214 , we define @xmath265 , with the corresponding phase - space operator @xmath266 ( see appendix [ app : conditions ] ) , yielding @xmath267 . the above measurement strategy can be straightforwardly implemented with single photons passing through balanced mach - zehnder interferometers , as depicted in fig . [ fig_4](b ) . therein , the spatial distribution of the single photons represent the cv system and the path of the interferometer the state of the ancilla . controlled unitary operations are realized via linear optical elements placed in one of the arms of the interferometer , and measurements of the ancilla state by detecting photons that exit form one of the two output ports using single photon bucket detectors . a slm , with the option of additionally placing it in the fourier plane between to lenses , allows us to perform arbitrary position or momentum phase gates , @xmath268 or @xmath269 , where @xmath270 is user defined on the slm . as discussed previously @xmath25 and @xmath26 can be considered as the near- and far - field variables with respect to the output plane of the source . phase gates @xmath271 in terms of an arbitrary quadrature @xmath272 can be realized through fractional fourier transform before and after the slm @xcite . hence , we have the ability to implement a broad class of unitaries on the spatial distribution of the photons allowing us to measure expectation values through @xmath273}\rangle}$ ] , yielding @xmath274 . at this point we note that the direct measurement of the observables @xmath214 , as described in this section , is less expensive in terms of the number of measurements needed in order to determine the expectation values @xmath274 , as compared to the indirect measurement strategy introduced in the previous section @xcite . moreover , there is no need for a spatially resolving detection of the photons . we presented a general framework which allows us to encode , manipulate and readout discrete quantum information in phase space in terms of continuous variables states . this is possible by using the modular variables formalism that naturally leads to an intuitive definition of a qubit and the necessary universal manipulations . we demonstrate its strong relationship with the gkp formalism , and show that as far as one is interested in performing quantum protocols involving expectation values of bounded periodic observables , so called modular variables , it is possible to encode binary quantum information in more general states than the gkp ones . a possible experimental implementation using the transverse degrees of freedom of single photons was discussed , as well . from a fundamental point of view , our framework shows how to reveal naturally discrete structures of states and operations written in a continuous variable representation . furthermore , it provides a unifying formalism that shows how , in general , measurements of modular variables can be employed in quantum information protocols . finally , an application of our ideas in hybrid quantum systems , which use cv besides some discrete degree of freedom , as is the case for single photons , could be advantageous for future experimental implementations of quantum information protocols . the authors are indebted to f. de melo , o. jimnez faras , l. aolita , a. d. ribeiro , n. menicucci , r. n. alexander , g. ferrini , a. laversanne - finot and t. douce for inspiring discussions . the authors acknowledge financial support by anr / cnpq hide , anr comb , faperj , cnpq and capes / cofecub project ph 855 - 15 . to show , that every state in the modular representation can indeed be written as in eq . ( [ eq : generalstate ] ) , we start from eq . ( [ app : eq : modvarstate ] ) and split the integration over @xmath8 into two equally sized domains : @xmath275 next , we redefine the complex modular wave function @xmath276 in the following way : @xmath277 where @xmath278 is complex amplitude , and @xmath279 , @xmath280 and @xmath63 are real functions , all defined on the domain @xmath281 . these functions directly relate to the modular wave function @xmath282 through : @xmath283 and @xmath284 hence , we can express every state as : @xmath285 with @xmath286 later on we will , for technical reasons , express certain operators in a slightly modified basis defined as @xmath287 , which involve additional @xmath9-dependent phase factors which can be absorbed in the definition of the wave function @xmath60 . in the case of the above discussed example of a comb of gaussian spikes with gaussian enelope @xmath60 becomes a gaussians with periodic boundary conditions . note that it is equivalently possible to define such a qubit structure in terms of the modular momentum @xmath9 . in this case , one splits the integration over @xmath9 , in eq . ( [ eqn : arbstate ] ) , into two parts and obtains a similar result to eqs . ( [ app : eq : contqubitrep1 ] ) and ( [ app : eq : contqubitrep2 ] ) , now with @xmath288 where the @xmath26-subscripts in eq . ( [ app : eq : contqubitrep3 ] ) refer to the splitting with respect to @xmath9 . furthermore , this intrinsic qubit structure can be generalized to qudit systems by splitting the integration in eq . ( [ eqn : arbstate ] ) into @xmath124-parts instead of two , as discussed also in @xcite . however , in the context of this work we will restrict ourselves to the above presented case of @xmath289 . the readout of the encoded logical state can be performed using the observables defined in eq . ( [ eq : gammas ] ) of the main text . for convenience , we reproduce their definition here : @xmath290 where @xmath291 where @xmath140 are real functions defined on the domain @xmath64 . in sec . [ app : conditions ] we will show which class of general phase space operators @xmath154 obey such a representation , however , for the moment we take their form as granted to discuss several important properties . first , we note that the matrix elements of the operators ( [ app : eq : sigx ] ) , ( [ app : eq : sigz ] ) and ( [ app : eq : sigy ] ) read : @xmath292 , \label{eqapp : sigzxpmatel}\\ \langle\bar x ' , \bar p ' | \hat\sigma_{x}(\bar x_0 , \bar p_0 ) |\bar x , \bar p\rangle&=\delta(\bar p-\bar p_0 ) \delta(\bar p'-\bar p_0 ) \nonumber \\ & \times \left[\delta(\bar x'-\bar x_0)\delta(\bar x-\frac{\ell}{2}-\bar x_0)e^{-i\bar p\ell/2 } + \delta(\bar x'-\frac{\ell}{2}-\bar x_0)\delta(\bar x-\bar x_0)e^{i\bar p\ell/2}\right],\label{eqapp : sigxxpmatel } \\ \langle\bar x ' , \bar p ' | \hat\sigma_{y}(\bar x_0 , \bar p_0 ) |\bar x , \bar p\rangle&=-i\delta(\bar p-\bar p_0 ) \delta(\bar p'-\bar p_0 ) \nonumber \\ & \times\left[\delta(\bar x'-\bar x_0)\delta(\bar x-\frac{\ell}{2}-\bar x_0)e^{-i\bar p\ell/2 } -\delta(\bar x'-\frac{\ell}{2}-\bar x_0)\delta(\bar x-\bar x_0)e^{i\bar p\ell/2}\right].\label{eqapp : sigyxpmatel}\end{aligned}\ ] ] now , using eqs . ( [ app : eq : sigx])-([eqapp : sigyxpmatel ] ) , we can show that the @xmath293-dependent pauli matrices @xmath294 , with @xmath295 , fulfill the relation : @xmath296 \label{eq : xppaulirelation}\end{aligned}\ ] ] where @xmath297 and @xmath298 . the relation ( [ eq : xppaulirelation ] ) resembles the one of a real pauli algebra with additional @xmath31 functions ensuring that the products of pauli operators corresponding to different subspaces , labeled by @xmath293 and @xmath299 , respectively , vanish . further on , we can calculate the expectation value of the observables ( [ app : eq : gammas ] ) with respect to an arbitrary cv state expressed in the modular representation ( [ app : eq : approxgkp ] ) , yielding : in the second step of the computation ( [ eqapp : expecgammax ] ) we dropped terms that are proportional to cross products of @xmath31 functions as , for instance , @xmath301 , because , upon integration of @xmath302 and @xmath303 over the interval @xmath55 , such terms are only nonzero in a single point being a set of measure zero , and thus the integration vanishes . equivalently , the expectation values of the observables @xmath304 and @xmath187 read : @xmath305 or in vector notation we can write @xmath306 with @xmath307 further on , by summing over the squares of the expectation values ( [ eqapp : expecgammax ] ) , ( [ eq : expecgammay ] ) and ( [ eq : expecgammaz ] ) we can show : @xmath308 \nonumber \\ \leq & \big ( \max_{\bar x,\bar p,\beta}|\zeta_\beta{(\bar x,\bar p)}|\big)^2 \iint_{-\ell/4}^{\ell/4}d\bar x \bar x ' \iint_{-\pi/\ell}^{\pi/\ell } d\bar p d\bar p ' |f(\bar x , \bar p ) |^2 |f(\bar x ' , \bar p ' ) |^2 \nonumber \\ & \times\underbrace { \frac{1}{2}\left[\sum_{\beta = x , y , z}v_\beta^2 ( \bar x,\bar p)+\sum_{\beta = x , y , z } v_\beta^2 ( \bar x',\bar p ' ) \right]}_{=1 } \nonumber \\ = & \big ( \max_{\bar x,\bar p,\beta}|\zeta_\beta{(\bar x,\bar p)}|\big)^2 \left(\int_{-\ell/4}^{\ell/4}d\bar x \int_{-\pi/\ell}^{\pi/\ell } d\bar p \ |f ( \bar x , \bar p)|^2 \right)^2 \leq \big ( \max_{\bar x,\bar p,\beta}|\zeta_\beta{(\bar x,\bar p)}|\big)^2 , \label{eqn : expecgammaapp}\end{aligned}\ ] ] where we use that @xmath309 ^ 2\geq 0 $ ] and that the bloch vector of a pure qubit state is normalized to @xmath310 . for the example discussed in the main text , we have @xmath311 , which shows that @xmath312 is contained in a unit sphere . let s consider an arbitrary observable in phase space , _ i.e. _ a valid function of the position and momentum operator , expressed in the modular basis : @xmath313 with the matrix elements @xmath314 , which , by using the definition of the modular eigenstates in eq . ( [ app : eq : modvareigenstates ] ) , can be expressed as @xmath315 further on , if we assume that the function @xmath154 is periodic with respect to @xmath171 and @xmath172 with period @xmath33 and @xmath10 , respectively , we can rewrite it as a double fourier series : @xmath316 where @xmath157 are the complex fourier coefficients obeying the normalization condition @xmath158 , and we have , by definition , @xmath317 . in the following , we discuss how one can construct the observables ( [ app : eq : gammas ] ) from periodic operators of the form @xmath154 . to start , we assume a modular operator that is diagonal in the modular position and momentum with the following matrix elements : @xmath318 which fulfill the periodicity condition @xmath319 . then we obtain @xmath320 where we defined ( for @xmath321 and @xmath322 ) @xmath323 with the phases @xmath324 which , up to now , can assume any value . now , if we assume a phase space operator of the form ( [ app : eq : doublefourier ] ) , with @xmath325 , and use eq . ( [ app : eq : genmodvarop2 ] ) we get @xmath326 where we used @xmath327 . we thus find that periodic phase space operators with periodicity @xmath33 and @xmath10 in @xmath171 and @xmath172 , respectively , lead to diagonal operators in the modular representation with matrix elements @xmath328 . moreover , to obtain the operator @xmath187 , the condition @xmath329 needs to be enforced as well . the latter is true if @xmath330 , for all even @xmath199 , leading to the following form of the diagonal elements in eq . ( [ eq : fzdiagel ] ) : @xmath331 which correspond to the phase space observable : @xmath332 hence , with eq . ( [ eq : fzobservable ] ) we provide a specific class of modular variables which in the modular representation can be expressed in the form of @xmath187 , with @xmath333 chosen according to eq . ( [ eq : f_z ] ) . a particular example of the observable ( [ eq : fzobservable ] ) is given by choosing only two nonzero coefficients @xmath334 and @xmath335 , leading to @xmath336 and @xmath337 . further on , we want to find the conditions on the general periodic phase space operator ( [ app : eq : doublefourier ] ) , such that it can be brought in the form ( [ app : eq : gammaxcond ] ) . therefore , we assume @xmath344 and @xmath330 , for all even @xmath211 , which yields : @xmath345 , \label{app : eq : matrixelex}\end{aligned}\ ] ] where we split up the domain of @xmath346 with the two rectangular functions @xmath347 and @xmath348 , defined in terms of the heaviside stepfunction @xmath349 . with this eq . ( [ app : eq : matrixelex ] ) becomes : @xmath350 \nonumber \\ = & \tilde f_x(\bar x,\bar p ) \delta(\bar p-\bar p ' ) \nonumber \\ & \times\left\{\begin{array}{llrl } e^{+i\bar p\ell/2}\ , \delta(\bar x+\ell/2-\bar x ' ) , & \mathrm{for } & -\frac{\ell}{4}&\le \bar{x } \le \frac{\ell}{4 } \\ e^{-i\bar p\ell/2}\ , \delta(\bar x-\ell/2-\bar x ' ) , & \mathrm{for } & \frac{\ell}{4}&\le \bar{x } \le \frac{3\ell}{4 } \end{array}\right . , \label{app : eq : matrixelex1}\end{aligned}\ ] ] where @xmath351 we thus find that all operators of the form @xmath352 where we set @xmath344 and @xmath330 , for all even @xmath211 , can be expressed as @xmath188 with @xmath353 and @xmath354 . an example of eq . ( [ eq : fxobservable ] ) is given by the operator @xmath355 , where only @xmath356 and @xmath357 are nonzero , which is equal to @xmath188 with @xmath358 . finally , we consider a modular operator defined by the matrix elements @xmath359 with the periodicity properties @xmath360 and @xmath361 , leading to @xmath362 \nonumber \\ & = \int_{-\ell/4}^{\ell/4 } d\bar x \int_{-\pi/\ell}^{\pi/\ell } d\bar p \ , \tilde f_y(\bar x,\bar p ) \hat\sigma_x(\bar x , \bar p ) \equiv \hat\gamma_y.\end{aligned}\ ] ] where we defined @xmath363 with @xmath364 . further on , consider the phase space operator ( [ app : eq : doublefourier ] ) , with @xmath365 and @xmath366 , yielding the matrix elements @xmath367 , \label{app : eq : matrixeley}\end{aligned}\ ] ] with @xmath368 we thus find that all operators of the form @xmath369 where we set @xmath365 , @xmath366 and @xmath330 , for all even @xmath199 and @xmath211 , can be expressed as @xmath304 with @xmath370 and @xmath371 . an example of eq . ( [ eq : fyobservable ] ) is given by @xmath372 , corresponding to the case where only @xmath373 and @xmath374 are nonzero , which is equal to @xmath304 with @xmath375 .

the yokonuma hecke algebras ( of type gl ) , denoted @xmath1 , have been used by j. juyumaya and s. lambropoulou to construct invariants for various types of links , in the same spirit as the construction of the homflypt polynomial from usual hecke algebras . we refer to @xcite and references therein . in particular , the algebras @xmath1 provide invariants for classical links and the natural question was to decide if these invariants were equivalent , or not , to the homflypt polynomial . this study culminated in the recent discovery @xcite that these invariants are actually topologically stronger than the homflypt polynomial ( _ i.e. _ they distinguish more links ) . in @xcite , another approach to study invariants coming from yokonuma hecke algebras was developed . the starting point was the fact that the algebra @xmath1 is isomorphic to a direct sum of matrix algebras with coefficients in tensor products of usual hecke algebras . this allowed an explicit construction of markov traces on @xmath2 from the known markov trace on hecke algebras ( on hecke algebras , there is a unique markov trace up to normalisation , and it gives the homflypt polynomial ) . in addition to its usefulness for the construction of markov traces , the approach via the isomorphism also helps to study the resulting invariants . indeed some properties of the invariants follow quite immediately from a precise understanding of the isomorphism ( see paragraph * 4 * below ) . independently of which approach is used , another ingredient was added in @xcite : a third parameter in the invariants . while the first two parameters come from the algebra @xmath1 , this third parameter @xmath3 has its origin in the framed braid group , and corresponds to a certain degree of freedom one has when going from the framed braid group to the algebra @xmath1 . more precisely , we can deform the standard surjective morphism from the framed braid group algebra to its quotient @xmath1 into a family of morphisms ( depending on @xmath3 ) respecting the braid relations and the markov conditions . another way of interpreting the parameter @xmath3 is that it modifies the quadratic relation satisfied by the generators of @xmath1 . its existence explains ( or is reflected in ) the fact that different presentations for @xmath1 were used before . lambropoulou invariants correspond to certain specialisations of this parameter @xmath3 , depending on the chosen presentation . so the parameter @xmath3 unifies every possible choices and yields more general invariants . it is indicated in ( * ? ? ? * remark 8.5 ) that changing the presentation seems to give a non - equivalent topological invariant . .2 cm * 2 . * in this paper , we consider the affine yokonuma hecke algebras ( of type gl ) , denoted @xmath4 . they were introduced in @xcite in connections with the representation theory and the jucys murphy elements of the classical yokonuma hecke algebras . our main goal here is to generalise for @xmath4 the whole approach to link invariants via the isomorphism theorem . the invariants are in general for links in the solid torus . the classical links are naturally contained in the solid torus links and , restricted to them , the obtained invariants correspond to the invariants obtained in @xcite from @xmath1 ( naturally seen as a subalgebra of @xmath4 ) . specialising the parameter @xmath3 , we identify the juyumaya lambropoulou invariants among them . for those invariants , we emphasize that we recover some known results @xcite by a different method and furthermore obtain some new results already in this particular case . we start with an isomorphism between the algebra @xmath4 and a direct sum of matrix algebras with coefficients in tensor products of affine hecke algebras . as done in @xcite , the isomorphism can be proved repeating the same arguments as for @xmath1 ( see @xcite where the proof for @xmath1 is presented , as a particular case of a more general result by g. lusztig @xcite ) . here we sketch a short different proof for @xmath4 using the known result for @xmath1 . we also prove the analogous theorem for the cyclotomic quotients of @xmath4 ( with ariki koike algebras replacing affine hecke algebras ) . useful for concrete use , the formulas for the generators are simple and given explicitly . concerning links , s. lambropoulou constructed invariants , analogues of the homflypt polynomial , for links in the solid torus from affine hecke algebras @xcite . then , it was explained in @xcite how to obtain invariants for those links from the algebras @xmath4 , unifying the methods of j. juyumaya and s. lambropoulou for @xmath1 and the construction of s. lambropoulou for affine hecke algebras . due to the recent results of @xcite , it is expected that the invariants obtained from @xmath4 are stronger than the ones obtained from affine hecke algebras . here we follow the alternative approach which uses the isomorphism to construct markov traces on the family of algebras @xmath5 . to sum up , the markov traces are constructed and can be calculated with the following steps : for an element of @xmath4 , apply first the isomorphic map to obtain an element of the direct sum of matrix algebras ; then , for each matrix , apply the usual trace which results in an element of a tensor product of affine hecke algebras ; finally apply a tensor product of markov traces on affine hecke algebras . our result consists in obtaining the compatibility conditions relating the markov traces appearing in different matrix algebras so that the preceding procedure eventually results in a genuine markov trace on @xmath5 . with the definition used here , for a given @xmath6 , the set of markov traces on @xmath5 forms a vector space . from the isomorphism , a set of distinguished markov traces appears naturally , which spans the set of all markov traces constructed here . thus , our study of markov traces ( and of invariants ) is reduced to the study of these basic " markov traces ( and of the corresponding basic " invariants ) . it turns out that these basic markov traces are indexed , for a given @xmath6 , by the non - empty subsets @xmath7 together with a choice , denoted formally by @xmath8 , of @xmath9 arbitrary markov traces on affine hecke algebras . we note that if we restrict to @xmath1 , the parameter @xmath8 disappears and the basic markov traces on @xmath2 are indexed , for a given @xmath6 , only by the non - empty subsets @xmath7 . this recovers a result of @xcite . .2 cm * 3 . * throughout the paper , we intended to give in details the connections between the two approaches , so that one would be able to pass easily from one to the other . this will allow in particular to specialise and translate all our results on the invariants to juyumaya lambropoulou invariants as well . roughly speaking , j. juyumaya and s. lambropoulou constructed invariants from @xmath1 in two steps @xcite . the same approach was followed in @xcite for @xmath4 . first a certain trace map , analogous to the ocneanu trace and satisfying a certain positive markov condition , was constructed . then a rescaling procedure was implemented , in order to produce genuine invariants . the rescaling procedure amounts to two things : a renormalisation of the generators and a renormalisation , depending on @xmath10 , of the trace . in the approach presented here , the first step is included from the beginning in a more general quadratic relation for the generators . the second step is already included in the definition of a markov trace , namely that it is a family , on @xmath10 , of trace maps satisfying the two markov conditions . as a consequence , to obtain invariants here , one directly applies the markov trace and no rescaling procedure is needed . for the comparison , our first task is to explain that juyumaya lambropoulou approach is equivalent to considering certain markov traces ( with the definition used here ) and to relate their variables with the parameters considered here . then we need to identify these markov traces in terms of the ones constructed via the isomorphism theorem . we obtain finally the explicit decomposition of these markov traces in terms of the basic markov traces indexed by @xmath7 and @xmath8 as above . in particular , for @xmath1 , this results in an explicit formula for the juyumaya lambropoulou invariants , as studied in @xcite , in terms of the basic invariants constructed here . we note that , in this case , the parameter @xmath8 is not present , and that juyumaya lambropoulou invariants are also parametrised , for a given @xmath6 , by non - empty subsets of @xmath11 . nevertheless , they do not coincide with the basic invariants and the comparison formula is not trivial ( see formulas ( [ compare2 ] ) in section [ sec - inv ] ) . in general , for @xmath4 , we obtain the expression of the invariants constructed in @xcite in terms of the basic invariants constructed here . concerning the third parameter @xmath3 , we recall that it was not present in the previous approach . actually , one need to specialise it to a certain value in our invariants to recover the juyumaya lambropoulou invariants . the two different presentations of @xmath1 that were used , as in @xcite , correspond to two different values of @xmath3 that we give explicitly . similarly , for @xmath4 , the invariants constructed in @xcite correspond to a certain specialisation of @xmath3 . .2 cm * 4 . * we conclude this introduction by describing the main properties obtained for the invariants . as explained before , they follow quite directly from a precise understanding of the isomorphism , and are expressed easily in terms of the basic invariants defined here . the main results are : * for @xmath6 and a non - empty subset @xmath7 , the corresponding invariants coincide with invariants corresponding to @xmath12 and the full set @xmath13 . therefore , we only have to consider the full sets @xmath11 for different @xmath6 . * further , given a number @xmath0 of connected components of a link , the invariants corresponding to @xmath11 are zero if @xmath14 . so , given @xmath0 , we only have to consider @xmath15 . moreover , with the comparison results explained in paragraph * 3 * , it is easy to deduce the similar properties for invariants obtained via juyumaya lambropoulou approach . the first item remains true as it is . the second item results in an explicit formula expressing , if @xmath14 , the invariants corresponding to @xmath11 in terms of the invariants corresponding to @xmath13 with @xmath16 . specialising @xmath17 to the appropriate values and restricting to classical links , we recover with the first item a result of @xcite . the second item in this case was proved only for @xmath18 also in @xcite . let @xmath19 and @xmath20 and @xmath21 be indeterminates . we work over the ring @xmath22 $ ] . the properties of the affine yokonuma hecke algebras recalled here can be found in @xcite . we use @xmath23 to denote the symmetric group on @xmath10 elements , and @xmath24 to denote the transposition @xmath25 . the affine yokonuma hecke algebra @xmath4 is generated by elements @xmath26 subject to the following defining relations ( [ def - aff1])([def - aff3 ] ) : @xmath27 g_ig_{i+1}g_i & = & g_{i+1}g_ig_{i+1 } & & \mbox{for $ i=1,\ldots , n-2$,}\\[0.1em ] x_1\,g_1x_1g_1 & = & g_1x_1g_1\,x_1 & & \\[0.1em ] x_1g_i & = & g_ix_1 & & \mbox{for $ i=2,\ldots , n-1$,}\\[0.1em ] \end{array}\ ] ] @xmath28 g_it_j & = & t_{s_i(j)}g_i & & \mbox{for $ i=1,\ldots , n-1 $ and $ j=1,\ldots , n$,}\\[0.1em ] t_j^d & = & 1 & & \mbox{for $ j=1,\ldots , n$,}\\[0.2em ] x_1t_j & = & t_jx_1 & & \mbox{for $ j=1,\ldots , n$ , } \end{array}\ ] ] @xmath29 where @xmath30 . the elements @xmath31 are idempotents and we have : @xmath32 let @xmath33 and let @xmath34 be a reduced expression for @xmath35 . since the generators @xmath36 of @xmath4 satisfy the same braid relations as the generators @xmath24 of @xmath23 , matsumoto s lemma implies that the following element does not depend on the reduced expression of @xmath35 : @xmath37 elements @xmath38 of @xmath4 are defined inductively by @xmath39 the elements @xmath40 commute with each other . they also commute with the generators @xmath41 and they satisfy @xmath42 if @xmath43 . for @xmath44 , we set @xmath45 . the following set of elements forms a basis of @xmath4 : @xmath46 this fact has the following consequences : @xmath47 recall that the yokonuma hecke algebra @xmath1 is presented by generators @xmath48 , @xmath41 and defining relations those in ( [ def - aff1])([def - aff3 ] ) which do not involve the generator @xmath49 . we have that @xmath1 is isomorphic to the subalgebra of @xmath4 generated by @xmath50 ( hence the common names for the generators ) . @xmath47 in particular , the commutative subalgebra @xmath51 of @xmath4 generated by @xmath41 is isomorphic to the group algebra of @xmath52 . @xmath47 by definition , the affine hecke algebra ( of type gl ) is @xmath53 . we have , for any @xmath6 , that the quotient of @xmath4 by the relations @xmath54 , @xmath55 , is isomorphic to @xmath56 . we denote by @xmath57 the corresponding surjective morphism from @xmath4 to @xmath56 , and the generators of @xmath56 are denoted @xmath58 . @xmath47 the subalgebra of @xmath56 generated by @xmath59 is the usual hecke algebra , denoted @xmath60 . we also have @xmath61 . let @xmath62 be the set of _ @xmath63-compositions _ of @xmath10 , that is the set of @xmath63-tuples @xmath64 such that @xmath65 . we denote @xmath66 . for @xmath66 , the young subgroup @xmath67 is the subgroup @xmath68 of @xmath69 , where @xmath70 acts on the letters @xmath71 , @xmath72 acts on the letters @xmath73 , and so on . the subgroup @xmath67 is generated by the transpositions @xmath24 with @xmath74 . we denote by @xmath75 the algebra @xmath76 ( by convention @xmath77 $ ] ) . it is isomorphic to the subalgebra of @xmath56 generated by @xmath78 and @xmath79 , with @xmath80 , and is a free submodule with basis @xmath81 . similarly , we have a subalgebra @xmath82 of the hecke algebra @xmath60 . it is naturally a subalgebra of @xmath75 ( generated only by @xmath79 , with @xmath80 ) . for @xmath66 , let @xmath83 be the index of the young subgroup @xmath67 in @xmath23 , that is , @xmath84 we define the _ socle _ @xmath85 of a @xmath63-composition @xmath86 by @xmath87 0 & \text{if $ \mu_a=0 $ , } \end{array}\right.\ \ \ \ \ \ \text{for $ a=1,\dots , d$.}\ ] ] the composition @xmath85 belongs to @xmath88 where @xmath0 is the number of non - zero parts in @xmath86 . we denote by @xmath89 the set of all socles of @xmath63-compositions , or in other words , @xmath89 is the set of @xmath63-compositions whose parts belong to @xmath90 . we note that there is a one - to - one correspondence between the set @xmath89 and the set of non - empty subsets of @xmath11 , given by @xmath91 0 & \text{if $ a\notin s$. } \end{array}\right.\ ] ] let @xmath93 be the set of roots of unity of order @xmath63 . a complex character @xmath94 of the group @xmath52 is characterised by the choice of @xmath95 for each @xmath96 . we denote by @xmath97 the set of complex characters of @xmath52 , extended to the subalgebra @xmath98 of @xmath4 . for each @xmath99 , we denote by @xmath100 the primitive idempotent of @xmath92 associated to @xmath94 . then the set @xmath101 is a basis of @xmath92 . therefore , from the basis ( [ basis ] ) of @xmath4 , we obtain the following other basis of @xmath4 : @xmath102 [ [ permutations - pi_chi . ] ] * permutations @xmath103 . * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + let @xmath104 . for @xmath105 , we let @xmath106 be the number of elements @xmath107 such that @xmath108 . then the sequence @xmath109 is a @xmath63-composition of @xmath10 which we denote by @xmath110 . for a given @xmath66 , we consider a particular character @xmath111 such that @xmath112 . the character @xmath113 is defined by @xmath114 \chi_0^{\mu } ( t_{\mu_1 + 1})&=&\ldots & = & \chi_0^{\mu } ( t_{\mu_1+\mu_2})&= & \xi_2\ , \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \chi_0^{\mu } ( t_{\mu_1+\dots+\mu_{d-1}+1})&=&\ldots & = & \chi_0^{\mu } ( t_{n})&= & \xi_d\ .\\ \end{array}\right.\ ] ] the symmetric group @xmath23 acts on the set @xmath97 by the formula @xmath115 . the stabilizer of @xmath113 under the action of @xmath23 is the young subgroup @xmath67 . in each left coset in @xmath116 , there is a unique representative of minimal length . so , for any @xmath104 such that @xmath117 , we define a permutation @xmath118 by requiring that @xmath103 is the element of minimal length such that : @xmath119 we present isomorphism theorems for the algebras @xmath4 and their cyclotomic quotients . we sketch a short proof , which uses the corresponding result for @xmath1 ( see ( * ? ? ? * section 3.1 ) ) . we are still working over @xmath22 $ ] . for @xmath66 , we consider the algebra @xmath120 of matrices of size @xmath83 with coefficients in @xmath75 . we recall that @xmath83 , given by ( [ mmu ] ) , is the number of characters @xmath104 such that @xmath117 . so we index the rows and columns of a matrix in @xmath120 by such characters . moreover , for two characters @xmath121 such that @xmath122 , we denote by @xmath123 the matrix in @xmath120 with 1 in line @xmath94 and column @xmath124 , and 0 everywhere else . [ theo - iso ] the affine yokonuma hecke algebra @xmath4 is isomorphic to @xmath125 , the isomorphism being given on the elements of the basis ( [ e - basis ] ) by @xmath126 where @xmath104 , @xmath127 and @xmath128 ( @xmath129 is the length function on @xmath23 ) . we start with explicit formulas for the images of the generators of @xmath4 given below in ( [ form - t])([form - g ] ) , and we check that the images of the generators satisfy all the defining relations ( [ def - aff1])([def - aff3 ] ) of @xmath4 . for the relations not involving the generator @xmath49 , this is already known from the isomorphism theorem for @xmath1 . we omit the remaining straightforward verifications . thus , formulas ( [ form - t])([form - g ] ) induce a morphism of algebras , and we check that it coincides with @xmath130 given by ( [ iso - aff ] ) . again , for the images of the elements of the basis of the form @xmath131 , this is already known from the @xmath1 situation . the multiplication by @xmath132 is straightforward . it remains to check that @xmath130 is bijective . the surjectivity follows from a direct inspection of formula ( [ form - x ] ) together with the already known fact that every @xmath133 is in the image of @xmath130 . the injectivity can be checked directly . indeed , assume that a certain linear combination @xmath134 is in the kernel of @xmath130 . then for every @xmath135 , we obtain that @xmath136 where the sum is over @xmath137 such that @xmath138 . for @xmath139 satisfying this condition , we have @xmath140 if and only if @xmath141 , and therefore every coefficients in the above sum are @xmath142 . [ [ formulas - for - the - generators . ] ] * formulas for the generators . * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + here , we give the images under the isomorphism @xmath130 of the generators of @xmath4 . we recall that @xmath143 in @xmath4 , the sum being over @xmath97 . let @xmath104 and set @xmath144 . then , by definition of @xmath103 , it is straightforward to see that @xmath145 , where @xmath146 and @xmath147 . * let @xmath148 . we have : @xmath149 it follows that the image of @xmath31 , @xmath150 , is a sum of diagonal matrices ; the coefficient in position @xmath94 is 1 if @xmath151 , and 0 otherwise . * let @xmath148 . we have : @xmath152 * let @xmath153 . we have : @xmath154 { \boldsymbol{1}}_{\chi,\chi}\,\og_{\pi_{\chi}^{-1}(i ) } & \text{if $ s_i(\chi)=\chi$\ . } \end{array}\right.\ ] ] the first line follows from @xmath155 if @xmath156 . the second line follows from @xmath157 if @xmath158 . let @xmath159\backslash\{0\}$ ] be an @xmath160-tuple of non - zero parameters for a certain @xmath161 ( equivalently , one could consider @xmath162 as indeterminates and work over the extended ring @xmath163 $ ] ) . the cyclotomic yokonuma hecke algebra @xmath164 is the quotient of the affine yokonuma hecke algebra @xmath4 by the relation @xmath165 it is shown in @xcite that the algebra @xmath164 is a free @xmath22$]-module with basis @xmath166 in particular , if @xmath167 , @xmath164 is isomorphic to the yokonuma hecke algebra @xmath1 . similarly , the cyclotomic hecke algebra @xmath168 ( or the ariki koike algebra ) is the quotient of the affine hecke algebra @xmath169 by the relation @xmath170 . equivalently , it is the quotient of the cyclotomic yokonuma hecke algebra @xmath164 by the relations @xmath54 , @xmath55 . it is a free @xmath22$]-module with basis @xmath171 . for @xmath66 , we set @xmath172 . by definition , @xmath173 is the quotient of the algebra @xmath75 by the relations @xmath174 the cyclotomic yokonuma hecke algebra @xmath164 is isomorphic to the direct sum @xmath175 . let @xmath176 be the ( two - sided ) ideal of @xmath4 generated by the left hand side of the relation ( [ rel - cyc ] ) . for @xmath66 , let @xmath177 be the ideal of @xmath75 generated by the left hand sides of the relations ( [ rel - cyc - mu ] ) . the corollary follows from theorem [ theo - iso ] together with the fact that @xmath178 . it remains to check this fact . the inclusion @xmath179 " follows at once from formula ( [ form - x ] ) for @xmath180 . for the other inclusion , let @xmath66 . let @xmath181 such that @xmath182 , so that there is a character @xmath94 with @xmath117 and @xmath183 . again , formula ( [ form - x ] ) for @xmath180 gives @xmath184 . therefore , for every generators of @xmath177 , we have in @xmath185 a matrix in @xmath186 with the generator as one diagonal element and @xmath142 everywhere else . as @xmath185 is an ideal , this shows that @xmath187 is included in @xmath185 . from now on , we extend the ground ring @xmath22 $ ] to @xmath188 $ ] , and we consider our algebras over this extended ring . a markov trace on the family of algebras @xmath5 is a family of linear functions @xmath190\}_{n\geq1}$ ] satisfying : @xmath191 \rho_{d , n+1}(xg_n)=\rho_{d , n+1}(xg_n^{-1})=\rho_{d , n}(x)\ , \quad & \text{$n\geq1 $ and $ x\in \hy_{d , n}$. } \end{array}\ ] ] a markov trace on the family of algebras @xmath192 is a family of linear functions @xmath193\}_{n\geq1}$ ] satisfying : @xmath194 \tau_{n+1}(x\og_n)=\tau_{n+1}(x\og_n^{-1})=\tau_n(x)\ , \quad & \text{$n\geq1 $ and $ x\in \hh_{n}$. } \end{array}\ ] ] recall the definition of @xmath89 from section [ sec - def ] . for each @xmath195 and each @xmath181 such that @xmath196 , we choose a markov trace @xmath197 on @xmath189 . by convention , @xmath198 $ ] and maps of the form @xmath199 are identities on @xmath200 . below in ( [ rho - n ] ) , each term in the sum over @xmath66 acts on @xmath120 . we skip the proof of the following theorem . it can be done exactly as in ( * ? ? ? * lemma 5.4 ) . [ theo - mark ] the following maps form a markov trace on @xmath5 : @xmath201 roughly speaking , to construct a markov trace on @xmath4 , after having applied the isomorphism @xmath130 and the usual trace of a matrix , we must choose and apply a markov trace on each component of @xmath202 for each @xmath86 . this choice of markov traces is restricted : if @xmath86 and @xmath203 have the same number of non - zero components , the chosen markov traces must coincide ; otherwise , they can be chosen independently . [ [ basic - markov - traces . ] ] * basic markov traces . * + + + + + + + + + + + + + + + + + + + + + + recall the bijection ( [ bij - soc ] ) between @xmath89 and non - empty subsets of @xmath204 . following the theorem , we define some distinguished markov traces as follows : * choose a non - empty @xmath7 and consider the associated @xmath205 . choose a markov trace @xmath206 on @xmath189 for each @xmath207 , and set @xmath208 , @xmath209 , in ( [ rho - n ] ) . * then , in ( [ rho - n ] ) , set all other markov traces @xmath197 with @xmath210 to be @xmath142 . we denote formally the choice of markov traces in the first item by @xmath8 and denote by @xmath211 the resulting markov trace on @xmath5 . we call it a _ basic _ markov trace . every markov traces constructed in the preceding theorem is a linear combination of basic markov traces @xmath211 , where @xmath212 and @xmath8 vary . a map on @xmath4 can be seen , up to @xmath130 , as acting on the direct sum of matrix algebras . this way , for a given @xmath212 , the maps @xmath213 are non - zero only on the summands @xmath120 such that @xmath214 , that is , such that @xmath215 if and only if @xmath207 . as examples : * if @xmath216 then @xmath213 is non - zero only on @xmath120 , for @xmath217 with @xmath10 in position @xmath218 . in this case , @xmath219 ; * we will see that it is enough to consider the situation @xmath220 . in this case , @xmath213 is non - zero only on @xmath120 , for @xmath86 with all parts different from 0 . [ rem - mark ] * ( i ) * by restriction to the subalgebra @xmath1 of @xmath4 , a markov trace on @xmath5 reduces to a markov trace on @xmath2 ( and similarly for @xmath56 and @xmath60 ) . on @xmath221 , there is a unique markov trace up to a normalisation factor . therefore , the choice of the markov traces @xmath222 in the theorem above reduces , for @xmath1 , to a choice of an overall factor @xmath223 for each @xmath195 . this is the result proved in @xcite . in other words , for @xmath1 , the basic markov traces are parametrised by @xmath89 , or equivalently , by the non - empty subsets of @xmath11 . * ( ii ) * let @xmath224 be the space of markov traces on @xmath225 . the space spanned by the basic markov traces @xmath213 is isomorphic to @xmath226 if we restrict to a subspace of @xmath224 of dimension @xmath227 , we obtain a space of markov traces on @xmath228 of dimension @xmath229 . in particular , for @xmath1 , the dimension is @xmath230 . we note that a full description of the space @xmath224 does not seem to be known ( and similarly for the cyclotomic quotients @xmath168 others than @xmath60 ) . let @xmath3 be another indeterminate . we work from now on over the ring @xmath231 $ ] and we consider now all algebras over this extended ring @xmath232 . we sketch a construction of invariants with values in @xmath232 for @xmath233-framed solid torus links . we refer to @xcite for definitions and fundamental results ( as the analogues of alexander and markov theorems ) concerning solid torus links and their @xmath233-framed versions . note that any invariant for @xmath233-framed links is also an invariant of non - framed links , simply by considering links with all framings equal to 0 . the set of classical links is naturally included in the set of solid torus links ( in other words , the braid group is naturally a subgroup of the affine braid group ) . the construction here includes , by restriction to the subalgebras @xmath1 , the construction for @xmath233-framed classical links explained in ( * ? ? ? * section 6 ) . as the construction and the results equally apply to the classical and the solid torus situations , we will simply use the word _ link _ to refer to both types of links . as in @xcite , we denote by @xmath234 the affine braid group on @xmath10 strands , and by @xmath235 the @xmath233-framed affine braid group . the generators of @xmath235 are denoted @xmath236 . the defining relations are ( [ def - aff1])-([def - aff2 ] ) with @xmath36 replaced by @xmath237 and @xmath49 by @xmath238 . the algebra @xmath4 is thus a quotient of the group algebra of @xmath235 by the relation ( [ def - aff3 ] ) . the subgroup of @xmath235 generated by @xmath239 is @xmath234 . the algebra @xmath169 is a quotient of the group algebra of @xmath234 by the relation @xmath240 , @xmath241 . finally , the subgroup @xmath242 of @xmath234 is the classical braid group . [ [ invariants - ptau_luv - from - hh_n . ] ] * invariants @xmath243 from @xmath169 . * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + let @xmath244 be a markov trace on @xmath189 . from the alexander and markov theorems for non - framed links ( see @xcite ) , we construct the invariant @xmath245 , for a link @xmath246 , as follows : @xmath247\ , \ ] ] where @xmath248 is a braid closing to @xmath246 and @xmath249 is the natural morphism from @xmath250 to @xmath56 , given on the generators by @xmath251 , @xmath241 and @xmath252 . from the fact that there is a unique markov trace ( up to normalisation ) on the usual hecke algebras @xmath253 , all the invariants @xmath254 , restricted to the set of classical links , reduce ( up to normalisation ) to the unique invariant coming from @xmath60 , the homflypt polynomial . [ [ invariants - p_ldsboldsymboltauuvgamma - from - hy_dn . ] ] * invariants @xmath255 from @xmath4 . * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + we consider the following map from @xmath256 $ ] to @xmath4 given on the generators by : @xmath257 one proves as in ( * ? ? ? * section 6 ) that , first , @xmath258 extends to a morphism of algebras , and moreover , that the following procedure defines invariants for @xmath233-framed links . let @xmath259 be a markov trace on @xmath5 . for a @xmath233-framed link @xmath246 , the invariant @xmath260 is defined as follows @xmath261 where @xmath248 is a @xmath233-framed braid closing to @xmath246 . from the preceding section , it is enough to consider the basic markov traces @xmath262 . we denote by @xmath263 the corresponding invariant and refer to it as a _ basic _ invariant . if we restrict a basic invariant to the set of classical @xmath233-framed links , it does not depend on @xmath8 , and coincides with the invariants constructed in @xcite . [ quadr - gamma ] in the definition of the maps @xmath258 , a rule @xmath264 would be enough to give a morphism of algebras . the condition @xmath265 is necessary for the construction of invariants . we note that , considering the map @xmath258 is equivalent to changing the quadratic relations @xmath266 to @xmath267 . the role of @xmath3 is therefore to interpolate between different presentations of @xmath4 . [ [ for - non - framed - links - from - hh_n- . ] ] * for non - framed links from @xmath56 @xcite . * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + define @xmath268 , @xmath269 , by the following formulas : @xmath270 similarly , we have the images @xmath271 of @xmath272 in @xmath56 . let @xmath273 $ ] be a set of parameters with @xmath274 . from the results in @xcite ( see remark [ rem - compare0 ] below ) , we have a unique markov trace on @xmath189 , which satisfies in addition @xmath275 the corresponding invariant of non - framed links is denoted @xmath276 . [ rem - compare0 ] in @xcite , the quadratic relation of @xmath56 was @xmath277 and a certain trace @xmath278 , depending on another parameter @xmath279 was constructed . setting @xmath280 , an invariant @xmath281 was obtained , by rescaling the generators , @xmath282 , and then rescaling the trace . this is equivalent , in our approach , to setting @xmath283 in conclusion , we have @xmath284 . [ [ for - zdz - framed - links - from - hy_dn- . ] ] * for @xmath233-framed links from @xmath4 @xcite . * + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + recall that @xmath93 is the set of @xmath63-th roots of unity . fix a non - empty subset @xmath285 and , for each @xmath286 , a set @xmath287 of parameters in @xmath188 $ ] with @xmath288 . denote formally @xmath289 the set @xmath290 with @xmath286 . from the results of @xcite ( and @xcite in the non - affine case ) , we have a unique markov trace , denoted @xmath291 , on @xmath5 which satisfies in addition , for all @xmath209 , @xmath292 where the parameters @xmath293 are given by @xmath294 the corresponding invariant of @xmath233-framed links , we denote @xmath295 . by restriction to classical @xmath233-framed links , the parameters @xmath289 do not appear , and we obtain invariants labelled by @xmath63 and @xmath227 , denoted @xmath296 ( see remarks below ) . [ rem - compare1 ] * ( i ) * in @xcite , the quadratic relation of @xmath4 was @xmath297 and a certain trace @xmath278 , depending on another parameter @xmath279 was constructed . an invariant @xmath298 was obtained . with @xmath299 , this is equivalent , in our approach , to setting @xmath300 in conclusion , we have @xmath301 . note that the construction in @xcite corresponds to the particular case @xmath302 here . * ( ii ) * the restriction of the above procedure to the subalgebra @xmath1 ( the non - affine case ) gives the comparison of invariants constructed here with the invariants studied in @xcite . the invariants only depend on @xmath63 and @xmath227 ( not on @xmath289 ) , and were denoted in @xcite by @xmath303 . * ( iii ) * originally , in @xcite , invariants ( in the non - affine case ) were constructed using @xmath1 with a different quadratic equation , namely with @xmath304 . a certain trace @xmath305 , depending on a parameter @xmath306 was constructed and invariants , denoted @xmath307 in @xcite , were obtained . with the same @xmath308 as above , the rescaling of the generators is now @xmath309 . a short calculation and comparison with the formula in remark [ quadr - gamma ] shows that @xmath20 and @xmath21 are as in item * ( i ) * , while now @xmath310 . as a conclusion , we have , for any classical @xmath233-framed link @xmath246 , @xmath311 we fix as above @xmath227 and @xmath289 . it remains to identify the markov trace @xmath291 in terms of the basic markov traces @xmath211 constructed after theorem [ theo - mark ] . for each @xmath313 , let @xmath8 be obtained by taking , for each @xmath314 , the markov trace @xmath315 in position @xmath218 . then we have @xmath316 denote @xmath317 the markov trace on @xmath5 defined by the right hand side of ( [ compare1 ] ) . to prove the proposition , we need to check that condition ( [ cond - x ] ) is satisfied . for @xmath318 , it is a straightforward verification . for @xmath319 , one may check the equivalent condition @xmath320 . we note that it is enough to take @xmath321 , where @xmath322 is such that @xmath117 where @xmath214 for some @xmath323 , and such that @xmath324 ( otherwise the condition is @xmath325 ) . then the condition can be checked by a straightforward calculation of both sides . one may use : an explicit description of the embedding @xmath326 on the matrix algebras side ( see ( * ? ? ? * section 3.4 ) ) ; and the fact that @xmath327 for @xmath55 ( induction on @xmath328 ) . we note that formula ( [ compare1 ] ) is a triangular change of basis , with inverse @xmath329 in particular , restricting to @xmath1 , we can forget the parameters @xmath289 on one hand , and the choice @xmath8 on the other . the proposition expresses the juyumaya lambropoulou invariant associated to @xmath285 in terms of our basic invariants associated to @xmath7 . the formulas are ( with notations as in remark [ rem - compare1 ] and coefficients @xmath330 as in ( [ compare1 ] ) ) : @xmath331 as consequences of the construction using the isomorphism theorem , we prove several properties of the constructed invariants , focusing essentially on the non - framed links . we emphasize that these properties are valid for all non - framed links ( classical and solid torus ) . let @xmath6 and @xmath212 a non - empty subset of @xmath11 . we denote @xmath332 . let @xmath8 be any choice of @xmath333 markov traces on @xmath189 . the following result says that , for non - framed links , it is enough to consider the situation @xmath220 for each @xmath6 . let @xmath66 be such that @xmath214 , that is , such that @xmath215 if and only if @xmath207 . to @xmath86 , we associate the composition @xmath338 , where @xmath339 are the non - zero parts of @xmath86 and @xmath340 . we have @xmath341 and , in turn , @xmath342 and @xmath343 . let @xmath104 with @xmath117 . for every @xmath55 , by hypothesis on @xmath86 and @xmath94 , there exists @xmath344 such that @xmath345 . then we set @xmath346 . this defines a bijection between the characters @xmath104 with @xmath117 and the characters @xmath347 with @xmath348 . this bijection allows to identify the spaces @xmath120 and @xmath349 . we have moreover @xmath350 . recall the formulas ( [ form - x])-([form - g ] ) giving the images of the generators @xmath351 under @xmath130 . it is then immediate to see that @xmath352 and @xmath353 coincide in the summand @xmath354 , for any @xmath355 in the subalgebra of @xmath4 generated by @xmath351 . [ rem - compare2 ] here we restrict to a classical non - framed link @xmath246 . with the comparison formulas ( [ compare2 ] ) of the preceding section , a straightforward consequence of proposition [ prop1 ] is the corresponding result for the juyumaya lambropoulou invariants . namely , we have @xmath356 where @xmath357 . in this case , this was proved in ( * ? ? ? * proposition 4.6 ) by a different approach . in particular if @xmath358 , we recover the homflypt polynomial . let @xmath66 and @xmath359 for some @xmath360 . we will say that @xmath361 is a _ refinement _ of @xmath86 if @xmath362 can be partitioned into @xmath63 disjoint subsets ( possibly empty ) : @xmath363 , such that @xmath364 for all @xmath365 . as examples , every composition @xmath366 is a refinement of the composition @xmath367 , while the composition @xmath368 is a refinement of every composition @xmath369 . for a permutation @xmath370 , we denote @xmath371 the collection of lengths of the cycles of @xmath372 and we consider it as a composition of @xmath10 ( the order is not relevant here ) . then , it is immediate that a permutation @xmath370 is conjugate to an element of @xmath373 if and only if @xmath371 is a refinement of @xmath86 . for a @xmath233-framed affine braid @xmath374 , we define its _ underlying permutation _ @xmath375 as the image of @xmath376 by the natural group homomorphism from @xmath235 to @xmath377 ( defined by @xmath378 , @xmath379 and @xmath380 ) . note that @xmath381 , for @xmath55 . let @xmath383 . as @xmath258 is a group homomorphism , we have @xmath384 . in @xmath352 , in the matrix corresponding to @xmath66 , the coefficient on the diagonal in position @xmath94 is @xmath385 . and we have @xmath386 . this is equal to 0 if @xmath387 . now @xmath388 if and only if @xmath389 , and this is impossible if @xmath390 is not a refinement of @xmath86 . now we will combine this general result on the isomorphism @xmath130 with two elementary facts . first , if a composition @xmath86 has strictly more non - zero parts than another composition @xmath361 , then @xmath361 can not be a refinement of @xmath86 . second , the closure of a ( @xmath233-framed , affine ) braid @xmath376 is a ( @xmath233-framed ) link with @xmath0 connected components if and only if @xmath390 has exactly @xmath0 non - zero parts . thus we have obtained the following result . finally , combining this corollary with proposition [ prop1 ] for a non - framed link @xmath246 , we conclude our study by determining which basic invariants it is enough to consider given the number of connected components of @xmath246 . [ cor - fin2 ] let @xmath246 be a non - framed link with @xmath0 connected components . every invariant for @xmath246 obtained here from ( affine ) yokonuma hecke algebras is a combination of the following basic invariants : @xmath394 in particular , for a non - framed knot ( @xmath395 ) , it is enough to consider the algebras @xmath396 . [ rem - compare3 ] here we restrict to a classical @xmath391-framed link @xmath246 with @xmath0 connected components . in this particular case , we give the translation of corollary [ cor - fin1 ] in terms of juyumaya lambropoulou invariants , following formula [ compare2 ] ( we write the formulas for the invariants @xmath397 ; the same formulas hold for @xmath398 ) . a straightforward analysis leads to @xmath399 note that the rescaling of the variable @xmath279 comes from the fact that the expressions relating @xmath400 with @xmath401 depend on @xmath402 . if moreover @xmath246 is a classical non - framed link with @xmath0 connected components , with remark [ rem - compare2 ] , it is enough to consider @xmath403 , and we obtain @xmath404 this formula is the generalisation , for @xmath405 , of ( * ? ? ? * theorem 5.8 ) . as a consequence , the analogue of corollary [ cor - fin2 ] holds as well for these invariants : it is enough to consider @xmath406 , @xmath407 , @xmath408 . this generalises ( * ? ? ? * theorem 7.1 ) for @xmath409 , obtained by a different method .

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ since all men have imperfections + hanging bones inside their closets + that they trust no one will notice + absent tips on where to find them , + it will shock no one to learn that + even mighty hiawatha + famous chief of myth and legend + did some things he was not proud of + while a brother in a frat house + with a surly reputation + at an unknown little college + that his father helped to finance + so that he would get admitted + by the shores of gitche - gumee . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ far from loving fields and flowers + and the odor of the forest + as one reads in all the textbooks + hiawatha hated woodlands + and the animals one finds there , + whom he felt were always pooping , + and the plants the critters fed on + down in dank and swampy bottoms , + nearly perfect grounds for breeding + mighty hordes of great mosquitoes + who were always lean and hungry + and equipped with maps and radar + could detect where you were hiding + to inflict their bites and torments , + with their sneaky friends the black flies , + and their angry friends the green flies , + and the rocks ensnared by tree roots + that existed just to trip you + and would look improved as concrete + in foundation for a condo . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus the kindly , thoughtful image + of a noble man of nature + was a total fabrication + of a team of gifted spin docs + hired later for this purpose . + he was really just a tech nerd + who cared only for equations + and explaining all behavior + from the basic laws of physics + armed with only mathematics . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha hated woodlands . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus , instead of lakes and forests , + hiawatha worshipped newton , + whose account of kepler s orbits + built on rules that galileo + had inferred from observation + plus the innocent assumption + of a law of gravitation + was a cosmic inspiration ; + and the brilliant sadie carnot , + whose insightful laws of heat flow + were deduced from working engines + absent microscopic theories ; + and the tragic ludwig boltzmann + who ascribed these laws to counting + but fell victim to depression + when he found no one believed him + and so killed himself by jumping + from an adriatic tower . + hiawatha saw that maxwell s + guessing missing laws of motion + needed for predicting light waves , + was the most transcendent genius , + as was albert einstein s insight + that the speed of light being constant + must mean time was not consistent + and that mass could be converted + into heat and vice versa . + just as clear was that the planck law + must imply debroglie s wavelength + was in force in any matter + so that sharp atomic spectra + and distinct atomic sizes + and the laws of bond formation + came from quantum interference . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus it was that hiawatha + came to be infatuated + with the laws of quantum matter , + which means liquid noble gases , + neutrons in a burnt - out star core , + or just rocks so cryogenic + they can not get any colder , + even with improved equipment , + like the state of too much sliding + on the ice of gitche - gumee + after dark in dead of winter + in an inexpensive loincloth . + pain and danger notwithstanding + quantum matter s simple structure + makes the eager physics tyro + quite unable to resist it . + hiawatha learned how atoms + self - assemble into crystals , + how electrons move right through them , + waving past the rigid ions + thereby making them metallic + in the absence of a bandgap + which arises from diffraction + and prevents the charge from moving + thereby causing insulation , + but by means of wires and doping + with atomic imperfections + when the bandgap is a small one + can be used to make transistors . + in addition to the basics + he learned how electric forces + like those seen in clinging woolens + cause some things to be magnetic + up until the lowly phonon , + quantum particle of sound wave , + storing heat the way that light does + mediates a strong attraction + that can pair up two electrons + causing them to move together + overcoming all resistance + and producing other magic + such as quantum oscillations . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ they were little more than con men . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ at this quite untimely moment + of his fragile student history + when his mind was most suggestive + our poor hapless hiawatha + had the terrible misfortune + to fall in with wicked people + who were little more than con men + and advanced in their profession + making theories of such matter + that were not at all deductive + but instead used mathematics + as a way to sow confusion + so that no one would discover + that their stuff was pure opinion + spiced with politics and chutzpah + so it looked somewhat like science + even though it really was nt . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ how they did this was ingenious + for it s not a simple matter + to produce concrete equations + that are absolutely hokum + and escape without detection + when they represent relations + of some quantities one measures + written down as abstract symbols + that could easily be tested . + what they did was deftly prey on + prejudicial ways of thinking + that their colleagues thought were reasoned + but were simply misconceptions , + generated during training + they had all received as students , + that the properties one wanted + were completely universal + so details did not matter . + but the data did not say this + and , moreover , had they done so + there would have been no good reason + to think any more about it . + so , while everyone was watching , + they swapped in some new equations + that they said would solve the problem + on account of being much simpler + but in fact described a system + very different from the first one + and , moreover , was unstable , + balanced at competing phases , + so that nobody could solve it + thus betraying the deception . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ adding to the dazzling brilliance + of this coldly thought - out swindle + they declared it _ fundamental _ + so that all the strange creations + made by people trying to solve it + and quite clearly not succeeding + proved it was a fount of deepness + one should struggle to unravel + even if it took a lifetime . + as a nifty added bonus + any hint you dropped in public + that it might have no solution + simply meant you were nt a genius , + told the world that you were stupid , + that you were a hopeless failure + who should not command a pencil . + no one wanted to admit this + so they d cover up their failure + and pretend that they had solved it + even though they clearly had nt . + this succeeded , for the most part , + but in one respect it did nt , + for their desperate need to publish + and thereby maintain their funding + caused a massive flood of papers , + each quite different from the others , + to descend upon the journals + and to overwhelm and clog them . + this would have been very funny + had it not been so pathetic . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ it just meant you were nt a genius . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha bought the story + took the bait , hook , line , and sinker + and , like many other students + who d been victimized before him , + got convinced that his strong math skills , + far exceeding those of others , + would reveal nature s mysteries + when he solved the hubbard model + and its child the t - j model + and the lattice kondo model + and the quantum spin glass model , + all of which possessed the feature + that no human being could solve them . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ nature has a sense of humor , + as one learns by working with it , + but it is an opportunist , + so that life s most bitter lessons + often wind up learned the hard way + when it moves to take advantage + of a single bad decision + and compound it with some mischief + custom made for the occasion . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ just when he d resolved to strike out + on his suicidal mission + there occurred a bold announcement + in a well - known german journal + that a tiny lab near zrich + had discovered a material + with the structure of perovskite + made of oxygen and copper + and some other stuff like strontium + that when cooled to thirty kelvin + lost all traces of resistance . + this event was simply shocking + for existing quantum theory + said it had to get much colder + for this special thing to happen , + as did all the careful surveys + of the properties of metals , + which were very comprehensive + and agreed well with the theory . + since the chemists were ambitious + to somehow transcend this limit , + which they thought too academic , + and someday kill all resistance + using no refrigeration , + there ensued a feeding frenzy + worthy of a horror movie , + like what happens when a trawler + dumps its hold of tuna entrails + off a reef in north australia . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ one example of this madness + was the _ physics woodstock _ conference + that took place in mid - manhattan + shortly after the announcement + where attendees got together , + comandeered a giant ballroom , + and gave talks not on the program + in a special all - night session + dedicated to the cuprates + which was packed to overflowing . + there was talk of maglev transport , + new kinds of computer circuit , + mighty , compact little motors + and efficient power cables , + all of which would soon be coming + thanks to this momentous breakthrough . + but it turns out we do nt have them + for they were nt a big improvement + over things we had already + and were hopelessly expensive . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ then there were the frantic searches + to find compounds that were better , + which one knew could be accomplished + if one spent enough time looking , + since this stuff had lots of phases + subtly different from each other , + and there had to be a best one . + there was very rapid progress + culminating in a patent + for a more complex material + in the same broad class of structure + which performed at ninety kelvin , + so much higher than the theory + would allow to ever happen + even with extreme assumptions + that one knew it was in trouble . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ almost overnight one found that + every spectrum known to science + had been taken on a cuprate . + their alleged profound importance + was , of course , a major factor , + but what mattered most was tactics . + without need to tell one s funders , + since it could be done so quickly , + one could telephone a chemist , + cut a deal to get some samples , + put them in one s apparatus + presto ! out would come a paper + that would instantly get published + even if it was a stinker . + this produced a pile of data , + growing without bound , like cancer , + that completely overwhelmed you + by being mostly unimportant , + like the growing list of options + coming from your cable service . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ presto ! out would come a stinker . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ often spectra were nt consistent , + but , instead of getting angry + as one would have in the old days , + one would handle it maturely + and just chalk it up to errors + that occur when one is hasty + or has had bad luck with samples . + but this tolerance , it turns out , + was a bargain with the devil + for it later was discovered + that enormous variation + was endemic to the cuprates , + and that things not reproducing + due to complex phase inclusions , + foreign atoms in the sample , + careless oxygen annealing , + surface preparation methods , + and a thousand other factors + was essential to their nature . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ sadly , by the time this surfaced + shameful habits of denying + that the differences existed + had become enshrined in writing , + and so wedded to the culture , + that they could not be corrected . + it was now accepted practice + in a public presentation + of experimental findings + not to mention other data + even if your own group took them . + grounds for this were rarely stated , + other than the innuendo + that one s sorry competition + were a hopeless bunch of bozos + who did not know how to measure + and therefore could not be trusted . + it was likewise viewed as kosher + to make up a little theory + or adopt somebody else s + that gave all your findings meaning + although not those of your colleagues , + which were , sadly , so imperfect + they were simply inconsistent . + but one never heard recanting , + since it would have meant admission + that one s judgement had been faulty . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus the cuprates weird caprices + long escaping understanding + transformed into pseudotheories + that , like gods on mount olympus , + were political creations + that could not be killed with reason + and , empowered as immortals , + took control of their creators , + warred among themselves for power , + schemed to have a lot of children , + and , in general , made a circus + of the scientific method . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha , being a student , + and , quite frankly , rather callow + did not have the slightest inkling + that such nonsense ever happened . + he believed the claims of science + to be rather more objective + than competing kinds of knowledge + on account of its precision + and the fact that you could test it . + rather than the yawning snake pit + seething with disinformation + that was really there before him , + certain death for young beginners , + he saw just a chance for glory + something of immense importance , + judging from the acrimony + coursing through the talks and papers , + and a vast supply of data + on which one could build a theory + and thereby become a hero , + much the way the dumber brother + of the famous brave odysseus + that no one has ever heard of , + sure he could outfox the sirens , + ordered that the men unbind him + and , of course , succumbing quickly + dove right in and bashed his brains out . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha s misconceptions were not shared by everybody . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha s misconceptions + of the nature of the problem + he was setting out to conquer + were not shared by everybody . + just as buzzards , with keen noses , + circling high above their breakfast + wait until it can not hurt them , + to swoop down and get to business , + and ichneuman wasps impregnate + larval caterpillar victims + with some eggs that grow to eat them , + thus not let them reach adulthood + when they might be hard to handle , + hiawatha s crafty mentors + sensed that science had stopped working + in the sub - field of the cuprates , + as it had before in others + where their scams had been successful . + smelling death was close upon it , + they resolved the time was ready . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ what ensued was simply awesome , + destined to go down in legend . + they proposed a cuprate theory + so magnificent in concept , + so much bolder than the others + that it blasted them to pieces + like some big atomic warhead , + so outshined them in its glory + like a nova in the heavens + that it blinded any person + who would dare to gaze upon it . + cuprates did these things , it stated , + just because a quirk of nature + made them like the _ hubbard model _ , + which , as had been long established , + did some things quite fundamental , + not yet known to modern science , + which explained the crazy data , + so to understand the cuprates + one would have to solve this model . + how colossal ! how stupendous ! + it was absolutely foolproof ! + no one could disprove this theory + with existing mathematics + or experimental data + for exactly the same reasons + nor could they admit they could nt , + so they d spend their whole lives trying , + blame themselves for being so stupid , + and pay homage in each paper + with the requisite citation ! + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ they left clues in great abundance + that they d made a vast deception + far surpassing p. t. barnum s + most creative whims and musings + trusting that no one would catch them + on account of being so guileless , + which they knew was part of science , + rather like the clever killer , + sure he can outsmart columbo , + leaving marks upon the crime scene + then in later verbal sparring + hints at them in brazen taunting . + one was that its short description , + resonating bonds of valence , + was the name that linus pauling + used for common bonds of benzene , + something so profoundly different + from the physics of the cuprates + that its use on this occasion + seemed to show a lousy word sense . + but , in fact , it was inspired , + for the permanent confusion + left by its uncertain meaning + like the data it reflected , + was defense against attackers , + made it very hard to target , + left its enemies bewildered . + and the thoughtful usurpation + of a well - established brand name + had the lovely added feature + of dispatching pesky pauling , + who had always been a nuisance , + down to davy jones s locker + in the minds of younger people . + _ getting rid of pesky pauling . _ there was also the assertion + running rampant through the theory + that the essence of the cuprates + was coulombic insulation , + which , on close inspection , turned out + no one could define precisely , + with a few concrete equations , + but was nonetheless a concept + people thought they comprehended , + like the fancy secret contents + of competing brands of toothpaste + that , of course , are total fictions + made up during lunch by ad guys . + but the best clue by some margin + was the _ deus ex machina _ + known as gutzwiller projection , + which began life as a method + for controlling the equations + but was morphed on this occasion + to a monsterous distortion + of the basic mathematics + on the grounds it was insightful . + but , in fact , it came from nowhere , + and was just a simple dictat + that an off - the - shelf conductor + could not be a quantum magnet + while one forced it to become one + thus creating awful conflict + when , in fact , there simply was none . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha , being clever , + quickly saw that he could do this , + saw that such manipulations + were , in fact , extremely easy , + that a high school kid could do them , + once he got the key idea + that one should evade the problem + of deducing the behavior + from the actual equations + by declaring that some answer + was correct because one said so + and proceeding to defend it + with a lot of complex symbols + simply cooked up to confuse things . + thus emboldened to abandon + his perverse outdated fear of + uncontrolled approximations + hiawatha bit the bullet + and jumped into cuprate theory + with the fury of a madman , + doing reckless calculations + based on nothing but some gas fumes + that produced some fine predictions , + as one was inclined to call them , + matching some existing data + but , of course , not matching others , + since they were not all consistent . + he would then just pick and choose them + as one would an orange or lemon + in the local supermarket + and declare the rest defective . + then he wrote up his conclusions + in a little physics paper + loaded up with fearsome symbols + proving that he had credentials + to make all these speculations , + sent it in for publication + and then found an awful problem + he had not anticipated . + for the paper to be published + it must get past refereeing + which , in theory , was for stopping + false results from being reported + but , in practice , was to censor + anyone whose work you hated , + somewhat of a sticky wicket + for someone who s main objective + was to publish speculation . + hiawatha soon discovered + though the process of rejection + that his papers could not make it + if they championed new ideas + or in any way conflicted + with the viewpoints of the experts + which , of course , were simply made up . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus the mighty hiawatha + found his plans to be a scholar + had an unexpected down side + that would later prove quite fatal + in that he was forced to pander + in his writing for the public + to a set of flakey concepts + that he d found extremely useful + but had not had time to question , + in exchange for recognition + needed for career advancement . + for a while it did not matter + but the problem slowly festered + and one day poor hiawatha , + waking to a huge disaster , + found himself up to his eyeballs + in a soup of black corruption . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ he would simply pick and choose them . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha s revelation + took a while to find its footing + for , as happens in such cases , + many awful misconceptions + were embedded in his thinking + where they had been put on purpose + and could only be uncovered , + if at all , through painful hours + scrutinizing tiny details , + contemplating reams of data , + finding out who s stuff was careful , + tracking down suspicious rumors , + reading through a mass of papers , + slowly tossing out the bad ones , + racking up the airline mileage + going to humongous meetings , + thereby building up a fact base + cleansed of all manipulations . + over time , as things got clearer , + hiawatha grew unhappy + trying to reconcile his viewpoint + with the facts that he had winnowed , + always finding that he could nt . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha studied transport + both electrical and thermal + that , one argued , showed the absence + of the landau fermi surface + symptomatic of a metal + thereby proving one was dealing + with a strange new state of matter . + but he found in every instance + that a sample made its coldest , + so one knew what one was doing , + either showed disorder problems + generated by the chemists + or agreed with classic theory . + thus , like all those dot - com profits + that they claimed would make you wealthy , + but , in fact , were nonexistent , + arguments for novel physics + built upon the facts of transport + did not hold up on inspection . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha studied optics + by and large his favorite spectrum + for he knew that light reflection + measured dielectric functions + in a way that used no theory , + and it showed how loose electrons + moved about and caused the bonding . + but , alas , the data varied + from one sample to another + even after years of efforts + to ensure that they were stable ! + this left lack of clear consensus + even over things that mattered . + understanding why this happened + was not really rocket science , + for the kramers - krnig process + amplified the defect signals + that were there in great abundance , + even though they all denied it , + and depended on the process + by which one prepared the sample , + something different for each grower + and a closely - guarded secret . + also , things would change with doping , + something very hard to measure + and which often was nt constant + as one moved across the sample + due to troubles in the furnace + which they claimed they d licked but had nt . + thus the stories of new physics + built upon results of optics , + like the troubled u. s. census + or the the streets of downtown boston + after weeks of too much snowing , + were polluted by disorder , + and , moreover , were deceptive + in that aspects of the spectra + that were reasonably stable + like the strange non - drude lineshapes + happened at such tiny wavelengths + one could plausibly ascribe them + to a nearby phase transition + rather than the state in question . + thus the stories were fantastic , + and , like those that richard nixon + told while he was in the white house , + or that pop star michael jackson + claimed occurred in los olivos + for the pleasure of the children , + in the end would not hold water . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus the stories were fantastic . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha studied neutrons + which he found he liked immensely + since they flowed from a reactor + with big purple signs upon it + warning you of radiation + that would kill you if allowed to , + since the neutrons went right through you + but would sometimes choose to stop there + and decay like little time bombs , + thus inducing stomach cancer . + but they went through cuprates also , + and that made them very useful , + since a few of them would scatter , + and detecting those that did so + gave you lots of information + from down deep inside the sample , + such as how the atoms ordered , + how they moved when something hit them + and if they were little magnets . + but the bad news was the signal + was quite small and hard to measure , + so one needed a detector + bigger than a dempsey dumpster + and a truly mammoth sample , + leading to big compromises + in the sample growing process + they preferred deemphasizing + but one knew was wreaking havoc + on the meaning of the data . + they would also never tell you + what the measurement itself was , + since the neutron kinematics + made it sensitive to factors + like the speed spread of the neutrons + and the tip of the detector + and the path on which one moved things + to survey deflection angles + that were messy and annoying , + so they d first massage the data + using big computer programs + to remove these nasty factors + and report the program output , + representing you should trust it + just because they were the experts . + but , of course , there were those upgrades + and the quiet little tweaking + that one always did at run time + that one never heard reported . + once he caught these key omissions + hiawatha got suspicious , + and quite quickly found the practice + of reporting neutron spectra + in some secret custom units + given names like `` counts '' to fool you , + like those helpful content labels + found on packs of sandwich slices + listing salt and beef by - products , + thus preventing one from telling + there was very poor agreement . + all this made a clearer picture + but it also meant the data + like the air - brushed prints in _ playboy _ + were , in fact , manipulated , + and that many strange behaviors + like the famous funny phonon + dogma said was nonexistent + got removed as standard practice + on the grounds they should not be there . + thus his plan to use those spectra + to pin down the magnetism + present sometimes in the cuprates + on account of all the errors + ended up a dismal failure . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ever helpful content labels . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha studied currents + made when cold electrons tunnel + right across an insulator + where they should have been forbidden , + something very close to magic + rather like the twinkly transport + people undergo on star trek , + and it s also quite revealing + of important quantum pairing + that goes on inside the cuprate . + in the old days one would simply + oxidize a thin - film sample , + coat the oxide with another , + solder on two tiny contacts , + dunk the whole thing into vapors + made so cold that they were liquid , + then just measure plain resistance + of the two protruding wires , + which would vary with the voltage + thus producing useful data . + hiawtha read these papers + with a mounting sense of horror , + for the wild disagreement + even in the basic features + from one sample to another + was so large it left one breathless . + and , of course , the accusations + that the other guys were morons + who just could not make good junctions + rose to unmatched heights of grandeur + even though the real villain , + obvious from spectral sharpness , + was the sample variation . + hiawatha s indignation + escalated when he found that + over time this fact got buried + since each group soon found a method + of preparing stable samples + different from that used by others + and producing different spectra + that they marketed as products , + thus evading any need to + answer penetrating questions . + an important fact , however , + that emerged from all these studies , + was that steady lossless currents + could indeed be made to flow from + films of lead into the cuprates + if one made a pitted surface , + proving that the state of matter + operating in the cuprates + was not new and was not different . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha studied spectra + made when light shined on a sample + causes it to lose electrons + which fly out in all directions + and one can detect by counting , + thus obtaining information + of their status in the sample + just before the light removed them . + hiawatha saw at once that + peaks for plain undressed electrons + that were not supposed to be there + in this great new state of matter + always were and had a sharpness + at the resolution limit + of the latest new detector + for the special ones at threshold , + where one knew what one was doing . + in addition they were beaming + in a lovely fourfold pattern + with the symmetry of d - wave , + something that had been suggested + they might do if they were simple , + just like those in other metals . + thus the arguments for strangeness + based on counting these electrons + lost their force as things got better , + and in time were proved a failure . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha studied muons , + which he thought were even neater + than the more prosaic neutrons , + since they came from atom smashers + that could also quickly kill you + if you chose to be so careless , + but they d stop inside much better + and once there , decay to gammas + that were easily detected + since they d even go through concrete , + and , moreover , they d be beaming + in the muon s spin direction + just before it went to heaven . + thus , implanted in a cuprate + they d arrest at some location + known to no one but their maker + and precess like little searchlights , + if there was some magnetism , + thus allowing you to see it + way deep down inside the sample . + thus with knowledge of their trapping + and a batch of big detectors + one could then back out the distance + of magnetic penetration . + hiawatha found this distance + shortened with increasing doping + just as theory said should happen , + if one forced the hubbard model + not to be a quantum magnet + by just saying that it was nt , + which might well have been important + had it not been for the problem + that this depth would not continue + to decline with increased doping + but instead would turn and lengthen . + this effect was quite perplexing , + since no theory of the cuprates + even twisted hubbard models , + could account for such behavior , + for it violated sum rules , + hence one just did not discuss it . + but the meaning was transparent + if one faced the facts with courage , + for the samples were degrading + in extremes of overdoping + in some ways that were nt predicted + and , moreover , were nt detected + by techniques except for this one . + this , in turn , implied these problems + might occur at other dopings + and likewise escape detection + or , what s worse , be used to argue + that new physics was occurring + when , in fact , it was just garbage . + thus the trail blazed by muons + led out in the woods to nowhere . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ dental pamphlets make you tired . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha studied spin flips + that the nuclei of atoms + undergo in great big magnets + near a radio transmitter + causing them to be antennas , + which absorb with complex lineshapes + one can read if one s a genius + but not , sadly , if one is nt , + since they , by and large , consist of + just a simple blobby bell curve + with a width and displaced center , + to which one must give some meaning + not a simple undertaking . + thus the all - important knight shift + and spin - lattice relaxation , + noms de plume for width and center , + vastly different for the copper + and the oxygen of cuprates , + were the source of endless theories , + often very thought - provoking , + stunning in sophistication , + but , like all those glossy pamphlets + found in waiting rooms of dentists + urging you to practice flossing , + soon began to make you tired , + since the data mainly showed you + that the stuff was not a metal + in the sense of gold or iron + which , in fact , one knew already + and was not a revelation . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha studied structure + of the surfaces of cuprates + freshly cleaved inside a vacuum + so that air would not get on them + and then probed with tiny needles + one could move with great precision , + by adjusting some piezos + on which everything was standing . + what he found was quite disturbing , + for while atoms at the surface + all had unperturbed positions , + showing that the cleave succeeded , + there were also complex patterns + on the scale of twenty atoms + that appeared to be diffraction . + this behavior might have come from + atoms underneath the surface + that were missing or defective + or some novel magnetism + of a kind unknown to science , + but the thing that so upset him + was that quantum interference + of the kind that he was seeing + could not happen if the lifetimes + were as short as he had thought them , + and which had been used to argue + for a brand new state of matter . + thus he soberly concluded + that this matter was nt different + and the whole confounded story + was a misinterpretation + of a plain materials problem . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus the mighty hiawatha + through the patient application + of the practices of science + tested over generations + slowly sloughed off misconceptions + and , in face of mounting failure , + sadly came to the conclusion + he d been taken to the cleaners . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ given all the clever swindles + lurking there to take our money , + that , of course , are part of living , + like a virus for pneumonia + or a hungry venus fly trap , + we must all be very thankful + that the celebrated law of murphy + strikes at random without warning + causing even brilliant concepts , + that appear completely foolproof + like distributing tobacco + or the business plan of enron , + to sometimes become derailed + due to something unexpected + one was sure could never happen , + like a lawsuit from consumers , + that requires intervention + of the most creative nature + to prevent strategic meltdown . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ sure enough , that s just what happened . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ as it turns out , the idea + that the conflict in the models + one was using for the cuprates + due to nearby phase transitions + would both hamper their solution + and engender rampant fibbing , + thus enshrining mass confusion + one would then call proof of meaning + with no need to fear exposure + had the unexpected weakness + that someone might _ solve _ the model + using tons and tons of money + and some capable computers + to a crude degree sufficient + to unmask the real problem + thus revealing the deception . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ sure enough , that s just what happened . + when the cuprates were discovered + and the whole endeavor started + one had not the slightest worry + that these guys would ever solve it , + since the accuracy needed + was not clear in the beginning , + so they uniformly low - balled + with the too - familiar outcome + that results were inconsistent . + so they quarrelled over method + and who had convergence problems + and whose code was most clairvoyant + even though a child could see that + they were different apparati , + so the test that they were working + was agreement with each other . + but , unlike the other issues + that had come and gone before it , + cuprates lingered on as timely + long enough to cause a shake - out , + for the money kept increasing + even as machines got cheaper + and their power kept on growing + due , of course , to needs of gaming , + rather than the ones of lanczos + or the quantum monte carlo + that one used for basic physics . + so the robots kept on plugging + as their owners upped the ante + very slowly , as did wagner + when composing _ ring _ and _ tristan _ + and their stuff began converging ! + there , of course , was no agreement + over matters of the phases + such as whether it conducted + when one cooled it down to zero , + since a crystal of electrons + was one state in competition . + but at length scales one could access + there was clearly dissipation + of a most peculiar nature + in the dielectric function + and the quantum magnetism , + just exactly as predicted + by an ancient bunch of papers + over quantum phase transitions , + which these guys had never studied + since it was too esoteric + and had not been seen in nature + and was hated by their funders . + but the thing that really clinched it + was the endless disagreement , + that got worse as things proceeded + and was very clearly cronic , + over type and shape of edges + that would best produce convergence , + since one found that subtle changes + in the way one built the model + would turn on and off the striping + and therefore the insulation , + so that whether it was present + in the limit of large sample + simply could not be determined + with the codes that they had written . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ soon their stuff began converging . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ this , of course , was a disaster + for the plan to keep things murky + and required drastic action + to somehow repair the damage + all this progress had created , + and prevent these guys from seeing + what was right beneath their noses . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ and one was not disappointed . + once again a flash of brilliance + like a great big city - buster + brighter than the sun at midday , + blazed across the dome of heaven + toward its final destination + in the guinness book of records . + they declared the problem _ over _ ! + the computer guys had solved it ! + for their codes had proved the cuprates + were indeed the hubbard model , + and that s why the stuff conducted . + thus there was no urgent reason + to pursue the matter further ! + one could zero out their budgets + with no loss to human knowledge + and , in fact , perhaps improve it + since this money was incentive + to continue calculations + that were clearly unimportant + and report them in the journals + thus just adding to the clutter . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha , now much wiser + through his labors as a scholar + and , quite frankly , some maturing + watched these things unfold before him , + as he had on past occasions , + but this time with eyes wide open + and was filled with understanding . + it was not a happy moment , + for it meant that his own judgement + as to what was good and worthy + had been faulty from the outset , + something for which he must answer . + but instead of indignation + and a passion to get even + that he might have felt when younger + hiawatha , deep in thinking , + found himself consumed with sadness . + he was not the only victim , + for the guys who manned those robots , + and were heroes of the cuprates + for through focussed dedication + they had stumbled on the answer + that the models were unstable + and did _ not _ describe the cuprates , + since a modest perturbation + would profoundly change their nature + were about to have their triumph + snatched from them by clever scoundrels + who , pretending to befriend them , + would then redefine their output + to mean something that it did nt , + thus protecting their investment , + but , of course , destroying others . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha s knowing sadness , + like the darkening at twilight + or a gathering storm in winter , + slowly gained in strength and deepened + as he spent time in reflection , + working through the implications + of the things that he had witnessed + for the cause of noble science + that thus far had so beguiled him . + it would simply not be manly + to pretend he was nt guilty + of ignoring frequent warnings + that the needed path to nature + was obscured or nonexistent . + it was clear that he d been foolish + to have bought this awful fiction + and that blame must fall quite squarely + on himself and not on others . + but this candid _ mea culpa _ , + made in silence where it mattered , + while it comforted his conscience , + did not quite assuage the wounding , + for it begged the nagging question + of how they could have succeeded + in hoodwinking all the people + for so long without some doubting . + it was simply not an option + to presume these guys were stupid , + since the instruments they dealt with , + often built by hand from nothing , + needed great sophistication + to deploy and mine for data . + there was clearly something larger + and extremely fundamental + working in the group dynamic + that involved access to funding + and the policy of journals + and the need to service markets + and the mythos of the subject + one must use to make a living + that these crooks had first deciphered , + then reduced with understanding , + then usurped to do their bidding . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ hiawatha , turning inward , + thought for weeks about this problem + during which he was obnoxious + due to his preoccupation . + but at last he got an answer + that made sense and was quite simple , + thus withstanding occam s razor , + so he thought that he believed it . + when he d set out on his mission + he had understood the challenge + of the mastery of nature + but not basic economics + and the fact that art and science + both require sacrifices + of a clear financial nature + that one sometimes just ca nt handle + nor , in fairness , should one do so + since a good guy pays the mortgage + and supports the kids in college + and the other things a body + has to do to keep the lights on . + but , in fact , the compromises + that one makes as part of living + such as saying what one has to + for maintaining healthy cash flow + often toss big monkey wrenches + in the fine machine of science + and can stop it altogether + in conflicted situations . + then the body , badly weakened , + barely able to keep breathing , + gets exploited by diseases , + such as villains lacking scruples + who descend on it like termites + to a house that s been neglected , + wreaking terrible destruction + on the lives of those affected . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the conclusion of this story + is well known from all the textbooks . + hiawatha never wavered + in his deep respect for physics , + but he came by this adventure + to the deeper understanding + that to get things done that mattered + often was a social question , + not just logical abstraction , + and , as well , a part of nature , + just the thing he thought he d hated + and had thrilled at desecrating + as a tender freshman student + in the little private college + by the shores of gitchee - gumee . + it was true that all the creatures + living in those swamps and woodlands + generated lots of pooping , + but then so did real people , + and the people poop was stronger , + so that one could not ignore it . + but one really would not want to , + for the lesson of the cuprates + was that lack of understanding + of these basic group dynamics , + was a recipe for failure + since they were the central issue + for most things that were essential . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ thus the mighty hiawatha + turned his mind to other problems + such as how to use resources + that were his by luck and birthright + through the power of his father + which he d been inclined to squander , + but now realized he should nt . + thus he studied like a madman + to acquire the skills of statecraft , + such as how to plan a project , + how to give effective orders , + how to make sure they were followed , + how to get things done with meetings , + and to leave the money grubbing + up to folks his father hired + such as all those gifted spin docs + who created key revisions + necessary for his image + to be something people honored . + thus the pain of too much sliding + on the ice in dead of winter + in an inexpensive loincloth + and his other misadventures + got removed , as did the cuprates , + from his long official story . + but the memory persisted + and it helped to make him wiser + for , of course , as he got older + he had many bad encounters + not so different from the cuprates . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ but whenever he was troubled + with a problem that would vex him + he would cheer himself by thinking + of the special room in hades + into which these happy people + on account of their transgressions + would be ushered when they bagged it + and be stuck in there forever , + forced to listen to each other + giving lectures on the cuprates . + it would always leave him smiling . + _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

in this paper we study the optimal stopping problems in an uncertain environment . the classical solution to the optimal stopping problems based on the dynamic programming principle assumes that there is a unique subjective prior distribution driving the reward process . however , for example , in incomplete financial markets , we have to deal with multiple equivalent martingale measures not being sure which one underlies the market . in fact under the presence of the multiple possible distributions , a solution of the optimal stopping problem by maximization with respect to some subjective prior can not be reliable . instead , it is reasonable to view the multitude of possible distributions as a kind of _ model uncertainty risk _ which should be taken into account while formulating an optimal stopping problem . here one may draw on concepts from the theory of risk measures . as the established generic notion for static risk assessment at present time @xmath14 , convex risk measures are specific functionals @xmath15 on vector spaces of random variables viewed as financial risks ( see @xcite and @xcite ) . they typically have the following type of robust representation @xmath16 - \gamma_{0}({\mathrm{q}})\bigr\},\end{aligned}\ ] ] where @xmath17 denotes the set of probability measures which are absolutely continuous w.r.t . a given reference probability measure @xmath18 and @xmath19 is some penalty function ( see e.g. @xcite and @xcite ) . in this way , model uncertainty is incorporated , as no specific probability measure is assumed . moreover , the penalty function scales the plausibility of models . turning over from static to dynamic risk assessment , convex risk measures have been extended to the concept of conditional convex risk measures @xmath20 at a future time @xmath21 which are specific functions on the space of financial risks with random outcomes ( see @xcite , @xcite and @xcite ) . under some regularity conditions , they have a robust representation of the form ( see e.g. @xcite , @xcite or ( * ? ? ? 11 ) ) @xmath22-\gamma_t({\mathrm{q}})\bigr\},\end{aligned}\ ] ] where @xmath23 is a ( random ) penalty function and @xmath24 consists of all @xmath25 with @xmath26 as in ( [ rhostatic ] ) , the robust representation mirrors the model uncertainty , but now at a future time @xmath8 in recent years the optimal stopping with families @xmath27}$ ] of conditional convex risk measures was subject of several studies . for example , the works @xcite and @xcite are settled within a time - discrete framework , where in addition the latter one provides some dual representations extending the well - known ones from the classical optimal stopping . optimal stopping in continuous time was considered in @xcite , @xcite , @xcite , @xcite . all these contributions restrict their analysis to the families @xmath27}$ ] satisfying the property of time consistency , sometimes also called recursiveness , defined to mean @xmath28 hence the results of the above papers can not be , for example , used to solve optimal stopping problems under such very popular convex risk measure as average value at risk . the only paper which tackled the case of non time - consistent families of conditional convex risk measures so far is @xcite , where the authors considered the so - called distorted mean payoff functionals . however , the analysis of @xcite excludes the case of average value at risk as well . moreover , the class of processes to be stopped is limited to the functions of a one - dimensional geometric brownian motion . the main probabilistic tool used in @xcite is the skorokhod embedding . in this paper we consider a rather general class of conditional convex risk measures having representation with @xmath29 $ ] for some lower semicontinuous convex mapping @xmath1.$ ] the related class of risk measures @xmath15 known as the class of _ divergence risk measures _ or _ optimized certainty equivalents _ was first introduced in @xcite , @xcite . any divergence risk measure has the representation @xmath30\ ] ] with @xmath31,\ , y\mapsto\sup_{x\geq 0}(xy - \phi(x)).\ ] ] ( cf . @xcite , @xcite , @xcite , or appendix [ appendixaa ] ) . here we study the problem of optimally stopping the reward process @xmath32 where @xmath9}$ ] is an adapted nonnegative , right - continuous stochastic process with @xmath33 } y_t$ ] satisfying some suitable integrability condition . we do not assume any time - consistency for the family @xmath34 and basically impose no further restrictions on @xmath35 . our main result is the representation @xmath36\right\},\end{aligned}\ ] ] which allows one to apply the well known methods from the theory of ordinary optimal stopping problems . in particular , we derive the so - called additive dual representation of the form : @xmath37}\big(\phi^{*}(x+ y_{t } ) - x - m_{t}\big)\right],\end{aligned}\ ] ] where @xmath38 is the class of adapted martingales vanishing at time 0 . this dual representation generalizes the well - known dual representation of rogers , @xcite . the representation together with can be used to efficiently construct lower and upper bounds for the optimal value by monte carlo . the paper is organised as follows . in section [ setup ] we introduce notation and set up the optimal stopping problem . the main results are presented in section [ main_results ] where in particular a criterion ensuring the existence of a saddle - point in is formulated . section [ discussion ] contains some discussion on the main results and on their relation to the previous literature . a monte carlo algorithm for computing lower and upper bounds for the value function is formulated in section [ mc ] , where also an example of optimal stopping under average value at risk is numerically analized . the crucial idea to derive representation is to consider the optimal stopping problem @xmath39 where @xmath40 denotes the set of all randomized stopping times on @xmath11.$ ] it will be studied in section [ optimalrandomizedstoppingtimes ] , where in particular it will turn out that this optimal stopping problem has the same optimal value as the originial one . finally , the proofs are collected in section [ proofs ] . let @xmath41 be a probability space and denote by @xmath42 the class of all finitely - valued random variables ( modulo the @xmath4-a.s . equivalence ) . let @xmath43 be a young function , i.e. , a left - continuous , nondecreasing convex function @xmath44 $ ] such that @xmath45 and @xmath46 . the orlicz space associated with @xmath43 is defined as @xmath47<\infty~ \mbox { for some $ c>0$}\big\}.\ ] ] it is a banach space when endowed with the luxemburg norm @xmath48\leq 1\right\}.\ ] ] the orlicz heart is @xmath49<\infty~ \mbox { for all $ c>0$}\big\}.\ ] ] for example , if @xmath50 for some @xmath51 then @xmath52 is the usual @xmath53space . in this case @xmath54 where @xmath55 stands for @xmath53norm . if @xmath43 takes the value @xmath56 , then @xmath57 and @xmath58 is defined to consist of all @xmath59essentially bounded random variables . by jensen inequality , we always have @xmath60 in the case of finite @xmath61 we see that @xmath62 is a linear subspace of @xmath63 which is dense w.r.t . @xmath64 ( see theorem 2.1.14 in @xcite ) . let @xmath65 and let @xmath66 be a filtered probability space , where @xmath67}$ ] is a right - continuous filtration with @xmath68 containing only the sets with probability @xmath14 or @xmath69 as well as all the null sets of @xmath70 . furthermore , consider a lower semicontinuous convex mapping @xmath1 $ ] satisfying @xmath71 for some @xmath72 @xmath73 and @xmath74 its fenchel - legendre transform @xmath75 is a finite nondecreasing convex function whose restriction @xmath76 to @xmath77 is a finite young function ( cf . lemma [ optimizedcertaintyequivalent ] in appendix [ appendixaa ] ) . we shall use @xmath78 to denote the orlicz heart w.r.t . @xmath79 then we can define a conditional convex risk measure @xmath80}$ ] via @xmath81- { \mathbb{e}}\left[\left.\phi\left(\frac{d{\mathrm{q}}}{d{\mathrm{p}}}\right)\right |{\mathcal f}_{t}\right]\right)\ ] ] for all @xmath82 where @xmath83 denotes the set of all probability measures @xmath3 which are absolutely continuous w.r.t . @xmath4 such that @xmath84 is @xmath59integrable and @xmath5 on @xmath6 note that @xmath85 is @xmath59integrable for every @xmath86 and any @xmath87 due to the young s inequality . consider now a right - continuous nonnegative stochastic process @xmath88 adapted to @xmath89 furthermore , let @xmath10 contain all finite stoping times @xmath90 w.r.t . @xmath89 the main object of our study is the following optimal stopping problem @xmath91 if we set @xmath92 for @xmath93 and @xmath94 otherwise , we end up with the classical stopping problem @xmath95.\ ] ] it is well known that the optimal value of the problem may be viewed as a risk neutral price of the american option with the discounted payoff @xmath96}$ ] at time @xmath97 however , in face of incompleteness , it seems to be not appropriate to assume the uniqueness of the risk neutral measure . instead , the uncertainty about the stochastic process driving the payoff @xmath98 should be taken into account . considering the optimal value of the problem ( [ stoppproblem ] ) as an alternative pricing rule , model uncertainty risk is incorporated by taking the supremum over @xmath83 where the penalty function is used to assess the plausibility of possible models . the more plausible is the model , the lower is the value of the penalty function . let us illustrate our setup in the case of the so called average value at risk risk measure . the average value at risk risk measure at level @xmath990,1]$ ] is defined as the following functional : @xmath100 where @xmath101 and @xmath102 denotes the left - continuous quantile function of the distribution function @xmath103 of @xmath104 defined by @xmath105 for @xmath990,1[$ ] . note that @xmath106 $ ] for any @xmath107 moreover , it is well known that @xmath108\quad\mbox{for}\ , x\in l^{1},\ ] ] where @xmath109 is the young function defined by @xmath110 for @xmath111 and @xmath112 otherwise ( cf . * theorem 4.52 ) and @xcite ) . observe that the set @xmath113 consists of all probability measures on @xmath70 with @xmath114a.s .. hence the optimal stopping problem ( [ stoppproblem ] ) reads as follows @xmath115 the family @xmath116}$ ] of conditional convex risk measure associated with @xmath109 is also known as the conditional av@r @xmath117}$ ] at level @xmath118 ( cf . * definition 11.8 ) ) . [ entropicriskmeasure ] let us consider , for any @xmath119 the continuous convex mapping @xmath120 } : [ 0,\infty[\rightarrow{\mathbb{r}}$ ] defined by @xmath120}(x ) = ( x\ln(x ) - x + 1)/\gamma$ ] for @xmath121 and @xmath120}(0 ) = 1/\gamma.$ ] the fenchel - legendre transform of @xmath120}$ ] is given by @xmath120}^{*}(y ) = ( \exp(\gamma y ) - 1)/\gamma$ ] for @xmath122 in view of lemma [ optimizedcertaintyequivalent ] ( cf . appendix [ appendixaa ] ) the corresponding risk measure @xmath123}}$ ] has the representation @xmath124}}(x ) = \inf_{x\in{\mathbb{r}}}{\mathbb{e}}\left[\frac{\exp(\gamma x - \gamma x ) - 1}{\gamma } - x\right ] = \frac{\ln\big({\mathbb{e}}[\exp ( - \gamma x)]\big)}{\gamma}\ ] ] for @xmath125}^{*}}.$ ] this is the well - known entropic risk measure . optimal stopping with the entropic risk measures is easy to handle , since it can be reduced to the standard optimal stopping problems via @xmath126}}(-y_{\tau } ) = \frac{1}{\gamma}\cdot\ln\big(\sup_{\tau\in{\mathcal t}}{\mathbb{e}}\big[\exp(\gamma y_{\tau})\big]\big).\ ] ] [ polynomial ] set @xmath127 } = x^{p}/p$ ] for any @xmath1281,\infty[,$ ] then the set @xmath129},0}$ ] contains all probability measures @xmath3 on @xmath70 with @xmath130 and @xmath131}}(x ) = \sup\limits_{{\mathrm{q}}\in { \mathcal q}_{\phi^{[p]},0}}\left({\mathbb{e}}_{{\mathrm{q}}}[- x ] - \frac{1}{p}~{\mathbb{e}}\left[\left(\frac{d{\mathrm{q}}}{d{\mathrm{p}}}\right)^{p}\right]\right)\quad\mbox{for}~x\in l^{p/(p - 1)}.\ ] ] let @xmath132 denote the topological interior of the effective domain of the mapping @xmath1.$ ] we shall assume @xmath133 to be a lower semicontinuous convex function satisfying @xmath134 the following theorem is our main result . [ new_representation ] let @xmath135 be atomless with countably generated @xmath136 for every @xmath137 furthermore , let ( [ annahmen young function ] ) be fulfilled , and let @xmath33}y_{t}\in h^{\phi^{*}},$ ] then @xmath138\\ & = & \inf_{x\in{\mathbb{r}}}\sup\limits_{\tau\in{\mathcal t}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x ] < \infty.\end{aligned}\ ] ] [ optimalstoppingoptimizedcertainty ] the functional @xmath139\ ] ] is known as the optimized certainty equivalent w.r.t . @xmath140 ( cf . @xcite,@xcite ) . thus the relationship @xmath141 = \inf_{x\in{\mathbb{r}}}\sup\limits_{\tau\in{\mathcal t}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x ] \end{aligned}\ ] ] may also be viewed as a representation result for optimal stopping with optimized certainty equivalents . let us illustrate theorem [ new_representation ] for the case @xmath142 with some @xmath990,1].$ ] the young function @xmath109 satisfies the conditions of theorem [ new_representation ] if and only if @xmath143 the fenchel - legendre transform @xmath144 of @xmath133 is given by @xmath145 and it fullfills the inequality @xmath146 for @xmath147 then , as an immediate consequence of theorem [ new_representation ] , we obtain the following primal representation for the optimal stopping problem ( [ av@rstopping ] ) . [ representation av@r ] let @xmath148 be atomless with countably generated @xmath136 for every @xmath137 if @xmath33}y_{t}\in l^{1},$ ] then it holds for @xmath990,1[$ ] @xmath149\\ & = & \inf_{x\leq 0}\sup\limits_{\tau\in{\mathcal t}}{\mathbb{e}}\left[\frac{1}{\alpha}\ , ( x + y_{\tau})^{+ } - x\right ] < \infty.\end{aligned}\ ] ] let us now consider the case @xmath150}$ ] for some @xmath1511,\infty[.$ ] this mapping meets all requirements of theorem [ new_representation ] , and @xmath127^{*}}(x ) = \phi^{\left[p/(p-1)\right]}(x^{+}).$ ] then by theorem [ new_representation ] , we have the following primal representation of the corresponding optimal stopping problem . [ primalrepresentation ] let @xmath148 be atomless with countably generated @xmath136 for every @xmath137 if @xmath33}y_{t}\in l^{p/(p-1)}$ ] for some @xmath1511,\infty[,$ ] then @xmath152}}(-y_{\tau } ) = \inf_{x\in{\mathbb{r}}}\sup\limits_{\tau\in{\mathcal t}}{\mathbb{e}}\left[\frac{(p-1)\ , \big((x + y_{\tau})^{+}\big)^{p/(p-1)}}{p } - x\right ] < \infty.\end{aligned}\ ] ] a natural question is whether we can find a real number @xmath153 and a @xmath154-stopping time @xmath155 which solve . we may give a fairly general answer within the context of discrete time optimal stopping problems . in order to be more precise , let @xmath156 denote all stopping times from @xmath10 with values in @xmath157 where @xmath158 is any finite subset of @xmath11 $ ] containing @xmath159 consider now the stopping problem @xmath160 turning over to the filtration @xmath161}$ ] defined by @xmath162}$ ] with @xmath163 : = { \max}\{s\in{\mathbb{t}}\mid s\leq t\},$ ] we see that @xmath164}$ ] with @xmath165}$ ] describes some @xmath166adapted process . hence we can apply theorem [ new_representation ] to get @xmath167\\ & = & \inf_{x\in{\mathbb{r}}}\sup_{\tau\in{\mathcal t}_{{\mathbb{t}}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x]\end{aligned}\ ] ] in this section we want to find conditions which guarantee the existence of a saddle point for the optimization problems @xmath168\,\mbox{over}\,\tau\in{\mathcal t}_{{\mathbb{t}}}\,\ ] ] and @xmath169\,\mbox{over}\ , x\in{\mathbb{r}}.\ ] ] to this end , we shall borrow some arguments from the theory of lyapunoff s theorem for infinite - dimensional vector measures . a central concept in this context is the notion of _ thin subsets _ of integrable mappings . so let us first recall it . for a fixed probability space @xmath170 a subset @xmath171 is called thin if for any @xmath172 with @xmath173 there is some nonzero @xmath174 vanishing outside @xmath175 and satisfying @xmath176 = 0 $ ] for every @xmath177 ( cf . @xcite , or @xcite ) . best known examples are finite subsets of @xmath178 or finite - dimensional linear subspaces of @xmath178 if @xmath179 is atomless ( cf . @xcite , or @xcite ) . [ saddle - point ] let the assumptions of theorem [ new_representation ] be fulfilled , and let @xmath180 with @xmath181 moreover , let @xmath182\mid x\in{\mathbb{r}}\right\}$ ] be a thin subset of @xmath183 for @xmath184 with @xmath185 and @xmath186 then there are @xmath187 and @xmath188 satisfying @xmath189 & = & \sup\limits_{\tau\in{\mathcal t}_{{\mathbb{t}}}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x]\\ & = & \inf_{x\in{\mathbb{r}}}\sup_{\tau\in{\mathcal t}_{{\mathbb{t}}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x]\\ & = & \sup_{\tau\in{\mathcal t}_{{\mathbb{t}}}}{\mathbb{e}}[\phi^{*}(x^ { * } + y_{\tau } ) - x^{*}].\end{aligned}\ ] ] in particular , it holds @xmath190 \leq { \mathbb{e}}[\phi^{*}(x^ { * } + y_{\tau^ { * } } ) - x^ { * } ] \leq { \mathbb{e}}[\phi^{*}(x + y_{\tau^ { * } } ) - x]\end{aligned}\ ] ] for any @xmath191 and @xmath192 the proof of proposition [ saddle - point ] can be found in section [ proof of saddle - point ] . let the mapping @xmath193 be defined by @xmath194 for some @xmath195 obviously , @xmath196 is convex , nondecreasing , and satisfies @xmath197 as well as @xmath198 hence @xmath199 defines a lower semicontinuous convex function which satisfies , and whose fenchel - legendre transform coincides with @xmath200 since @xmath196 is continuous . moreover , for any @xmath184 such that @xmath201 and @xmath202 the set @xmath203\mid x\in{\mathbb{r}}\right\}$ ] is contained in the finite - dimensional linear subspace of @xmath183 spanned by the sequence of r. v. @xmath204 ~\big|~ \ , k=0,\dots , n\right\},\ ] ] where by definition @xmath205 as a result , @xmath203\mid x\in{\mathbb{r}}\right\}$ ] is a thin subset of @xmath183 in the case of atomless @xmath148 ( cf . e.g. ( * ? ? ? * proposition 2.6 ) ) . in this section we generalize the celebrated additive dual representation for optimal stopping problems ( see @xcite ) to the case of optimal stopping under uncertainty . the result in @xcite is formulated in terms of martingales @xmath206 with @xmath207 satisfying @xmath208}|m_{t}|\in l^{1}.$ ] the set of all such adapted martingales will be denoted by @xmath209 [ dualrepresentation ] let @xmath210 $ ] be the snell envelope of an integrable right - continuous stochastic process @xmath211}$ ] adapted to @xmath212 if @xmath208}|z_{t}|\in l^{p}$ ] for some @xmath213 then @xmath214 = \inf_{m\in { \mathcal m}_{0}}{\mathbb{e}}\left[\sup_{t\in [ 0,t]}(z_{t } - m_{t})\right],\ ] ] where the infimum is attained for @xmath215 with @xmath216 being the martingale part of the doob - meyer decomposition of @xmath217}.$ ] even more it holds @xmath218 = \sup_{t\in [ 0,t]}({z}_{t } - m^{*}_{t})\quad{\mathrm{p}}-\mbox{a.s.}.\ ] ] [ relaxed ] by inspection of the proof of theorem 2.1 in @xcite , one can see that the assumption @xmath208}{\mathbb{e}}[z_{t}]\in l^{p}$ ] for some @xmath219 is only used to guarantee the existence of the doob - meyer decomposition of the snell envelope @xmath217}.$ ] therefore this assumption may be relaxed , if we consider discrete time optimal stopping problems on the set @xmath158 for some finite @xmath220 $ ] containing @xmath159 in this case , the doob - meyer decomposition always exists if @xmath221 is integrable , and theorem [ dualrepresentation ] holds with @xmath10 replaced by @xmath222 and @xmath11 $ ] replaced by @xmath158 ( see also ( * ? ? ? * theorem 5.5 ) ) . theorem [ new_representation ] allows us to extend the additive dual representation to the case of stopping problems . we shall use the following notation . for a fixed @xmath133 and @xmath191 we shall denote by @xmath223}$ ] the snell - envelope w.r.t . to @xmath224}$ ] defined via @xmath225.\ ] ] the application of theorem [ new_representation ] together with theorem [ dualrepresentation ] provides us with the following additive dual representation of the stopping problem . [ dualrepresentation_utility ] under assumptions on @xmath133 and @xmath226 of theorem [ new_representation ] and under the condition @xmath33}|\phi^{*}(x + y_{t})|\in l^{p}$ ] for some @xmath219 and any @xmath227 the following dual representation holds @xmath228}\big(\phi^{*}(x + y_{t } ) - x - m_{t}\big)\big]\\ & = & \inf_{x\in{\mathbb{r}}}{\mathbb{e}}\big[\sup_{t\in [ 0,t]}\big(\phi^{*}(x + y_{t } ) - x - m^{*,\phi , x}_{t}\big)\big]\\ & = & { \mathop{\mathrm{ess\,inf}}\displaylimits}_{x\in{\mathbb{r}}}\sup_{t\in [ 0,t]}\big(\phi^{*}(x + y_{t } ) - x - m^{*,\phi , x}_{t}\big ) \quad { \mathrm{p}}-\mbox{a.s.}.\end{aligned}\ ] ] here @xmath229 stands for the martingale part of the doob - meyer decomposition of the snell - envelope @xmath230 [ relaxed2 ] under the assumptions of theorem [ new_representation ] , we have that @xmath33}y_{t}\in h^{\phi^{*}}.$ ] furthermore , @xmath140 is convex and nondecreasing with @xmath231 ( see lemma [ optimizedcertaintyequivalent ] in appendix [ appendixaa ] ) so that for any @xmath232 @xmath233 where @xmath234 denotes the right - sided derivative of @xmath235 using the monotonicity of @xmath140 again , we conclude that @xmath236}y_{t})\in l^{1}\ ] ] for all @xmath191 and @xmath237.$ ] hence the application of theorem [ dualrepresentation_utility ] to is already possible under the assumptions of theorem [ new_representation ] . the dual representation for the optimal stopping problem under average value at risk reads as follows . [ cor_dual_avar ] let the assumptions on @xmath133 and @xmath226 be as in theorem [ new_representation ] . if @xmath33}y_{t}\in l^{p}$ ] for some @xmath213 then it holds @xmath4-a.s . @xmath238}\left(\frac{1}{\alpha}\ , ( x + y_{t})^{+ } - x - m_{t}\right)\right]\\ & = & \nonumber \inf_{x\leq 0}{\mathbb{e}}\left[\sup_{t\in [ 0,t]}\left(\frac{1}{\alpha}\ , ( x + y_{t})^{+ } - x - m^{*,\alpha , x}_{t}\right)\right]\\ & = & \label{dual_avr } { \mathop{\mathrm{ess\,inf}}\displaylimits}_{x\leq 0}\sup_{t\in [ 0,t]}\left(\frac{1}{\alpha}\ , ( x + y_{t})^{+ } - x - m^{*,\alpha , x}_{t}\right)\quad { \mathrm{p}}-\mbox{a.s.}. \ ] ] here @xmath239 denotes the martingale part of the doob - meyer decomposition of the snell - envelope @xmath240 [ relax3 ] let us consider a discrete time optimal stopping problem @xmath241 for some finite @xmath220 $ ] with @xmath242 in view of remark [ relaxed2 ] , the assumptions of theorem [ new_representation ] are already sufficient to obtain the dual representation with @xmath10 replaced by @xmath243 and @xmath11 $ ] replaced by @xmath244 in @xcite the optimal stopping problems of the type @xmath245 were studied , where for any @xmath246 the functional @xmath247 maps a linear subspace @xmath248 of the space @xmath249 into @xmath250 and satisfies @xmath251 for @xmath252a.s .. in fact there is a one - to - one correspondence between conditional convex risk measures @xmath27}$ ] and dynamic utility functionals @xmath253}$ ] satisfying the following two properties : * * conditional translation invariance : * + @xmath254 for @xmath255 and @xmath256 * * conditional concavity : * + @xmath257 for @xmath258 and @xmath259 with @xmath260 more precisely , any conditionally translation invariant and conditionally concave dynamic utility functional @xmath261}$ ] defines a family @xmath262}$ ] of conditional convex risk measures via @xmath263 and vice versa . the results of @xcite essentially rely on the following additional assumptions * * regularity : * + @xmath264 for @xmath265 and @xmath256 * * recursiveness : * + @xmath266 for @xmath267 recursiveness is often also referred to as time consistency . obviously , the dynamic utility functional @xmath268},$ ] defined by @xmath269 satisfies the regularity and the conditional translation invariance , but it fails to be recursive ( cf . * example , 11.13 ) ) . even worse , according to theorem 1.10 in @xcite for any @xmath270 there is in general no regular conditionally translation invariant and recursive dynamic utility functional @xmath271 such that @xmath272 this means that we can not in general reduce the stopping problem ( [ av@rstopping ] ) to the stopping problem ( [ dmu ] ) with a regular , conditionally translation invariant and recursive dynamic utility functional @xmath271 . note that this conclusion can be drawn from theorem 1.10 of @xcite , because @xmath273 is law - invariant , i.e. , @xmath274 for identically distributed @xmath104 and @xmath275 , and satisfies the properties @xmath276 as well as @xmath277 for any @xmath278 and @xmath279 with @xmath280 the stopping problem ( [ av@rstopping ] ) may also be viewed as a special case of the following stopping problem : @xmath281 where @xmath282\mapsto[0,1]$ ] is a so - called distortion function , i.e. , @xmath283 is nondecreasing and satisfies @xmath284 @xmath285 indeed , if for @xmath990,1[$ ] the distortion function @xmath286 is defined by @xmath287 then the stopping problems ( [ av@rstopping ] ) and ( [ osf_distortion ] ) coincide . recalling theorem 1.10 of @xcite again , we see that the stopping problem ( [ osf_distortion ] ) is not in general representable in the form ( [ dmu ] ) with some regular , conditionally translation invariant and recursive dynamic utility functional . the stopping problem ( [ osf_distortion ] ) was recently considered by @xcite . however , the analysis in @xcite relies on some additional assumptions . first of all , the authors allow for all finite stopping times w.r.t . to some filtered probability space @xmath288 instead of restricting to those which are bounded by a fixed number . secondly , they assume a special structure for the process @xmath289 namely it is supposed that @xmath290 for an absolutely continuous nonnegative function @xmath291 on @xmath77 and for a one - dimensional geometric brownian motion @xmath292 . thirdly , the authors focus on strictly increasing absolutely continuous distortion functions @xmath283 so that their analysis does not cover the case of average value at risk . more precisely , in @xcite the optimal stopping problems of the form @xmath293 are studied , where @xmath294 denotes the set of all finite stopping times . a crucial step in the authors argumentation is the reformulation of the optimal stopping problem as @xmath295 where @xmath296 and @xmath297 are derivatives of @xmath291 and @xmath298 respectively , and @xmath299 denotes the set of all distribution functions @xmath300 with a nonnegative support such that @xmath301 the main idea of the approach in @xcite is that any such distribution function may be described as the distribution function of @xmath302 for some finite stopping time @xmath303 and this makes the application of the skorokhod embedding technique possible . hence , the results essentially rely on the special structure of the stochastic process @xmath304 and seem to be not extendable to stochastic processes of the form @xmath305 where @xmath306 is a multivariate markov process . moreover , it remains unclear whether the analysis of @xcite can be carried over to the case of bounded stopping times , as the skorokhod embedding can not be applied to the general sets of stopping times @xmath10 ( see e.g. @xcite ) . in this section we illustrate how our results can be used to price bermudan - type options in uncertain environment . specifically , we consider the model with @xmath307 identically distributed assets , where each underlying has dividend yield @xmath308 . the dynamic of assets is given by @xmath309where @xmath310 , are independent one - dimensional brownian motions and @xmath311 are constants . at any time @xmath312 the holder of the option may exercise it and receive the payoff @xmath313if we are uncertain about our modelling assumption and if the average value at risk is used to measure the risk related to this uncertainty , then the risk - adjusted price of the option is given by @xmath314}av@r_{\alpha}(-y_{\tau})&= & { \sup_{\tau\in{\mathcal t}[t_0,\ldots , t_j]}\sup\limits_{{\mathrm{q}}\in{\mathcal q}_{\phi_{\alpha},0}}{\mathbb{e}}_{{\mathrm{q}}}[-y_{\tau } ] } \\ \label{opt_stop_ex } & = & \inf_{x\leq 0}\sup\limits_{\tau\in{\mathcal t}[t_0,\ldots , t_j]}{\mathbb{e}}\left[\frac{1}{\alpha}\ , ( x + y_{\tau})^{+ } - x\right],\end{aligned}\ ] ] where @xmath315 consists of all probability measures @xmath316 on @xmath70 with @xmath317 if we restrict our attention to the class of generalised black scholes models of the type @xmath318 with adapted processes @xmath319 @xmath320 and independent brownian motions @xmath321 then @xmath322 with @xmath323 and the condition transforms to @xmath324 due to corollary [ representation av@r ] , one can use the standard methods based on dynamic programming principle to solve and @xmath325 $ ] stands for a set of stopping times with values in @xmath326 indeed , for any fixed @xmath327 the optimal value of the stopping problem @xmath328}{\mathbb{e}}\left[\frac{1}{\alpha}\ , ( x + y_{\tau})^{+ } - x\right]\end{aligned}\ ] ] can be , for example , numerically approximated via the well known regression methods like longstaff - schwartz method . in this way one can get a ( suboptimal ) stopping rule @xmath329 where @xmath330 are continuation values estimates . then @xmath331 is a low - biased estimate for @xmath332 . note that the infimum in can be easily computed using a simple search algorithm . an upper - biased estimate can be constructed using the well known andersen - broadie dual approach ( see @xcite ) . for any fixed @xmath333 this approach would give us a discrete time martingale @xmath334 which in turn can be used to build an upper - biased estimate via the representation ( [ dual_avr ] ) : @xmath335\right\}.\end{aligned}\ ] ] note that remains upper biased even if we replace the infimum of the objective function in by its value at a fixed point @xmath336 in table [ max_call_2d ] we present the bounds @xmath337 and @xmath338 together with their standard deviations for different values of @xmath339 as to implementation details , we used @xmath340 basis functions for regression ( see @xcite ) and @xmath341 training paths to compute @xmath342 in the dual approach of andersen and broadie , @xmath343 inner simulations were done to approximate @xmath344 in both cases we simulated @xmath345 testing paths to compute the final estimates . for comparison let us consider a problem of pricing the above bermudan option under entropic risk measure . due to , we need to solve the optimal stopping problem @xmath346}{\mathbb{e}}\big[\exp(\gamma y_{\tau})\big].\ ] ] the latter problem can be solved via the standard dynamic programming combined with regression as described above . in table [ max_call_2d_entrop ] the upper and lower mc bounds for @xmath347 are presented for different values of the parameter @xmath348 unfortunately for larger values of @xmath349 the corresponding mc estimates become unstable due to the presence of exponent in . in figure [ fig : bounds ] the lower bounds for av@r and the entropic risk measure are shown graphically . as can be seen the quality of upper and lower bounds are quite similar . however due to above mentioned instability , av@r should be preferred under higher uncertainty . [ max_call_2d ] .bounds ( with standard deviations ) for @xmath350-dimensional bermudan max - call with parameters @xmath351 , @xmath352 @xmath353 under av@r at level @xmath118 [ cols="^,^,^",options="header " , ] in order to prove theorem [ new_representation ] we shall proceed as follows . first , by lemma [ optimizedcertaintyequivalent ] ( cf . appendix [ appendixaa ] ) , we obtain immediately @xmath354- { \mathbb{e}}\left[\phi\left(\frac{d{\mathrm{p}}}{d{\mathrm{q}}}\right)\right]\right ) = \sup\limits_{\tau\in{\mathcal t}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x].\ ] ] the proof of theorem [ new_representation ] would be completed , if we can show that @xmath355 = \inf_{x\in{\mathbb{r}}}\sup\limits_{\tau\in{\mathcal t}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x].\ ] ] using fubini s theorem , we obtain for any @xmath356 and every @xmath191 @xmath357 = \int_{x^{-}}^{\infty}\phi^{*'}(x + z)[1 - f_{y_{\tau}}(z)]~dz + \phi^{*}(x^{+ } ) - x,\ ] ] where @xmath358 stands for the distribution function of @xmath359 and @xmath234 denotes the right - sided derivative of the convex function @xmath235 in the same way we may also find @xmath360 = -\int_{0}^{x^{-}}\phi^{*'}(x + z)f_{y_{\tau}}(z)~dz.\ ] ] hence the property @xmath361 for @xmath191 yields @xmath362 = \int_{0}^{\infty}\phi^{*'}(x + z)[1 - f_{y_{\tau}}(z)]~dz + \phi^{*}(x ) - x\ ] ] for @xmath356 and @xmath363 since the set @xmath364 of distribution functions @xmath358 of @xmath359 is not , in general , a convex subset of the set of distribution functions on @xmath365 we can not apply the known minimax results . the idea is to first establish ( [ minimaxrelationship ] ) for the larger class of randomized stopping times , and then to show that the optimal value coincides with the optimal value @xmath366.$ ] let us recall the notion of randomized stopping times . by definition ( see e.g. @xcite ) , a randomized stopping time w.r.t . @xmath66 is a mapping @xmath367\rightarrow [ 0,\infty]$ ] which is nondecreasing and left - continuous in the second component such that @xmath368 is a stopping time w.r.t . @xmath67}$ ] for any @xmath369.$ ] notice that any randomized stopping time @xmath370 is also an ordinary stopping time w.r.t . the enlarged filtered probability space @xmath371,{\mathcal f}\otimes { \mathcal b}([0,1 ] ) , \big({\mathcal f}_{t}\otimes { \mathcal b}([0,1])\big)_{t\in [ 0,t]},{\mathrm{p}}\otimes { \mathrm{p}}^{u}\big).$ ] here @xmath372 denotes the uniform distribution on @xmath373,$ ] defined on @xmath374),$ ] the usual borel @xmath375algebra on @xmath373.$ ] we shall call a randomized stopping time @xmath370 to be degenerated if @xmath376 is constant for every @xmath377 there is an obvious one - to - one correspondence between stopping times and degenerated randomized stopping times . consider the stochastic process @xmath378 defined by @xmath379\rightarrow{\mathbb{r}},\ , ( \omega , u)\mapsto y_{t}(\omega).\ ] ] which is adapted w.r.t . the enlarged filtered probability space . denoting by @xmath380 the set of all randomized stopping times @xmath381 we shall study the following new stopping problem @xmath382~\mbox{over}~\tau^r\in{\mathcal t^{r}}.\ ] ] obviously , @xmath383 = \inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y^{r}_{{\tau^{r } } } ) - x]$ ] is valid for every stopping time @xmath384 where @xmath385 is the corresponding degenerated randomized stopping time such that @xmath386 @xmath369.$ ] thus , in general the optimal value of the stopping problem ( [ randomstop ] ) is at least as large as the one of the original stopping problem ( [ stoppproblem ] ) due to ( [ hilfsstoppproblem ] ) . one reason to consider the new stopping problem is that it has a solution under fairly general conditions . [ solution ] let @xmath96}$ ] be quasi - left - continuous , defined to mean @xmath387 @xmath59a.s . whenever @xmath388 is a sequence in @xmath10 satisfying @xmath389 for some @xmath356 . if @xmath390 is countably generated , then there exists a randomized stopping time @xmath391 such that @xmath392 = \sup_{{\tau^{r}}\in { \mathcal t^{r}}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}\left[\phi^{*}(x + y^{r}_{{\tau^{r } } } ) - x\right].\ ] ] proposition will be proved in section [ proof of solution ] . moreover the following important minimax result for the stopping problem ( [ randomstop ] ) holds . [ minimax ] if ( [ annahmen young function ] ) is fulfilled , and if @xmath33}y_{t}\in h^{\phi^{*}},$ ] then @xmath393 = \inf_{x\in{\mathbb{r}}}\sup\limits_{\tau^{r}\in{\mathcal t}^{r}}{\mathbb{e}}[\phi^{*}(x + y_{{\tau^{r}}}^{r } ) - x].\end{aligned}\ ] ] moreover , if @xmath96}$ ] is quasi - left - continuous and if @xmath390 is countably generated , then there exist @xmath394 and @xmath188 such that @xmath395 \leq { \mathbb{e}}[\phi^{*}(x^ { * } + y^{r}_{\tau^{r * } } ) - x^ { * } ] \leq { \mathbb{e}}[\phi^{*}(x + y^{r}_{\tau^{r * } } ) - x]\end{aligned}\ ] ] for @xmath191 and @xmath396 the proof of proposition [ minimax ] can be found in section [ beweis minimax ] . in the next step we shall provide conditions ensuring that the stopping problems ( [ stoppproblem ] ) and ( [ randomstop ] ) have the same optimal value . [ derandomize2 ] let @xmath148 be atomless with countably generated @xmath136 for every @xmath137 if ( [ annahmen young function ] ) is fulfilled , and if @xmath33}y_{t}$ ] belongs to @xmath397 then @xmath393 & = & \sup\limits_{\tau\in{\mathcal t}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x]\\ & = & \sup\limits_{\tau\in{\mathcal t}}\sup_{{\mathrm{q}}\in{\mathcal q}_{\phi,0}}\left({\mathbb{e}}_{{\mathrm{q}}}[y_{\tau}]- { \mathbb{e}}\left[\phi\left(\frac{d{\mathrm{p}}}{d{\mathrm{q}}}\right)\right]\right ) \end{aligned}\ ] ] the proof of proposition [ derandomize2 ] is delegated to section [ beweis derandomize2 ] . we shall start with some preparations which also will turn out to be useful later on . let us recall ( cf . @xcite ) that every @xmath385 induces a stochastic kernel @xmath398)\rightarrow [ 0,1 ] $ ] with @xmath399 being the distribution of @xmath376 under @xmath372 for any @xmath377 here @xmath400)$ ] stands for the usual borel @xmath375algebra on @xmath11.$ ] this stochastic kernel has the following properties : @xmath401)~\mbox{is}~{\mathcal f}_{t}-\mbox{measurable for every}~t\geq 0,\\ & & k_{{\tau^{r}}}(\omega,[0,t ] ) = \sup\{u\in [ 0,1]\mid { \tau^{r}}(\omega , u)\leq t\}. \ ] ] the associated stochastic kernel @xmath402 is useful to characterize the distribution function @xmath403 of @xmath404 [ stopped distribution ] for any @xmath385 with associated stochastic kernel @xmath405 the distribution function @xmath406 of @xmath407 may be represented in the following way @xmath408\mid y_{t}\leq x\})]\quad\mbox{for}~ x\in{\mathbb{r}}.\ ] ] let @xmath409 and let us fix @xmath363 then @xmath410-\infty , x]}(y_{{\tau^{r}}}^{r } ) ] & = & \int_{0}^{1}{\mathbb{e}}[{\mathbbm{1}}_{]-\infty , x]}(y_{{\tau^{r}}(\cdot , u)}^{r})]\,du\\ & = & { \mathbb{e}}\left[\int_{0}^{1}{\mathbbm{1}}_{]-\infty , x]}(y_{{\tau^{r}}(\cdot , u)}^{r})\ , du\right]\end{aligned}\ ] ] holds ( cf . * theorem 4.5 ) ) , where the last equation on the right hand side is due to fubini - tonelli theorem . then by definition of @xmath405 we obtain for every @xmath411 @xmath412-\infty , x]}(y^{r}_{{\tau^{r}}}(\omega , u))\ , du & = & { \mathbb{e}}_{{\mathrm{p}}^{u}}\left[{\mathbbm{1}}_{]-\infty , x]}(y^{r}_{{\tau^{r}}(\omega,\cdot)}(\omega))\right]\\ & = & { \mathrm{p}}^{u}\left(\left\{y^{r}_{{\tau^{r}}(\omega,\cdot)}(\omega)\leq x\right\}\right)\\ & = & k_{{\tau^{r}}}(\omega,\{t\in [ 0,t]\mid y_{t}(\omega)\leq x\ } ) . \end{aligned}\ ] ] this completes the proof . following a suggestion by one referee we placed the proof of proposition [ solution ] in front of that of proposition [ minimax ] . let us introduce the filtered probability space @xmath413 defined by @xmath414 we shall denote by @xmath415 the set of randomized stopping times according to @xmath416 furthermore , we may extend the processes @xmath96}$ ] and @xmath417}$ ] to right - continuous processes @xmath418}$ ] and @xmath419}$ ] in the following way @xmath420 recall that we may equip @xmath415 with the so called baxter - chacon topology which is compact in general , and even metrizable within our setting because @xmath390 is assumed to be countably generated ( cf . theorem 1.5 in @xcite and discussion afterwards ) . next , consider the mapping @xmath421.\ ] ] by assumption on @xmath96}$ ] , the processes @xmath418}$ ] and @xmath419}$ ] are quasi - left - continuous . moreover , @xmath140 is continuous due to lemma [ optimizedcertaintyequivalent ] , ( i ) in appendix [ appendixaa ] , so that @xmath422}$ ] is a quasi - left - continuous and right - continuous adapted process . hence in view of ( * ? ? ? * theorem 4.7 ) , the mapping @xmath423 is continuous w.r.t . the baxter - chacon topology for every @xmath424 and thus @xmath425 is upper semicontinuous w.r.t . the baxter - chacon topology . then by compactness of the baxter - chacon topology , we may find some randomized stopping time @xmath426 such that @xmath427 this completes the proof because @xmath428 and @xmath429 belongs to @xmath380 for every @xmath430 @xmath431 let us define the mapping @xmath432 by @xmath433.\ ] ] since @xmath33}y_{t}$ ] is assumed to belong to @xmath397 the mapping @xmath434 is finite and convex , and thus continuous . moreover , by lemma [ optimizedcertaintyequivalent ] ( cf . appendix [ appendixaa ] ) @xmath435 hence @xmath436 } \sup\limits_{{\tau^{r}}\in{\mathcal t^{r}}}h({\tau^{r}},x)$ ] for some @xmath437 thus @xmath438 attains its minimum at some @xmath439 due to continuity of @xmath440 moreover , if @xmath96}$ ] is quasi - left - continuous and if @xmath390 is countably generated , then @xmath441 for some @xmath394 due to proposition [ solution ] . it remains to show that @xmath442 following the same line of reasoning as for the derivation of , we may rewrite @xmath443 in the following way . @xmath444~dz + \phi^{*}(x ) - x,\ ] ] where @xmath403 stands for the distribution function of @xmath445 and @xmath234 denotes the right - sided derivative of the convex function @xmath235 obviously , we have @xmath446 set @xmath447 + 1 { = \inf\limits_{x\in{\mathbb{r}}}\sup\limits_{\tau^{r}\in{\mathcal t}^{r}}h({\tau^{r}},x ) + 1}$ ] which is a real number because @xmath448 has been already proved to be a finite function which attains its minimum on some compact interval of @xmath449 . furthermore , we may conclude from @xmath450 for @xmath191 that @xmath451 and @xmath452 by ( [ infcompact ] ) we verify @xmath453 and @xmath454 we want to apply fan s minimax theorem ( cf . * theorem 2 ) or @xcite ) to @xmath455 . in view of and it remains to show that for every @xmath456 and any @xmath4570,1[$ ] there exists some @xmath385 such that @xmath458 to this end let @xmath459 with associated stochastic kernels @xmath460 and @xmath4570,1[.$ ] first , @xmath461)\rightarrow [ 0,1]$ ] defines a stochastic kernel satisfying @xmath462)~\mbox{is}~{\mathcal f}_{t}-\mbox{measurable for every}~t\in [ 0,t],\\ & & k(\omega,[0,t ] ) = 1.\end{aligned}\ ] ] then @xmath463\mid k(\omega,[0,t])\geq u\}\ ] ] defines some @xmath385 with @xmath464 furthermore , we obtain @xmath465 due to lemma [ stopped distribution ] . in view of ( [ rewrite ] ) this implies ( [ concave ] ) and the proof of proposition [ minimax ] is completed . @xmath431 the starting idea for proving proposition [ derandomize2 ] is to reduce the stopping problem ( [ randomstop ] ) to suitably discretized random stopping times . the choice of the discretized randomized stopping times is suggested by the following lemma . [ discretize ] for @xmath385 the construction @xmath466(\omega , u ) : = \min\{k/2^{j}\mid k\in{\mathbb{n } } , { \tau^{r}}(\omega , u)\leq k/2^{j}\}\wedge t\ ] ] defines a sequence @xmath467)_{j\in{\mathbb{n}}}$ ] in @xmath380 satisfying the following properties . 1 . @xmath468\searrow{\tau^{r}}$ ] pointwise , in particular it follows @xmath469(\omega , u)}(\omega , u ) = y^{r}_{{\tau^{r}}(\omega , u)}(\omega , u)\ ] ] for any @xmath411 and every @xmath369.$ ] 2 . @xmath470}}(x ) = f^{r}_{y_{{\tau^{r}}}}(x)$ ] holds for any continuity point @xmath471 of @xmath472 3 . for any @xmath191 and every @xmath473 we have @xmath474}}(x ) = { \mathbb{e}}\left[\widehat{y}_{t_{1j}}^{x } k_{{\tau^{r}}}(\cdot,[0,t_{1j}])\right ] + \sum\limits_{k=2}^{\infty}{\mathbb{e}}\left[\widehat{y}_{t_{kj}}^{x}\ , k_{{\tau^{r}}}(\cdot,]t_{(k-1)j } , t_{kj}])\right],\ ] ] where @xmath475 for @xmath476 and @xmath477-\infty , x]}\circ y_{t}$ ] for @xmath237.$ ] statements ( i ) and ( ii ) are obvious , so it remains to show ( iii ) . to this end recall from lemma [ stopped distribution ] @xmath478}}(x ) = { \mathbb{e}}[k_{{\tau^{r}}[j]}(\cdot,\{t\in [ 0,t]\mid y_{t}\leq x\})]\quad\mbox{for}~ x\in{\mathbb{r}}.\ ] ] since @xmath479}(\omega,\cdot)$ ] is a probability measure , we also have @xmath480}(\omega,\{t\in [ 0,t]\mid y_{t}(\omega)\leq x\})\\ \nonumber & = & k_{{\tau^{r}}[j]}(\omega,\{t\in [ 0,t_{1j}]\mid y_{t}(\omega)\leq x\})\\ \nonumber & & \qquad + \sum_{k = 2}^{\infty } k_{{\tau^{r}}[j]}(\omega,\{t\in ] t_{(k-1)j},t_{kj}]\mid y_{t}(\omega)\leq x\})\\ \nonumber & = & k_{{\tau^{r}}[j]}(\omega,\{t\in [ 0,t_{1j}]\mid \widehat{y}_{t}^{x}(\omega ) = 1\})\\ & & \qquad + \sum_{k = 2}^{\infty } k_{{\tau^{r}}[j]}(\omega,\{t\in ] t_{(k-1)j},t_{kj}]\mid \widehat{y}_{t}^{x}(\omega ) = 1\})\end{aligned}\ ] ] for every @xmath377 then by definitions of @xmath479}$ ] and @xmath405 @xmath481}(\omega,\{t\in ] t_{(k-1)j},t_{kj}]\mid \widehat{y}_{t}^{x}(\omega ) = 1\ } ) \\ \nonumber & = & { \mathrm{p}}^{u}(\{{\tau^{r}}[j](\omega,\cdot)\in ] t_{(k-1)j},t_{kj}],\,\widehat{y}_{{\tau^{r}}[j](\omega,\cdot)}^{x}(\omega ) = 1\})\\ \nonumber & = & { \mathrm{p}}^{u}(\{{\tau^{r}}[j](\omega,\cdot ) = t_{kj},\,\widehat{y}_{t_{kj}}^{x}(\omega ) = 1\})\\ \nonumber & = & \widehat{y}^{x}_{t_{kj}}(\omega)\,{\mathrm{p}}^{u}(\{{\tau^{r}}[j](\omega,\cdot ) = t_{kj}\})\\ \nonumber & = & \widehat{y}^{x}_{t_{kj}}(\omega)\,{\mathrm{p}}^{u}(\{{\tau^{r}}(\omega,\cdot)\in ] t_{(k-1)j},t_{kj}]\})\\ & = & \widehat{y}^{x}_{t_{kj}}(\omega)\ , k_{{\tau^{r}}}(\omega,]t_{(k-1)j},t_{kj}])\end{aligned}\ ] ] for @xmath411 and @xmath482 with @xmath483 analogously , we also obtain @xmath484}(\omega,\{t\in [ 0,t_{1j}]\mid \widehat{y}_{t}^{x}(\omega ) = 1\ } ) = \widehat{y}_{t_{1j}}(\omega)\ , k_{{\tau^{r}}}(\omega,[0,t_{1j}]).\ ] ] then statement ( iii ) follows from ( [ ausgangspunkt ] ) combining ( [ auseinanderziehen ] ) with ( [ anwendungkerndefinition ] ) and ( [ anwendungkerndefinition2 ] ) . the proof is finished . we shall use the discretized randomized stopping times , as defined in lemma [ discretize ] , to show that we can restrict ourselves to discrete randomized stopping times in the stopping problem ( [ randomstop ] ) . [ discretizedstop ] if ( [ annahmen young function ] ) is fulfilled , then for any @xmath409 we have 1 . @xmath485 } ) - x_{j } ] = { \mathbb{e}}[\phi^{*}(x + y^{r}_{{\tau^{r } } } ) - x ] $ ] for any sequence @xmath486 in @xmath487 converging to some @xmath488 2 . @xmath489 } ) - x ] = \inf\limits_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y^{r}_{{\tau^{r } } } ) - x ] . $ ] let the mapping @xmath490 be defined by @xmath491.$ ] for every @xmath409 the mapping @xmath492 is convex and thus continuous . recalling that @xmath493 a direct application of lemma [ discretize ] , ( i ) , along with the dominated convergence theorem yields part ( i ) . using terminology from @xcite ( see also @xcite ) , statement ( i ) implies that the sequence @xmath494,\cdot))_{j\in{\mathbb{n}}}$ ] of continuous mappings @xmath495,\cdot)$ ] epi - converges to the continuous mapping @xmath496 moreover , in view of ( [ infcompactneu ] ) and ( [ infcompact ] ) , we may conclude @xmath497,x ) = \inf\limits_{x\in{\mathbb{r}}}h({\tau^{r}},x),\ ] ] drawing on theorem 7.31 in @xcite ( see also satz b 2.18 in @xcite ) . the following result provides the remaining missing link to prove proposition [ derandomize2 ] . [ missinglink ] let ( [ annahmen young function ] ) be fulfilled . furthermore , let @xmath409 and let us for any @xmath473 denote by @xmath498 $ ] the set containing all nonrandomized stopping times from @xmath10 taking values in @xmath499 with probability @xmath500 if @xmath148 is atomless with countably generated @xmath136 for every @xmath501 and if @xmath502 for @xmath501 then @xmath503 } ) - x ] \leq \sup_{\tau\in{\mathcal t}[j]}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x].\end{aligned}\ ] ] let @xmath504 if @xmath505 then the statement of lemma [ missinglink ] is obvious . so let us assume @xmath506 set @xmath475 and let the mapping @xmath507 be defined via @xmath508.$ ] we already know from lemma [ discretize ] that @xmath509}}(x ) = { \mathbb{e}}\left[\widehat{y}_{t_{1j}}^{x } k_{{\tau^{r}}}(\cdot,[0,t_{1j}])\right ] + \sum\limits_{k=2}^{k_{j}}{\mathbb{e}}\left[\widehat{y}_{t_{kj}}^{x } k_{{\tau^{r}}}(\cdot,]t_{(k-1)j},t_{kj}])\right]\ ] ] holds for any @xmath363 here @xmath477-\infty , x]}\circ y_{t}$ ] for @xmath237.$ ] next @xmath510)&k = 1\\ k_{{\tau^{r}}}(\cdot,]t_{(k-1 ) j},t_{kj}])&k\in\{2, ... ,k_{j}\ } \ecswitch\ ] ] defines a random variable on @xmath511 which satisfies @xmath512 @xmath59a.s .. in addition , we may observe that @xmath513 holds @xmath59a.s .. since the probability spaces @xmath514 @xmath515 are assumed to be atomless and countably generated , we may draw on corollary [ dichtheit ] ( cf . appendix [ appendixc ] ) along with lemma [ banachalaoglu ] ( cf . appendix [ appendixc ] ) and proposition [ angelic ] ( cf . appendix [ appendixa ] ) to find a sequence @xmath516 in @xmath517{$\scriptstyle k=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle k_{j}$ } } \end{picture } } } { \mathcal f}_{t_{kj}}$ ] such that @xmath518 is a partition of @xmath519 for @xmath520 and @xmath521 = { \mathbb{e}}\left[z_{k}\cdot g\right]\ ] ] holds for @xmath522 and @xmath523 in particular we have by ( [ eins ] ) @xmath474}}(x ) = \lim\limits_{n\to\infty } \sum\limits_{k=1}^{k_{j}}{\mathbb{e}}\left[\widehat{y}_{t_{kj}}^{x } { \mathbbm{1}}_{b_{kn}}\right]\ , \mbox{for}\ , x\in{\mathbb{r}}.\ ] ] so by fatou s lemma along with , @xmath524,x ) \leq \liminf\limits_{n\to\infty } \int_{0}^{\infty}\phi^{*'}(x + z)\,\big ( 1 - \sum\limits_{k=1}^{k_{j}}{\mathbb{e}}\left[\widehat{y}_{t_{kj}}^{z } { \mathbbm{1}}_{b_{kn}}\right]\big)\ , dz + \phi^{*}(x ) - x\ ] ] for @xmath363 here @xmath234 denotes the right - sided derivative of @xmath235 next we can define a sequence @xmath388 of nonrandomized stopping times from @xmath498 $ ] via @xmath525 the distribution function @xmath526 of @xmath527 satisfies @xmath528\ , \mbox{for}\ , x\in{\mathbb{r}}\ ] ] so that by ( [ rewrite ] ) @xmath529\big)\ , dz + \phi^{*}(x ) - x\ ] ] for @xmath363 the crucial point now is to show that ( @xmath530 ) : : @xmath531\big\}$ ] is equicontinuous , where @xmath532 is the interval defined in . note that @xmath533 is a sequence in @xmath534 and that @xmath535\big\}$ ] is bounded for every @xmath363 thus , in view of ( [ convex ] ) the statement ( @xmath530 ) together with arzela - ascoli theorem implies that we can find a subsequence @xmath536 such that @xmath537 for some continuous mapping @xmath538 hence , we may conclude from ( [ drei ] ) and ( [ zwei ] ) @xmath539,x)\ , \mbox { for } \ , x\in i_{\beta}.\ ] ] for any @xmath540 we may find some @xmath541 such that @xmath542 which implies by ( [ vier ] ) together with ( [ infcompact ] ) : @xmath543,x ) - \varepsilon\\ & \geq & \inf\limits_{x\in{\mathbb{r}}}h({\tau^{r}}[j],x ) - \varepsilon\end{aligned}\ ] ] and is proved . therefore it remains to show the statement ( @xmath530 ) . [ [ proof - of - star ] ] proof of ( @xmath530 ) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + first , observe that for @xmath544 $ ] and real numbers @xmath545 the inequality @xmath546 holds . hence @xmath547 - { \mathbb{e}}\left[\phi^{*}(x + y_{\tau})\right ] + & = & \sum\limits_{k = 1}^{k_{j } } { \mathbb{e}}\big[{\mathbbm{1}}_{\{t_{kj}\}}\circ \tau\ , \underbrace{\big(\phi^{*}\big(y + y_{t_{kj}}\big ) - \phi^{*}\big(x + y_{t_{kj}}\big)\big)}_{\geq 0}\big ] + |x - y| \nonumber\\ & \leq & \sum\limits_{k = 1}^{k_{j } } { \mathbb{e}}\big [ \phi^{*}\big(y + y_{t_{kj}}\big ) - \phi^{*}\big(x + y_{t_{kj}}\big)\big ] + |x - y| \nonumber\\ & \leq & \sum\limits_{k = 1}^{k_{j}}|h(t_{kj},x ) - h(t_{kj},y)| + ( k_{j } + 1)\ , |x - y|\end{aligned}\ ] ] by convexity , the mappings @xmath548 @xmath549 are also locally lipschitz continuous . thus , in view of ( [ fuenf ] ) , it is easy to verify that @xmath550 is equicontinuous at every @xmath551 this proves ( @xmath530 ) . now , we are ready to prove proposition [ derandomize2 ] . by ( [ hilfsstoppproblem ] ) we have @xmath552 & \geq & \sup\limits_{\tau\in{\mathcal t}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x]\\ & = & \sup\limits_{\tau\in{\mathcal t}}\sup_{{\mathrm{q}}\in{\mathcal q}_{\phi,0}}\left({\mathbb{e}}_{{\mathrm{q}}}[y_{\tau}]- { \mathbb{e}}\left[\phi\left(\frac{d{\mathrm{p}}}{d{\mathrm{q}}}\right)\right]\right ) \end{aligned}\ ] ] moreover , due to ( ii ) of corollary [ discretizedstop ] and lemma [ missinglink ] we conclude that for any @xmath385 @xmath553 & = & \lim_{j\to\infty}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y^{r}_{{\tau^{r}}[j ] } ) - x]\\ & \leq & \sup_{\tau\in{\mathcal t}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x].\end{aligned}\ ] ] thus proposition [ derandomize2 ] is proved . @xmath431 first , we get from propositions [ minimax ] and [ derandomize2 ] @xmath554 & = & \sup\limits_{\tau^{r}\in{\mathcal t}^{r}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y^{r}_{{{\tau^{r } } } } ) - x]\\ & = & \sup\limits_{\tau\in{\mathcal t}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x ] . \end{aligned}\ ] ] furthermore , @xmath554 & \geq & \inf_{x\in{\mathbb{r}}}\sup\limits_{\tau\in{\mathcal t}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x]\\ & \geq & \sup\limits_{\tau\in{\mathcal t}}\inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x].\end{aligned}\ ] ] thus @xmath555 = \inf_{x\in{\mathbb{r}}}\sup\limits_{\tau\in{\mathcal t}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x]\ ] ] which completes the proof of theorem [ new_representation ] . @xmath431 just simplifying notation , we assume that @xmath556 with @xmath557 being a positive integer . by we have @xmath558 = \inf_{x\in{\mathbb{r}}}\sup\limits_{\tau\in{\mathcal t}_{{\mathbb{t}}}}{\mathbb{e}}[\phi^{*}(x + y_{\tau } ) - x].\ ] ] so it is left to show that there exists a solution @xmath559 of the maximization problem and a solution @xmath439 of the minimization problem . indeed such a pair @xmath560 would be as required . in view of , we may find some compact interval @xmath561 of @xmath449 such that @xmath562 = \sup_{\tau\in{\mathcal t}_{\mathbb{t}}}\inf_{x\in i}{\mathbb{e}}\left[\phi^{*}(x + y_{\tau } ) - x\right].\ ] ] let @xmath563 denote the space of continuous real - valued mappings on @xmath564 this space will be equipped with the sup - norm @xmath565 whereas the product @xmath566 is viewed to be endowed with the norm @xmath567 defined by @xmath568 the key in solving the maximization problem is to show that @xmath569 is a weakly compact subset of @xmath566 w.r.t . the norm @xmath570 here @xmath571 stands for the set of all @xmath572 satisfying @xmath573 for @xmath574 as well as @xmath575 for @xmath576 and @xmath577 furthermore , define @xmath578\quad\mbox{for}~t\in\{1,\dots , t\},~a_{t}\in{\mathcal f}_{t}.\ ] ] notice that any mapping @xmath579 is extendable to a real - valued convex function on @xmath449 , and therefore also continuous . before proceeding , we need some further notation , namely @xmath580 denoting the set of all @xmath581 satisfying @xmath582 with @xmath583a.s . for @xmath584 and @xmath585a.s .. obviously , the subset @xmath586 consists of extreme points of @xmath587 any @xmath582 may be associated with the mapping @xmath588\quad(t \in\{1,\dots , t\}).\ ] ] it is extendable to a real - valued convex function on @xmath449 , and thus also continuous . hence , the mapping @xmath589{$\scriptstyle t=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle t$ } } \end{picture } } } l^{\infty}(\omega,{\mathcal f}_{t},{\mathrm{p}}|_{{\mathcal f}_{t}})\rightarrow { \mathcal c}(i)^{t},~ ( f_{1},\dots , f_{t})\mapsto ( h_{1,f_{1}},\dots , h_{t , f_{t}})\ ] ] is well - defined , and obviously linear . in addition it satisfies the following convenient continuity property . [ kompaktheitparti ] let @xmath517{$\scriptstyle t=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle t$ } } \end{picture } } } \sigma(l^{\infty}_{t},l^{1}_{t})$ ] be the product topology of @xmath590 @xmath591 on @xmath517{$\scriptstyle t=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle t$ } } \end{picture } } } l^{\infty}(\omega,{\mathcal f}_{t},{\mathrm{p}}|_{{\mathcal f}_{t}}),$ ] where @xmath590 denotes the weak * topology on @xmath592 then , @xmath580 is compact w.r.t . @xmath517{$\scriptstyle t=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle t$ } } \end{picture } } } \sigma(l^{\infty}_{t},l^{1}_{t}),$ ] and the mapping @xmath593 is continuous w.r.t . @xmath517{$\scriptstyle t=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle t$ } } \end{picture } } } \sigma(l^{\infty}_{t},l^{1}_{t})$ ] and the weak topology induced by @xmath570 in particular the image @xmath594 of @xmath580 under @xmath593 is weakly compact w.r.t . @xmath570 the continuity of @xmath593 follows in nearly the same way as in the proof of proposition 3.1 from @xcite . moreover , @xmath580 is obviously closed w.r.t . the product topology @xmath517{$\scriptstyle t=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle t$ } } \end{picture } } } \sigma(l^{\infty}_{t},l^{1}_{t}),$ ] and even compact due to banach - alaoglu theorem . then by continuity of @xmath595 the set @xmath594 is weakly compact w.r.t . @xmath570 this completes the proof . we need some further preparation to utilize lemma [ kompaktheitparti ] . [ thin ] let @xmath596 with @xmath201 and let @xmath186 if @xmath148 is atomless and if @xmath182\mid x\in{\mathbb{r}}\right\}$ ] is a thin subset of @xmath183 , then @xmath597\mid x\in{\mathbb{r}}\right\}$ ] is a thin subset of @xmath598 let @xmath265 with @xmath280 since @xmath148 is atomless , we may find disjoint @xmath599 contained in @xmath175 with @xmath600 then by assumption there exist nonzero @xmath601 such that @xmath602 vanishes outside @xmath603 as well as @xmath604~\right ] = 0 $ ] for @xmath191 and @xmath605 moreover , we may choose @xmath606 with @xmath607 for at least one @xmath608 and @xmath609 = 0.$ ] finally , @xmath610 and , setting @xmath611 @xmath612~\right]\\ & = & \sum_{i=1}^{2}\lambda_{i}~{\mathbb{e}}\left[~f_{i}\cdot{\mathbb{e}}\left[{\mathbbm{1}}_{a}\cdot\phi^{*}(x + y_{s } ) ~|~{\mathcal f}_{t}\right]~\right ] - x~{\mathbb{e}}\left[\left(\lambda_{1 } f_{1 } + \lambda_{2 } f_{2}\right)\cdot{\mathbbm{1}}_{a}\right ] = 0\end{aligned}\ ] ] for @xmath363 this completes the proof . the missing link in concluding the desired compactness of the set @xmath613 from is provided by the following auxiliary result . [ schlussel ] let @xmath148 be atomless for @xmath584 and furthermore let the subset @xmath182\mid x\in{\mathbb{r}}\right\}$ ] of @xmath183 be thin for arbitrary @xmath596 with @xmath185 and @xmath186 then for any @xmath614 there exist @xmath615 and mappings @xmath616 @xmath617 such that @xmath618 and @xmath619 let @xmath596 with @xmath185 and @xmath186 we may draw on lemma [ thin ] to observe that @xmath620\mid x\in{\mathbb{r}}\}$ ] is a thin subset of @xmath598 then the statement of lemma [ schlussel ] follows immediately from proposition [ kreinmilman ] ( cf . appendix [ appendixc ] ) applied to the sets @xmath621 ( @xmath622 ) , where @xmath623 under the assumptions of lemma [ schlussel ] , the set @xmath613 defined in coincides with @xmath624 which in turn is weakly compact w.r.t . @xmath625 due to lemma [ kompaktheitparti ] . [ kompaktheitpartii ] under the assumptions of lemma [ schlussel ] , the set @xmath613 ( cf . ) is weakly compact w.r.t . @xmath570 now we are ready to select a solution of the maximization problem . : + let the assumptions of proposition [ saddle - point ] be fulfilled . in view of it suffices to solve @xmath626\,\mbox{over}\,\tau\in{\mathcal t}_{{\mathbb{t}}}.\ ] ] let us assume that @xmath627 > \inf_{x\in i}{\mathbb{e}}[\phi^{*}(x + y_{0 } ) - x]$ ] because otherwise @xmath628 would be optimal . since @xmath629 for @xmath630 by assumption , any stopping time @xmath631 is concentrated on @xmath632 by corollary [ kompaktheitpartii ] , the set @xmath613 ( cf . ) is weakly compact w.r.t . the norm @xmath570 furthermore , the concave mapping @xmath633 defined by @xmath634 is continuous w.r.t . @xmath570 this means that @xmath635 is convex as well as well as @xmath636continuous , and thus also weakly lower semicontinuous because @xmath636closed convex subsets are also weakly closed . hence @xmath637 is weakly upper semicontinuous , and therefore its restriction to @xmath613 attains a maximum . in particular , the set @xmath638\mid \tau\in{\mathcal t}_{{\mathbb{t}}}\setminus\{0\}\right\ } = l(k)\ ] ] has a maximum . this shows that we may find a solution of .@xmath431 : + by @xmath639 $ ] we may define a convex , and therefore also continuous mapping @xmath640 moreover by lemma [ optimizedcertaintyequivalent ] ( cf . appendix [ appendixaa ] ) , @xmath641 this means that @xmath642}l(x)$ ] for some @xmath437 hence @xmath643 attains its minimum at some @xmath644 $ ] because @xmath643 is continuous . any such @xmath439 is a solution of the problem . @xmath431 [ optimizedcertaintyequivalent ] let @xmath1 $ ] be a lower semicontinuous , convex mapping satisfying @xmath73 and @xmath645 furthermore , let @xmath646 denote the set of all probability measures @xmath3 on @xmath70 which are absolutely continuous w.r.t . @xmath4 such that the radon - nikodym derivative @xmath647 satisfies @xmath648 < \infty.$ ] then the following statements hold true . 1 . if @xmath71 for some @xmath72 then the fenchel - legendre transform @xmath649 of @xmath133 is a nondecreasing , convex finite mapping . in particular its restriction @xmath650 to @xmath77 is a finite young - function , which in addition satisfies the condition @xmath651 if @xmath652 and @xmath653 in the case of @xmath654 2 . if @xmath655 for some @xmath656 then for any @xmath104 from @xmath657 we obtain @xmath658- { \mathbb{e}}\left[\phi\left(\frac{d{\mathrm{q}}}{d{\mathrm{p}}}\right)\right]\right ) = \inf_{x\in{\mathbb{r}}}{\mathbb{e}}[\phi^{*}(x + x ) - x],\ ] ] where the supremum on the left hand side of the equality is attained for some @xmath659 let @xmath660 for some @xmath661 obviously , @xmath140 is a nondecreasing convex function satisfying the properties @xmath662 next , we want to verify the finiteness of @xmath235 since @xmath140 is nondecreasing , and @xmath663 holds for any @xmath664 it suffices to show that @xmath665 for every @xmath666 for that purpose consider the mapping @xmath667 by assumption on @xmath668 we have @xmath669 hence for any @xmath670 we may find some @xmath671 such that we obtain @xmath672 moreover , @xmath673 is upper semicontinuous for @xmath666 hence , for every @xmath670 there is some @xmath674 $ ] with @xmath675 as a finite convex function @xmath140 is continuous . since it is also nondecreasing , we may conclude from that its restriction to @xmath77 is a finite young function . let us now assume that @xmath676 then @xmath677 analogously , @xmath678 may be derived in the case of @xmath654 thus we have proved the full statement ( i ) . let us turn over to the proof of statement ( ii ) , and let us consider the mapping @xmath679\ ] ] then , due to convexity of @xmath680 we may apply jensen s inequality along with statement ( i ) to conclude @xmath681\geq \lim_{x\to - \infty}([\phi^{*}(x - { \mathbb{e}}[x])]-x ) = \infty\quad\mbox{for}\ , x\in h^{\phi^{*}},\ ] ] and @xmath682\geq \lim_{x\to \infty}[\phi^{*}(x - { \mathbb{e}}[x ] ) - x ] = \infty\quad\mbox{for}\ , x\in h^{\phi^{*}}.\ ] ] thus , for any @xmath82 we find some @xmath683 such that @xmath684}{\mathbb{e}}[\phi^{*}(x - x ) - x].\ ] ] in addition , for @xmath82 the mapping @xmath685 $ ] is a convex mapping on @xmath365 hence its restriction to @xmath686 $ ] is continuous . this implies that @xmath687 is a real - valued function . moreover , it is easy to check that @xmath687 is a so called convex risk measure , defined to mean that it satisfies the following properties . * monotonicity : @xmath688 for all @xmath689 with @xmath690 , * cash - invariance : @xmath691 for all @xmath87 and @xmath692 , * convexity : @xmath693 for all @xmath694 @xmath695.$ ] then we obtain from theorem 4.3 in @xcite that @xmath696- \rho^{*}({\mathrm{q}})\right)\ ] ] holds for all @xmath82 where @xmath697 - \rho(x)\right).\ ] ] by routine procedures we may verify @xmath698 - { \mathbb{e}}[\phi^{*}(x)]\right)\ ] ] for @xmath659 since @xmath699 = \lim_{x\to \infty}[\phi^{*}(x ) - x ] = \infty$ ] due to statement ( i ) , we may conclude from ( * ? ? ? * ( 5.23 ) ) @xmath700 - { \mathbb{e}}[\rho^{*}(x)]\right ) = { \mathbb{e}}\left[\phi\left(\frac{d{\mathrm{q}}}{d{\mathrm{p}}}\right)\right]\quad\mbox{for all}\ , { \mathrm{q}}\in{\mathcal q}_{\phi,0}.\ ] ] this completes the proof . let @xmath701 be a filtered probability space , and let the product space @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] be endowed with the product topology @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i})$ ] of the weak * topologies @xmath702 on @xmath703 ( for @xmath704 ) . [ angelic ] let @xmath705 be separable w.r.t . the weak topology @xmath706 for @xmath707 and let @xmath708{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] be relatively compact w.r.t . @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i}).$ ] then for any @xmath104 from the @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i})-$]closure of @xmath709 we may find a sequence @xmath710 in @xmath711 which converges to @xmath104 w.r.t . the @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i}).$ ] setting @xmath712{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}}),$ ] we shall denote by @xmath713 the topological dual of @xmath714 w.r.t . @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i}).$ ] it is easy to check that @xmath715,\ ] ] where @xmath716 and @xmath717 ( for @xmath704 ) defines a linear operator from @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{1}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] onto @xmath713 which is continuous w.r.t . the product topology @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{1}_{i},l^{\infty}_{i})$ ] of the weak topologies @xmath718 and the weak topology @xmath719 since @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{1}_{i},l^{\infty}_{i})$ ] is separable by assumption , we may conclude that @xmath720 is separable too . then the statement of the proposition [ angelic ] follows immediately from @xcite , p.30 . let for @xmath721 denote by @xmath722 a filtered probability space , and let the set @xmath723 gather all sets @xmath724 from @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } { \overline{{\mathcal f}}}_{i}$ ] satisfying @xmath725 for @xmath726 and @xmath727 we shall endow respectively the product spaces @xmath517{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] with the product topologies @xmath517{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i})$ ] of the weak * topologies @xmath702 on @xmath703 ( for @xmath728 and @xmath729 ) . fixing @xmath728 and nonnegative @xmath730 the subset @xmath731{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] is defined to consist of all @xmath732{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] such that @xmath733 @xmath734a.s . for any @xmath735 and @xmath736 @xmath734a.s .. for abbreviation we shall use notation @xmath737 [ banachalaoglu ] @xmath738 is a compact subset of @xmath517{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] w.r.t . the topology @xmath517{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i})$ ] for @xmath728 and arbitrary nonnegative @xmath739 the statement of lemma [ banachalaoglu ] is obvious in view of the banach - alaoglu theorem . [ kreinmilmanallgemein ] let @xmath740 be nonvoid for @xmath741 such that @xmath742\mid f\in m_{j}\}$ ] is a thin subset of @xmath705 for @xmath743 with @xmath744 and any @xmath745 furthermore , let us fix @xmath746 and consider the set @xmath747 consisting of all @xmath748 from @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] satisfying @xmath749 = { \mathbb{e}}\left[f_{i}\cdot \varphi_{i}\right]$ ] for any @xmath750 @xmath751 then the set @xmath752 has extreme points , and for each extreme point @xmath753 there exists some @xmath754 such that @xmath755 @xmath734 a.s . holds for @xmath756 we shall use ideas from the proof of proposition 6 in @xcite . first , let us , for any @xmath728 , denote by @xmath757 the set of all @xmath758 from @xmath517{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } l^{\infty}({\overline{\omega}},{\overline{{\mathcal f}}}_{i},{\overline{{\mathrm{p}}}}|_{{\overline{{\mathcal f}}}_{i}})$ ] satisfying @xmath749 = { \mathbb{e}}\left[f_{i}\cdot \varphi_{i}\right]$ ] for @xmath759 and @xmath760 it is closed w.r.t . @xmath517{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i}).$ ] hence by lemma [ banachalaoglu ] , the set @xmath761 is compact w.r.t . @xmath517{$\scriptstyle i = k$ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i})$ ] for every nonnegative @xmath739 since it is also convex , we may use the krein - milman theorem to conclude that each set @xmath762 has some extreme point if it is nonvoid . notice that @xmath763 contains at least @xmath764 so that it has some extreme point . we shall now show by backward induction that for any @xmath728 and any nonnegative @xmath765 with nonvoid @xmath762 * each of its extreme points @xmath766 satisfies @xmath767 @xmath734a.s . ( @xmath729 ) for some @xmath754 with @xmath768 if @xmath769 obviously , this would imply the statement of proposition [ kreinmilmanallgemein ] . for @xmath770 the set @xmath771 is nonvoid iff @xmath772 = { \mathbb{e}}\left[f_{m}\cdot\varphi_{m}\right]$ ] holds for every @xmath773 in this case , @xmath443 is the only extreme point , which has trivial representation @xmath774 corresponding to @xmath775 now let us assume that for some @xmath776 and every nonvoid @xmath762 statement @xmath777 is satisfied . let @xmath778 be nonnegative with @xmath779 and select any extreme point @xmath780 of @xmath781 then @xmath782 belongs to @xmath783 and is nonnegative . moreover , @xmath784 and it is easy to check that @xmath785 is even an extreme point of @xmath786 hence by assumption , there exists some @xmath754 satisfying @xmath768 if @xmath787 and @xmath788 @xmath734a.s . for @xmath789 setting @xmath790 we want to show @xmath791 this will be done by contradiction assuming @xmath792 then @xmath793 for some @xmath540 where @xmath794 we may observe by assumption that @xmath795\mid \varphi_{i } \in m_{i}\}$ ] ( with @xmath729 ) as well as @xmath796 are all thin subsets of @xmath797 since finite unions of thin subsets are thin subsets again ( cf . * proposition 2.1 ) ) , we may find some nonzero @xmath798 vanishing outside @xmath799 and satisfying @xmath800 = 0 $ ] for @xmath801 as well as @xmath802 = { \mathbb{e}}\left[g\cdot{\mathbb{e}}\left[{\mathbbm{1}}_{a_{i}}\cdot\varphi_{i}~|~{\overline{{\mathcal f}}}_{k-1}\right]\right ] = 0 \quad ( \varphi_{i}\in m_{i},~i\in\{k,\dots , m\}).\ ] ] according to theorem 2.4 in @xcite , we may choose @xmath803 such that @xmath804 holds . now , define @xmath805 and @xmath806 by @xmath807 since @xmath725 for @xmath726 and @xmath808 we obtain @xmath809 @xmath734a.s .. so by construction , @xmath810 differ , and belong both to @xmath781 moreover , @xmath811 for @xmath812 this contradicts the fact that @xmath780 is an extreme point of @xmath781 therefore , @xmath791 now define @xmath813{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } { \overline{{\mathcal f}}}_{i}$ ] by @xmath814 obviously , @xmath815 for @xmath726 follows from @xmath725 for @xmath816 moreover , @xmath817 in particular @xmath818 finally , it may be verified easily that @xmath819 @xmath734a.s . holds for @xmath820 hence @xmath821 fulfills statement @xmath777 completing the proof . [ kreinmilman ] let @xmath740 be nonvoid for @xmath741 such that @xmath742\mid f\in m_{j}\}$ ] is a thin subset of @xmath705 for @xmath743 with @xmath744 and any @xmath745 then for any @xmath822 there exist @xmath754 and @xmath823 @xmath824 such that @xmath825 = 0\quad\mbox{for}~\varphi_{i}\in m_{i}~\mbox{with}~i = 1,\dots , m,\ ] ] and @xmath826 let us fix any @xmath827 and let @xmath747 denote the set consisting of all @xmath828 where @xmath829 such that @xmath749 = { \mathbb{e}}\left[f_{i}\cdot \varphi_{i}\right]$ ] for @xmath830 by proposition [ kreinmilmanallgemein ] , we may select an extreme point @xmath748 of @xmath752 and some @xmath754 such that @xmath831 @xmath734a.s . holds for @xmath751 then @xmath832 and @xmath724 are as required . [ dichtheit ] if @xmath833 is atomless for every @xmath707 then @xmath834 is the @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i})-$]closure of @xmath835 let @xmath746 be arbitrary . consider the subsets @xmath836 < \varepsilon~\mbox{for}~f\in m_{i}\},\ ] ] where @xmath540 and @xmath837 any nonvoid , finite subset of @xmath838 the sets @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } u_{i\varepsilon}(m_{i})$ ] constitute a basis of the @xmath517{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } \sigma(l^{\infty}_{i},l^{1}_{i})-$]neighbourhoods of @xmath839 so let us select any @xmath840 and nonvoid finite subsets @xmath837 of @xmath705 for @xmath751 let @xmath743 with @xmath841 and @xmath745 then the set consisting of all @xmath842 $ ] with @xmath843 is a nonvoid finite subset of @xmath844 in particular it is thin because @xmath833 is assumed to be atomless ( cf . * lemma 2 ) ) . hence we may apply proposition [ kreinmilman ] to select some @xmath754 satisfying @xmath845 = 0 $ ] for @xmath846 and @xmath847 this means @xmath848{$\scriptstyle i=1 $ } } \put(5.00,18.00){\makebox(0,0)[cc]{$\scriptstyle m$ } } \end{picture } } } u_{i\varepsilon}(m_{i}),\ ] ] and completes the proof . the authors would like to thank alexander schied and mikhail urusov for fruitful discussions and helpful remarks . 0 anantharaman , r. 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the abundance patterns of extremely metal - poor ( emp ) stars are useful in studying nucleosynthesis in massive supernovae ( sne ) . population ( pop ) iii stars are usually considered to be massive stars . some of them might become black holes without supernova explosions , but some should have exploded as supernovae to initiate the first metal enrichment in the early universe . the stars born from the gas enriched by the pop iii sne are pop ii stars with low metallicity . low - mass pop ii stars have long lifetimes and might be observed as extremely metal poor ( emp ) stars with [ fe / h ] @xmath0 . ( here , [ a / b ] = @xmath1(@xmath2/@xmath3 ) - @xmath1(@xmath2/@xmath3)@xmath4 , where the subscript `` @xmath5 '' refers the solar value and @xmath2 and @xmath3 are the abundances of elements a and b , respectively . ) therefore , a emp star may reflect the nucleosynthetic result of a pop iii sn and constrain properties of the pop iii sn . there have been attempts to actually fit the abundance patterns of emp stars with the supernova nucleosynthesis models . for example , using the mixing - fallback model proposed by umeda @xmath6 nomoto ( 2002 ) ( hereafter un02 ) and umeda @xmath6 nomoto ( 2003 ) ( hereafter un03 ) mimicking aspherical explosion effects ( e.g. , tominaga 2009 ) , they showed that the abundance patterns of the elements from c to zn of carbon - normal emp stars and carbon - rich emp stars can be successfully reproduced by energetic core - collapse sn ( `` hypernova '' , hereafter hn ) , models and faint sn models , respectively ( un02 ; un05 ; tominaga et al . 2007 ) , while those of very metal poor ( vmp ) stars ( @xmath7 [ fe / h ] @xmath8 ) can be reproduced by normal core - collapse sn models or the imf - integration of hypernova and normal core - collapse sn models ( tominaga et al . it is important to note that the observed emp stars are so far all explained by the pollutions by core - collapse sne with initial stellar masses 11@xmath9 and no evidence of pair instability sne with initial stellar masses 140@xmath10 ( un02 , see also chieffi @xmath6 limongi 2002 ; un03 ; umeda @xmath6 nomoto 2005 , hereafter un05 ; heger @xmath6 woosley 2002 , 2008 ) . these previous sn models do not eject elements heavier than zn in a sizeable amount , and this is consistent with the abundance of some emp stars . however , there are also emp stars showing enhancements of neutron - capture elements . some of them show abundance patterns almost identical to the solar system r - process pattern for sr and heavier elements ( e.g. , sneden et al . 2000 ; hill et al . one example of such a star is cs22892 - 052 and called as a main `` r - process star '' ( sneden et al . the process producing heavy neutron capture elements ( @xmath11ba - u ) is referred to as `` main '' r - process ( e.g. , truran et al . 2002 ; wanajo & ishimaru 2006 ) . on the other hand , there are other emp stars that require another neutron - capture process referred sometimes as `` lepp '' ( lighter element primary process ) or `` weak r - process '' ( travaglio et al . 2004 ; wanajo @xmath6 ishimaru 2006 ) . travaglio et al . ( 2004 ) reported emp stars with abundances of sr , y , and zr which can not be explained by the s - process or main r - process . in the weak r - process stars , the elements with intermediate mass ( 37@xmath12z@xmath1247 , i.e. , from rb to ag ) elements show moderate enhancements with respect to heavy ones ( z@xmath1356 , i.e. , heavier than ba ) . more recently francois et al . ( 2007 ) showed several other examples of the weak r - process stars . there are evidences of the existence of weak r - process but its origin is unknown . several possible mechanisms to produce the weak r - process elements are proposed . wanajo et al . ( 2001 ) presented calculations of r - process nucleosynthesis in neutrino - driven winds from a proto - neutron stars . they showed that the abundance pattern of weak r - process is reproduced when the main r - process nucleosynthesis is failed . the nucleosynthesis in neutrino - driven winds was also studied in hoffman et al . ( 1996 ) by taking the electron fraction @xmath14 of the wind matter as a free parameter . they showed that weak r - process elements may be synthesized for low @xmath14 ( @xmath150.47 ) . although the production of weak r - process elements in neutrino - driven winds was suggested , it is difficult to give a detailed yield because the physical conditions and ejected mass depend on unknown supernova explosion mechanisms . explosive nucleosynthesis in low @xmath14 matter was also studied in the context of multi - dimensional explosion models ( janka et al . 2003 ; pruet et al . they showed that small amounts of low @xmath14 ( @xmath160.46 ) matter as well as high @xmath14 ( @xmath170.56 ) matter are ejected from a hot bubble just outside a proto - neutron star . the high @xmath14 ( @xmath170.56 ) matter is suggested to be ejected even in the one - dimensional cases ( e.g. , frlich et al . 2006 ) . on the other hand , the ejection of low @xmath14 matter is driven by the convection in the hot bubble , and thus essentially the multi - dimensional phenomenon . janka et al . ( 2003 ) suggested that the low @xmath14 matter contains the weak r - process elements ( sr , y and zr ) to explain the galactic abundances , but a detailed nucleosynthesis calculation did not confirm the production of these elements ( pruet et al . 2005 ) . in this paper , we investigate the physical conditions to produce sufficient amounts of the weak r - process elements ( sr , y and zr ) and discuss whether core - collapse sne with a slight modification can be compatible with the observed abundances of the weak r - process elements in the emp stars . in order to do this , we assume that small amount of low @xmath14 matter is ejected by the multi - dimensional effects , which may be driven by the convection in a hot bubble ( janka et al . 2003 ) or jets in a jet - like explosion ( e.g. , maeda @xmath6 nomoto 2003 ) or a collapsar model ( e.g. , pruet et al . 2003 , 2004 ; popham , woosley @xmath6 fryer 1999 ) . the entropy of the low @xmath14 matter flow may depend on the ejection mechanism . we assume that the matter flow has the same entropy as the supernova shock wave . jet - like explosion or collapsar models may describe hne . however , they contain many unknown parameters , and the innermost @xmath14 of the ejecta depends on those parameters . on the other hand , the simulations of janka et al . ( 2003 ) contain less input parameters , so the obtained @xmath14 profile is more reliable , though their simulations are about normal sne . we are interested in the emp stars , and their progenitor may be more massive and explode energetically , i.e. , may become hne . therefore , we vary @xmath14 beyond the range given by the simulation in janka et al . although we have multi - dimensional effects in mind , we only perform one - dimensional calculations in this paper , because it is often useful to make a large parametric search to disclose the essence of physics . in @xmath18 2 , we show observational trends of [ sr , y , zr / fe ] . in @xmath18 3 , we describe our progenitor and explosion models . in @xmath18 4 , we present weak r - process nucleosynthesis and specify conditions mention our assumption applying to our models in order to reproduce reproducing the observational [ sr , y , zr / fe ] . we also compare our yields with 4 emp stars which have peculiarly high [ sr , y , zr / fe ] . in @xmath18 5 , summaries and discussions are given . since we are interested in the weak r - process elements in the emp stars , we select stars with [ fe / h ] @xmath19 from cayrel et al . ( 2004 ) and use their data from carbon to zinc . observations of [ sr , y , zr / fe ] are taken from honda et al . ( 2006 ) for hd122563 and francois et al . ( 2007 ) for the other stars . taking the previous works on the weak r - process into consideration , we use two abundance ratios as a diagnostic to distinguish `` main r - process stars '' and `` weak r - process stars '' . they are relative numbers of sr and ba , sr / ba , and y and eu , y / eu . we use @xmath20 from anders @xmath6 grevesse ( 1989 ) . a main r - process star cs22892 - 052 has [ sr / ba ] = -0.57 and [ y / eu ] = -1.16 . therefore , if a emp star has [ sr / ba ] @xmath21 and [ y / eu ] @xmath22 , we consider the star as a weak r - process star , otherwise as a main r - process star ( see also aoki et al . figure 1 shows [ sr / fe ] , [ y / fe ] , and [ zr / fe ] vs. [ fe / h ] of the weak r - process stars . among 21 selected stars , 20 stars are with @xmath23 [ sr , y , zr / fe ] @xmath24 . the calculation method and other assumptions are the same as described in umeda et al . 2000 ( hereafter unn00 ) , un02 , un05 and tominaga et al . 2007 , except for the size of the nuclear reaction networks . in this paper , we adopt the pop iii progenitors as in un05 and apply the model with @xmath25 = 13 @xmath26 and @xmath27 = 1.5 ( hereafter model-1301 ) , the one with @xmath25 = 25 @xmath26 and @xmath27 = 1 ( hereafter model-2501 ) , and the one with @xmath25 = 25 @xmath26 and @xmath27 = 20 ( hereafter model-2520 ) . model-1301 and model-2501 are normal sn models , and model-2520 is a hn model . detailed nucleosynthesis is calculated as a postprocessing after the hydrodynamical calculation with a simple @xmath28-network . the isotopes included in the post process calculations are 809 species up to @xmath29pd ( see table 1 ) . we note that a neutrino process during explosive burning ( yoshida et al . 2008 ; woosley @xmath6 weaver 1995 ) is not taken into account . the abundance distributions after the sn explosion for model-1301 , model-2501 and model-2520 are shown in figure 2 . we obtain the final yields by setting the inner boundary of the ejected matter , a mass - cut ( @xmath30 ) as we describe in the next section . in this section we summarize the abundance pattern of sn ejecta when a `` conventional '' mass - cut is adopted . the `` conventional '' means that the mass - cut is chosen to eject a reasonable amount of @xmath31ni . although previously the mass - cut is often chosen to eject 0.07 @xmath26 of @xmath31ni reproducing the brightness of normal sne as sn1987a ( e.g. , nomoto et al . 2003b ) , we set @xmath30 = 1.59 @xmath26 for model-1301 , 1.76 @xmath26 for model-2501 and 2.31 @xmath26 for model-2520 to yield [ si / fe ] @xmath32 [ si/@xmath31ni ] @xmath11 0.4 . as a result , 13@xmath26 model eject a similar amount of @xmath31ni to sn 1987a ( @xmath11 0.07 @xmath26 ) , but 25@xmath26 models eject large amounts of @xmath31ni ( @xmath11 0.5 @xmath26)ni mass is smaller than this value . see e.g. , table 3 and section 4.4 below . ] . in figure 3 , the abundance patterns from si to ru are compared with those of emp stars . this figure shows that the adopted @xmath30s yield a rather good agreement between the predicted and the observed abundance ratios for most of the elements above si . on the contrary , [ sr , y , zr / fe ] in the models are much lower than those observed in the `` weak r - process stars '' . as mentioned in @xmath18 3 , the models with the `` conventional '' mass - cut do not reproduce @xmath33 [ sr , y , zr / fe ] @xmath34 . in this section , we study the conditions to produce weak r - process elements . we take into account of the uncertainty of @xmath14 and assume some ejection of matter from regions below @xmath30 . recent theoretical multi - d hydrodynamical simulations of core collapse supernova have shown that the presupernova value of @xmath14 can be modified during the explosion , even significantly , in the innermost zones of the exploding envelope . in figure 4 we schematically depict the value of @xmath14 before and after the explosion . the presupernova value shown is for the model when the central density is @xmath11 @xmath35 g @xmath36 . after that time , the electron capture significantly reduces @xmath14 ( @xmath170.4 ) in the inner part , but the very neutron - rich matter is rarely ejected . recent simulations have shown that not only neutron - rich(@xmath14 @xmath15 0.5 ) but also proton - rich(@xmath14 @xmath37 0.5 ) regions appear after explosion(e.g . , fr@xmath38lich et al . the @xmath14 distributions based on an actual 2d - simulations is , for example , given in figure 4 of pruet et al . figure 4 shows the assumed @xmath14 profile mimicking the results of such simulations . those simulations have shown that a density just above a proto - neutron star surface rapidly decreases after the supernova shockwave passes through a fe - core . this region is often called a hot bubble , in which @xmath14 is set by a competition between different lepton capture processes on free nucleons . at the beginning of the explosion , an excess of electron neutrinos over antineutrinos makes the matter tend to be proton - rich ( qian and woosley 1996 ) . recent detailed one- and two - dimensional simulations have shown that some of these proton - rich matter is actually ejected . in the later stages , the fluxes and spectral change of neutrinos make @xmath14 less than 0.5 . 2d simulations by janka et al . ( 2003 ) have shown that not only the proton - rich matter but also some neutron - rich matter ( @xmath14@xmath390.46 - 0.47 ) is ejected . this is because the hot bubble is convective and some of the inner matter can be dragged outside . although these previous calculations are for less massive normal supernovae , the similar mechanism may work for more massive supernova . even if the explosion mechanism is completely different , the inner matter may be carried outside along the jets in a jet - like explosions . therefore , in the following we calculate nucleosynthesis in a deep region of supernovae to estimate the total yield when the same amount of matter below the conventional mass cut , with @xmath14 arbitrarily changed with respect to the presupernova value , is ejected . in order to perform nucleosynthesis calculations , we need histories of temperature and density for a given mass element . strictly speaking , the histories depend on explosion models and how the matter is carried outside and can not be represented by a one - dimensional model . to avoid complication , however , we carry out the same calculations with section 3 but in the region below @xmath30 and arbitrarily changing the @xmath14 in the progenitor model according to figure 4 . this approach helps simplifying the complicated problems and clarifying the essence of physics . with this assumption , the entropy of the low @xmath14 flow is @xmath40@xmath113 for sne and @xmath1115 for hne , which is similar value as the matter just above @xmath30 . it has been long discussed that if a normal sn produces main r - process elements . this is because the sn has to eject quite high - entropy neutrino driven wind . as for the weak r - process , wanajo et al . ( 2001 ) , for example , showed that if the entropy of the neutrino driven wind is not high enough for the main r - process , a weak r - process - like pattern is obtained . however , whether a sn can eject such kind of mater is unpredictable because explosion simulations have not been succeeded . therefore , we consider an extreme case that no neutrino - driven high - entropy matter but the supernova - shocked matter is ejected . the ejection of high entropy matter are discussed in the last part of section 5 briefly and will be discussed in detail elsewhere . in this subsection , we show parameter dependences of the products of the complete - si burning . the complete si - burning takes place in the shocked matter attaining the maximum temperature of @xmath41 @xmath379.5 and thus no unburned si is left after the complete si - burning . since we are mainly interested in the ejection of low @xmath14 matter from the beneath of a conventional mass - cut , @xmath30 , we change @xmath14 in the region from 0.40 to 0.50 . although janka et al . ( 2003 ) showed the small amount of ejection of matter with @xmath14 @xmath110.56 - 0.46 from the hot bubble region , we consider @xmath14 as low as 0.40 because the explosion mechanism is quite uncertain especially for hypernovae . in this paper we do not consider @xmath14 @xmath370.5 matter . its effect is briefly discussed in section 5 . we use the temperature and density trajectories of the models 1301 , 2501 and 2520 as 3 . the difference from 3 is that we consider the deep region below @xmath30 . we denote the mass coordinate of the inner and outer boundaries of the region by @xmath42 and @xmath43 , respectively , and take mass average in the region . @xmath43 is defined by the location where @xmath44(@xmath45si)@xmath1110@xmath46 . @xmath42 is chosen at a point near fe core surface . we note that the result is not sensitive to @xmath42 because density and temperature trajectories during nucleosynthesis is almost the same around @xmath42 . the specific values taken are ( @xmath42,@xmath43)=(1.41@xmath26,1.64@xmath26),(1.52@xmath26,1.92@xmath26 ) and ( 1.61@xmath26,2.69@xmath26 ) for model-1301 , model-2501 , and model-2520 , respectively . figure 5 illustrates the @xmath14 profile for the three models we have assumed . nucleosynthesis of this region basically proceeds as @xmath28-rich freezeout . when the shock wave reaches the region , temperature rises rapidly and heavy elements are decomposed mainly into @xmath28-particles . as the star expands and the temperature drops , @xmath28-rich freezeout takes place with roughly constant @xmath14 . the mass fraction of @xmath28 at the maximum temperature and later stages depends on the entropy , i.e. , temperature and density . more @xmath28 is produced and heavier elements are synthesized for higher entropy explosions ( e.g. , un02 ) . in figures 6 - 8 , abundance ratios , [ x / fe ] , of the complete si - burning products integrated over @xmath42 and @xmath43 are shown . the abundance ratios of heavy elements as a function of @xmath14 show non - monotonic behavior in figure 6 - 8 . this behavior depends not only on the abundance of x , but also on that of fe . for example , in figure 6 and 7 , fe to heavy elements ratios , such as sr / fe , appear to be minimum for @xmath14 @xmath110.45 . this is because for @xmath14 @xmath110.45 the most abundant isotope which becomes fe is @xmath31fe and not @xmath31ni . @xmath31fe is produced relatively a lot for @xmath14 @xmath110.45 . therefore , the reason for this non - monotonic behavior is not simple . these figures also show that for @xmath14 @xmath39 0.49 , elements heavier than zn are not efficiently produced compared to fe . we provide some explanations of the result paying attention to the abundance of x in the following paragraphs . absolute amounts of the synthesized weak r - process elements are easily seen in figure 9 , which shows the mass fractions of sr , y and zr in the region . this figure shows that the abundances of these elements are relatively large for @xmath14@xmath170.48 and have a peak around @xmath14@xmath110.43 - 0.46 . for @xmath14@xmath110.42 - 0.43 , most abundant isotopes that become sr , y and zr are @xmath47kr , and @xmath48kr and @xmath49kr , respectively . for @xmath14@xmath110.45 - 0.46 , these are @xmath47sr , @xmath48y and @xmath49zr , respectively . since these isotopes are all neutron - rich having @xmath14@xmath50z / a@xmath110.41 - 0.45 , the weak r - process elements are tend to be produced most efficiently for this range of @xmath14 . we note that the exact processes complicatedly depend on entropy , @xmath14 , the properties of nuclear states , and the mass fraction of @xmath28 particles during @xmath28-rich freezeout . the abundance peaks seem to locate at @xmath14@xmath110.43 for the normal energy models ( 1301 and 2501 ) , and @xmath14@xmath110.45 - 0.46 for model-2520 . it is interesting that the peak is located at a larger value of @xmath14 for the higher energy model . the reason for this is not simple because the yields depend on the complicated properties of nuclear structures . for example , in the @xmath14=0.40 case in figure 8 ( model-2520 ) , the synthesized amounts of the weak r - process elements are small , but lighter elements , ge to kr , are quite abundant . for this specific case @xmath51ge , with @xmath14@xmath50z / a=0.390 , is quite abundant after the explosive synthesis . this decays into @xmath51se and the synthesis of heavier weak r - process elements are suppressed . since the density and temperature trajectories of the complete si - burning region are not so different for model-1301 and model-2501 , the abundance of weak r - process elements in model-1301 and model-2501 are similar ( figure 6 , 7 and 9 ) . as seen in figure 7 , 8 and 9 , high @xmath52 enhances the weak r - elements especially from @xmath14 = 0.45 to 0.47 . temperature of the complete si - burning region in model-2520 is much higher than that in the model-2501 . therefore , an entropy of model-2520 is much higher than that of model-2501 , and much more @xmath28-particles can be obtained in model-2520 than in model-2501 . this is why more sr , y and zr are produced in model-2520 than in model-2501 . the abundances of sr , y and zr are almost same in model-2501 and model-2520 when @xmath14=0.49 and 0.50 . when @xmath14 is 0.49 and 0.50 , the elements produced also have @xmath530.5 . since heavy nucleus with @xmath530.5 are less bound than those with neutron - rich ( hoffman et al . 1996 ) , when @xmath54 0.5 , heavier elements than zn are not produced even though @xmath52 is high . nucleosynthesis pattern in the complete si - burning region with constant @xmath14 is shown in the previous subsection . in this subsection , assuming various @xmath14 distributions in region below @xmath30 , we present abundance patterns of whole ejecta with mass ejection below @xmath30 . for the matter below @xmath30 , the yields are averaged for the @xmath14 values ranged from @xmath55 to @xmath56 . for example , the y@xmath57 distribution y@xmath57 = 0.45 - 0.50 means that the yields are the average of the six models with y@xmath57 = 0.45 , 0.46 , 0.47 , 0.48 , 0.49 and 0.50 . in this subsection again we do not consider the matter with @xmath14@xmath370.5 . the main parameters of our models are @xmath58 and @xmath55 . @xmath58 is the mass of the ejected matter from the region below @xmath30 , and @xmath55 is the lowest @xmath14 of the ejected matter . @xmath58 of the matter below @xmath30 is added to the matter above @xmath30 , and they are assumed to be ejected to the outer space all together . small amounts of matter with low @xmath14 could be ejected in two - dimensional or jet - like explosion models ( e.g. , janka et al . as shown in the previous subsection , large amounts of sr , y and zr are obtained when @xmath14 = 0.43 in model-1301 and model-2501 , and when @xmath14 = 0.45 in model-2520 . therefore , we take @xmath56 = 0.5 , and @xmath55 = 0.42 and 0.43 for model-1301 and model-2501 , and various values from @xmath55 = 0.45 to 0.49 for model-2520 . here , @xmath58 is selected to obtain 0 @xmath17 [ zn / fe ] @xmath17 0.5 . the abundance patterns of the ejected matter for model-1301 , model-2501 and model-2520 are shown in figure 10 , 11 and 12 - 13 , respectively . @xmath55 and @xmath58 and some related numbers for models ( a ) are summarized in table 2 . in the models ( a ) , @xmath59 is set to obtain [ zn / fe]@xmath320.5 for @xmath55@xmath170.46 . for @xmath600.47 ( model-2520 ) , the same value of @xmath58 with the @xmath55@xmath170.46 models is adopted because [ zn / fe ] is lower than 0.5 even if all the matter below @xmath30 is ejected . for the models ( b ) and ( c ) , @xmath58 of the model ( a ) is divided by 3 and 10 , respectively . as shown in figure 10 and 11 , [ sr , y , zr / fe ] are not improved to fit the observation in model-1301 and model-2501 . on the contrary , [ sr , y , zr / fe ] in model-2520 are high enough to be ranged in @xmath61 [ sr , y , zr / fe ] @xmath15 1 when @xmath55 is from 0.45 to 0.46 ( fig . note that the abundance patterns of [ sr , y , zr / fe ] are different depending on @xmath55 . when @xmath55 @xmath620.45 , those ratios have the relation of [ sr / fe]@xmath11[zr / fe]@xmath37[y / fe ] . when @xmath55 @xmath620.452 , [ sr / fe ] @xmath63 [ y / fe ] @xmath15 [ zr / fe ] . when @xmath55 @xmath64 , [ sr / fe ] @xmath15 [ y / fe ] @xmath15 [ zr / fe ] . in this subsection we compare our yields of model-2520 with abundance patterns of emp stars . among the 22 weak r - process stars mentioned in @xmath18 1 , 15 stars have [ sr / ba]@xmath37 1 and [ y / eu ] @xmath37 1 . we select sr - rich stars with [ sr / ba ] @xmath37 1 , [ y / eu ] @xmath37 1 and [ sr / fe ] @xmath37 0 : bs16477 - 003 ( [ fe / h]=-3.36 ) , cs22873 - 166 ( [ fe / h]=-2.97 ) , cs22897 - 008 ( [ fe / h]=-3.41 ) and cs29518 - 051 ( [ fe / h]=-2.78 ) . the observational data from he to zn and beyond zn are taken from cayrel et al . ( 2004 ) and francois et al . ( 2007 ) , respectively . @xmath14 distribution and @xmath58 have been chosen in order to obtain the best fit to the observed abundances of sr , y and zr . in order to adjust light to heavy element ratios , such as o / fe , we also assume the following mixing - fallback process to take place in hn model-2520 ( see e.g. , tominaga et al . 2007 for detail ) : + 1 . burned material below @xmath65(out ) is uniformly mixed . afterward only a fraction ( @xmath66 ) of the mixed material is ejected with the matter above @xmath65(out ) . + we set @xmath65(out ) = 3.60@xmath26 for the c - poor stars with [ c / mg]@xmath150 , such as cs22873 - 166 and cs29518 - 051 , and 5.76@xmath26 for c - normal stars with [ c / mg]@xmath110 , such as bs16477 - 003 and cs22897 - 008 . @xmath67 = 3.60@xmath26 is a point above which x(ca)@xmath15x(mg ) , and @xmath67 = 5.76@xmath26 is a point in the o - rich layer . @xmath55 , @xmath58 , @xmath66 , @xmath65(out ) and the other values for each comparison are summarized in table 3 . figure 14 shows comparisons between the yields of our mixing - fallback models and the abundance patterns of emp stars . in addition to the abundance ratios of the elements heavier than si , [ c / fe ] , [ mg / fe ] and [ al / fe ] show reasonable agreements with the observations . the nucleosynthesis yields in the ejecta for selected isotopes at the time around 150 seconds after the explosion are also given in tables 4 to 7 . to obtain these tables , the isotopes with their half - lives less than 30 days except @xmath31ni are radioactively decayed . in this paper we assume uncertainty of @xmath14 in the deep regions below @xmath30 and mass ejection from the regions for three models model-1301 , model-2501 and model-2520 . among those models , we obtain high [ sr , y , zr / fe ] ( ranged from -1 to 1 ) only in the model-2520 . the `` hypernova '' model-2520 can reproduce the observational data of sr , y and zr in addition to the elements from c to zn . we also find that the weak r - process elements are not contained in the `` normal '' supernova models 1301 @xmath6 2501 , even though low @xmath14 ( @xmath390.40 ) matter is ejected . in the normal supernova models , however , intermediate mass elements from ga to rb may be abundantly ejected ( figure 10 and 11 ) . it is interesting to note that there have been no observational evidences that ga - rb - rich stars exist in emp stars . it is possible that normal sne do not eject sufficient amounts of low @xmath14 matter or that we have observationally overlooked such ga - rb - rich stars . in comparisons with 4 emp stars in figure 14 , the ratios of some elements , i.e. , [ na / fe ] , [ k / fe ] , [ sc / fe ] , [ mn / fe ] and [ co / fe ] show deficiencies from the observation as our previous finding ( see e.g. , un05 ; tominaga et al . 2007 ) . in tominaga et al . ( 2007 ) , possible solutions are discussed as follows : na is mostly synthesized in the c - shell burning , and the produced amount of na depends on the overshooting at the edge of the convective c - burning shell ( iwamoto et al . since no overshooting is included in the present presupernova evolution models , the inclusion of the overshooting could enhance the na abundance . [ k / fe ] is slightly enhanced by the `` low - density '' modification ( un05 ; tominaga et al . 2007 ) and iwamoto et al . ( 2006 ) suggests that the matter with large @xmath14 ( @xmath370.5 ) can produce enough k. [ sc / fe ] and [ ti / fe ] can be enhanced by nucleosynthesis in high - entropy environments ( a low - density modification , un05 ; tominaga et al . 2007 ) or in a jet - like explosion ( nagataki et al . 2003 ; maeda @xmath6 nomoto 2003 ; tominaga 2009 ) , and further an enhancement of [ sc / fe ] can be realized if @xmath14 ( @xmath370.5 ) ( pruet et al . 2004a , 2005 ; frhlich et al . 2006b ; iwamoto et al . [ co / fe ] and [ mn / fe ] can be improved by the @xmath14 modification in the si - burning region ( un05 ; tominaga et al . 2007 ) and [ mn / fe ] can also be enhanced by a neutrino process ( woosley @xmath6 weaver 1995 ; yoshida et al . 2008 ) . many of the solutions discussed above include the the ejection of proton - rich ( @xmath14@xmath370.5 ) `` complete si - burning '' matter . this does not contradict with our assumption that the neutron - rich ( @xmath14@xmath150.5 ) `` complete si - burning '' matter is ejected . this is because both @xmath14@xmath370.5 matter and @xmath14@xmath150.5 matter could be ejected simultaneously from the `` hot - bubble region '' in the multi - dimensional simulations ( e.g. , janka et al . 2003 ; pruet et al . although we do not include the proton - rich matter , the inclusion of the matter does not change the present results because the contributions from the proton - rich matter can merely be added to the present results . the nucleosynthesis in the proton - rich matter as well as the neutrino process will be considered elsewhere . there also remains a possible problem in elements mo , ru and rh . the observational data of those elements is obtained in only two emp stars hd122563 and hd88609 ( see honda et al . the abundance patterns of them show continuously decreasing trends compared with the main r - process as a function of atomic number , from sr to yb ( z=38@xmath6870 ) . our models have [ mo / fe]@xmath695 , [ ru / fe]@xmath692 and [ rh / fe]@xmath693 , while hd122563 has [ mo / fe]=@xmath680.02 , [ ru / fe]=0.07 and [ rh / fe]@xmath150.45 , and hd88609 has [ mo / fe]=0.15 , [ ru / fe]=0.32 and [ rh / fe]@xmath150.70 . further observation will be needed to investigate whether high ratios of mo , ru and rh are typical in the weak r - process stars . since there two stars are relatively metal rich , [ fe / h ] @xmath70 ( hd88609 ) and [ fe / h ] @xmath71 ( hd122563 ) , their abundance patterns may be contaminated by several sne , r - process and s - process . there may be a solution in a high - entropy matter ejection . pruet et al . ( 2006 ) investigated the contribution of the proton - rich high - entropy winds using the two - dimensional 15@xmath26 core collapse model of janka et al . the origin of the so - called p - process nuclei from a=92 to 126 is an unsolved riddles of nuclear astrophysics , but they found synthesis of p - rich nuclei up to @xmath72pd in the proton - rich wind , although their calculations do not show an efficient production of @xmath73mo . @xmath14 of proton - rich neutrino wind in pruet et al . ( 2006 ) is ranged from 0.539 to 0.558 , and entropy ( @xmath74/@xmath75 ) is from 54.8 to 76.9 . the entropy in the supernova shock model ( @xmath74/@xmath75@xmath1715 ) is much smaller . since the properties of the neutrino driven wind is uncertain , the nucleosynthesis in the proton - rich wind is certainly interesting , especially if there are no other possibilities . the @xmath14 below the mass cut is very sensitive to the rates for the neutrino and positron captures on neutrons and for the inverse captures on protons . unfortunately the actual amount of neutrino flux depends on the unknown explosion mechanisms . therefore , the @xmath14 of ejecta for a specific model needs to be calculated in the future works . in figure 15 we show [ sr / fe ] vs. [ zn / fe ] because [ zn / fe ] is a rough barometer of the sn explosion energy ( e.g. , un02 , un05 ) . the implications obtained from this figure are as follows . we have shown that the weak r - process elements can be produced without introducing extra higher - entropy matter in high @xmath52 sn models . previous work ( un02 ) suggests that the abundance of emp stars with high [ zn / fe ] are reproduced by high @xmath52 sn models . the apparent no - correlation between [ sr / fe ] and [ zn / fe ] means that , if our interpretation is correct , only a portion of hne eject a large amount of sr but the rest of hne eject a small amount of sr . in other words high @xmath52 is just a necessary condition to eject the weak r - process elements and other factors determine the ejected mass of the weak r - process elements . our results show that normal @xmath52 models do not produce large amount of sr , y , and zr . however we should note that this is not the case if a normal sn ejects somehow higher - entropy matter than the supernova shock . figure 15 does not deny such a possibility because [ zn / fe]@xmath110 stars may be reproduced by a normal sne or the mixture of several sne ( tominaga et al . this figure show that all [ zn / fe]@xmath110 stars show high [ sr / fe ] . a possible interpretation is that the actual normal sne can produce [ sr / fe]@xmath11 0 . if this is the case , our results imply that a normal sn can be ejecting a higher entropy matter than the supernova shock , that is likely the neutrino driven wind . this fact may be used to constrain the neutrino driven wind of a normal sn , though we have to handle the @xmath760.5 matter before qualitatively constraining the model , because the high - entropy proton - rich matter may also produce the weak r - process elements as shown in pruet et al . ( 2006 ) . we would like to thank t. yoshida , w. aoki and s. wanajo for useful comments and discussions . this work has been supported in part by the grants - in - aid for scientific research ( 19840010 ) from the mext of japan . nomoto , k. , maeda , k. , umeda , h. , ohkubo , t. , deng , j. , & mazzali , p.a . 2003a , in iau symp 212 , a massive star odyssey , from main sequence to supernova , eds . hucht , a. herrero and c. esteban(san francisco : asp ) , 395(astro - ph/0209064 ) llcc n & 1 & v & 44 - 60 h & 1 - 3 & cr & 46 - 63 he & 3 - 4 & mn & 48 - 65 li & 6 - 7 & fe & 50 - 68 be & 7 - 9 & co & 51 - 71 b & 8 - 13 & ni & 54 - 73 c & 11 - 15 & cu & 56 - 76 n & 13 - 18 & zn & 59 - 78 o & 14 - 21 & ga & 60 - 81 f & 17 - 23 & ge & 59 - 84 ne & 18 - 26 & as & 64 - 86 na & 21 - 28 & se & 65 - 89 mg & 22 - 31 & br & 68 - 92 al & 25 - 34 & kr & 66 - 94 si & 26 - 36 & rb & 72 - 97p & 27 - 39 & sr & 69 - 100s & 30 - 42 & y & 76 - 102 cl & 32 - 44 & zr & 74 - 105ar & 34 - 47 & nb & 80 - 107 k & 36 - 50 & mo & 79 - 110ca & 38 - 52 & tc & 85 - 113sc & 40 - 55 & ru & 84 - 115ti & 42 - 57 & rh & 89 - 118 & & pd & 89 - 121 rlcccccccc 1301(a ) & 0.42 & 1.65e-03 & -2.38 & -3.38 & -3.20 & 1.02e-08 & 2.10e-10 & 8.02e-10 & 5.93e-021301(a ) & 0.43 & 1.65e-03 & -2.37 & -3.47 & -3.25 & 1.05e-08 & 1.69e-10 & 7.16e-10 & 5.93e-022501(a ) & 0.42 & 2.02e-02 & -1.99 & -3.18 & -3.12 & 9.12e-08 & 1.21e-09 & 3.55e-09 & 2.19e-012501(a ) & 0.43 & 2.02e-02 & -1.99 & -3.33 & -3.20 & 9.26e-08 & 8.62e-10 & 2.92e-09 & 2.19e-012520(a ) & 0.45 & 1.08e-02 & 0.54 & 0.22 & 0.43 & 6.56e-05 & 6.40e-06 & 2.59e-05 & 4.60e-012520(a ) & 0.452 & 1.08e-02 & 0.38 & 0.36 & 0.52 & 4.51e-05 & 8.70e-06 & 3.20e-05 & 4.60e-012520(a ) & 0.454&1.08e-02 & 0.15 & 0.35 & 0.60 & 2.65e-05 & 8.53e-06 & 3.68e-05 & 4.60e-012520(a ) & 0.456 & 1.08e-02 & -0.11 & 0.26 & 0.65 & 1.47e-05 & 7.00e-06 & 4.28e-05 & 4.60e-012520(a ) & 0.458 & 1.08e-02 & -0.33 & 0.15 & 0.66 & 8.73e-06 & 5.47e-06 & 4.44e-05 & 4.60e-012520(a ) & 0.46 & 1.08e-02 & -0.72 & -0.17 & 0.46 & 3.57e-06 & 2.61e-06 & 2.77e-05 & 4.61e-012520(a ) & 0.47&1.08e-02 & -2.24 & -2.32 & -0.51 & 1.07e-07 & 1.84e-08 & 2.98e-06 & 4.61e-012520(a ) & 0.48&1.08e-02 & -3.41 & -4.13 & -3.37 & 7.32e-09 & 2.84e-10 & 4.09e-09 & 4.62e-012520(a ) & 0.49&1.08e-02 & -3.67 & -4.51 & -3.67 & 4.01e-09 & 1.20e-10 & 2.07e-09 & 4.64e-01 llccccccccc 2520 - 1 & 0.452 & 2.70e-04 & 0.05 & 5.76 & 1.13e-06 & 2.17e-07 & 8.00e-07&2.30e-02 & bs16477 - 0032520 - 2 & 0.45 & 5.90e-04 & 0.05 & 5.76 & 3.61e-06 & 3.52e-07 & 1.43e-06 & 2.30e-02 & cs22897 - 0082520 - 3 & 0.45 & 1.08e-03 & 0.20 & 3.60&6.56e-06 & 6.40e-07 & 2.60e-06 & 9.19e-02 & cs22873 - 1662520 - 4 & 0.45 & 1.35e-03 & 0.25 & 3.60 & 8.19e-06&8.00e-07&3.24e-06 & 1.15e-01 & cs29518 - 051 llccccccccc @xmath77c & 1.755e-01 & @xmath78sc & 4.216e-13 & @xmath79ge & 3.551e-10 & @xmath80mo & 4.850e-09 @xmath81c & 5.045e-08 & @xmath82ti & 1.502e-05 & @xmath83ge & 8.559e-09 & @xmath84mo & 9.053e-11 @xmath85c & 1.369e-10 & @xmath78ti & 2.482e-07 & @xmath86ge & 2.588e-12 & @xmath87mo & 2.144e-10 @xmath85n & 6.300e-04 & @xmath88ti & 9.284e-07 & @xmath79as & 4.569e-09 & @xmath89mo & 5.496e-13 @xmath90n & 4.046e-07 & @xmath91ti & 4.214e-05 & @xmath92as & 3.377e-09 & @xmath93mo & 9.662e-13 @xmath94o & 4.086e-01 & @xmath95ti & 3.174e-10 & @xmath83se & 3.920e-08 & @xmath96mo & 2.373e-13 @xmath97o & 1.478e-06 & @xmath98ti & 2.678e-07 & @xmath92se & 1.748e-09 & @xmath99mo & 9.873e-14 @xmath100o & 5.506e-07 & @xmath95v & 4.535e-07 & @xmath86se & 1.596e-07 & @xmath101mo & 6.059e-15 @xmath102f & 8.379e-08 & @xmath98v & 1.184e-11 & @xmath103se & 1.744e-09 & @xmath89tc & 8.480e-11 @xmath104ne & 6.517e-02 & @xmath105v & 2.164e-06 & @xmath106se & 4.886e-08 & @xmath96tc & 5.534e-11 @xmath107ne & 1.754e-06 & @xmath98cr & 4.088e-07 & @xmath108se & 7.851e-11 & @xmath99tc & 1.019e-13 @xmath109ne & 7.150e-07 & @xmath110cr & 4.455e-04 & @xmath111se & 6.644e-10 & @xmath112tc & 7.414e-14 @xmath109na & 1.287e-06 & @xmath113cr & 9.406e-08 & @xmath114se & 2.263e-13 & @xmath93ru & 2.458e-10 @xmath115na & 1.496e-05 & @xmath116cr & 1.555e-06 & @xmath108br & 4.948e-09 & @xmath99ru & 1.099e-10 @xmath117 mg & 3.295e-02 & @xmath113mn & 1.134e-05 & @xmath118br & 2.286e-09 & @xmath112ru & 5.395e-11 @xmath119 mg & 8.304e-05 & @xmath116mn & 1.981e-09 & @xmath106kr & 9.330e-09 & @xmath101ru & 3.006e-10 @xmath120 mg & 5.831e-06 & @xmath121mn & 3.709e-07 & @xmath111kr & 2.412e-08 & @xmath122ru & 9.298e-14 @xmath120al & 2.346e-06 & @xmath116fe & 4.782e-06 & @xmath118kr & 5.302e-10 & @xmath72ru & 5.723e-14 @xmath123al & 2.605e-04 & @xmath121fe & 5.501e-06 & @xmath114kr & 3.516e-08 & @xmath124ru & 1.393e-13 @xmath125si & 4.050e-02 & @xmath126fe & 5.754e-06 & @xmath127kr & 5.357e-10 & @xmath128ru & 2.455e-14 @xmath129si & 9.614e-05 & @xmath130fe & 2.071e-07 & @xmath131kr & 1.470e-08 & @xmath132ru & 1.458e-15 @xmath133si & 3.956e-05 & @xmath134fe & 5.094e-06 & @xmath135kr & 3.664e-11 & @xmath122rh & 4.509e-11 @xmath136si & 2.587e-13 & @xmath137fe & 8.189e-10 & @xmath138kr & 2.902e-10 & @xmath72rh & 2.008e-14 @xmath139p & 1.844e-05 & @xmath140fe & 7.672e-10 & @xmath127rb & 1.383e-09 & @xmath124rh & 9.428e-12 @xmath136s & 1.471e-02 & @xmath126co & 4.064e-08 & @xmath131rb & 1.082e-10 @xmath141s & 1.725e-05 & @xmath130co & 3.451e-04 & @xmath135rb & 3.471e-09 @xmath142s & 3.206e-06 & @xmath134co & 2.239e-09 & @xmath143rb & 8.623e-09 @xmath144s & 4.812e-11 & @xmath137co & 1.883e-07 & @xmath131sr & 1.239e-09 @xmath145s & 3.057e-12 & @xmath140co & 1.308e-09 & @xmath135sr & 1.931e-10 @xmath144cl & 1.906e-06 & @xmath126ni & 2.297e-02 & @xmath138sr & 3.750e-09 @xmath145cl & 1.244e-09 & @xmath134ni & 1.897e-04 & @xmath143sr & 1.490e-09 @xmath146cl & 1.131e-09 & @xmath137ni & 5.428e-05 & @xmath147sr & 1.121e-06 @xmath145ar & 2.644e-03 & @xmath140ni & 5.189e-04 & @xmath148sr & 3.053e-11 @xmath146ar & 1.193e-06 & @xmath149ni & 9.478e-06 & @xmath150sr & 9.241e-14 @xmath151ar & 6.251e-07 & @xmath152ni & 4.693e-05 & @xmath147y & 7.221e-11 @xmath153ar & 1.520e-12 & @xmath154ni & 2.101e-08 & @xmath148y & 2.174e-07 @xmath155ar & 7.346e-13 & @xmath156ni & 2.509e-07 & @xmath157y & 3.240e-13 @xmath158ar & 3.761e-17 & @xmath154cu & 2.266e-06 & @xmath147zr & 2.081e-10 @xmath153k & 6.151e-07 & @xmath159cu & 1.583e-07 & @xmath150zr & 7.952e-07 @xmath155k & 3.358e-11 & @xmath156zn & 6.511e-05 & @xmath157zr & 9.083e-10 @xmath160k & 1.288e-11 & @xmath159zn & 2.625e-07 & @xmath80zr & 4.393e-11 @xmath155ca & 2.605e-03 & @xmath161zn & 1.272e-05 & @xmath84zr & 4.954e-14 @xmath160ca & 2.355e-07 & @xmath162zn & 7.894e-08 & @xmath87zr & 2.988e-14 @xmath158ca & 1.772e-08 & @xmath163zn & 3.639e-07 & @xmath89zr & 7.698e-14 @xmath164ca & 1.459e-08 & @xmath165zn & 4.206e-11 & @xmath93zr & 1.295e-14 @xmath82ca & 3.043e-11 & @xmath166ga & 5.433e-08 & @xmath157nb & 3.611e-09 @xmath167ca & 3.966e-14 & @xmath168ga & 4.531e-09 & @xmath80nb & 1.732e-11 @xmath78ca & 3.965e-12 & @xmath163ge & 1.131e-06 & @xmath84nb & 2.300e-12 @xmath91ca & 2.440e-12 & @xmath165ge & 1.089e-06 & @xmath87nb & 6.665e-14 @xmath167sc & 1.662e-08 & @xmath169ge & 4.058e-07 & @xmath89nb & 2.658e-13 llccccccccc @xmath77c & 1.755e-01 & @xmath78sc & 9.068e-13 & @xmath79ge & 1.385e-09 & @xmath80mo & 1.046e-08 @xmath81c & 5.045e-08 & @xmath82ti & 1.508e-05 & @xmath83ge & 6.408e-08 & @xmath84mo & 1.500e-10 @xmath85c & 1.369e-10 & @xmath78ti & 2.600e-07 & @xmath86ge & 9.684e-11 & @xmath87mo & 2.518e-10 @xmath85n & 6.300e-04 & @xmath88ti & 9.410e-07 & @xmath79as & 8.465e-09 & @xmath89mo & 6.159e-13 @xmath90n & 4.046e-07 & @xmath91ti & 4.224e-05 & @xmath92as & 1.002e-08 & @xmath93mo & 9.878e-13 @xmath94o & 4.086e-01 & @xmath95ti & 1.117e-09 & @xmath83se & 8.518e-08 & @xmath96mo & 2.383e-13 @xmath97o & 1.478e-06 & @xmath98ti & 1.168e-06 & @xmath92se & 3.689e-09 & @xmath99mo & 9.912e-14 @xmath100o & 5.506e-07 & @xmath95v & 4.614e-07 & @xmath86se & 2.947e-07 & @xmath101mo & 6.227e-15 @xmath102f & 8.379e-08 & @xmath98v & 2.573e-11 & @xmath103se & 4.212e-09 & @xmath89tc & 8.516e-11 @xmath104ne & 6.517e-02 & @xmath105v & 2.287e-06 & @xmath106se & 1.252e-07 & @xmath96tc & 5.567e-11 @xmath107ne & 1.754e-06 & @xmath98cr & 4.174e-07 & @xmath108se & 4.423e-10 & @xmath99tc & 1.022e-13 @xmath109ne & 7.150e-07 & @xmath110cr & 4.468e-04 & @xmath111se & 7.106e-09 & @xmath112tc & 7.460e-14 @xmath109na & 1.287e-06 & @xmath113cr & 2.320e-07 & @xmath114se & 1.243e-11 & @xmath93ru & 2.472e-10 @xmath115na & 1.496e-05 & @xmath116cr & 5.295e-06 & @xmath108br & 8.292e-09 & @xmath99ru & 1.105e-10 @xmath117 mg & 3.295e-02 & @xmath113mn & 1.135e-05 & @xmath118br & 8.530e-09 & @xmath112ru & 5.432e-11 @xmath119 mg & 8.304e-05 & @xmath116mn & 3.734e-09 & @xmath106kr & 1.820e-08 & @xmath101ru & 3.021e-10 @xmath120 mg & 5.831e-06 & @xmath121mn & 8.530e-07 & @xmath111kr & 5.220e-08 & @xmath122ru & 9.375e-14 @xmath120al & 2.347e-06 & @xmath116fe & 4.860e-06 & @xmath118kr & 1.098e-09 & @xmath72ru & 5.789e-14 @xmath123al & 2.605e-04 & @xmath121fe & 5.526e-06 & @xmath114kr & 6.539e-08 & @xmath124ru & 1.403e-13 @xmath125si & 4.050e-02 & @xmath126fe & 1.081e-05 & @xmath127kr & 2.574e-09 & @xmath128ru & 2.507e-14 @xmath129si & 9.614e-05 & @xmath130fe & 4.053e-07 & @xmath131kr & 4.073e-08 & @xmath132ru & 2.162e-15 @xmath133si & 3.956e-05 & @xmath134fe & 1.482e-05 & @xmath135kr & 2.244e-10 & @xmath122rh & 4.534e-11 @xmath136si & 2.758e-13 & @xmath137fe & 4.922e-09 & @xmath138kr & 3.441e-09 & @xmath72rh & 2.011e-14 @xmath139p & 1.844e-05 & @xmath140fe & 8.019e-09 & @xmath127rb & 2.995e-09 & @xmath124rh & 9.514e-12 @xmath136s & 1.471e-02 & @xmath126co & 4.292e-08 & @xmath131rb & 2.291e-10 @xmath141s & 1.725e-05 & @xmath130co & 3.476e-04 & @xmath135rb & 6.494e-09 @xmath142s & 3.210e-06 & @xmath134co & 4.357e-09 & @xmath143rb & 2.652e-08 @xmath144s & 4.816e-11 & @xmath137co & 3.613e-07 & @xmath131sr & 2.651e-09 @xmath145s & 6.989e-12 & @xmath140co & 4.607e-09 & @xmath135sr & 4.205e-10 @xmath144cl & 1.913e-06 & @xmath126ni & 2.302e-02 & @xmath138sr & 6.693e-09 @xmath145cl & 1.246e-09 & @xmath134ni & 2.436e-04 & @xmath143sr & 2.464e-09 @xmath146cl & 1.132e-09 & @xmath137ni & 5.601e-05 & @xmath147sr & 3.593e-06 @xmath145ar & 2.644e-03 & @xmath140ni & 5.788e-04 & @xmath148sr & 2.271e-10 @xmath146ar & 1.194e-06 & @xmath149ni & 1.067e-05 & @xmath150sr & 1.103e-12 @xmath151ar & 6.295e-07 & @xmath152ni & 9.771e-05 & @xmath147y & 1.393e-10 @xmath153ar & 2.135e-12 & @xmath154ni & 1.066e-07 & @xmath148y & 3.515e-07 @xmath155ar & 1.633e-12 & @xmath156ni & 2.265e-06 & @xmath157y & 1.904e-12 @xmath158ar & 1.188e-16 & @xmath154cu & 3.344e-06 & @xmath147zr & 3.741e-10 @xmath153k & 6.229e-07 & @xmath159cu & 5.006e-07 & @xmath150zr & 1.418e-06 @xmath155k & 3.891e-11 & @xmath156zn & 8.389e-05 & @xmath157zr & 1.582e-09 @xmath160k & 2.182e-11 & @xmath159zn & 4.592e-07 & @xmath80zr & 1.178e-10 @xmath155ca & 2.605e-03 & @xmath161zn & 2.364e-05 & @xmath84zr & 1.000e-13 @xmath160ca & 2.359e-07 & @xmath162zn & 1.572e-07 & @xmath87zr & 3.159e-14 @xmath158ca & 2.588e-08 & @xmath163zn & 1.622e-06 & @xmath89zr & 7.723e-14 @xmath164ca & 1.619e-08 & @xmath165zn & 9.605e-10 & @xmath93zr & 1.357e-14 @xmath82ca & 6.262e-11 & @xmath166ga & 1.067e-07 & @xmath157nb & 7.695e-09 @xmath167ca & 1.711e-13 & @xmath168ga & 1.207e-08 & @xmath80nb & 3.452e-11 @xmath78ca & 2.337e-11 & @xmath163ge & 1.900e-06 & @xmath84nb & 3.862e-12 @xmath91ca & 1.134e-11 & @xmath165ge & 2.229e-06 & @xmath87nb & 6.996e-14 @xmath167sc & 1.798e-08 & @xmath169ge & 8.084e-07 & @xmath89nb & 2.675e-13 llccccccccc @xmath77c & 2.417e-01 & @xmath78sc & 1.775e-12 & @xmath79ge & 2.519e-09 & @xmath80mo & 1.938e-08 @xmath81c & 7.670e-08 & @xmath82ti & 6.014e-05 & @xmath83ge & 1.165e-07 & @xmath84mo & 3.594e-10 @xmath85c & 3.219e-10 & @xmath78ti & 1.028e-06 & @xmath86ge & 1.761e-10 & @xmath87mo & 8.444e-10 @xmath85n & 6.309e-04 & @xmath88ti & 3.714e-06 & @xmath79as & 1.631e-08 & @xmath89mo & 2.131e-12 @xmath90n & 5.305e-07 & @xmath91ti & 1.686e-04 & @xmath92as & 1.821e-08 & @xmath93mo & 3.853e-12 @xmath94o & 2.043e+00 & @xmath95ti & 2.032e-09 & @xmath83se & 1.557e-07 & @xmath96mo & 9.491e-13 @xmath97o & 1.487e-06 & @xmath98ti & 2.123e-06 & @xmath92se & 6.738e-09 & @xmath99mo & 3.949e-13 @xmath100o & 5.541e-07 & @xmath95v & 1.819e-06 & @xmath86se & 5.390e-07 & @xmath101mo & 2.424e-14 @xmath102f & 8.399e-08 & @xmath98v & 4.735e-11 & @xmath103se & 7.826e-09 & @xmath89tc & 3.392e-10 @xmath104ne & 1.646e-01 & @xmath105v & 8.739e-06 & @xmath106se & 2.277e-07 & @xmath96tc & 2.213e-10 @xmath107ne & 8.490e-06 & @xmath98cr & 1.732e-06 & @xmath108se & 8.043e-10 & @xmath99tc & 4.076e-13 @xmath109ne & 1.462e-06 & @xmath110cr & 1.782e-03 & @xmath111se & 1.292e-08 & @xmath112tc & 2.966e-13 @xmath109na & 5.343e-06 & @xmath113cr & 4.218e-07 & @xmath114se & 2.261e-11 & @xmath93ru & 9.833e-10 @xmath115na & 2.018e-04 & @xmath116cr & 9.627e-06 & @xmath108br & 1.520e-08 & @xmath99ru & 4.396e-10 @xmath117 mg & 1.477e-01 & @xmath113mn & 4.536e-05 & @xmath118br & 1.551e-08 & @xmath112ru & 2.158e-10 @xmath119 mg & 1.102e-04 & @xmath116mn & 6.796e-09 & @xmath106kr & 3.737e-08 & @xmath101ru & 1.202e-09 @xmath120 mg & 2.944e-05 & @xmath121mn & 1.551e-06 & @xmath111kr & 9.522e-08 & @xmath122ru & 3.719e-13 @xmath120al & 3.404e-06 & @xmath116fe & 1.958e-05 & @xmath118kr & 2.033e-09 & @xmath72ru & 2.289e-13 @xmath123al & 1.036e-03 & @xmath121fe & 2.202e-05 & @xmath114kr & 1.194e-07 & @xmath124ru & 5.570e-13 @xmath125si & 2.094e-01 & @xmath126fe & 1.968e-05 & @xmath127kr & 4.680e-09 & @xmath128ru & 9.821e-14 @xmath129si & 5.847e-04 & @xmath130fe & 7.369e-07 & @xmath131kr & 7.406e-08 & @xmath132ru & 5.831e-15 @xmath133si & 9.516e-05 & @xmath134fe & 2.695e-05 & @xmath135kr & 4.080e-10 & @xmath122rh & 1.804e-10 @xmath136si & 3.666e-13 & @xmath137fe & 8.949e-09 & @xmath138kr & 6.256e-09 & @xmath72rh & 8.031e-14 @xmath139p & 6.653e-05 & @xmath140fe & 1.458e-08 & @xmath127rb & 5.462e-09 & @xmath124rh & 3.771e-11 @xmath136s & 7.671e-02 & @xmath126co & 1.632e-07 & @xmath131rb & 4.166e-10 @xmath141s & 1.964e-04 & @xmath130co & 1.380e-03 & @xmath135rb & 1.181e-08 @xmath142s & 2.550e-05 & @xmath134co & 7.945e-09 & @xmath143rb & 4.823e-08 @xmath144s & 5.278e-10 & @xmath137co & 6.570e-07 & @xmath131sr & 4.984e-09 @xmath145s & 5.323e-11 & @xmath140co & 8.377e-09 & @xmath135sr & 7.702e-10 @xmath144cl & 1.503e-05 & @xmath126ni & 9.188e-02 & @xmath138sr & 1.225e-08 @xmath145cl & 2.379e-08 & @xmath134ni & 7.591e-04 & @xmath143sr & 4.552e-09 @xmath146cl & 2.221e-08 & @xmath137ni & 2.171e-04 & @xmath147sr & 6.533e-06 @xmath145ar & 1.234e-02 & @xmath140ni & 2.070e-03 & @xmath148sr & 4.130e-10 @xmath146ar & 1.576e-05 & @xmath149ni & 3.764e-05 & @xmath150sr & 2.066e-12 @xmath151ar & 7.734e-06 & @xmath152ni & 1.880e-04 & @xmath147y & 2.534e-10 @xmath153ar & 2.101e-11 & @xmath154ni & 1.939e-07 & @xmath148y & 6.392e-07 @xmath155ar & 3.820e-12 & @xmath156ni & 4.118e-06 & @xmath157y & 3.541e-12 @xmath158ar & 5.836e-16 & @xmath154cu & 8.746e-06 & @xmath147zr & 8.341e-10 @xmath153k & 4.466e-06 & @xmath159cu & 9.101e-07 & @xmath150zr & 2.578e-06 @xmath155k & 5.160e-10 & @xmath156zn & 2.602e-04 & @xmath157zr & 2.876e-09 @xmath160k & 1.293e-10 & @xmath159zn & 1.016e-06 & @xmath80zr & 2.143e-10 @xmath155ca & 1.075e-02 & @xmath161zn & 4.431e-05 & @xmath84zr & 2.681e-13 @xmath160ca & 1.993e-06 & @xmath162zn & 3.278e-07 & @xmath87zr & 1.224e-13 @xmath158ca & 1.826e-07 & @xmath163zn & 2.972e-06 & @xmath89zr & 3.079e-13 @xmath164ca & 5.851e-08 & @xmath165zn & 1.747e-09 & @xmath93zr & 5.192e-14 @xmath82ca & 1.883e-10 & @xmath166ga & 1.988e-07 & @xmath157nb & 1.415e-08 @xmath167ca & 3.175e-13 & @xmath168ga & 2.199e-08 & @xmath80nb & 6.286e-11 @xmath78ca & 4.255e-11 & @xmath163ge & 4.524e-06 & @xmath84nb & 7.268e-12 @xmath91ca & 2.025e-11 & @xmath165ge & 4.055e-06 & @xmath87nb & 2.663e-13 @xmath167sc & 7.514e-08 & @xmath169ge & 1.481e-06 & @xmath89nb & 1.064e-12 llccccccccc @xmath77c & 2.417e-01 & @xmath78sc & 2.184e-12 & @xmath79ge & 3.149e-09 & @xmath80mo & 2.423e-08 @xmath81c & 7.670e-08 & @xmath82ti & 7.515e-05 & @xmath83ge & 1.456e-07 & @xmath84mo & 4.492e-10 @xmath85c & 3.219e-10 & @xmath78ti & 1.274e-06 & @xmath86ge & 2.201e-10 & @xmath87mo & 1.056e-09 @xmath85n & 6.309e-04 & @xmath88ti & 4.642e-06 & @xmath79as & 2.039e-08 & @xmath89mo & 2.664e-12 @xmath90n & 5.306e-07 & @xmath91ti & 2.107e-04 & @xmath92as & 2.277e-08 & @xmath93mo & 4.817e-12 @xmath94o & 2.046e+00 & @xmath95ti & 2.540e-09 & @xmath83se & 1.946e-07 & @xmath96mo & 1.186e-12 @xmath97o & 1.487e-06 & @xmath98ti & 2.654e-06 & @xmath92se & 8.422e-09 & @xmath99mo & 4.936e-13 @xmath100o & 5.541e-07 & @xmath95v & 2.272e-06 & @xmath86se & 6.737e-07 & @xmath101mo & 3.030e-14 @xmath102f & 8.399e-08 & @xmath98v & 5.902e-11 & @xmath103se & 9.783e-09 & @xmath89tc & 4.240e-10 @xmath104ne & 1.646e-01 & @xmath105v & 1.092e-05 & @xmath106se & 2.846e-07 & @xmath96tc & 2.767e-10 @xmath107ne & 8.490e-06 & @xmath98cr & 2.135e-06 & @xmath108se & 1.005e-09 & @xmath99tc & 5.095e-13 @xmath109ne & 1.462e-06 & @xmath110cr & 2.227e-03 & @xmath111se & 1.615e-08 & @xmath112tc & 3.707e-13 @xmath109na & 5.343e-06 & @xmath113cr & 5.273e-07 & @xmath114se & 2.826e-11 & @xmath93ru & 1.229e-09 @xmath115na & 2.018e-04 & @xmath116cr & 1.203e-05 & @xmath108br & 1.901e-08 & @xmath99ru & 5.495e-10 @xmath117 mg & 1.477e-01 & @xmath113mn & 5.670e-05 & @xmath118br & 1.939e-08 & @xmath112ru & 2.697e-10 @xmath119 mg & 1.102e-04 & @xmath116mn & 8.493e-09 & @xmath106kr & 4.669e-08 & @xmath101ru & 1.503e-09 @xmath120 mg & 2.944e-05 & @xmath121mn & 1.939e-06 & @xmath111kr & 1.190e-07 & @xmath122ru & 4.649e-13 @xmath120al & 3.417e-06 & @xmath116fe & 2.434e-05 & @xmath118kr & 2.541e-09 & @xmath72ru & 2.861e-13 @xmath123al & 1.036e-03 & @xmath121fe & 2.752e-05 & @xmath114kr & 1.493e-07 & @xmath124ru & 6.963e-13 @xmath125si & 2.261e-01 & @xmath126fe & 2.459e-05 & @xmath127kr & 5.850e-09 & @xmath128ru & 1.228e-13 @xmath129si & 5.859e-04 & @xmath130fe & 9.211e-07 & @xmath131kr & 9.257e-08 & @xmath132ru & 7.289e-15 @xmath133si & 9.565e-05 & @xmath134fe & 3.369e-05 & @xmath135kr & 5.100e-10 & @xmath122rh & 2.255e-10 @xmath136si & 3.753e-13 & @xmath137fe & 1.119e-08 & @xmath138kr & 7.820e-09 & @xmath72rh & 1.004e-13 @xmath139p & 6.763e-05 & @xmath140fe & 1.823e-08 & @xmath127rb & 6.827e-09 & @xmath124rh & 4.714e-11 @xmath136s & 8.853e-02 & @xmath126co & 2.038e-07 & @xmath131rb & 5.207e-10 @xmath141s & 1.983e-04 & @xmath130co & 1.725e-03 & @xmath135rb & 1.477e-08 @xmath142s & 2.648e-05 & @xmath134co & 9.927e-09 & @xmath143rb & 6.028e-08 @xmath144s & 5.279e-10 & @xmath137co & 8.212e-07 & @xmath131sr & 6.221e-09 @xmath145s & 5.534e-11 & @xmath140co & 1.047e-08 & @xmath135sr & 9.622e-10 @xmath144cl & 1.610e-05 & @xmath126ni & 1.149e-01 & @xmath138sr & 1.531e-08 @xmath145cl & 2.384e-08 & @xmath134ni & 9.488e-04 & @xmath143sr & 5.690e-09 @xmath146cl & 2.223e-08 & @xmath137ni & 2.714e-04 & @xmath147sr & 8.166e-06 @xmath145ar & 1.486e-02 & @xmath140ni & 2.587e-03 & @xmath148sr & 5.163e-10 @xmath146ar & 1.626e-05 & @xmath149ni & 4.706e-05 & @xmath150sr & 2.583e-12 @xmath151ar & 8.030e-06 & @xmath152ni & 2.350e-04 & @xmath147y & 3.168e-10 @xmath153ar & 2.157e-11 & @xmath154ni & 2.424e-07 & @xmath148y & 7.990e-07 @xmath155ar & 4.532e-12 & @xmath156ni & 5.147e-06 & @xmath157y & 4.427e-12 @xmath158ar & 6.299e-16 & @xmath154cu & 1.093e-05 & @xmath147zr & 1.042e-09 @xmath153k & 4.955e-06 & @xmath159cu & 1.138e-06 & @xmath150zr & 3.222e-06 @xmath155k & 5.257e-10 & @xmath156zn & 3.253e-04 & @xmath157zr & 3.594e-09 @xmath160k & 1.372e-10 & @xmath159zn & 1.271e-06 & @xmath80zr & 2.679e-10 @xmath155ca & 1.334e-02 & @xmath161zn & 5.539e-05 & @xmath84zr & 3.352e-13 @xmath160ca & 2.163e-06 & @xmath162zn & 4.097e-07 & @xmath87zr & 1.527e-13 @xmath158ca & 1.934e-07 & @xmath163zn & 3.715e-06 & @xmath89zr & 3.849e-13 @xmath164ca & 7.309e-08 & @xmath165zn & 2.183e-09 & @xmath93zr & 6.485e-14 @xmath82ca & 2.157e-10 & @xmath166ga & 2.485e-07 & @xmath157nb & 1.768e-08 @xmath167ca & 3.950e-13 & @xmath168ga & 2.748e-08 & @xmath80nb & 7.857e-11 @xmath78ca & 5.316e-11 & @xmath163ge & 5.655e-06 & @xmath84nb & 9.085e-12 @xmath91ca & 2.474e-11 & @xmath165ge & 5.068e-06 & @xmath87nb & 3.329e-13 @xmath167sc & 9.122e-08 & @xmath169ge & 1.851e-06 & @xmath89nb & 1.329e-12 = 1.59@xmath26 , 1.76@xmath26 and 2.31@xmath26 for model-1301 , model-2501 and model-2520 , respectively . bs16477 - 003 is the weak r - process star with the highest abundance of sr , y and zr , and cs22172 - 002 is the weak r - process star with the lowest abundance of these elements.,title="fig:",width=302 ] = 1.59@xmath26 , 1.76@xmath26 and 2.31@xmath26 for model-1301 , model-2501 and model-2520 , respectively . bs16477 - 003 is the weak r - process star with the highest abundance of sr , y and zr , and cs22172 - 002 is the weak r - process star with the lowest abundance of these elements.,title="fig:",width=302 ] = 1.59@xmath26 , 1.76@xmath26 and 2.31@xmath26 for model-1301 , model-2501 and model-2520 , respectively . bs16477 - 003 is the weak r - process star with the highest abundance of sr , y and zr , and cs22172 - 002 is the weak r - process star with the lowest abundance of these elements.,title="fig:",width=302 ] , @xmath43 , and @xmath14 in section 4.2 . panels ( a ) , ( b ) and ( c ) are model-1301 , model-2501 and model-2520 , respectively . @xmath42 and @xmath43 are the inner and outer boundaries of the calculated region , i.e. , the complete si - burning region , respectively . we carry out the nucleosynthesis calculation in the region @xmath42@xmath15@xmath25@xmath15@xmath43 with constant @xmath14 from 0.40 to 0.50 as shown in solid lines.,width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 and @xmath170 = 0.43 , 0.44 , 0.45 ( from the top ) . the right panels : @xmath170 = 0.46 , 0.47 , 0.48 and @xmath170 = 0.49 , 0.50 ( from the top ) . , title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 and @xmath170 = 0.43 , 0.44 , 0.45 ( from the top ) . the right panels : @xmath170 = 0.46 , 0.47 , 0.48 and @xmath170 = 0.49 , 0.50 ( from the top ) . , title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 and @xmath170 = 0.43 , 0.44 , 0.45 ( from the top ) . the right panels : @xmath170 = 0.46 , 0.47 , 0.48 and @xmath170 = 0.49 , 0.50 ( from the top ) . , title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 and @xmath170 = 0.43 , 0.44 , 0.45 ( from the top ) . the right panels : @xmath170 = 0.46 , 0.47 , 0.48 and @xmath170 = 0.49 , 0.50 ( from the top ) . , title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 and @xmath170 = 0.43 , 0.44 , 0.45 ( from the top ) . the right panels : @xmath170 = 0.46 , 0.47 , 0.48 and @xmath170 = 0.49 , 0.50 ( from the top ) . , title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 and @xmath170 = 0.43 , 0.44 , 0.45 ( from the top ) . the right panels : @xmath170 = 0.46 , 0.47 , 0.48 and @xmath170 = 0.49 , 0.50 ( from the top ) . , title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 and @xmath170 = 0.43 , 0.44 , 0.45 ( from the top ) . the right panels : @xmath170 = 0.46 , 0.47 , 0.48 and @xmath170 = 0.49 , 0.50 ( from the top ) . , title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 and @xmath170 = 0.43 , 0.44 , 0.45 ( from the top ) . the right panels : @xmath170 = 0.46 , 0.47 , 0.48 and @xmath170 = 0.49 , 0.50 ( from the top ) . , title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 , @xmath170 = 0.43 , 0.44 , 0.45 and @xmath170 = 0.452 , 0.454 , 0.456 ( from the top ) . the right panels : @xmath170 = 0.458 , 0.46 , 0.47 and @xmath170 = 0.48 , 0.49 , 0.50 ( from the top).,title="fig:",width=302 ] . the left panels : @xmath170 = 0.40 , 0.41 , 0.42 , @xmath170 = 0.43 , 0.44 , 0.45 and @xmath170 = 0.452 , 0.454 , 0.456 ( from the top ) . the right panels : @xmath170 = 0.458 , 0.46 , 0.47 and @xmath170 = 0.48 , 0.49 , 0.50 ( from the top).,title="fig:",width=302 ] distribution and @xmath58 . here , the abundance below @xmath30 is averaged for @xmath14 = @xmath171 0.50 . the parameters and some related numbers for models ( a ) are shown in table 2 . for the models ( b ) and ( c ) , @xmath58 of model ( a ) is divided by 3 and 10 , respectively . , title="fig:",width=302 ] distribution and @xmath58 . here , the abundance below @xmath30 is averaged for @xmath14 = @xmath171 0.50 . the parameters and some related numbers for models ( a ) are shown in table 2 . for the models ( b ) and ( c ) , @xmath58 of model ( a ) is divided by 3 and 10 , respectively . , title="fig:",width=302 ]

barely fifteen years after the invention of the laboratory maser in 1953 which required a population inversion in the ammonia molecule to be carefully _ engineered _ interstellar masers were _ discovered _ as a remarkable and naturally - occurring phenomenon . the @xmath3 transition of water vapor , with a frequency near 22 ghz , was among the first masing transitions detected from an astrophysical source ( the orion molecular cloud ; cheung et al . 1969 ) , and has proven to be an extraordinarily useful probe in environments as diverse as interstellar shock waves , circumnuclear gas in active galactic nuclei ( agn ) , and the envelopes of evolved stars ( elitzur 1992 ; lo 2005 ; and references therein ) . with brightness temperatures that often exceed @xmath5 k and in extreme cases can exceed @xmath6 k ( e.g. garay , moran & haschick ( 1989 ) , and with linewidths that are often extremely narrow , 22 ghz water masers can be observed with very long baseline interferometry ( vlbi ) . the very high angular resolution of such vlbi observations enables proper motion studies of the kinematics of gas outflowing from proto- and evolved stars . moreover , trigonometric and geometric / rotational parallaxes can be determined , yielding the distances and motions of star forming regions in our galaxy , in local group galaxies , and in more distant agn ( e.g. brunthaler et al . 2007 ; braatz et al . such observations have led to revised estimates of the size , shape and kinematics of the milky way ( e.g. reid et al . 2009 ) , as well as the best evidence yet obtained for the existence of supermassive black holes ( e.g. miyoshi et al . 1995 ) . masers directly probe extreme environments ( of density and temperature ) , whose properties can often only be inferred indirectly from other lines . over the 40 years since the first detection of interstellar water maser emission in the 22 ghz transition , several additional interstellar maser transitions have been detected at higher frequencies , some in response to a specific prediction of a population inversion ( menten et al . 1990a ) . in the case of pure rotational emissions , these comprise the 183 ghz ( waters et al . 1980 ) , 321 ghz ( menten et al . 1990b ) , 325 ghz ( menten et al . 1990a ) , 380 ghz ( phillips et al . 1980 ) , 439 ghz ( melnick et al . 1993 ) and 471 ghz ( melnick et al . 1993 ) lines , associated respectively with the @xmath7 , @xmath8 , @xmath9 , @xmath10 , @xmath11 , and @xmath12 transitions . these transitions are shown in the energy level diagram in figure 1 . in addition , several water transitions within the the @xmath131 and 2 vibrationally - excited states have been observed toward the circumstellar envelopes of evolved stars , as have the 437 ghz @xmath14 and 475 ghz 5@xmath15 pure rotational transitions ( melnick et al . 1993 ; menten et al . 2008 ) . the detection of multiple masing transitions has been crucial in elucidating the pumping mechanism responsible for the population inversions and the physical conditions in the masing gas . understanding the latter , of course , is crucial to the astrophysical interpretation of maser observations . while the interpretation of the emission observed in a single line ( e.g. the 22 ghz transition ) is very geometry dependent , because the non - linear amplification of the radiation strongly favors those sight - lines that happen to possess the greatest velocity coherence , the interpretation of multiline observations is more robust and permits important constraints to be placed upon ( 1 ) the excitation mechanism and ( 2 ) the conditions of temperature and density in the emitting gas . the pattern of pure rotational maser transitions observed from interstellar sources appears to confirm an excitation model ( e.g. neufeld & melnick 1991 ) in which collisional pumping , combined with spontaneous radiative decay , leads to the inversion of exactly those transitions that are observed to mase . in evolved stars , however , the additional presence of the 437 ghz @xmath14 maser transition suggests that collisional pumping is not the entire story and that radiative pumping by dust continuum radiation is also important . quantitative measurements of maser line ratios provide additional constraints . for example , the predicted dependence of a maser line ratio upon the temperature and density of the masing gas was used by melnick et al . ( 1993 ) to derive a lower limit of 1000 k upon the gas temperature in several observed sources ; this value argued against a model ( elitzur , hollenbach & mckee 1989 ) in which the masers were excited in 400 k material behind a dissociative shock wave , and favored a model ( kaufman & neufeld 1996 ) in which non - dissociative magnetohydrodynamic shocks were the source of the emission . the analysis makes the assumption that the maser beam angle is similar for the two transitions that are being compared . prior to the launch of the _ herschel space observatory _ ( pilbratt et al . 2010 ) , the observational data on pure rotational water masers had been limited to transitions of frequency less than 500 ghz , largely because of atmospheric absorption . the hifi spectrometer on _ herschel _ ( de grauuw et al . 2010 ) , however , provides the opportunity of expanding the available data set by the addition of higher frequency transitions that promise to increase our leverage on the pumping mechanism and the physical conditions in the maser - emitting gas . to date , _ herschel _ observations of oxygen - rich evolved stars have led to the detection of two additional water maser transitions : ( 1 ) the @xmath0 transition at 620.701 ghz ( detected toward vy cma by harwit et al . 2010 , and toward w hya , ik tau and irc+10011 by justtanont et al . 2012 ) ; and ( 2 ) the @xmath16 transition at 970.315 ghz ( detected toward w hya and ik tau by justtanont et al . to our knowledge , neither maser has previously been observed from interstellar gas . in this paper , we report the first detection of 621 ghz water maser emission from _ interstellar _ gas , obtained with _ herschel_/hifi in mapping observations of the orion - kl region . in addition , we report the detection of 621 ghz water maser emission obtained serendipitously in single pointings toward the w49n star - forming region and the orion south molecular cloud . our _ herschel _ observations of orion - kl were carried out on 2011 march 11 as part of the `` orion small maps '' subprogram within the hexos guaranteed time key program ( gtkp ; p.i . , e. bergin ) . we used the `` heterodyne instrument for the infrared '' ( hifi ) , in `` on - the - fly mapping '' ( otf ) mode , to obtain a nyquist - sampled map consisting of a 6 by 8 rectangular array of pointings spaced by @xmath17 in r.a . and the map center was located at offset ( @xmath18 relative to orion - kl . ( all offsets given in this paper are relative to @xmath19 ( j2000 ) , the position we adopt for orion - kl . ) the reference position for the otf mapping observations , at offset @xmath20 , was chosen to be devoid of known molecular emission . the observations were carried out in the upper sideband of mixer band 1b , using the wbs spectrometer , which provides an oversampled channel spacing of 0.5 mhz ( 0.27 km / s at a frequency of 621 ghz ) , roughly one - half the effective resolution . the absolute frequency calibration is accurate to 100 khz ( roefsema et al . the beam size was 34@xmath21 ( hpbw ) , and the absolute pointing accuracy is @xmath22 . two concatenated astronomical observation requests ( aors ) were used to obtain two separate maps , with a small relative offset ( @xmath23 ) chosen to make the center of the h polarization beam for one map coincident with the center of the v polarization beam for the other map . as discussed in 3 , the goal of acquiring two separate maps in this manner was to obtain a measurement of any linear polarization in the maser feature . as summarized in table 1 , the total duration of each aor was 1987 s , with an on - source integration time of 420 s , which yielded individual spectra with an r.m.s . noise of 62 mk ( on the scale of antenna temperature and for a 1.1 mhz channel width ) . the _ herschel _ data on orion - kl were reduced using standard methods in the herschel interactive processing environment ( hipe ; ott 2010 ) , version 10.0 on 2011 march 21 , ten days after the _ herschel _ observations of orion - kl were performed , we used the effelsberg 100 m telescope to carry out observations of the 22.23508 ghz @xmath3 transition , with the goal of determining the 621 ghz / 22 ghz line flux ratio as a constraint upon models for the maser emission mechanism . here , we obtained a map , centered at offset @xmath24 , consisting of a 9 by 9 square array of pointings spaced by @xmath25 in r.a . and the beam size was 41@xmath21 ( hpbw ) , and the frequency resolution was 6.1 khz , corresponding to a velocity resolution of 0.082 km / s . the observations were performed using the k - band receiver at the prime focus of the 100 m telescope , with an observing time of 1.0 min per position . in calibrating the spectra , we applied corrections for the atmospheric attenuation and for the dependence of the telescope gain on the elevation . the calibration factor was determined by the observation of suitable calibration sources like ngc7027 and 3c286 ( taking into account the significant linear polarization of the latter ) . in addition to the mapping observations of orion - kl that are the primary subject of this paper , we have also identified two additional _ spectra that show narrow and/or time variable 621 ghz features suggestive of maser action . our observations of high - mass star - forming region w49n were carried out at three separate epochs , as part of the prismas gtkp ( p.i , m. gerin ) . these observations had the primary goal of measuring foreground absorption in nearby spectral lines of @xmath26 , but included the 621 ghz line within the bandpass . for each epoch , the relevant data were acquired in 3 aors obtained with slightly different lo settings . the aor numbers , dates of observation , beam center position , duration of the observations , and rms noise achieved are listed in table 1 . the data reduction methods that we adopted within the prismas gtkp have been described , for example , by neufeld et al . our observations of the hot core in the orion south molecular cloud were performed as part of a spectral line survey in the hexos gtkp . the data of present interest were acquired in a full spectral scan of band 1b , in which the 621 ghz transition was observed ( in either the upper or the lower sideband ) at 19 separate lo settings . the data reduction methods that we adopted for spectral scans within the hexos gtkp have been described by bergin et al . ( 2010 ) , and further details of the observations appear in table 1 . ancillary 22 ghz maser observations were performed using the effelsberg 100 m telescope toward w49n and orion s on 2012 oct 1 and 2012 sep 28 , respectively . because water masers are well - known to exhibit significant variability on timescales of months , our comparison of these non - contemporaneous 22 ghz and 621 ghz spectra must be interpreted with caution ; indeed , as described in 3 below , time variability in the maser emission is readily inferred from a comparison of the w49n 621 ghz data acquired at the 3 epochs . in figure 2 , we show the 621 ghz @xmath0 line spectra obtained toward orion - kl . the spectra are autoscaled , with the color of each border indicating the vertical scaling in accordance with the color bars on the left , and the two linear polarizations are shown in separate panels . the lsr velocity range shown in each panel is 20 to + 30 km / s . analogous results are shown for the 22 ghz @xmath3 transition in figure 3 . clearly , the strongest 621 ghz line emission originates in the vicinity of orion - kl , and exhibits a broad profile characteristic of high - lying ( and non - masing ) rotational transitions of water observed by _ ( melnick et al . 2010 ) . however , to the northwest of orion - kl , close to the shocked material associated with orion h@xmath1 peak 1 ( beckwith et al . 1978 ) , a narrow emission feature is clearly present . this feature is most apparent in the four spectra obtained closest to offset @xmath27 , the average of which is shown in figure 4 ( e.g. upper panel , blue histogram ) . the 22 ghz maser emission , by contrast , is dominated by multiple narrow emission components , the strongest lying in the 7 to 13 km / s lsr velocity range ; see figure 4 ( upper panel , red histogram ) . in the lower panel of figure 4 , we compare the 621 ghz and 22 ghz line emissions observed in the vicinity of @xmath27 for the 9 to 14 km / s lsr velocity range . in this panel , the two polarizations are shown separately for each transition , and a broad emission component has been subtracted from the 621 ghz line profile . although the 621 ghz and 22 ghz spectra are dissimilar outside this velocity range the 22 ghz spectrum exhibiting a strong narrow feature near @xmath28 km / s that is absent in the 621 ghz spectrum , for example the spectra are very similar within the 10 13 km / s velocity range . and which are expected to account for more than @xmath29 of the observed emission , ) lie at frequency shifts of @xmath30 , @xmath31 and @xmath32 khz relative to the mean @xmath3 line frequency , corresponding to velocity shifts of @xmath33 , @xmath34 , and @xmath35 . thus , even if departures from lte lead to large changes in the hyperfine line ratios , the maximum resulting velocity shift is at most 0.5 km / s ; hyperfine splitting of the higher - frequency 621 ghz transitions is completely negligible . ] furthermore , for both lines , the spectra obtained in the two polarizations are very similar . given the systematic uncertainties ( associated , for example , with possible differences between the telescope beam profiles for the two polarizations ) , these data provide no compelling evidence for polarization . a more detailed study of polarization in this source , which makes use of additional observations obtained at multiple spacecraft roll angles to measure the linear polarization of the 621 ghz maser transition , is currently under way ( jones et al . 2013 , private communication ) the upper panel of figure 5 shows the 22 ghz spectrum obtained toward w49n ( red ) , along with the 621 ghz spectra obtained at three epochs spanning two years ( see table 1 ) . because the 621 ghz transition was not specifically targeted but rather observed serendipitously in observations with a different primary purpose , we did not perform near - contemporaneous 22 ghz observations . thus , the 22 ghz spectrum shown in figure 5 was acquired 170 days after the last 621 ghz observation , making a detailed comparison between the transitions difficult . on the other hand , the availability of 621 ghz line observations at three separate epochs has allowed us to detect unequivocal variability in the line profile . most notably , a narrow emission feature near @xmath36 km / s was present in the 2011 observation but absent in the 2010 and 2012 observations ; and the line profile in the @xmath37 km / s range shows clear variability throughout the period covered by the three epochs . this variability , which can result from small changes in the ( negative ) optical depth along the sight - line to the observer , is characteristic of high - gain maser emission from a turbulent medium . four interferometric studies provide information about the spatial distribution of the 22 ghz water maser emission in w49n over a 30-year time period ( moran et al . 1973 ; walker , matsakis , & garcia - barreto 1982 ; gwinn , moran & reid 1992 ; mcgrath , goss & de pree 2004 ; performed in 197071 , 1978 , 198082 , and 199899 , respectively ) . they indicate that 99% of the 22 ghz maser flux is associated with the ultracompact hii regions designated g1 and g2 ( dreher et al . 1984 ) , and originates within 5@xmath38 of the beam center position adopted for our _ herschel _ ( and effelsberg ) observations . the remainder of the observed 22 ghz maser emission originates from a region @xmath39 northeast of the beam center . given these offsets , the _ herschel _ beam size ( 34@xmath38 ) , and the absolute pointing accuracy ( 2@xmath38 ) , we find it very unlikely that the observed 621 ghz variability could be the result of pointing errors . the only way that pointing errors could falsely indicate variability is if a very strong 621 ghz maser emission feature were located in the far wing of the _ herschel _ beam profile ( where the beam response is a strong function of offset ) ; such an emission feature would lie outside the region from which 22 ghz emission has been detected in the interferometric studies discussed above . in the case of the 621 ghz spectra obtained toward w49n , the relative contribution of maser and non - maser emission is hard to disentangle , at least based on the data currently in hand . while it is tempting to interpret the spectra as a superposition of narrow maser features on top of a broad ( @xmath40 km / s wide ) plateau of thermal line emission , the 22 ghz spectrum has an entirely similar appearance . in the case of the 22 ghz transition , however , the expected contribution of thermal emission is negligible ; indeed , the interferometric studies referenced above show that the broad pedestal apparent in single - dish observations is - in reality - a superposition of numerous compact and narrow emission features within the beam . the lower panel of figure 5 shows the spectra obtained toward orion south in ( non - contemponeous ) observations of the 621 ghz and 22 ghz transitions . the 621 ghz spectrum shows a single prominent narrow feature of width @xmath41 km / s ( fwhm ) in near coincidence with one of three narrow 22 ghz features . the 621 ghz linewidth is considerably narrower than that ( @xmath42 km / s ) measured at this position for non - masing spectral lines such as those of h@xmath1cl@xmath43 ( neufeld et al . 2012 ) and cn ( rodriguez - franco et al . 2001 ) , for example and has its centroid at a lower lsr velocity ( @xmath44 km / s versus of 7.2 km / s ) . the strongest 22 ghz emission in orion south was detected in to 0 to + 17 km s@xmath45 lsr velocity range , but weaker features are present over a wider interval from @xmath46 to @xmath47 km s@xmath45 ; the latter range is somewhat wider than that obtained by gaume et al . 1998 ( @xmath48 to @xmath49 km s@xmath45 ; see their fig . 5 and table 2 ) in sensitive very large array observations performed in 1991 . the phenomenon of maser amplification in w49n and orion s is not unique to water . maser emission from hydroxyl radicals has been long been studied in w49n , but is believed to require different excitation conditions ( e.g. mader et al . 1975 ) . in orion south , voronkov et al . ( 2005 ) have reported 6.7 ghz @xmath50 emissions that are likely masing . in table 2 , we present the 22 ghz and 621 ghz photon luminosities for the three sources we have observed , each computed with the assumption that the emission is isotropic and for adopted distances of 414 pc ( orion - kl and orion - s ; menten et al . 2007 ) and 11400 pc ( w49n : gwinn , moran & reid 1992 ) . as we have noted in 3 above , however , particular velocity components that are clearly present in the 22 ghz spectra are sometimes unaccompanied by detectable 621 ghz line emission ( though the converse is never true ) , suggesting that the requirements for strong 621 ghz maser amplification are more stringent than those for strong 22 ghz maser action . thus , the source - averaged 621 ghz/22 ghz line ratios may be considerable smaller than those applying to specific 621 ghz emission features . accordingly , we have also computed luminosities and line ratios for specific velocity ranges in which strong 621 ghz emission is observed from orion - kl and orion - s . furthermore , as will be discussed further in 4.1.2 below , our mapping observations of orion - kl indicate that _ even when integrated over a narrow range of lsr velocities _ , the 621 ghz / 22 ghz line ratio varies spatially . in figure 6 , we present channel maps for the 22 ghz line emissions detected from orion - kl , with the location of the peak intensity marked by a red diamond , and the channel - averaged peak flux ( in jy / beam ) listed in red near the top of each panel . the two strongest velocity components , at @xmath28 km / s and @xmath51 km / s show peak intensities near @xmath52 and @xmath53 respectively , positions separated by 26@xmath38 , more than one - half the effelberg hpbw ( 41@xmath38 ) . km / s maser emission feature can be identified with a bursting maser feature observed using vlbi by hirota et al . ( 2011 ) and located at offset @xmath54 . this feature , which brightened dramatically in 2011 february ( tolmachev 2011 ) , is associated the orion compact ridge . hirota et al . suggested that the bursting maser emission arises in shocked gas associated with the interaction of an outflow with ambient material , and identified radio source i ( plambeck et al . 2009 ) as the likely origin of the outflow . the latter is located roughly 3400 au northeast of the bursting maser source , and is a site of well - studied sio maser emission ( e.g. matthews et al . ] moreover , as shown in figure 7 ( yellow contours ) , the peak 22 ghz line intensity integrated over the 10 13 km / s velocity range ( upper panel ) is clearly separated also from the location of the peak 621 ghz emission ( lower panel ) in the same velocity range , despite the similarity of the spectral line profiles shown in figure 4 ( lower panel ) . one intepretation of this separation is that the 22 ghz maser emission in this velocity range is actually the sum of two ( or more ) spatially distinct components , the stronger of which is unaccompanied by 621 ghz maser emission . to test this hypothesis , we have carefully compared the 10 13 km / s 22 ghz channel map with the beam profile for the effelsberg telescope . the latter was measured at 22 ghz by observing the compact continuum source 3c84 at the prime focus . the beam response function is shown in figure 8 ( black contours , upper panel ) , superposed on red contours indicating the best fit gaussian ( with a hpbw of @xmath55 ) . in each case , contours are labeled as a percentage of the peak response . the beam profile is reasonably well - approximated by a gaussian , with deviations less than @xmath56 of the peak intensity . the lower panel shows a grayscale representation of the residuals relative to the best - fit gaussian , with gray levels separated by 1@xmath57 of the peak intensity and the blue contour indicating a residual of zero . having determined the best - fit to the 22 ghz beam profile , we subtracted that beam profile centered at the centroid of the strongest 22 ghz maser emission ( at offset [ @xmath58,@xmath59 ) and rotated as appropriate for the mean parallactic angle of the orion - kl observations ( 19@xmath60 ) to 27@xmath60 ] from the observed profile of the 22 ghz maser emission for @xmath62 km / s . the grayscale background in figure 7 ( both panels ) represents the result of that subtraction , which we will henceforth refer to as the 22 ghz residual " ( i.e. the residual with respect to the intensity expected for a point source at offset [ @xmath58,@xmath59 ) . here , adjacent grayscale levels are separated by 1000 k km / s , corresponding to @xmath56 of the peak 22 ghz line intensity integrated over the 10 13 km / s velocity range . the spatial distribution of the 22 ghz residual " matches that of the 621 ghz line emission reasonably well , supporting the interpretation introduced at the of end the previous paragraph . thus , the ratio of the luminosity of the 621 ghz line to that of the 22 ghz _ residual _ provides our best estimate of the true maser line ratio in the 621 ghz - emitting gas : 0.28 ( table 2 ) . it is difficult to provide a quantitative estimate of the systematic uncertainty in the 22 ghz residual ; the latter depends upon the reproducibility of the effelsberg beam profile shown in figure 8 , an issue that is not presently constrained by observations . we note , however , that the peak flux in the residual is @xmath63 of the peak flux in the 10 13 km / s velocity range . by comparison , the maximum deviations from a gaussian in the beam response function ( blue contours in figure 8) are only 5@xmath57 of the peak response . this suggests that the likely error in our estimate of the residual flux is at most 25@xmath57 , even under the conservative assumption that the deviations from a gaussian profile have locations that vary from one observation to the next . in addition to comparing our observations of 621 ghz water emission with single - dish 22 ghz observations ( performed nearly contemporaneously ) , we may also compare them with interferometric observations of the 22 ghz maser transition . unfortunately , we are unaware of any such observations of the orion region performed in recent years . the only published map of 22 ghz maser features to cover the region of narrow 621 ghz emission is that presented by genzel et al . ( 1981 ) , based upon observations performed in 197779 ; more recent observations reported by gaume et al . 1998 do not extend far enough to the north . we have , however , located a set of more recent unpublished data , acquired using very large array ( vla ) on 1996 march 15 and covering the region of interest , in the national radio astronomy observatory ( nrao ) archive . our reduction of these data is described in the appendix , where we tabulate the velocities and positions of 339 maser spots of peak flux density exceeding 1 jy that were identified in our analysis . in figure 7 , blue , cyan and red circles show the locations of maser spots with lsr velocities in the 10 11 , 11 12 , and 12 13 km / s ranges , with the size of each circle indicating the peak flux on a logarithmic scale . figure 7 indicates that a strong 22 ghz maser counterpart was indeed present at a location and velocity close to that of the narrow 621 ghz emission , despite the 15 year time period that separates the two observing epochs . to help interpret the observed maser line ratios , we have updated the maser excitation models described by ( neufeld & melnick 1991 ; hereafter nm91 ) to make use of recent calculations of the rate coefficients for excitation of @xmath64 by h@xmath1 ( daniel , dubernet , & grosjean 2011 , extrapolated for additional states of h@xmath1o using the artificial neural network method introduced by neufeld 2010 ) . as in nm91 , we performed a parameter study in which the equations of statistical equilibrium for the h@xmath1o level populations were solved for a grid of temperatures , @xmath65 , h@xmath1 densities , @xmath66 , and water column densities , @xmath67 . these calculations were performed for a medium with a large velocity gradient , with use of a standard escape probability method to account for radiative trapping . these calculations yield two key quantities for any masing transition : ( 1 ) the line optical depth in the unsaturated limit , and ( 2 ) the line luminosity in the saturated limit . we computed the negative sobolev optical depth , along the direction of the velocity gradient , that would be obtained without the effect of maser amplification . these values are plotted in figure 9 , as a function of @xmath68 and @xmath65 , for four values of @xmath67 . here , red contours show results for the 22 ghz transition , and blue contours those for the 621 ghz transition . for both transitions , the sobolev optical depth is more negative than 3 within large regions of the parameter space plotted in figure 9 . at high densities , the level populations inevitably tend to their local thermodynamic equilibrium ( lte ) values , maser action is quenched , and the optical depths become positive ( in regions to the right of the contours labeled zero . ) as expected , the lower - frequency 22 ghz transition can maintain its population inversion up to higher gas densities than the 621 ghz transition , allowing for the possibility of maser amplification of the 22 ghz transition unaccompanied by maser emission in the 621 ghz transition . as discussed in 3 , _ this possibility is clearly realized in the observations reported here_. for example , the strong 14.5 km / s 22 ghz feature in orion south shows no detectable 621 ghz emission ; the implied upper limit on the 621 ghz/22 ghz line ratio is @xmath69 ) . at high densities ( e.g. @xmath70 at a temperature of 1000 k ) , the maser line ratio depends strongly upon temperature and density : thus , relatively modest variations in the physical conditions can lead to large changes in the emergent line ratio . the optical depths plotted in figure 9 are the values along sightlines _ parallel _ to the direction of the velocity gradient ( i.e. in the direction of _ minimum _ velocity coherence . ) in plane - parallel geometry , the optical depth of a masing transition formally approaches @xmath71 as the inclination of the ray approaches @xmath72 , although in reality the magnitude of the optical depth is limited by departures from plane - parallel geometry ( such as curvature in a shocked molecular shell , for example ) . nevertheless , the maximum optical depth in a typical interstellar region can easily exceed the sobolev optical depth by an order of magnitude : thus , a sobolev optical depth of 3 can easily produce a maser gain of @xmath73 . gain factors of this magnitude will reduce the population inversion , leading to maser saturation . we computed the maximum photon luminosity that can be obtained under conditions of saturation , with the use of the formalism discussed in nm91 . in essence , this is the maximum rate of stimulated emission that can occur without eliminating the population inversion . taking the ratio of these luminosities for the two transitions yields the results shown in figure 10 . if the 22 ghz and 621 ghz transition are both saturated and are similarly beamed , this luminosity ratio can be compared with the observed values in rightmost column of table 2 . this comparison indicates that the observed luminosity of the 621 ghz maser transition can be accounted for in the same collisional pumping model that has proved successful in explaining other masing transitions of interstellar water . in no velocity range , in any source , does the 621 ghz/22 ghz maser luminosity ratio exceed 1.16 ; given the likelihood of time variability ( in w49n and orion south where the observations were non - contemporaneous ) and other uncertainties associated with maser beaming , _ this value can be regarded as consistent with the range of values ( up to 0.8 ) predicted by the collisional excitation model_. unfortunately , and unlike some of the other submillimeter maser transitions ( e.g. melnick et al . 1993 ) , the 621 ghz transition does not provide strong constraints on the gas temperature ; thus the observed 621 ghz/22 ghz line ratios are consistent with either 400 k gas behind dissociative j - type shocks ( e.g. elitzur , hollenbach & mckee 1989 ) , or the hotter gas that can arise behind non - dissociative shocks ( kaufman & neufeld 1993 ) . a search for maser emission in the @xmath12 ( 471 ghz ) and @xmath11 ( 439 ghz ) transitions of water could provide a stronger constraint on the gas temperature , as these transitions can only be pumped significantly at temperatures @xmath74 k or higher ( melnick et al . we have performed _ herschel_/hifi mapping observations of the 621 ghz @xmath0 transition of ortho - h@xmath1o within a @xmath2 region encompassing the kleinmann - low nebula in orion , and pointed observations of that transition toward the orion south condensation and the w49n region of high - mass star formation . we also obtained ancillary observations of the 22.23508 ghz @xmath3 water maser transition using the effelsberg 100 m telescope . in the case of orion - kl , the 621 ghz and 22 ghz observations were carried out nearly contemporaneously ( within 10 days of each other ) . the key results of these observations are as follows : \1 . in all three sources , the 621 ghz water line emission shows clear evidence for strong maser amplification . in both orion - kl and orion south , the observations reveal a narrow ( @xmath4 km / s fwhm ) emission feature , superposed in the case of orion - kl on broad thermal line emission , with a velocity centroid that is consistent with that of an observed 22 ghz maser feature . in the case of w49n - for which observations were available at three epochs spanning a two year period - observed variability in the 621 ghz line profile provides further evidence for maser action . \2 . in all three sources , 621 ghz maser emission is apparently always accompanied by 22 ghz maser emission . our mapping observations of orion - kl reveal a spatial offset between the 621 ghz maser emission and the 22 ghz maser emission occuring at the same velocity . however , a careful analysis of the spatial distribution of the observed 22 ghz maser emission reveals two spatially - distinct components , the weaker of which is spatially coincident with the 621 ghz maser emission . the observed 621 ghz/22 ghz line ratios are always consistent with a maser pumping model in which the population inversions arise from the combined effects of collisional excitation and spontaneous radiative decay . the observed line ratios do not place strong constraints on the gas temperature , and thus the inferred physical conditions can plausibly arise behind either dissociative or non - dissociative shocks . the collisional excitation model predicts that the 22 ghz population inversion will be quenched at higher densities than that of the 621 ghz transition , providing a natural explanation for the observational fact that 22 ghz maser emission appears to be a necessary but insufficient condition for 621 ghz maser emission . hifi has been designed and built by a consortium of institutes and university departments from across europe , canada and the united states under the leadership of sron netherlands institute for space research , groningen , the netherlands and with major contributions from germany , france and the us . consortium members are : canada : csa , u. waterloo ; france : cesr , lab , lerma , iram ; germany : kosma , mpifr , mps ; ireland , nui maynooth ; italy : asi , ifsi - inaf , osservatorio astrofisico di arcetri- inaf ; netherlands : sron , tud ; poland : camk , cbk ; spain : observatorio astronmico nacional ( ign ) , centro de astrobiolog ( csic - inta ) . sweden : chalmers university of technology - mc2 , rss & gard ; onsala space observatory ; swedish national space board , stockholm university - stockholm observatory ; switzerland : eth zurich , fhnw ; usa : caltech , jpl , nhsc . support for this work was provided by nasa through an award issued by jpl / caltech . this study was partly based on observations with the 100-m telescope of the mpifr ( max - planck - institut fr radioastronomie ) at effelsberg . we are grateful to m. gerin for useful comments on the manuscript . we thank s. jones and collaborators for providing us with a draft of their results prior to publication . we have downloaded , from the nrao archive , a set of 22 ghz h@xmath1o maser data targeting orion - kl . the observations were performed on 1996 march 15 with the vla in its c configuration , as part of a project ( ac443 ) that targeted seven sources . the total observing time for that project was 3 hours , of which 16 minutes was devoted to on - source observations of orion - kl . the extragalactic radio source 0605 - 085 was used as a phase calibrator . in the absence of any non - variable radio source that could be used for absolute flux calibration , we established the flux density scale by assuming a flux density for 0605 - 085 of 1.7 jy , the value measured in 1997 december by ( kovalev et al . the spectral - line observations carried out with the 1.3 cm receivers employed a single intermediate frequency band ( if ) mode with total of 127 channels , each of width 49 khz , corresponding to 0.7 km s@xmath45 . the central channel was set to a local standard of rest ( lsr ) velocity of 10 km s@xmath45 for the 22 ghz transition , providing a total lsr velocity coverage from @xmath75 to + 52 km / s . the data were reduced with astronomical image processing system ( aips ) software package . after a standard flux and phase calibration , we used the strongest maser emission feature - which has a flux of 1500 jy and a v@xmath76 of 4.6 km s@xmath45 - for self calibration , and then applied the solutions to all the other channels . the fwhm primary beam of individual vla antennas is @xmath77 2@xmath78 . we imaged and cleaned the data by applying the natural weighting of the uv data , which resulted in a synthesized beam of size 1.3 @xmath79 1.0 ( fwhm ) at position angle 23@xmath80 . we produced a 2048 @xmath792048 maser image cube , with a cell size of 0.1 @xmath79 0.1 . we then used the aips task sad to detect maser emissions with peak intensity higher than 5 times the r.m.s . noise level . for line channels in the orion - kl region that are not limited by dynamic range , the r.m.s . noise level was @xmath77 0.12 jy . in table 3 , we list the parameters obtained for 339 maser emission spots , all selected to satisfy the following criteria : flux @xmath81 1 jy , signal noise ratio ( snr ) @xmath81 10 , and dynamic range @xmath81 0.01 . ( in other words , for channels with peak flux greater than 100 jy , we used a flux cutoff of 0.01 @xmath79 the peak flux ) . figure 11 shows distribution of maser spots , with the color and symbol size representing lsr velocity and peak flux respectively . for comparision with the 621 ghz maser feature observed toward orion - kl , we show in figure 12 the 22 ghz maser emission integrated over the 10 13 km s@xmath45 velocity range . the letters a , b , c , d , e and f mark subregions from which the spectra presented in figure 13 were extracted . these spectra were obtained at the peak of each subregion , after hanning smoothing the data with a kernel of fwhm @xmath82 . bergin , e. a. , phillips , t. g. , comito , c. , et al . 2010 , , 521 , l20 braatz , j. a. , reid , m. j. , humphreys , e. m. l. , et al . 2010 , , 718 , 657 beckwith , s. , persson , s. e. , neugebauer , g. , & becklin , e. e. 1978 , , 223 , 464 brunthaler , a. , reid , m. j. , falcke , h. , henkel , c. , & menten , k. m. 2007 , , 462 , 101 cheung , a. c. , rank , d. m. , townes , c. h. , thornton , d. d. , & welch , w. j. 1969 , , 221 , 626 daniel , f. , dubernet , m .- l . , & grosjean , a. 2011 , , 536 , a76 de graauw , t. , et al . 2010 , , 518 , l6 dreher , j. w. , johnston , k. j. , welch , w. j. , & walker , r. c. 1984 , , 283 , 632 elitzur , m. , hollenbach , d. j. , & mckee , c. f. 1989 , , 346 , 983 elitzur , m. 1992 , , 30 , 75 garay , g. , moran , j. m. , & haschick , a. d. 1989 , , 338 , 244 gaume , r. a. , wilson , t. l. , vrba , f. j. , johnston , k. j. , & schmid - burgk , j. 1998 , , 493 , 940 genzel , r. , reid , m. j. , moran , j. m. , & downes , d. 1981 , , 244 , 884 gwinn , c. r. , moran , j. m. , & reid , m. j. 1992 , , 393 , 149 harwit , m. , houde , m. , sonnentrucker , p. , et al . 2010 , , 521 , l51 hirota , t. , tsuboi , m. , fujisawa , k. , et al . 2011 , , 739 , l59 justtanont , k. , khouri , t. , maercker , m. , et al . 2012 , , 537 , a144 kaufman , m. j. , & neufeld , d. a. 1996 , , 456 , 250 kovalev , y. y. , nizhelsky , n. a. , kovalev , y. a. , et al . 1999 , , 139 , 545 kukolich , s. g. 1969 , , 50 , 3751 lo , k. y. 2005 , , 43 , 625 mader , g. l. , johnston , k. j. , knowles , s. h. , et al . 1975 , , 200 , l111 matthews , l. d. , greenhill , l. j. , goddi , c. , et al . 2010 , , 708 , 80 mcgrath , e. j. , goss , w. m. , & de pree , c. g. 2004 , , 155 , 577 melnick , g. j. , menten , k. m. , phillips , t. g. , & hunter , t. 1993 , , 416 , l37 melnick , g. j. , tolls , v. , neufeld , d. a. , et al . 2010 , , 521 , l27 menten , k. m. , melnick , g. j. , & phillips , t. g. 1990a , , 350 , l41 menten , k. m. , melnick , g. j. , phillips , t. g. , & neufeld , d. a. 1990b , , 363 , l27 menten , k. m. , lundgren , a. , belloche , a. , thorwirth , s. , & reid , m. j. 2008 , , 477 , 185 menten , k. m. , reid , m. j. , forbrich , j. , & brunthaler , a. 2007 , , 474 , 515 miyoshi , m. , moran , j. , herrnstein , j. , et al . 1995 , , 373 , 127 moran , j. m. , papadopoulos , g. d. , burke , b. f. , et al . 1973 , , 185 , 535 neufeld , d. a. , & melnick , g. j. 1991 , , 368 , 215 ( nm91 ) neufeld , d. a. , sonnentrucker , p. , phillips , t. g. , et al . 2010 , , 518 , l108 neufeld , d. a. 2010 , , 708 , 635 neufeld , d. a. , roueff , e. , snell , r. l. , et al . 2012 , , 748 , 37 ott , s. 2010 , astronomical data analysis software and systems xix , 434 , 139 pilbratt , g. l. , riedinger , j. r. , passvogel , t. , et al . 2010 , , 518 , l1 phillips , t. g. , kwan , j. , & huggins , p. j. 1980 , iaus , 87 , 21 plambeck , r. l. , wright , m. c. h. , friedel , d. n. , et al . 2009 , , 704 , l25 reid , m. j. , menten , k. m. , zheng , x. w. , brunthaler , a. , & xu , y. 2009 , , 705 , 1548 rodrguez - franco , a. , wilson , t. l. , martn - pintado , j. , & fuente , a. 2001 , , 559 , 985 roelfsema , p. r. , helmich , f. p. , teyssier , d. , et al . 2012 , , 537 , a17 tolmachev , a. 2011 , the astronomer s telegram , 3177 , 1 voronkov , m. a. , sobolev , a. m. , ellingsen , s. p. , & ostrovskii , a. b. 2005 , , 362 , 995 walker , r. c. , matsakis , d. n. , & garcia - barreto , j. a. 1982 , , 255 , 128 waters , j. w. , kakar , r. k. , kuiper , t. b. h. , et al . 1980 , , 235 , 57 r.a . ( j2000 ) & 5h 35 m 14.31s @xmath83 & 5h 35 m 13.44s & 19h 10 m 13.20s + declination ( j2000 ) & 5d 22@xmath78 23.2@xmath38 @xmath83 & 5d 24@xmath78 8.1@xmath38 & 9d 06@xmath78 12.0@xmath38 + + date & 2011 - 03 - 11 & 2010 - 08 - 31 & 2010 - 04 - 11 ( 1st epoch ) + & & & 2011 - 10 - 08 ( 2nd epoch ) + & & & 2012 - 04 - 14 ( 3rd epoch ) + total duration ( s ) & 3964 & 7482 & 226 @xmath84 ( 1st epoch ) + & & & 158 @xmath84 ( 2nd epoch ) + & & & 282 @xmath84 ( 3rd epoch ) + r.m.s . noise ( mk ) & 65 @xmath85 & 30 @xmath86 & 16 @xmath86 ( 1st epoch ) + & & & 16 @xmath86 ( 2nd epoch ) + & & & 12 @xmath86 ( 3rd epoch ) + hifi mode & otf mapping & full spectral scan & pointed + hifi key program & hexos & hexos & prismas / ddt + herschel aors # s & 134221591819 & 1342204001 & 134219452325 ( 1st epoch ) + & & & 134223037880 ( 2nd epoch ) + & & & 134224440204 ( 3rd epoch ) + + + date & 2011 - 03 - 21 & 2012 - 9 - 28 & 2012 - 10 - 01 + receiver & prime focus & secondary focus & prime focus + observing mode & raster map & on - off & on - off + gain ( @xmath87 ) & 0.9 k / jy & 0.9 k / jy & 0.9 k / jy + system temperature ( k)@xmath88 & 56 & 97 & 72 + orion - kl & [ 40 , 60 ] & 4.17(16)@xmath83 & 5.37(17 ) & 2.08(+45 ) & 7.50(+45 ) & 0.28 + orion - kl & [ 40 , 60 ] & 1.86(17)@xmath84 & 5.37(17 ) & 9.27(+43)@xmath84 & 7.50(+45 ) & 0.012 + orion - kl & [ 10 , 13 ] & 1.86(17)@xmath84 & 1.25(17 ) & 9.27(+43)@xmath84 & 1.74(+45 ) & 0.053 + orion - kl & [ 10 , 13 ] & 1.86(17)@xmath84 & 2.85(18)@xmath85 & 9.27(+43)@xmath84 & 3.96(+44)@xmath85 & 0.23 + orion s & [ 30 , 30 ] & 1.67(17 ) & 2.07(18 ) & 8.33(+43 ) & 2.88(+44 ) & 0.29 + & [ 4 , 7 ] & 8.95(18 ) & 2.77(19 ) & 4.46(+43 ) & 3.85(+43 ) & 1.16 + w49n & [ 80 , 100 ] & 5.22(15 ) & 2.09(16 ) & 1.97(+49 ) & 2.21(+49 ) & 0.89 + & [ 10 , 5 ] & 1.54(16 ) & 4.52(17 ) & 5.84(+47 ) & 4.77(+48 ) & 0.12 + & [ 5,0 ] & 1.79(16 ) & 1.43(17 ) & 6.75(+47 ) & 1.51(+48 ) & 0.45 + & [ 0 , 5 ] & 2.11(16 ) & 1.50(17 ) & 7.99(+47 ) & 1.58(+48 ) & 0.50 + & [ 5 , 10 ] & 2.80(16 ) & 2.18(17 ) & 1.06(+48 ) & 2.30(+48 ) & 0.46 + & [ 10 , 15 ] & 2.49(16 ) & 2.36(17 ) & 9.41(+47 ) & 2.49(+48 ) & 0.38 + & [ 15 , 20 ] & 2.04(16 ) & 6.89(18 ) & 7.71(+47 ) & 7.27(+47 ) & 1.06 + -31.8 & -2.4 & -2.8 & 7.9 & -21.2 & -0.2 & -2.4 & 7.1 & -12.6 & 3.5 & 3.1 & 10.7 & -6.6 & -2.5 & -2.8 & 20.2 + -31.8 & -0.2 & -2.3 & 4.0 & -20.5 & 12.2 & 17.0 & 15.2 & -11.9 & -2.5 & -2.8 & 14.9 & -5.9 & 3.5 & 3.2 & 163.4 + -31.1 & 12.2 & 17.0 & 13.2 & -19.9 & -2.5 & -2.8 & 10.9 & -11.9 & -0.2 & -2.4 & 13.2 & -5.9 & 6.4 & 11.1 & 152.9 + -30.5 & -2.4 & -2.8 & 8.1 & -19.9 & -0.2 & -2.4 & 7.7 & -11.2 & 12.2 & 17.0 & 18.2 & -5.9 & 12.2 & 17.0 & 21.5 + -30.5 & -0.2 & -2.4 & 4.3 & -19.2 & 12.2 & 17.0 & 15.5 & -11.2 & 3.5 & 3.1 & 13.4 & -5.3 & 3.5 & 3.2 & 333.6 + -29.8 & 12.2 & 17.0 & 13.4 & -19.2 & 3.5 & 3.1 & 4.3 & -10.6 & -2.5 & -2.8 & 15.9 & -5.3 & 6.4 & 11.1 & 101.3 + -29.1 & -2.4 & -2.8 & 8.4 & -18.5 & -2.5 & -2.8 & 11.4 & -10.6 & -0.2 & -2.4 & 14.9 & -5.3 & -0.2 & -2.4 & 27.8 + -29.1 & -0.2 & -2.4 & 4.6 & -18.5 & -0.2 & -2.4 & 8.3 & -9.9 & 12.2 & 17.0 & 18.9 & -5.3 & -2.5 & -2.7 & 22.6 + -28.5 & 12.2 & 17.0 & 13.6 & -17.9 & 12.2 & 17.0 & 15.9 & -9.9 & 3.5 & 3.1 & 17.5 & -4.6 & 3.5 & 3.1 & 1532.0 + -27.8 & -2.4 & -2.8 & 8.7 & -17.9 & 3.5 & 3.1 & 5.1 & -9.9 & -7.5 & -2.7 & 8.4 & -3.9 & 3.5 & 3.1 & 1379.0 + -27.8 & -0.2 & -2.4 & 5.0 & -17.2 & -2.5 & -2.8 & 12.0 & -9.2 & -2.5 & -2.8 & 17.2 & -3.3 & 3.5 & 3.2 & 153.3 + -27.2 & 12.2 & 17.0 & 13.8 & -17.2 & -0.2 & -2.4 & 9.1 & -9.2 & -0.2 & -2.4 & 16.9 & -3.3 & 12.2 & 17.0 & 23.8 + -26.5 & -2.4 & -2.8 & 9.0 & -16.5 & 12.2 & 17.0 & 16.2 & -9.2 & -7.6 & -2.7 & 6.6 & -2.6 & 3.4 & 3.0 & 148.6 + -26.5 & -0.2 & -2.4 & 5.3 & -16.5 & 3.5 & 3.1 & 6.1 & -8.6 & 3.6 & 3.2 & 35.5 & -2.6 & -0.2 & -2.4 & 44.5 + -25.8 & 12.2 & 17.0 & 14.0 & -15.9 & -2.5 & -2.8 & 12.6 & -8.6 & 12.2 & 17.0 & 19.7 & -2.6 & -2.5 & -2.8 & 28.2 + -25.2 & -2.4 & -2.8 & 9.3 & -15.9 & -0.2 & -2.4 & 9.9 & -8.6 & -7.6 & -2.7 & 11.8 & -2.6 & 6.4 & 11.1 & 5.6 + -25.2 & -0.2 & -2.4 & 5.7 & -15.2 & 12.2 & 17.0 & 16.7 & -7.9 & -7.6 & -2.7 & 34.4 & -1.9 & 12.2 & 17.0 & 26.3 + -24.5 & 12.2 & 17.0 & 14.3 & -15.2 & 3.5 & 3.1 & 7.3 & -7.9 & -0.2 & -2.4 & 19.3 & -1.3 & -0.2 & -2.4 & 63.8 + -23.8 & -2.4 & -2.8 & 9.7 & -14.6 & -2.5 & -2.8 & 13.2 & -7.9 & -2.5 & -2.8 & 18.6 & -1.3 & 3.5 & 3.0 & 56.4 + -23.8 & -0.2 & -2.4 & 6.2 & -14.6 & -0.2 & -2.4 & 10.8 & -7.3 & 3.5 & 3.2 & 47.0 & -1.3 & -2.5 & -2.7 & 33.3 + -23.2 & 12.2 & 17.0 & 14.6 & -13.9 & 12.2 & 17.0 & 17.2 & -7.3 & 12.2 & 17.0 & 20.5 & -1.3 & -3.2 & -0.8 & 17.4 + -22.5 & -2.5 & -2.8 & 10.1 & -13.9 & 3.5 & 3.1 & 8.8 & -7.3 & 6.4 & 11.1 & 8.5 & -0.6 & 12.2 & 17.0 & 28.3 + -22.5 & -0.2 & -2.4 & 6.6 & -13.2 & -2.5 & -2.8 & 14.0 & -7.3 & -7.6 & -2.7 & 4.2 & 0.0 & -0.2 & -2.4 & 113.8 + -21.8 & 12.2 & 17.0 & 14.9 & -13.2 & -0.2 & -2.4 & 11.9 & -6.6 & 3.5 & 3.5 & 29.7 & 0.7 & -3.1 & -0.8 & 37.3 + -21.2 & -2.5 & -2.8 & 10.5 & -12.6 & 12.2 & 17.0 & 17.7 & -6.6 & -0.2 & -2.4 & 22.8 & 0.7 & 12.2 & 17.0 & 31.2 + 1.4 & -0.2 & -2.4 & 1207.0 & 6.7 & -3.6 & -5.8 & 457.9 & 11.3 & -2.5 & -2.7 & 57.3 & 14.6 & -7.7 & -5.3 & 175.2 + 1.4 & -2.6 & -2.7 & 36.7 & 7.3 & -2.5 & -2.8 & 1372.0 & 12.0 & 9.7 & 14.5 & 50.4 & 14.6 & 22.8 & -6.9 & 98.6 + 2.0 & -0.2 & -2.4 & 1158.0 & 7.3 & 12.2 & 17.0 & 55.7 & 12.0 & -0.2 & -6.6 & 41.9 & 14.6 & 12.2 & 17.0 & 18.6 + 2.0 & 12.2 & 17.0 & 33.0 & 8.0 & -3.6 & -5.8 & 890.1 & 12.0 & -7.6 & -5.4 & 14.0 & 14.6 & -5.8 & -9.7 & 8.6 + 2.7 & -0.2 & -2.2 & 520.2 & 8.0 & 0.6 & -4.4 & 173.6 & 12.0 & 8.5 & 17.2 & 8.9 & 15.3 & 22.8 & -6.9 & 272.4 + 2.7 & -2.7 & -2.7 & 190.6 & 8.0 & 6.2 & 6.0 & 51.9 & 12.0 & 22.8 & -6.9 & 6.6 & 15.3 & -7.7 & -5.2 & 157.9 + 2.7 & 12.0 & -5.2 & 131.1 & 8.7 & -3.5 & -5.9 & 1163.0 & 12.7 & -11.8 & 34.0 & 307.3 & 15.3 & 3.4 & 3.6 & 44.8 + 3.4 & -0.2 & -2.3 & 626.0 & 8.7 & 12.2 & 17.0 & 151.9 & 12.7 & 9.7 & 14.5 & 67.8 & 15.3 & -3.7 & -5.6 & 34.5 + 3.4 & 12.0 & -5.2 & 64.7 & 8.7 & -2.5 & -2.7 & 151.3 & 12.7 & 12.4 & 16.9 & 39.2 & 15.3 & -2.5 & -2.8 & 24.4 + 3.4 & 12.2 & 17.0 & 40.3 & 8.7 & 6.2 & 6.0 & 112.2 & 12.7 & -2.5 & -2.8 & 37.5 & 15.3 & 18.1 & 1.2 & 16.1 + 4.0 & -3.6 & -5.8 & 607.0 & 8.7 & 0.5 & -4.4 & 71.5 & 12.7 & -0.2 & -6.6 & 19.4 & 15.3 & -0.1 & -2.4 & 16.0 + 4.0 & -0.5 & -2.0 & 110.9 & 8.7 & 20.6 & -3.9 & 62.3 & 12.7 & -0.2 & -2.4 & 16.8 & 16.0 & -7.6 & -5.4 & 295.0 + 4.0 & 3.4 & 1.5 & 67.7 & 9.3 & -4.1 & -5.1 & 683.6 & 13.3 & 8.5 & 17.2 & 212.5 & 16.0 & 22.8 & -6.8 & 28.9 + 4.0 & -2.4 & -2.8 & 63.8 & 9.3 & 20.7 & -4.0 & 215.9 & 13.3 & -11.9 & 34.1 & 121.4 & 16.0 & 12.2 & 17.0 & 15.2 + 4.0 & 3.6 & 3.2 & 21.9 & 9.3 & 2.9 & 3.3 & 81.1 & 13.3 & -7.9 & -5.3 & 77.3 & 16.6 & -7.6 & -5.4 & 110.6 + 4.7 & -0.3 & -2.2 & 555.0 & 9.3 & 6.2 & 6.0 & 53.7 & 13.3 & 12.3 & 16.9 & 29.1 & 16.6 & -3.7 & -5.6 & 27.2 + 4.7 & -3.5 & -5.9 & 439.5 & 9.3 & -0.2 & -6.6 & 34.6 & 13.3 & 22.8 & -6.9 & 12.8 & 16.6 & -2.5 & -2.8 & 21.7 + 4.7 & 12.2 & 17.0 & 49.6 & 9.3 & -11.8 & 34.0 & 30.0 & 13.3 & 17.2 & 13.9 & 12.3 & 16.6 & -0.1 & -2.4 & 14.2 + 5.4 & -0.4 & -2.1 & 218.6 & 10.0 & -4.0 & -5.1 & 1331.0 & 13.3 & -0.1 & -6.6 & 11.4 & 16.6 & 22.8 & -6.8 & 7.4 + 5.4 & -2.4 & -2.8 & 135.0 & 10.0 & 12.2 & 17.0 & 548.2 & 14.0 & 2.9 & 3.1 & 324.7 & 17.3 & -7.7 & -5.4 & 49.2 + 5.4 & 3.5 & 3.1 & 17.9 & 10.0 & -36.6 & -49.2 & 101.8 & 14.0 & 8.5 & 17.2 & 177.0 & 17.3 & 12.2 & 16.9 & 11.6 + 6.0 & 12.2 & 17.0 & 64.1 & 10.7 & 12.2 & 17.0 & 1573.0 & 14.0 & -7.9 & -5.2 & 47.2 & 18.0 & -2.5 & -2.8 & 19.0 + 6.0 & -0.2 & -2.3 & 54.4 & 11.3 & 12.2 & 17.0 & 736.1 & 14.0 & -2.5 & -2.8 & 30.8 & 18.0 & 5.8 & 22.7 & 12.8 + 6.0 & 20.9 & 8.8 & 34.4 & 11.3 & -11.8 & 34.1 & 593.2 & 14.0 & 17.2 & 13.8 & 21.7 & 18.6 & -7.7 & -5.4 & 170.6 + 6.7 & -2.5 & -2.8 & 767.3 & 11.3 & -0.2 & -6.6 & 67.1 & 14.0 & -0.2 & -2.4 & 18.0 & 18.6 & 0.7 & -3.9 & 41.3 + 18.6 & 12.2 & 16.9 & 8.4 & 22.6 & -14.0 & 13.3 & 10.7 & 27.3 & -7.7 & -5.4 & 7.0 & 34.6 & 3.5 & 3.0 & 6.3 + 19.3 & -7.7 & -5.4 & 141.6 & 22.6 & 3.5 & 3.0 & 8.3 & 27.3 & 0.8 & -4.5 & 3.9 & 35.2 & -0.2 & -2.4 & 6.0 + 19.3 & -3.0 & -5.8 & 94.4 & 22.6 & -8.4 & 14.7 & 5.9 & 27.9 & 3.5 & 3.0 & 7.2 & 35.2 & -2.6 & -2.8 & 5.2 + 19.3 & 0.6 & -3.9 & 73.6 & 22.6 & 12.3 & 16.9 & 3.7 & 28.6 & -2.5 & -2.7 & 9.0 & 35.2 & -7.7 & -5.4 & 4.1 + 19.3 & -2.5 & -2.8 & 17.3 & 23.3 & -3.6 & -5.7 & 13.0 & 28.6 & -3.6 & -5.8 & 8.0 & 35.2 & 12.1 & 17.0 & 3.1 + 19.3 & -0.2 & -2.3 & 10.9 & 23.3 & -2.5 & -2.7 & 11.8 & 28.6 & -0.2 & -2.4 & 7.6 & 35.9 & -10.6 & 11.6 & 35.0 + 20.0 & -2.8 & -5.8 & 136.2 & 23.3 & -7.7 & -5.4 & 11.7 & 28.6 & -7.7 & -5.4 & 6.2 & 35.9 & 3.5 & 3.0 & 6.2 + 20.0 & 0.6 & -3.9 & 36.1 & 23.3 & -0.2 & -2.4 & 9.6 & 29.2 & 3.5 & 3.0 & 7.0 & 36.5 & -10.6 & 11.6 & 203.0 + 20.0 & 3.5 & 3.0 & 9.1 & 23.9 & 3.5 & 3.0 & 7.9 & 29.9 & -2.5 & -2.7 & 8.6 & 36.5 & -2.6 & -2.8 & 6.2 + 20.0 & 12.2 & 16.9 & 7.1 & 23.9 & 12.3 & 16.9 & 2.5 & 29.9 & -0.2 & -2.4 & 7.3 & 36.5 & -0.1 & -2.3 & 5.3 + 20.6 & -3.0 & -5.8 & 63.5 & 24.6 & -3.6 & -5.8 & 11.3 & 29.9 & -3.6 & -5.9 & 7.1 & 37.2 & -10.6 & 11.6 & 122.8 + 20.6 & -7.7 & -5.4 & 27.4 & 24.6 & -2.5 & -2.8 & 10.4 & 29.9 & -7.7 & -5.4 & 5.6 & 37.2 & 3.5 & 3.0 & 6.2 + 20.6 & -2.5 & -2.7 & 15.8 & 24.6 & -7.7 & -5.4 & 9.4 & 30.6 & 3.5 & 3.0 & 6.8 & 37.2 & -10.5 & 9.3 & 3.3 + 20.6 & -0.2 & -2.4 & 10.9 & 24.6 & -0.2 & -2.4 & 9.0 & 31.2 & -2.5 & -2.7 & 10.8 & 37.9 & -0.2 & -2.4 & 5.5 + 21.3 & 0.8 & -4.4 & 11.9 & 24.6 & 0.9 & -4.6 & 5.2 & 31.2 & -0.2 & -2.4 & 6.9 & 37.9 & -2.6 & -2.8 & 4.4 + 21.3 & 3.5 & 3.0 & 8.7 & 25.3 & 3.5 & 3.0 & 7.6 & 31.2 & -7.7 & -5.4 & 5.1 & 37.9 & 12.1 & 17.0 & 3.8 + 21.3 & 12.3 & 16.9 & 5.1 & 25.3 & 12.4 & 16.8 & 1.5 & 31.9 & 3.5 & 3.0 & 6.6 & 37.9 & -7.7 & -5.4 & 3.7 + 21.3 & -5.2 & 9.5 & 1.8 & 25.9 & -3.6 & -5.8 & 10.0 & 32.6 & -2.6 & -2.7 & 7.2 & 38.5 & 3.5 & 3.0 & 5.8 + 21.9 & -14.1 & 13.3 & 23.0 & 25.9 & -2.5 & -2.8 & 9.5 & 32.6 & -0.2 & -2.4 & 6.6 & 38.5 & -10.6 & 11.6 & 3.7 + 21.9 & -3.5 & -5.7 & 17.3 & 25.9 & -0.2 & -2.4 & 8.5 & 32.6 & -7.7 & -5.4 & 4.7 & 39.2 & -0.2 & -2.4 & 5.3 + 21.9 & -2.6 & -2.7 & 16.1 & 25.9 & -7.7 & -5.4 & 8.0 & 33.2 & 3.5 & 3.0 & 6.4 & 39.2 & 12.1 & 17.0 & 4.1 + 21.9 & -7.7 & -5.4 & 15.8 & 26.6 & 3.5 & 3.0 & 7.4 & 33.9 & -0.2 & -2.4 & 6.3 & 39.2 & -2.6 & -2.8 & 3.9 + 21.9 & -0.2 & -2.4 & 10.2 & 27.3 & -3.6 & -5.8 & 8.9 & 33.9 & -2.6 & -2.8 & 5.7 & 39.2 & -7.7 & -5.4 & 3.5 + 21.9 & 0.8 & -4.6 & 5.9 & 27.3 & -2.5 & -2.8 & 8.6 & 33.9 & -7.7 & -5.4 & 4.4 & 39.9 & 3.5 & 3.0 & 5.7 + 21.9 & -8.3 & 14.8 & 5.2 & 27.3 & -0.2 & -2.4 & 8.1 & 33.9 & 12.1 & 17.0 & 2.7 & 39.9 & -10.6 & 11.6 & 2.2 + 40.5 & -0.2 & -2.4 & 5.1 & 42.5 & 3.5 & 3.0 & 5.5 & 45.8 & 12.2 & 17.0 & 5.4 & 48.5 & -0.2 & -2.4 & 4.0 + 40.5 & 12.2 & 17.0 & 4.4 & 43.2 & 12.2 & 17.0 & 4.9 & 45.8 & -0.2 & -2.4 & 4.4 & 49.1 & 3.5 & 3.0 & 5.0 + 40.5 & -2.6 & -2.8 & 3.6 & 43.2 & -0.2 & -2.4 & 4.7 & 45.8 & -7.7 & -5.4 & 2.9 & 49.8 & 12.2 & 17.0 & 5.9 + 40.5 & -7.7 & -5.4 & 3.4 & 43.2 & -7.7 & -5.4 & 3.2 & 45.8 & -2.6 & -2.8 & 2.5 & 49.8 & -0.2 & -2.4 & 3.8 + 41.2 & 3.5 & 3.0 & 5.6 & 43.2 & -2.6 & -2.8 & 3.0 & 46.5 & 3.5 & 3.0 & 5.2 & 50.5 & 3.5 & 3.0 & 4.9 + 41.2 & -10.6 & 11.6 & 1.5 & 43.8 & 3.5 & 3.0 & 5.4 & 47.2 & 12.2 & 17.0 & 5.6 & 51.1 & 12.2 & 17.0 & 6.1 + 41.8 & -0.2 & -2.4 & 4.9 & 44.5 & 12.2 & 17.0 & 5.1 & 47.2 & -0.2 & -2.4 & 4.2 & 51.1 & -0.2 & -2.4 & 3.7 + 41.8 & 12.2 & 17.0 & 4.6 & 44.5 & -0.2 & -2.4 & 4.5 & 47.2 & -7.7 & -5.4 & 2.9 & 51.1 & -2.7 & -2.8 & 1.7 + 41.8 & -2.6 & -2.8 & 3.3 & 44.5 & -2.6 & -2.8 & 2.7 & 47.8 & 3.5 & 3.0 & 5.1 & 51.8 & 3.5 & 3.0 & 4.8 + 41.8 & -7.7 & -5.4 & 3.3 & 45.2 & 3.5 & 3.0 & 5.3 & 48.5 & 12.2 & 17.0 & 5.8 + , @xmath89(average of four closest positions for the 621 ghz line or 2 closest positions for the 22 ghz line ) . lower panel : the 9 14 km / s velocity range , after subtraction of a broad pedestal ( 621 ghz line ) , and with two linear polarizations shown separately . , width=14 ] , @xmath90 ) . successive grayscale values are separated by 1000 jy km / s per beam , with the lowest value corresponding to negative residuals in the interval [ 1000 jy km / s per beam , 0 ] . open circles show the locations of 22 ghz maser spots observed in vla observations performed on 1996 march 15 ( see appendix ) . blue , cyan and red circles indicate lsr velocities in the 10 11 , 11 12 , and 12 13 km / s ranges , respectively , with the size of each circle indicating the flux on a logarithmic scale . offsets are shown in arcsec , computed relative to orion - kl at position @xmath19 ( j2000).,width=7 ] , where @xmath92 is the sobolev optical depth . these optical depths apply in the direction of the velocity gradient ( for which the magnitude of the optical depth is smallest ) . red and blue contours apply to the 22 ghz and 621 ghz maser transitions respectively.,width=12 ] o maser spectra extracted from the vla image cube . the six spectra were extracted from the peak positions labeled a , b , c , d , e and f in figure 12 , by hanning smoothing the data in spatial coordinates ( with a kernel of fwhm @xmath82 ) . some spectra show ringing artifacts associated with the gibbs phenomenon ; these can occur when a strong maser feature is narrower than the channel width.[fig - spectra],title="fig:",width=5]o maser spectra extracted from the vla image cube . the six spectra were extracted from the peak positions labeled a , b , c , d , e and f in figure 12 , by hanning smoothing the data in spatial coordinates ( with a kernel of fwhm @xmath82 ) . some spectra show ringing artifacts associated with the gibbs phenomenon ; these can occur when a strong maser feature is narrower than the channel width.[fig - spectra],title="fig:",width=5]o maser spectra extracted from the vla image cube . the six spectra were extracted from the peak positions labeled a , b , c , d , e and f in figure 12 , by hanning smoothing the data in spatial coordinates ( with a kernel of fwhm @xmath82 ) . some spectra show ringing artifacts associated with the gibbs phenomenon ; these can occur when a strong maser feature is narrower than the channel width.[fig - spectra],title="fig:",width=5 ] o maser spectra extracted from the vla image cube . the six spectra were extracted from the peak positions labeled a , b , c , d , e and f in figure 12 , by hanning smoothing the data in spatial coordinates ( with a kernel of fwhm @xmath82 ) . some spectra show ringing artifacts associated with the gibbs phenomenon ; these can occur when a strong maser feature is narrower than the channel width.[fig - spectra],title="fig:",width=5]o maser spectra extracted from the vla image cube . the six spectra were extracted from the peak positions labeled a , b , c , d , e and f in figure 12 , by hanning smoothing the data in spatial coordinates ( with a kernel of fwhm @xmath82 ) . some spectra show ringing artifacts associated with the gibbs phenomenon ; these can occur when a strong maser feature is narrower than the channel width.[fig - spectra],title="fig:",width=5]o maser spectra extracted from the vla image cube . the six spectra were extracted from the peak positions labeled a , b , c , d , e and f in figure 12 , by hanning smoothing the data in spatial coordinates ( with a kernel of fwhm @xmath82 ) . some spectra show ringing artifacts associated with the gibbs phenomenon ; these can occur when a strong maser feature is narrower than the channel width.[fig - spectra],title="fig:",width=5 ]

the field of _ computational learning theory _ deals with the abilities and limitations of algorithms that learn functions from data . many models of how learning algorithms access data have been considered in the literature . among these , two of the most prominent are via _ membership queries _ and via _ random examples_. membership queries are `` black - box '' queries ; in a membership query , a learning algorithm submits an input @xmath9 to an oracle and receives the value of @xmath10 . in models of learning from random examples , each time the learning algorithm queries the oracle it receives a labeled example @xmath11 where @xmath9 is independently drawn from some fixed probability distribution over the space of all possible examples . ( we give precise definitions of these , and all the learning models we consider , in section [ sec : jprelim ] . ) in recent years a number of researchers have considered quantum variants of well - studied models in computational learning theory , see e.g. @xcite . as we describe in section [ sec : jprelim ] , models of learning from quantum membership queries and from fixed quantum superpositions of labeled examples ( we refer to these as _ quantum examples _ ) have been considered ; such oracles have been studied in the context of _ quantum property testing _ as well @xcite . one common theme in the existing literature on quantum computational learning and testing is that these works study algorithms whose only access to the function is via some form of quantum oracle such as the quantum membership oracle or quantum example oracles mentioned above . for instance , @xcite modifies the classical harmonic sieve algorithm of @xcite so that it uses only uniform quantum examples to learn @xmath12 formulas . @xcite considers the problem of quantum property testing using quantum membership queries to give an exponential separation between classical and quantum testers for certain concept classes . @xcite studies the information - theoretic requirements of exact learning using quantum membership queries and probably approximately correct ( pac ) learning using quantum examples . many other articles such as @xcite could further extend this list . as the problem of building large scale quantum computers remains a major challenge , it is natural to question the technical feasibility of large scale implementation of the quantum oracles considered in the literature . it is desirable to minimize the number of quantum ( as opposed to classical ) oracle queries or examples required by quantum algorithms . thus motivated , in this paper we are interested in designing testing and learning algorithms with access to both quantum and classical sources of information ( with the goal of minimizing the quantum resources required ) . all of our positive results are based on a quantum subroutine due to @xcite , which we will refer to as an @xmath2 ( fourier sample ) oracle call . as explained in section [ sec : jprelim ] , a call to the @xmath2 oracle yields a subset of @xmath13 ( this set should be viewed as a subset of the input variables @xmath14 of @xmath3 ) drawn according to the fourier spectrum of the boolean function @xmath3 . as demonstrated by @xcite , such an oracle can be implemented using @xmath15 uniform quantum examples from a uniform distribution quantum example oracle . in fact , all of our algorithms will be purely classical apart from their use of the @xmath2 oracle . thus , all of our algorithms can be implemented within the ( uniform distribution ) quantum pac model first proposed by @xcite . this model is a natural quantum extension of the classical pac model introduced by valiant @xcite , as described in section [ sec : jprelim ] . we emphasize that no membership queries , classical or quantum , are used in our algorithms , only uniform quantum superpositions of labeled examples , and we recall that such uniform quantum examples can not efficiently simulate even classical membership queries in general ( see @xcite ) . our approach of focusing only on the @xmath2 oracle allows us to abstract away from the intricacies of quantum computation , and renders our results useful in any setting in which an @xmath2 oracle can be provided to the user . in fact , learning and testing with @xmath2 oracle queries may be regarded as a new distinct model ( which may possibly be weaker than the uniform distribution quantum example model ) . we are primarily interested in the information theoretic requirements ( i.e. the number of oracle calls needed ) of the learning and testing problems that we discuss . we give upper and lower bounds for a range of learning and testing problems related to _ @xmath0-juntas _ ; these are boolean functions @xmath16 that depend only on ( an unknown subset of ) at most @xmath0 of the @xmath1 input variables @xmath14 . juntas have been the subject of intensive research in learning theory and property testing in recent years , see e.g. @xcite . our first result , in section [ sec : testjuntas ] , is a @xmath0-junta testing algorithm which uses @xmath17 @xmath2 oracle calls . our algorithm uses fewer queries than the best known classical junta testing algorithm due to fischer _ @xcite , which uses @xmath18 membership queries . however , since the best lower bound known for classical membership query based junta testing ( due to chockler and gutfreund @xcite ) is @xmath19 , our result does not rule out the possibility that there might exist a classical membership query algorithm with the same query complexity . to complement our @xmath2 based testing algorithm , we establish a new lower bound : any @xmath0-junta testing algorithm that uses only a @xmath2 oracle requires @xmath6 calls to the @xmath2 oracle this shows that our testing algorithm is not too far from optimal . finally , we consider algorithms that can both make @xmath2 queries and also access classical random examples . in section [ sec : learnjuntas ] we give an algorithm for learning @xmath0-juntas over @xmath20 that uses @xmath7 @xmath2 queries and @xmath21 random examples . since any classical learning algorithm requires @xmath22 examples ( even if it is allowed to use membership queries ) , this result illustrates that it is possible to reduce the classical query complexity substantially ( in particular , to eliminate the dependence on @xmath1 ) if the learning algorithm is also permitted to have some very limited quantum information . moreover most of the consumption of our algorithm is from classical random examples which are considered quite `` cheap '' relative to quantum examples . from another perspective , our result shows that for learning @xmath0-juntas , almost all the quantum examples used by the algorithm of bshouty and jackson @xcite can in fact be converted into ordinary classical random examples . we show that our algorithm is close to best possible by giving a nearly matching lower bound . in section [ sec : jprelim ] we describe the models and problems we will consider and present some useful preliminaries from fourier analysis and probability . section [ sec : testjuntas ] gives our results on testing juntas and section [ sec : learnjuntas ] gives our results on learning juntas . in keeping with standard terminology in learning theory , a _ concept _ @xmath3 over @xmath23 is a boolean function @xmath24 , where @xmath25 stands for true and @xmath26 stands for false . concept class _ @xmath27 is a set of concepts where @xmath28 consists of those concepts in @xmath29 whose domain is @xmath30 for ease of notation throughout the paper we will omit the subscript in @xmath28 and simply write @xmath31 to denote a collection of concepts over @xmath20 . the concept class we will chiefly be interested in is the class of _ @xmath0-juntas_. a boolean function @xmath32 is a @xmath0-junta if @xmath3 depends only on @xmath0 out of its @xmath1 input variables . we are interested in the following computational problems : pac learning under the uniform distribution : : : given any _ target concept _ @xmath33 , an _ @xmath34-learning algorithm for concept class @xmath31 _ under the uniform distribution outputs a _ hypothesis _ function @xmath35 which , with probability at least @xmath36 , agrees with @xmath37 on at least a @xmath38 fraction of the inputs in @xmath30 this is a widely studied framework in the learning theory literature both in classical ( see for instance @xcite ) and in quantum ( see @xcite ) versions . property testing : : : let @xmath3 be any boolean function @xmath39 . a _ property testing algorithm for concept class @xmath31 _ is an algorithm which , given access to @xmath3 , behaves as follows : + * if @xmath33 then the algorithm outputs accept with probability at least @xmath36 ; * if @xmath3 is _ @xmath34-far _ from any concept in @xmath31 ( i.e. for every concept @xmath40 , @xmath3 and @xmath41 differ on at least an @xmath34 fraction of all inputs ) , then the algorithm outputs reject with probability at least @xmath36 . + the notion of property testing was first developed by @xcite and @xcite . quantum property testing was first studied by buhrman _ et al . _ @xcite , who first gave an example of an exponential separation between the query complexity of classical and quantum testers for a particular concept class . note that a learning or testing algorithm for @xmath31 `` knows '' the class @xmath31 but does not know the identity of the concept @xmath3 . while our primary concern is the number of oracle calls that our algorithms use , we are also interested in _ time efficient _ algorithms for testing and learning ; for the concept class of @xmath0-juntas , these are algorithms running in poly@xmath42 time steps . in order for learning and testing algorithms to gather information about the unknown concept @xmath3 , they need an information source called an _ oracle_. the number of times an oracle is queried by an algorithm is referred to as the _ query complexity_. sometimes our algorithms will be allowed access to more than one type of oracle in our discussion . in this paper we will consider the following types of oracles that provide classical information : membership oracle @xmath43 : : : for @xmath3 a boolean function , a _ membership oracle _ @xmath44 is an oracle which , when queried with input @xmath9 , outputs the label @xmath10 assigned by @xmath3 to @xmath45 uniform random example oracle @xmath46 : : : a query @xmath47 of the random example oracle returns an ordered pair @xmath48 where @xmath9 is drawn uniformly random from the set @xmath49 of all possible inputs . clearly a single call to an @xmath43 oracle can simulate the random example oracle @xmath46 . indeed @xmath46 oracle queries are considered `` cheap '' compared to membership queries . for example , in many settings it is possible to obtain random labeled examples but impossible to obtained the label of a particular desired example ( consider prediction problems dealing with phenomena such as weather or financial markets ) . we note that the set of concept classes that are known to be efficiently pac learnable from uniform random examples only is rather limited , see e.g. @xcite . in contrast , there are known efficient algorithms that use membership queries to learning important function classes such as @xmath12 ( disjunctive normal form ) formulas @xcite . we will consider the following quantum oracles , which are the natural quantum generalizations of membership queries and uniform random examples respectively . quantum membership oracle @xmath50 : : : the quantum membership oracle @xmath51 is the quantum oracle whose query acts on the computational basis states as follows : @xmath52 uniform quantum examples @xmath53 : : : the uniform quantum example oracle @xmath54 is the quantum oracle whose query acts on the computational basis state @xmath55 as follows : @xmath56 the action of a @xmath54 query is undefined on other basis states , and an algorithm may only invoke the @xmath54 query on the basis state @xmath57 . it is clear that a @xmath50 oracle can simulate a @xmath53 oracle or an @xmath43 oracle , and a @xmath53 oracle can simulate an @xmath46 oracle . the model of pac learning with a uniform quantum example oracle was introduced by bshouty and jackson in @xcite . several researchers have also studied learning from a more powerful @xmath51 oracle , see e.g. @xcite . turning to property testing , we are not aware of prior work on quantum testing using only the @xmath54 oracle ; instead researchers have considered quantum testing algorithms that use the more powerful @xmath51 oracle , see e.g. @xcite . we will make use of the fourier expansion of real valued functions over @xmath23 . we write @xmath58 $ ] to denote the set of variables @xmath59 . consider the set of real valued functions over @xmath23 endowed with the inner product @xmath60 = { \frac 1 { 2^n } } \sum_x f(x ) g(x)\ ] ] and induced norm @xmath61 . for each @xmath62 $ ] , let @xmath63 be the parity function @xmath64 it is a well known fact that the @xmath65 functions @xmath66\}$ ] form an orthonormal basis for the vector space of real valued functions over @xmath23 with the above inner product . consequently , every @xmath67 can be expressed uniquely as : @xmath68 } { \hat{f}(s)}\chi_s(x)\ ] ] which we refer to as the _ fourier expansion _ or _ fourier transform _ of @xmath3 . alternatively , the values @xmath69 \ } $ ] are called the _ fourier coefficients _ or the _ fourier spectrum _ of @xmath3 . _ parseval s identity _ , which is an easy consequence of orthonormality of the basis functions , relates the values of the coefficients to the values of the function : [ bparseval ] for any @xmath67 , we have @xmath70 } |{\hat{f}(s)}|^2= { { \bf e}}[f^2]$ ] . thus for a boolean valued function @xmath70 } |{\hat{f}(s)}|^2=1 $ ] . we will use the following simple and well - known fact : [ kmfact ] for any @xmath32 and any @xmath71 , we have @xmath72\leq{{\bf e}}_{x}[{(f(x)-g(x))}^{2}]=\sum_{s\subseteq[n]}|{\hat{f}(s)}-\hat{g}(s)|^{2}\ ] ] recall that the _ influence _ of a variable @xmath73 on a boolean function @xmath3 is the probability ( taken over a uniform random input @xmath9 for @xmath3 ) that @xmath3 changes its value when the @xmath74-th bit of @xmath9 is flipped , i.e. @xmath75.\ ] ] it is well known ( see e.g. @xcite ) that @xmath76 [ dpi]let @xmath77 be two random variables over the same domain . for any ( possibly randomized ) algorithm @xmath78 , one has that @xmath79 let @xmath80 be random variables corresponding to sequences of draws taken from two different distributions over the same domain . by the above inequality , if @xmath81 is known to be small , then the probability of success must be small for any algorithm designed to distinguish if the draws are made according to @xmath82 or @xmath83 . we will also use standard chernoff bounds on tails of sums of independent random variables : let @xmath84 be i.i.d . random variables with mean @xmath85 taking values in the range @xmath86 $ ] . then for all @xmath87 we have @xmath88\leq 2 \exp(\frac{-2\lambda^2 m}{(b - a)^2})$ ] . let @xmath89 be a boolean function . the _ fourier sampling oracle _ @xmath90 is the classical oracle which , at each invocation , returns each subset of variables @xmath91 with probability @xmath92 , where @xmath93 denotes the fourier coefficient corresponding to @xmath94 as defined in section [ sec : bfour ] . this oracle will play an important role in our algorithms . note that by parseval s identity we have @xmath95 } @xmath96 indeed has total weight 1 . in @xcite bshouty and jackson describe a simple constant - size quantum network ` qsamp ` , which has its roots in an idea from @xcite . ` qsamp ` allows sampling from the fourier spectrum of a boolean function using @xmath15 @xmath53 oracle queries : [ bjfact ] for any boolean function @xmath3 , it is possible to simulate a draw from the @xmath90 oracle with probability @xmath97 using @xmath98 queries to @xmath54 . all the algorithms we describe are actually classical algorithms that make @xmath2 queries . fischer _ et al . _ @xcite studied the problem of testing juntas given black - box access ( i.e. , classical membership query access ) to the unknown function @xmath3 using harmonic analysis and probabilistic methods . they gave several different algorithms with query complexity independent of @xmath1 , the most efficient of which yields the following : [ s1thm1]there is an algorithm that tests whether an unknown @xmath89 is a @xmath0-junta using @xmath18 membership queries . fischer _ et al . _ also gave a lower bound on the number of queries required for testing juntas , which was subsequently improved by chockler _ _ to the following : [ s1thm2 ] any algorithm that tests whether @xmath3 is a @xmath0-junta or is @xmath99-far from every @xmath0-junta must use @xmath19 membership queries . we emphasize that that both of these results concern algorithms with classical membership query access . in this section we describe a new testing algorithm that uses the @xmath2 oracle and prove the following theorem about its performance : [ s1qthm1]there is an algorithm that tests the property of being a @xmath0-junta using @xmath5 calls to the @xmath2 oracle . as described in section [ sec : jprelim ] , the algorithm can thus be implemented using @xmath100 uniform quantum examples from @xmath54 . consider the following algorithm @xmath101 which has @xmath2 oracle access to an unknown function @xmath32 . algorithm @xmath101 first makes @xmath102 calls to the @xmath2 oracle ; let @xmath103 denote the union of all the sets of variables received as responses to these oracle calls . algorithm @xmath101 then outputs `` accept '' if @xmath104 and outputs `` reject '' if @xmath105 . it is clear that if @xmath3 is a @xmath0-junta then @xmath101 outputs `` accept '' with probability 1 . to prove correctness of the test it suffices to show that if @xmath3 is @xmath4-far from any @xmath0-junta then @xmath106 outputs `` reject''@xmath107 \geq { \frac 2 3}.$ ] the argument is similar to the standard analysis of the coupon collector s problem let us view the set @xmath103 as growing incrementally step by step as successive calls to the @xmath2 oracle are performed . let @xmath108 be a random variable which denotes the number of @xmath2 queries that take place starting immediately after the @xmath109-st new variable is added to @xmath103 , up through the draw when the @xmath74-th new variable is added to @xmath103 . if the @xmath109-st and @xmath74-th new variables are obtained in the same draw then @xmath110 . ( for example , if the first three queries to the @xmath2 oracle are @xmath111 @xmath112 , @xmath113 , then we would have @xmath114 , @xmath115 , @xmath116 , @xmath117 , @xmath118 . ) since @xmath3 is @xmath4-far from any @xmath0-junta , we know that for any set @xmath119 of @xmath120 variables , it must be the case that @xmath121 ( since otherwise if we set @xmath122 and use fact [ kmfact ] , we would have @xmath123\leq{{\bf e}}_{x}[{(f(x)-g(x))}^{2 } ] = \sum_{s \not\subseteq \mathcal{t } } \hat{f}(s)^2 < \epsilon\ ] ] which contradicts the fact that @xmath3 is @xmath4-far from any @xmath0-junta ) . it follows that for each @xmath124 , if at the current stage of the construction of @xmath103 we have @xmath125 , then the probability that the next @xmath2 query yields a new variable outside of @xmath103 is at least @xmath4 . consequently we have @xmath126 \leq { \frac 1 { \epsilon}}$ ] for each @xmath127 , and hence @xmath128 \leq { \frac { ( k+1 ) } { \epsilon}}.\ ] ] by markov s inequality , the probability that @xmath129 is at least @xmath130 , and therefore with probability at least @xmath130 it will be the case after @xmath102 draws that @xmath105 and the algorithm will consequently output `` reject . '' note that the @xmath100 uniform quantum examples required for algorithm @xmath101 improves on the @xmath131 query complexity of the best known classical algorithm . however our result does not conclusively show that @xmath53 queries are more powerful than classical membership queries for this problem since it is conceivable that there could exist an as yet undiscovered @xmath100 classical membership query algorithm . as a first attempt to obtain a lower bound on the number of @xmath2 oracle calls required to test @xmath0-juntas , it is natural to consider the approach of chockler _ et al . _ from @xcite . to prove theorem [ s1thm2 ] , chockler _ et al . _ show that any classical algorithm which can successfully distinguish between the following two probability distributions over black - box functions must use @xmath19 queries : * * scenario i : * the distribution @xmath132 is uniform over the set of all boolean functions over @xmath1 variables which do not depend on variables @xmath133 * * scenario ii : * the distribution @xmath134 is defined as follows : to draw a function @xmath3 from this distribution , first an index @xmath74 is chosen uniformly from @xmath135 , and then @xmath3 is chosen uniformly from among those functions that do not depend on variables @xmath136 or on variable @xmath74 . the following observation shows that this approach will not yield a strong lower bound for algorithms that have access to a @xmath2 oracle : with @xmath137 queries to a @xmath2 oracle , it is possible to determine w.h.p . whether a function @xmath3 is drawn from scenario i or scenario ii . it is easy to see that a function drawn from scenario i is simply a random function on the first @xmath138 variables . the fourier spectrum of random boolean functions is studied in @xcite , where it is shown that sums of squares of fourier coefficients of random boolean functions are tightly concentrated around their expected value . in particular , proposition 6 of @xcite directly implies that for any fixed variable @xmath139 we have : @xmath140 < \exp(-2^{k+1}/2592).\ ] ] thus with overwhelmingly high probability , if @xmath3 is drawn from scenario i then each @xmath2 query will `` expose '' variable @xmath74 with probability at least @xmath99 . it follows that after @xmath137 queries all @xmath138 variables will have been exposed ; so by making @xmath137 @xmath2 queries and simply checking whether or not @xmath138 variables have been exposed , one can determine w.h.p . whether @xmath3 is drawn from scenario i or scenario ii . thus we must adopt a more sophisticated approach to prove a strong lower bound on @xmath2 oracle algorithms . our main result in this section is the following theorem : [ s1qthm2 ] any algorithm that has @xmath2 oracle access to an unknown @xmath3 must use @xmath6 oracle calls to test whether @xmath3 is a @xmath0-junta . let @xmath0 be such that @xmath141 for some positive integer @xmath142 we let @xmath143 denote @xmath144 the _ addressing function _ on @xmath145 variables has @xmath146 `` addressing variables , '' which we shall denote @xmath147 and @xmath148 `` addressee variables '' which we denote @xmath149 the output of the function is the value of variable @xmath150 where the `` address '' @xmath151 is the element of @xmath152 whose binary representation is given by @xmath153 . figure 1 depicts a decision tree that computes the addressing function in the case @xmath154 . formally , the addressing function @xmath155 is defined as follows : @xmath156 [ fig : tree ] [ @xmath157[@xmath158 [ @xmath159 [ @xmath160 [ @xmath161 ] [ @xmath159 [ @xmath162 [ @xmath163 ] ] [ @xmath158 [ @xmath159 [ @xmath164 [ @xmath165 ] [ @xmath159 [ @xmath166 ] [ @xmath167 ] ] ] intuitively , the addressing function will be useful for us because as we will see the fourier spectrum is `` spread out '' over the @xmath143 addressee variables ; this will make it difficult to distinguish the addressing function ( which is not a @xmath0-junta since @xmath168 and as we shall see is in fact far from every @xmath0-junta ) from a variant which is a @xmath0-junta . let @xmath169 be the @xmath1 variables that our boolean functions are defined over . we now define two distributions @xmath170 , @xmath171 over functions on these variables . the distribution @xmath170 is defined as follows : to make a draw from @xmath170 , 1 . first uniformly choose a subset @xmath172 of @xmath143 variables from @xmath173 ; 2 . next , replace the variables @xmath174 in the function @xmath175 with the variables in @xmath172 ( choosing the variables from @xmath172 in a uniformly random order ) . return the resulting function . note that step ( 2 ) in the description of making a draw from @xmath170 above corresponds to placing the variables in @xmath172 uniformly at the leaves of the decision tree for @xmath176 ( see figure 1 ) . equivalently , if we write @xmath177 to denote the following function _ over @xmath1 variables _ @xmath178 a draw from @xmath170 is a function chosen uniformly at random from the set @xmath179 where @xmath180 ranges over all permutations of @xmath181 it is clear that every function in @xmath182 ( the support of @xmath170 ) depends on @xmath145 variables and thus is not a @xmath0-junta . in fact , every function in @xmath182 is far from being a @xmath0-junta : every @xmath3 that has nonzero probability under @xmath170 is @xmath183-far from any @xmath0-junta . fix any such @xmath3 and let @xmath41 be any @xmath0-junta . it is clear that at least @xmath184 of the `` addressee '' variables of @xmath3 are not relevant variables for @xmath41 . for a @xmath185 fraction of all inputs to @xmath3 , the value of @xmath3 is determined by one of these addressee variables ; on such inputs the error rate of @xmath41 relative to @xmath3 will be precisely @xmath186 fix any function @xmath177 in @xmath182 . we now give an expression for the fourier representation of @xmath177 . the expression is obtained by viewing @xmath177 as a sum of @xmath143 subfunctions , one for each leaf of the decision tree , where each subfunction takes the appropriate nonzero value on inputs which reach the corresponding leaf and takes value 0 on all other inputs : @xmath187 @xmath188 note that whenever @xmath189 , the sum on the rhs of equation has precisely one non - zero term which is @xmath190 . this is because the rest of the terms are annihilated since in each of these terms there is some index @xmath191 such that @xmath192 which makes @xmath193 . consequently this sum gives rise to exactly the addressing function in equation which is defined as @xmath194 and consequently the equality in equation follows . equation follows easily from rearranging . now we turn to @xmath195 the distribution @xmath171 is defined as follows : to make a draw from @xmath171 , 1 . first uniformly choose a subset @xmath172 of @xmath196 variables from @xmath173 ; 2 . next , replace the variables @xmath197 in the function @xmath175 with the variables in @xmath172 ( choosing the variables from @xmath172 in a uniformly random order ) . 3 . finally , for each @xmath198 do the following : if variable @xmath199 was used to replace variable @xmath200 in the previous step , let @xmath201 be a fresh uniform random @xmath202 value and replace variable @xmath203 with @xmath204 . return the resulting function . observe that for any integer @xmath205 with binary expansion @xmath206 , we have that the binary expansion of @xmath207 is @xmath208 . thus steps ( 2 ) and ( 3 ) in the description of making a draw from @xmath171 may be restated as follows in terms of the decision tree representation for @xmath176 : * place the variables @xmath209 randomly among the leaves of the decision tree with index less than @xmath196 . * for each variable @xmath209 placed at the leaf with index @xmath210 above , throw a @xmath202 valued coin @xmath201 and place @xmath211 at the antipodal leaf location with index : @xmath212 . equivalently , if we write @xmath213 to denote the following function _ over @xmath1 variables _ @xmath214 a draw from @xmath171 is a function chosen uniformly at random from the set @xmath215 where @xmath180 ranges over all permutations of @xmath216 and @xmath217 ranges over all of @xmath218 . it is clear that every function in @xmath219 depends on at most @xmath220 variables , and thus is indeed a @xmath0-junta . by considering the contribution to the fourier spectrum from each pair of leaves @xmath221 of the decision tree , we obtain the following expression for the fourier expansion of each function in the support of @xmath171 : @xmath222 @xmath223}\quad = \frac{1}{2^{r-1}}\sum_{\mathbf{i}=0}^{r/2 - 1}\begin{cases}\displaystyle \sum_{x\subseteq\{x_1,\ldots , x_r\ } , |x|\ \text{even } } ( -1)^{(\sum_{x_{j}\in x } i_{j } ) } y_{\tau(\mathbf{i } ) } \chi_{x } & \text{if $ s_{\mathbf{i}}=1$;}\\ \displaystyle \sum_{x\subseteq\{x_1,\ldots , x_r\ } , |x|\ \text{odd } } ( -1)^{(\sum _ { x_{j}\in x } i_{j } ) } y_{\tau(\mathbf{i } ) } \chi_{x } & \text{if $ s_{\mathbf{i}}=-1$.}\\ \end{cases}\ ] ] just as in the equation , whenever @xmath224 , the sum on the rhs of equation has precisely one non - zero term which is @xmath190 if @xmath225 and @xmath226 if @xmath227 . therefore this sum gives rise to exactly the addressing function in equation which is defined as @xmath213 and consequently the equality in equation follows . it follows that for each @xmath228 in the support of @xmath171 and for any fixed @xmath199 , all elements of the set @xmath229 will have the same parity . moreover , when draws from @xmath171 are considered , for every distinct @xmath199 this odd / even parity is independent and uniformly random . now we are ready to prove theorem [ s1qthm2 ] . recall that a @xmath2 oracle query returns @xmath96 with probability @xmath230 for every subset @xmath96 of input variables to the function . considering the equations and , for any @xmath3 in @xmath219 or @xmath182 its @xmath2 oracle will return a pair of the form @xmath231 . let us define a set @xmath232 of `` typical '' outcomes from @xmath2 oracle queries . fix any @xmath233 , and let @xmath232 denote the set of all sequences @xmath234 of length @xmath235 which have the property that _ no @xmath236 occurs more than once among @xmath237_. note that for any fixed @xmath238 , every non - zero fourier coefficient @xmath239 satisfies @xmath240 due to equation . therefore after @xmath194 is drawn , for any fixed @xmath199 the probability of receiving a response of the form @xmath241 as the outcome of a @xmath2 query is either @xmath242 , : : if @xmath194 is not a function of @xmath199 , i.e. @xmath243 ; or @xmath244 , : : if @xmath245 . this is because each of the @xmath246 responses @xmath241 occurs with probability @xmath247 . similarly , for any fixed @xmath248 , every non - zero fourier coefficient @xmath249 satisfies @xmath250 due to equation . therefore after @xmath228 is drawn , for any fixed @xmath199 the probability of receiving a response of the form @xmath241 as the outcome of a @xmath2 query is either @xmath242 , : : if @xmath228 is not a function of @xmath199 , i.e. @xmath251 ; or @xmath252 , : : if @xmath253 . this is because each of the @xmath254 responses @xmath241 occurs with probability @xmath255 . now let us consider the probability of obtaining a sequence from @xmath232 under each scenario . * if the function is drawn from @xmath170 : the probability is at least @xmath256}.\ ] ] * if the function is from @xmath171 : the probability is at least @xmath257}\ ] ] now the crucial observation is that whether the function is drawn from @xmath170 or from @xmath171 , each sequence in @xmath232 is equiprobable by symmetry in the construction . to see this , simply consider the probability of receiving a fixed @xmath258 for some new @xmath259 in the next @xmath2 query of an unknown function drawn from either one of these distributions . using the above calculations for @xmath260 , one can directly calculate that these probabilities are equal in either scenario . alternatively , for a function drawn from @xmath171 one can observe that since each successive @xmath259 is `` new '' , a fresh random bit determines whether the support is an @xmath258 with @xmath261 odd or even ; once this is determined , the choice of @xmath262 is uniform from all subsets with the correct parity . thus the overall draw of @xmath241 is uniform over all @xmath262 s . considering that the subset of relevant variables @xmath263 is uniformly chosen from @xmath264 , this gives the equality of the probabilities for each @xmath241 with a new @xmath259 when the function is drawn from @xmath171 . the argument for the case of @xmath170 is clear . consequently the statistical difference between the distributions corresponding to the sequence of outcomes of the @xmath235 @xmath2 oracle calls under the two distributions is at most @xmath265 . now fact [ dpi ] implies that no algorithm making only @xmath235 oracle calls can distinguish between these two scenarios with high probability . this gives us the result , and concludes the proof of theorem [ s1qthm2 ] . intuitively , under either distribution on functions , each element of a sequence of @xmath235 @xmath2 oracle calls will `` look like '' a uniform random draw @xmath262 from subsets of @xmath266 and @xmath191 from @xmath267 where @xmath191 and @xmath262 are independent . note that this argument breaks down at @xmath268 . this is because if the algorithm queried the @xmath2 oracle @xmath269 times it will start to see some @xmath270 s more than once with constant probability ( again by the birthday paradox ) . but when the functions are drawn from @xmath171 the corresponding @xmath108 s will always have a fixed parity for a given @xmath270 whereas for functions drawn from @xmath170 the parity will be random each time . this will provide the algorithm with sufficient evidence to distinguish with constant probability between these two scenarios . the problem of learning an unknown @xmath0-junta has been well studied in the computational learning theory literature , see e.g. @xcite . the following classical lower bound will be a yardstick against which we will measure our results . [ s2lem1 ] any classical membership query algorithm for learning @xmath0-juntas to accuracy @xmath271 must use @xmath22 membership queries . consider the restricted problem of learning an unknown function @xmath10 which is simply a single boolean variable from @xmath272 . since any two variables disagree on half of all inputs , any @xmath271-learning algorithm can be easily modified into an algorithm that exactly learns an unknown variable with no more queries . it is well known that any set of @xmath1 concepts requires @xmath273 queries for any exact learning algorithm that uses membership queries only , see e.g. @xcite . this gives the @xmath273 lower bound . for the @xmath274 lower bound , we may suppose that the algorithm `` knows '' that the junta has relevant variables @xmath275 . even in this case , if fewer than @xmath276 membership queries are made the learner will have no information about at least @xmath277 of the function s output values . a straightforward application of the chernoff bound shows that it is very unlikely for such a learner s hypothesis to be @xmath271-accurate , if the target junta is a uniform random function over the relevant variables . this establishes the result . learning juntas from uniform random examples @xmath47 is a seemingly difficult computational problem . simple algorithms based on exhaustive search can learn from @xmath278 examples but require @xmath279 runtime . the fastest known algorithm in this setting , due to mossel _ et al . _ , uses @xmath280 examples and runs in @xmath280 examples time , where @xmath281 is the matrix multiplication exponent @xcite . bshouty and jackson @xcite gave an algorithm using uniform quantum examples from the @xmath53 oracle to learn general @xmath12 formulas . their algorithm uses @xmath282 calls to @xmath53 to learn an @xmath217-term @xmath12 over @xmath1 variables to accuracy @xmath4 . since any @xmath0-junta is expressible as a @xmath12 with at most @xmath283 terms , their result immediately yields the following statement . [ s2thm2 ] there exists an @xmath34-learning quantum algorithm for @xmath0-juntas using @xmath284 quantum examples under the uniform distribution quantum pac model . note that @xcite did not try to optimize the quantum query complexity of their algorithms in the special case of learning juntas . in contrast , our goal is to obtain a more efficient algorithm for juntas . the lower bound of ( * ? ? ? * observation 6.3 ) for learning with quantum membership queries for an arbitrary concept class can be rephrased for the purpose of learning @xmath0-juntas as follows . [ s2fac1 ] any algorithm for learning @xmath0-juntas to accuracy @xmath285 with quantum membership queries must use @xmath274 queries . since we are proving a lower bound we may assume that the algorithm is told in advance that the junta depends on variables @xmath286 consequently we may assume that the algorithm makes all its queries with nonzero amplitude only on inputs of the form @xmath287 . now ( * ? ? ? * observation 6.3 ) states that any quantum algorithm which makes queries only over a shattered set ( as is the set of inputs @xmath288 for the class of @xmath0-juntas ) must make at least vc - dim(@xmath31)/100 @xmath50 queries to learn with error rate at most @xmath289 ; here vc - dim(@xmath31 ) is the vapnik - chervonenkis dimension of concept class @xmath31 . since the vc dimension of the class of all boolean functions over variables @xmath275 is @xmath290 , the result follows . this shows that a @xmath50 oracle can not provide sufficient information to learn a @xmath0-junta using @xmath291 queries to high accuracy . it is worth noting that there are other similar learning problems known where an @xmath235-query @xmath50 algorithm can exactly identify a target concept whose description length is @xmath292 bits . for instance , a single @xmath2 oracle call ( which can be implemented by a single @xmath50 query ) can potentially give up to @xmath0 bits of information ; if the concept class @xmath31 is the class of all @xmath290 parity functions over the first @xmath0 variables , then any concept in the class can be exactly learned by a single @xmath2 oracle call . note that all the results we have discussed in this subsection concern algorithms with access to only one type of oracle ; this is in contrast with the algorithm we present in the next section . the motivating question for this section is : `` is it possible to reduce the classical query / sample complexity drastically for the problem of junta learning if the learning algorithm is also permitted to have very limited quantum information ? '' we will give an affirmative answer to this question by describing a new algorithm that uses both @xmath2 queries ( i.e. quantum examples ) and classical uniform random examples . [ s1qlem1 ] let @xmath32 be a function whose value depends on the set of variables @xmath293 . then there is an algorithm querying the @xmath2 oracle @xmath294 times which w.h.p . outputs a list of variables such that * the list contains all the variables @xmath73 for which @xmath295 ; and * all the variables @xmath296 in the list have non - zero influence : @xmath297 . the algorithm simply queries the @xmath2 oracle @xmath298 many times and outputs the union of all the sets of variables received as responses to these queries . if @xmath295 then the probability that @xmath73 never occurs in any response obtained from the @xmath235 @xmath2 oracle calls is at most @xmath299 the union bound now yields that with probability at least @xmath130 , every @xmath73 with @xmath295 is output by the algorithm . [ s2thm3 ] there is an efficient algorithm @xmath34-learning @xmath0-juntas with @xmath7 queries of the @xmath2 oracle and @xmath300 random examples . we claim algorithm 1 satisfies these requirements . * input : * @xmath301 . * stage 1 : * construct a set containing all variables of @xmath3 with an influence at least @xmath302 using the algorithm in lemma [ s1qlem1 ] . let @xmath78 be the final result . * stage 2 : * @xmath304 draw from @xmath47 . let @xmath305 denote the projection of @xmath9 onto the variables in @xmath78 . @xmath306 . . output the hypothesis : @xmath307 assume we are given a boolean function @xmath3 whose value depends on the set of variables @xmath293 with @xmath308 . by lemma [ s1qlem1 ] , @xmath7 queries of the @xmath2 oracle will reveal all variables with influence at least @xmath302 with high probability during stage 1 . assuming the algorithm of lemma [ s1qlem1 ] was successful , we group the variables as follows : [ cols="^,^",options="header " , ] note that @xmath309 by lemma [ s1qlem1 ] and by the assumption that @xmath3 is a @xmath0-junta . we reorder the variables of @xmath3 so that the new order is @xmath310 for notational simplicity , i.e. @xmath3 is now considered to be over @xmath311 . we will denote an assignment to these variables by @xmath312 . in stage 2 the algorithm draws random examples until at least @xmath313 fraction of all assignments to the variables in @xmath78 are observed . let us call this set of assignments by @xmath103 , and for every @xmath314 , let us denote the first example @xmath315 drawn in stage 2 for which @xmath316 by @xmath317 . at the end of the algorithm , the following hypothesis is produced as the output : @xmath318 in other words , the value of the hypothesis only depends on the setting of the variables in @xmath78 . observe the probability that any given setting of a fixed set of variables in @xmath78 has not been seen can be made less than @xmath319 using @xmath320 uniform random examples . therefore the linearity of expectation implies that after @xmath320 random examples , the expected fraction of unseen assignments is @xmath321 . thus by markov s inequality the fraction of unseen assignments will be @xmath322 w.h.p . hence stage 2 will terminate w.h.p . after @xmath320 random examples . consequently , the whole algorithm terminates with high probability with the desired query consumption . all we need to verify is that the hypothesis constructed is @xmath34-accurate . * the hypothesis @xmath323 is @xmath34-accurate with high probability : * we introduce some notation : let @xmath324 ; and given two strings @xmath325 , let @xmath326 denote the bitwise multiplication between @xmath327 ; and let @xmath328 denote the total number of @xmath25 s in @xmath329 . also let @xmath330 denote the indicator function that takes value @xmath26 if @xmath331 holds and value @xmath332 if @xmath331 is false . we start with the following fact : [ fac : stupidfact ] for any @xmath333 , we have @xmath334 } < { \epsilon}/10 $ ] . given any string @xmath335 , clearly there exists a sequence of @xmath336 strings : @xmath337 therefore , @xmath338 } \\ & \leq\frac{1}{2^{n } } \displaystyle\sum_{\mathbf{a}\in\mathbb{b}^{|\mathcal{a}| } } \sum_{\mathbf{b}\in\mathbb{b}^{|\mathcal{b}| } } \sum_{\mathbf{c}\in\mathbb{b}^{|\mathcal{c}|}}\sum_{i=1}^{|s|}\mathbf{1}_{[f(\mathbf{a},\mathbf{b}\odot u^{i+1},\mathbf{c})\neq f(\mathbf{a},\mathbf{b}\odot u^{i},\mathbf{c } ) ] } \\ & = \sum_{i=1}^{|s|}\underbrace{\left(\frac{1}{2^{n } } \displaystyle\sum_{\mathbf{a}\in\mathbb{b}^{|\mathcal{a}| } } \sum_{\mathbf{b}\in\mathbb{b}^{|\mathcal{b}| } } \sum_{\mathbf{c}\in\mathbb{b}^{|\mathcal{c}|}}\mathbf{1}_{[f(\mathbf{a},\mathbf{b}\odot u^{i}\odot u^{i+1},\mathbf{c})\neq f(\mathbf{a},\mathbf{b},\mathbf{c})]}\right)}_{=\text{the influence of the unique variable $ b_{j(i)}$ that takes value $ -1 $ in $ u^{i+1}\odot u^{i}$}}\\ & < { \epsilon}/10.\quad [ \text{since every $ b_j\in\mathcal{b}$ has influence $ < \frac{{\epsilon}}{10k}$ and $ |\mathcal{b}|\leq k$ } ] \end{aligned}\ ] ] for each @xmath339 , consider a fixed setting of strings @xmath340 , @xmath341 . let us call the list of all these assignments @xmath342 , i.e. @xmath343 for any such `` list of assignments '' @xmath342 , we define the function @xmath344 as follows : @xmath345 . the error incurred by approximating @xmath3 by @xmath346 is : @xmath347 = { \mathbf{pr}}_{(\mathbf{a},\mathbf{b},\mathbf{c})}[f(\mathbf{a},\mathbf{b}^{\mathbf{a}},\mathbf{c}^{\mathbf{a}})\neq f(\mathbf{a},\mathbf{b},\mathbf{c})]\ ] ] @xmath348\quad [ \text{since $ f$ does not depend on the variables in $ \mathcal{c}$}]\ ] ] @xmath349 } = \frac{1}{2^{n } } \displaystyle\sum_{\mathbf{a}\in\mathbb{b}^{|\mathcal{a}| } } \sum_{s\in\mathbb{b}^{|\mathcal{b}| } } \sum_{\mathbf{c}\in\mathbb{b}^{|\mathcal{c}|}}\mathbf{1}_{[f(\mathbf{a},\mathbf{b}^{\mathbf{a}},\mathbf{c})\neq f(\mathbf{a},\mathbf{b}^{\mathbf{a}}\odot s,\mathbf{c})]}\label{eqn : fgamma}\ ] ] therefore if we consider the expected value of the incurred error @xmath350 $ ] over all `` lists of assignments '' @xmath342 , equation implies that : @xmath351&=\frac{1}{2^{|\mathcal{b}|}}\sum_{s\in\mathbb{b}^{|\mathcal{b}|}}\underbrace{\left(\frac{1}{2^{n } } \displaystyle\sum_{\mathbf{a}\in\mathbb{b}^{|\mathcal{a}| } } \sum_{\mathbf{b}^{\mathbf{a}}\in\mathbb{b}^{|\mathcal{b}| } } \sum_{\mathbf{c}\in\mathbb{b}^{|\mathcal{c}|}}\mathbf{1}_{[f(\mathbf{a},\mathbf{b}^{\mathbf{a}}\odot s,\mathbf{c})\neq f(\mathbf{a},\mathbf{b}^{\mathbf{a}},\mathbf{c})]}\right ) } _ { < { \epsilon}/10,\ \text{due to fact~\ref{fac : stupidfact}}}\\ & < { \epsilon}/10 . \end{aligned}\ ] ] consequently , the expected error of approximating @xmath3 by a uniformly chosen @xmath346 is less than @xmath352 . this also implies that for a uniformly chosen subset @xmath103 of assignments to variables in @xmath78 with size @xmath353 , the expected error over @xmath103 satisfies : @xmath354 < { \epsilon}/10 $ ] . therefore by markov s inequality , we obtain the following observation : [ obs : jlproof]for a uniformly chosen subset @xmath103 and @xmath346 as described above , @xmath346 will agree with @xmath3 on @xmath355 fraction of the coordinates @xmath356 with probability at least @xmath357 . now if we go back and recall what the algorithm does in stage 2 , we will observe that the generation of the hypothesis in stage 2 is equivalent to drawing a uniform @xmath346 and @xmath103 as described and resetting the values of @xmath346 at those coordinates @xmath358 to true . this is because the algorithm only draws classical random examples during stage 2 . therefore due to observation [ obs : jlproof ] , the hypothesis will disagree with @xmath3 on at most @xmath359 fraction of the inputs with overall probability at least @xmath36 . this gives the desired result . note that this algorithm * uses only a moderate number of quantum examples ; * has overall query complexity with no dependence on @xmath1 , in contrast with known lower bounds ( lemma [ s2lem1 ] ) for learning from classical membership queries ; * uses the @xmath46 oracle as its only source of classical information ( @xmath43 queries are not used ) ; and * is computationally efficient . one can compare this result to that of theorem [ s2thm2 ] which requires @xmath284 quantum examples to learn @xmath0-juntas . in contrast , our algorithm uses not only substantially fewer quantum examples but also fewer uniform random examples , which are considered quite cheap . intuitively , this means that for the junta learning problem , almost all the quantum queries used by the algorithm of bshouty and jackson @xcite can in fact be converted into ordinary classical random examples . the algorithm of theorem [ s2thm3 ] is optimal in the following sense : [ obsjlrn ] any @xmath360-learning quantum membership query algorithm for @xmath0-juntas that uses only @xmath361 classical @xmath43 queries must additionally use @xmath362 @xmath50 queries . this statement easily follows from fact [ s2fac1 ] since a classical membership query can be simulated by a @xmath50 query . contrasting our junta learning algorithm with observation [ obsjlrn ] , we see that if the allowed number of classical examples or queries is decreased even slightly from the @xmath363 used by our algorithm to @xmath361 , then an additional @xmath362 quantum queries are required , even if @xmath50 queries are allowed . we have given some results on learning and testing @xmath0-juntas using both quantum examples and classical random examples . it would be interesting to develop other testing and learning algorithms that combine these two sorts of oracles , with the goal of minimizing the number of quantum oracle calls required . another interesting goal for future work is to further explore the power of the @xmath2 oracle . can the gap between our @xmath100-query upper bound and our @xmath6-query lower bound for the @xmath2 oracle be closed ? a. blum , _ learning a function of @xmath146 relevant variables ( open problem ) _ , proceedings of the 16th annual conference on learning theory and 7th kernel workshop , pp . 731733 ( 2003 ) . h. buhrman , l. fortnow , i. newman , h. rhrig , _ quantum property testing _ , proceedings of 14th soda , pp . 480488 ( 2003 ) . n. bshouty , r. cleve , r. gavald , s. kannan and c. tamon . _ oracles and queries that are sufficient for exact learning _ , j. comput , vol * 52 * , no . 3 , pp . 421 - 433 ( 1996 ) . r. lipton , e. markakis , a. mehta , n. vishnoi , _ on the fourier spectrum of symmetric boolean functions with applications to learning symmetric juntas _ , proceedings of the 20th annual ieee conference on computational complexity , pp . 112119 ( 2005 ) . e. mossel , r. odonnell and r. a. servedio , _ learning functions of @xmath0 variables _ , journal of computer and system sciences , vol . * 69 * , no . 3 , pp . 421434 ( 2004 ) . m. nielsen and i. chuang , _ quantum computation and quantum information _ , cambridge university press ( 2000 ) . r. odonnell , r. a. servedio , _ extremal properties of polynomial threshold functions _ , journal of computer & system sciences , to appear . available at http://www.cs.columbia.edu/`~`rocco/papers/ccc03.html . preliminary version appeared in eighteenth annual ieee conference on computational complexity , pp . 312 ( 2003 ) . r. rubinfeld and m. sudan , _ robust characterizations of polynomials with applications to program testing _ , siam journal on computing , * 25*(2 ) : pp . 252271 ( 1996 ) .

this article proposes an approach to inferring the causal impact of a market intervention , such as a new product launch or the onset of an advertising campaign . our method generalises the widely used difference - in - differences approach to the time - series setting by explicitly modelling the counterfactual of a time series observed both before and after the intervention . it improves on existing methods in two respects : it provides a fully bayesian time - series estimate for the effect ; and it uses model averaging to construct the most appropriate synthetic control for modelling the counterfactual . the ` causalimpact ` r package provides an implementation of our approach ( http://google.github.io/causalimpact/ ) . inferring the impact of market interventions is an important and timely problem . partly because of recent interest in big data , many firms have begun to understand that a competitive advantage can be had by systematically using impact measures to inform strategic decision making . an example is the use of `` a@xmath0b experiments '' to identify the most effective market treatments for the purpose of allocating resources [ @xcite ] . here , we focus on measuring the impact of a discrete marketing event , such as the release of a new product , the introduction of a new feature , or the beginning or end of an advertising campaign , with the aim of measuring the event s impact on a response metric of interest ( e.g. , sales ) . the causal impact of a treatment is the difference between the observed value of the response and the ( unobserved ) value that would have been obtained under the alternative treatment , that is , the effect of treatment on the treated [ @xcite , @xcite ( @xcite ) ] . in the present setting the response variable is a time series , so the causal effect of interest is the difference between the observed series and the series that would have been observed had the intervention not taken place . a powerful approach to constructing the counterfactual is based on the idea of combining a set of candidate predictor variables into a single `` synthetic control '' [ @xcite ] . broadly speaking , there are three sources of information available for constructing an adequate synthetic control . the first is the time - series behaviour of the response itself , prior to the intervention . the second is the behaviour of other time series that were predictive of the target series prior to the intervention . such control series can be based , for example , on the same product in a different region that did not receive the intervention or on a metric that reflects activity in the industry as a whole . in practice , there are often many such series available , and the challenge is to pick the relevant subset to use as contemporaneous controls . this selection is done on the _ pre - treatment _ portion of potential controls ; but their value for predicting the counterfactual lies in their _ post - treatment _ behaviour . as long as the control series received no intervention themselves , it is often reasonable to assume the relationship between the treatment and the control series that existed prior to the intervention to continue afterwards . thus , a plausible estimate of the counterfactual time series can be computed up to the point in time where the relationship between treatment and controls can no longer be assumed to be stationary , for example , because one of the controls received treatment itself . in a bayesian framework , a third source of information for inferring the counterfactual is the available prior knowledge about the model parameters , as elicited , for example , by previous studies . ) with an intervention beginning in january 2014 . two other markets ( @xmath1 , @xmath2 ) were not subject to the intervention and allow us to construct a synthetic control [ cf . @xcite ] . inverting the state - space model described in the main text yields a prediction of what would have happened in @xmath3 had the intervention not taken place ( posterior predictive expectation of the counterfactual with pointwise 95% posterior probability intervals ) . the difference between observed data and counterfactual predictions is the inferred causal impact of the intervention . here , predictions accurately reflect the true ( gamma - shaped ) impact . a key characteristic of the inferred impact series is the progressive widening of the posterior intervals ( shaded area ) . this effect emerges naturally from the model structure and agrees with the intuition that predictions should become increasingly uncertain as we look further and further into the ( retrospective ) future . another way of visualizing posterior inferences is by means of a cumulative impact plot . it shows , for each day , the summed effect up to that day . here , the 95% credible interval of the cumulative impact crosses the zero - line about five months after the intervention , at which point we would no longer declare a significant overall effect . ] [ fig : intro : illustration ] we combine the three preceding sources of information using a state - space time - series model , where one component of state is a linear regression on the contemporaneous predictors . the framework of our model allows us to choose from among a large set of potential controls by placing a spike - and - slab prior on the set of regression coefficients and by allowing the model to average over the set of controls [ @xcite ] . we then compute the posterior distribution of the counterfactual time series given the value of the target series in the pre - intervention period , along with the values of the controls in the post - intervention period . subtracting the predicted from the observed response during the post - intervention period gives a semiparametric bayesian posterior distribution for the causal effect ( figure [ fig : intro : illustration ] ) . as with other domains , causal inference in marketing requires subtlety . marketing data are often observational and rarely follow the ideal of a randomised design . they typically exhibit a low signal - to - noise ratio . they are subject to multiple seasonal variations , and they are often confounded by the effects of unobserved variables and their interactions [ for recent examples , see @xcite , @xcite ( @xcite ) ] . rigorous causal inferences can be obtained through randomised experiments , which are often implemented in the form of geo experiments . many market interventions , however , fail to satisfy the requirements of such approaches . for instance , advertising campaigns are frequently launched across multiple channels , online and offline , which precludes measurement of individual exposure . campaigns are often targeted at an entire country , and one country only , which prohibits the use of geographic controls within that country . likewise , a campaign might be launched in several countries but at different points in time . thus , while a large control group may be available , the treatment group often consists of no more than one region or a few regions with considerable heterogeneity among them . a standard approach to causal inference in such settings is based on a linear model of the observed outcomes in the treatment and control group before and after the intervention . one can then estimate the difference between ( i ) the pre - post difference in the treatment group and ( ii ) the pre - post difference in the control group . the assumption underlying such _ difference - in - differences _ ( dd ) designs is that the level of the control group provides an adequate proxy for the level that would have been observed in the treatment group in the absence of treatment [ see @xcite ] . dd designs have been limited in three ways . first , dd is traditionally based on a static regression model that assumes i.i.d . data despite the fact that the design has a temporal component . when fit to serially correlated data , static models yield overoptimistic inferences with too narrow uncertainty intervals [ see also @xcite , @xcite ( @xcite ) , @xcite ] . second , most dd analyses only consider two time points : before and after the intervention . in practice , the manner in which an effect evolves over time , especially its onset and decay structure , is often a key question . third , when dd analyses _ are _ based on time series , previous studies have imposed restrictions on the way in which a synthetic control is constructed from a set of predictor variables , which is something we wish to avoid . for example , one strategy [ @xcite ] has been to choose a convex combination @xmath4 of @xmath5 predictor time series in such a way that a vector of pre - treatment variables ( not time series ) @xmath1 characterising the treated unit before the intervention is matched most closely by the combination of pre - treatment variables @xmath6 of the control units w.r.t . a vector of importance weights @xmath7 . these weights are themselves determined in such a way that the combination of pre - treatment outcome time series of the control units most closely matches the pre - treatment outcome time series of the treated unit . such a scheme relies on the availability of interpretable characteristics ( e.g. , growth predictors ) , and it precludes nonconvex combinations of controls when constructing the weight vector @xmath8 . we prefer to select a combination of control series without reference to external characteristics and purely in terms of how well they explain the pre - treatment outcome time series of the treated unit ( while automatically balancing goodness of fit and model complexity through the use of regularizing priors ) . another idea [ @xcite ] has been to use classical variable - selection methods ( such as the lasso ) to find a sparse set of predictors . this approach , however , ignores posterior uncertainty about both which predictors to use and their coefficients . the limitations of dd schemes can be addressed by using state - space models , coupled with highly flexible regression components , to explain the temporal evolution of an observed outcome . state - space models distinguish between a state equation that describes the transition of a set of latent variables from one time point to the next and an observation equation that specifies how a given system state translates into measurements . this distinction makes them extremely flexible and powerful [ see @xcite for a discussion in the context of marketing research ] . the approach described in this paper inherits three main characteristics from the state - space paradigm . first , it allows us to flexibly accommodate different kinds of assumptions about the latent state and emission processes underlying the observed data , including local trends and seasonality . second , we use a fully bayesian approach to inferring the temporal evolution of counterfactual activity and incremental impact . one advantage of this is the flexibility with which posterior inferences can be summarised . third , we use a regression component that precludes a rigid commitment to a particular set of controls by integrating out our posterior uncertainty about the influence of each predictor as well as our uncertainty about which predictors to include in the first place , which avoids overfitting . the remainder of this paper is organised as follows . section [ sec : theory ] describes the proposed model , its design variations , the choice of diffuse empirical priors on hyperparameters , and a stochastic algorithm for posterior inference based on markov chain monte carlo ( mcmc ) . section [ sec : apps : synthetic ] demonstrates important features of the model using simulated data , followed by an application in section [ sec : apps : empirical ] to an advertising campaign run by one of google s advertisers . section [ sec : disc ] puts our approach into context and discusses its scope of application . structural time - series models are state - space models for time - series data . they can be defined in terms of a pair of equations @xmath9 where @xmath10 and @xmath11 are independent of all other unknowns . equation ( [ eq : observation ] ) is the _ observation equation _ ; it links the observed data @xmath12 to a latent @xmath13-dimensional state vector @xmath14 . equation ( [ eq : transition ] ) is the _ state equation _ ; it governs the evolution of the state vector @xmath14 through time . in the present paper , @xmath12 is a scalar observation , @xmath15 is a @xmath13-dimensional output vector , @xmath16 is a @xmath17 transition matrix , @xmath18 is a @xmath19 control matrix , @xmath20 is a scalar observation error with noise variance @xmath21 , and @xmath22 is a @xmath23-dimensional system error with a @xmath24 state - diffusion matrix @xmath25 , where @xmath26 . writing the error structure of equation ( [ eq : transition ] ) as @xmath27 allows us to incorporate state components of less than full rank ; a model for seasonality will be the most important example . structural time - series models are useful in practice because they are flexible and modular . they are flexible in the sense that a very large class of models , including all arima models , can be written in the state - space form given by ( [ eq : observation ] ) and ( [ eq : transition ] ) . they are modular in the sense that the latent state as well as the associated model matrices @xmath28 , and @xmath25 can be assembled from a library of component sub - models to capture important features of the data . there are several widely used state - component models for capturing the trend , seasonality or effects of holidays . a common approach is to assume the errors of different state - component models to be independent ( i.e. , @xmath25 is block - diagonal ) . the vector @xmath14 can then be formed by concatenating the individual state components , while @xmath16 and @xmath18 become block - diagonal matrices . the most important state component for the applications considered in this paper is a regression component that allows us to obtain counterfactual predictions by constructing a synthetic control based on a combination of markets that were not treated . observed responses from such markets are important because they allow us to explain variance components in the treated market that are not readily captured by more generic seasonal sub - models . this approach assumes that covariates are unaffected by the effects of treatment . for example , an advertising campaign run in the united states might spill over to canada or the united kingdom . when assuming the absence of spill - over effects , the use of such indirectly affected markets as controls would lead to pessimistic inferences , that is , the effect of the campaign would be underestimated [ cf . @xcite ] . the first component of our model is a local linear trend , defined by the pair of equations @xmath29 \label{eq : local - linear - trend } \\[-8pt ] \nonumber \delta_{t+1 } & = & \delta_{t } + \eta_{\delta , t},\end{aligned}\ ] ] where @xmath30 and @xmath31 . the @xmath32 component is the value of the trend at time @xmath33 . the @xmath34 component is the expected increase in @xmath35 between times @xmath33 and @xmath36 , so it can be thought of as the _ slope _ at time @xmath33 . the local linear trend model is a popular choice for modelling trends because it quickly adapts to local variation , which is desirable when making short - term predictions . this degree of flexibility may not be desired when making longer - term predictions , as such predictions often come with implausibly wide uncertainty intervals . there is a generalisation of the local linear trend model where the slope exhibits stationarity instead of obeying a random walk . this model can be written as @xmath37 \label{eq : generalized - local - linear - trend } \\[-8pt ] \nonumber \delta_{t+1 } & = & d + \rho(\delta_{t } - d ) + \eta_{\delta , t},\end{aligned}\ ] ] where the two components of @xmath38 are independent . in this model , the slope of the time trend exhibits @xmath39 variation around a long - term slope of @xmath40 . the parameter @xmath41 represents the learning rate at which the local trend is updated . thus , the model balances short - term information with information from the distant past . there are several commonly used state - component models to capture seasonality . the most frequently used model in the time domain is @xmath42 where @xmath43 represents the number of seasons and @xmath44 denotes their joint contribution to the observed response @xmath12 . the state in this model consists of the @xmath45 most recent seasonal effects , but the error term is a scalar , so the evolution equation for this state model is less than full rank . the mean of @xmath46 is such that the total seasonal effect is zero when summed over @xmath43 seasons . for example , if we set @xmath47 to capture four seasons per year , the mean of the _ winter _ coefficient will be @xmath48 . the part of the transition matrix @xmath16 representing the seasonal model is an @xmath49 matrix with @xmath50 s along the top row , 1 s along the subdiagonal and 0 s elsewhere . the preceding seasonal model can be generalised to allow for multiple seasonal components with different periods . when modelling daily data , for example , we might wish to allow for an @xmath51 day - of - week effect , as well as an @xmath52 weekly annual cycle . the latter can be handled by setting @xmath53 , with zero variance on the error term , when @xmath33 is not the start of a new week , and setting @xmath16 to the usual seasonal transition matrix , with nonzero error variance , when @xmath33 is the start of a new week . control time series that received no treatment are critical to our method for obtaining accurate counterfactual predictions since they account for variance components that are shared by the series , including , in particular , the effects of other unobserved causes otherwise unaccounted for by the model . a natural way of including control series in the model is through a linear regression . its coefficients can be static or time - varying . a _ static _ regression can be written in state - space form by setting @xmath54 and @xmath55 . one advantage of working in a fully bayesian treatment is that we do not need to commit to a fixed set of covariates . the spike - and - slab prior described in section [ sec : posterior - sampling ] allows us to integrate out our posterior uncertainty about which covariates to include and how strongly they should influence our predictions , which avoids overfitting . all covariates are assumed to be contemporaneous ; the present model does not infer on a potential lag between treated and untreated time series . a known lag , however , can be easily incorporated by shifting the corresponding regressor in time . an alternative to the above is a regression component with _ dynamic _ regression coefficients to account for time - varying relationships [ e.g. , @xcite ] . given covariates @xmath56 , this introduces the dynamic regression component @xmath57 \label{eq : dynamic - regression } \\[-8pt ] \nonumber \beta_{j , t+1 } & = & \beta_{j , t } + \eta_{\beta , j , t},\end{aligned}\ ] ] where @xmath58 . here , @xmath59 is the coefficient for the @xmath60th control series and @xmath61 is the standard deviation of its associated random walk . we can write the dynamic regression component in state - space form by setting @xmath62 and @xmath63 and by setting the corresponding part of the transition matrix to @xmath64 , with @xmath65 . structural time - series models allow us to examine the time series at hand and flexibly choose appropriate components for trend , seasonality , and either static or dynamic regression for the controls . the presence or absence of seasonality , for example , will usually be obvious by inspection . a more subtle question is whether to choose static or dynamic regression coefficients . when the relationship between controls and treated unit has been stable in the past , static coefficients are an attractive option . this is because a spike - and - slab prior can be implemented efficiently within a forward - filtering , backward - sampling framework . this makes it possible to quickly identify a sparse set of covariates even from tens or hundreds of potential variables [ @xcite ] . local variability in the treated time series is captured by the dynamic local level or dynamic linear trend component . covariate stability is typically high when the available covariates are close in nature to the treated metric . the empirical analyses presented in this paper , for example , will be based on a static regression component ( section [ sec : apps : empirical ] ) . this choice provides a reasonable compromise between capturing local behaviour and accounting for regression effects . an alternative would be to use dynamic regression coefficients , as we do , for instance , in our analyses of simulated data ( section [ sec : apps : synthetic ] ) . dynamic coefficients are useful when the linear relationship between treated metrics and controls is believed to change over time . there are a number of ways of reducing the computational burden of dealing with a potentially large number of dynamic coefficients . one option is to resort to dynamic latent factors , where one uses @xmath66 with @xmath67 and uses @xmath68 instead of @xmath69 as part of @xmath15 in ( [ eq : observation ] ) , coupled with an ar - type model for @xmath68 itself . another option is latent thresholding regression , where one uses a dynamic version of the spike - and - slab prior as in @xcite . the state - component models are assembled independently , with each component providing an additive contribution to @xmath12 . figure [ fig : theory : model ] illustrates this process assuming a local linear trend paired with a static regression component . is modelled as the result of a latent state plus gaussian observation noise with error standard deviation @xmath70 . the state @xmath71 includes a local level @xmath32 , a local linear trend @xmath34 , and a set of contemporaneous covariates @xmath69 , scaled by regression coefficients @xmath72 . state components are assumed to evolve according to independent gaussian random walks with fixed standard deviations @xmath73 and @xmath74 ( conditional - dependence arrows shown for the first time point only ) . the model includes empirical priors on these parameters and the initial states . in an alternative formulation , the regression coefficients @xmath75 are themselves subject to random - walk diffusion ( see main text ) . of principal interest is the posterior predictive density over the unobserved counterfactual responses @xmath76 . subtracting these from the actual observed data @xmath77 yields a probability density over the temporal evolution of causal impact . ] let @xmath78 generically denote the set of all model parameters and let @xmath79 denote the full state sequence . we adopt a bayesian approach to inference by specifying a prior distribution @xmath80 on the model parameters as well as a distribution @xmath81 on the initial state values . we may then sample from @xmath82 using mcmc . most of the models in section [ sec : components - state ] depend solely on a small set of variance parameters that govern the diffusion of the individual state components . a typical prior distribution for such a variance is @xmath83 where @xmath84 is the gamma distribution with expectation @xmath85 . the prior parameters can be interpreted as a prior sum of squares @xmath86 , so that @xmath87 is a prior estimate of @xmath88 , and @xmath89 is the weight , in units of prior sample size , assigned to the prior estimate . we often have a weak default prior belief that the incremental errors in the state process are small , which we can formalise by choosing small values of @xmath89 ( e.g. , 1 ) and small values of @xmath87 . the notion of `` small '' means different things in different models ; for the seasonal and local linear trend models our default priors are @xmath90 , where @xmath91 is the sample variance of the target series . scaling by the sample variance is a minor violation of the bayesian paradigm , but it is an effective means of choosing a reasonable scale for the prior . it is similar to the popular technique of scaling the data prior to analysis , but we prefer to do the scaling in the prior so we can model the data on its original scale . when faced with many potential controls , we prefer letting the model choose an appropriate set . this can be achieved by placing a spike - and - slab prior over coefficients [ @xcite ( @xcite ) , @xcite ] . a spike - and - slab prior combines point mass at zero ( the `` spike '' ) , for an unknown subset of zero coefficients , with a weakly informative distribution on the complementary set of nonzero coefficients ( the `` slab '' ) . contrary to what its name might suggest , the `` slab '' is usually not completely flat , but rather a gaussian with a large variance . let @xmath92 , where @xmath93 if @xmath94 and @xmath95 otherwise . let @xmath96 denote the nonzero elements of the vector @xmath97 and let @xmath98 denote the rows and columns of @xmath99 corresponding to nonzero entries in @xmath100 . we can then factorise the spike - and - slab prior as @xmath101 the spike portion of ( [ eq : spike - slab - general ] ) can be an arbitrary distribution over @xmath102 in principle ; the most common choice in practice is a product of independent bernoulli distributions , @xmath103 where @xmath104 is the prior probability of regressor @xmath60 being included in the model . values for @xmath104 can be elicited by asking about the _ expected model size _ @xmath105 , and then setting all @xmath106 . an alternative is to use a more specific set of values @xmath104 . in particular , one might choose to set certain @xmath104 to either 1 or 0 to force the corresponding variables into or out of the model . generally , framing the prior in terms of expected model size has the advantage that the model can adapt to growing numbers of predictor variables without having to switch to a hierarchical prior [ @xcite ] . for the `` slab '' portion of the prior we use a conjugate normal - inverse gamma distribution , @xmath107 the vector @xmath108 in equation ( [ eq : slab - normal ] ) encodes our prior expectation about the value of each element of @xmath75 . in practice , we usually set @xmath109 . the prior parameters in equation ( [ eq : slab - variance ] ) can be elicited by asking about the expected @xmath110 $ ] as well as the number of observations worth of weight @xmath111 the prior estimate should be given . then @xmath112 . the final prior parameter in ( [ eq : slab - normal ] ) is @xmath99 , which , up to a scaling factor , is the prior precision over @xmath75 in the full model , with all variables included . the total information in the covariates is @xmath113 , and so @xmath114 is the average information in a single observation . zellner s @xmath115-prior [ @xcite ] sets @xmath116 , so that @xmath115 can be interpreted as @xmath115 observations worth of information . zellner s prior becomes improper when @xmath113 is not positive definite ; we therefore ensure propriety by averaging @xmath113 with its diagonal , @xmath117 with default values of @xmath118 and @xmath119 . overall , this prior specification provides a broadly useful default while providing considerable flexibility in those cases where more specific prior information is available . posterior inference in our model can be broken down into three pieces . first , we simulate draws of the model parameters @xmath120 and the state vector @xmath121 given the observed data @xmath122 in the training period . second , we use the posterior simulations to simulate from the posterior predictive distribution @xmath123 given the observed pre - intervention activity @xmath124 . third , we use the posterior predictive samples to compute the posterior distribution of the pointwise impact @xmath125 for each @xmath126 . we use the same samples to obtain the posterior distribution of cumulative impact . we use a gibbs sampler to simulate a sequence @xmath127 from a markov chain whose stationary distribution is @xmath128 . the sampler alternates between a _ data - augmentation _ step that simulates from @xmath129 and a _ parameter - simulation _ step that simulates from @xmath130 . the data - augmentation step uses the posterior simulation algorithm from @xcite , providing an improvement over the earlier forward - filtering , backward - sampling algorithms by @xcite , @xcite , and @xcite . in brief , because @xmath131 is jointly multivariate normal , the variance of @xmath132 does not depend on @xmath124 . we can therefore simulate @xmath133 and subtract @xmath134 to obtain zero - mean noise with the correct variance . adding @xmath135 restores the correct mean , which completes the draw . the required expectations can be computed using the kalman filter and a _ fast mean smoother _ described in detail by @xcite . the result is a direct simulation from @xmath136 in an algorithm that is linear in the total ( pre- and post - intervention ) number of time points ( @xmath137 ) and quadratic in the dimension of the state space ( @xmath13 ) . given the draw of the state , the parameter draw is straightforward for all state components other than the static regression coefficients @xmath138 . all state components that exclusively depend on variance parameters can translate their draws back to error terms @xmath22 and accumulate sums of squares of @xmath38 , and , because of conjugacy with equation ( [ eq : typical - variance ] ) , the posterior distribution will remain gamma distributed . the draw of the static regression coefficients @xmath138 proceeds as follows . for each @xmath139 in the pre - intervention period , let @xmath140 denote @xmath12 with the contributions from the other state components subtracted away , and let @xmath141 . the challenge is to simulate from @xmath142 . because of conjugacy , we can integrate out @xmath138 and @xmath143 and be left with @xmath144 where @xmath145 is an unknown normalizing constant . the sufficient statistics in equation ( [ eq : inclusion - indicator - posterior ] ) are @xmath146 to sample from ( [ eq : inclusion - indicator - posterior ] ) , we use a gibbs sampler that draws each @xmath147 given all other @xmath148 . each full - conditional is easy to evaluate because @xmath147 can only assume two possible values . it should be noted that the dimension of all matrices in ( [ eq : inclusion - indicator - posterior ] ) is @xmath149 , which is small if the model is truly sparse . there are many matrices to manipulate , but because each is small , the overall algorithm is fast . once the draw of @xmath150 is complete , we sample directly from @xmath151 using standard conjugate formulae . for an alternative that may be even more computationally efficient , see @xcite . while the posterior over model parameters and states @xmath152 can be of interest in its own right , causal impact analyses are primarily concerned with the posterior incremental effect , @xmath153 as shown by its indices , the density in equation ( [ eqn : theory : postpred ] ) is defined precisely for that portion of the time series which is unobserved : the counterfactual market response @xmath154 that would have been observed in the treated market , after the intervention , in the absence of treatment . it is also worth emphasising that the density is conditional on the observed data ( as well as the priors ) and only on these , that is , on activity in the treatment market before the beginning of the intervention as well as activity in all control markets both before and during the intervention . the density is _ not _ conditioned on parameter estimates or the inclusion or exclusion of covariates with static regression coefficients , all of which have been integrated out . thus , through bayesian model averaging , we commit neither to any particular set of covariates , which helps avoid an arbitrary selection , nor to point estimates of their coefficients , which prevents overfitting . the posterior predictive density in ( [ eqn : theory : postpred ] ) is defined as a coherent ( joint ) distribution over all counterfactual data points , rather than as a collection of pointwise univariate distributions . this ensures that we correctly propagate the serial structure determined on pre - intervention data to the trajectory of counterfactuals . this is crucial , in particular , when forming summary statistics , such as the cumulative effect of the intervention on the treatment market . posterior inference was implemented in c@xmath155 with an r interface . given a typically - sized data set with @xmath156 time points , @xmath157 covariates , and @xmath158 iterations ( see section [ sec : apps : empirical ] for an example ) , this implementation takes less than 30 seconds to complete on a standard computer , enabling near - interactive analyses . samples from the posterior predictive distribution over counterfactual activity can be readily used to obtain samples from the posterior causal effect , that is , the quantity we are typically interested in . for each draw @xmath159 and for each time point @xmath160 , we set @xmath161 yielding samples from the approximate posterior predictive density of the effect attributed to the intervention . in addition to its pointwise impact , we often wish to understand the cumulative effect of an intervention over time . one of the main advantages of a sampling approach to posterior inference is the flexibility and ease with which such derived inferences can be obtained . reusing the impact samples obtained in ( [ eqn : theory : causal_samples ] ) , we compute for each draw @xmath159 @xmath162 the preceding _ cumulative sum _ of causal increments is a useful quantity when @xmath163 represents a _ flow _ quantity , measured over an interval of time ( e.g. , a day ) , such as the number of searches , sign - ups , sales , additional installs or new users . it becomes uninterpretable when @xmath163 represents a _ quantity , usefully defined only for a point in time , such as the total number of clients , users or subscribers . in this case we might instead choose , for each @xmath159 , to draw a sample of the posterior _ running average _ effect following the intervention , @xmath164 unlike the cumulative effect in ( [ eqn : cumlift ] ) , the running average is always interpretable , regardless of whether it refers to a flow or a stock . however , it is more context - dependent on the length of the post - intervention period under consideration . in particular , under the assumption of a true impact that grows quickly at first and then declines to zero , the cumulative impact approaches its true total value ( in expectation ) as we increase the counterfactual forecasting period , whereas the average impact will eventually approach zero ( while , in contrast , the probability intervals diverge in both cases , leading to more and more uncertain inferences as the forecasting period increases ) . to study the characteristics of our approach , we analysed simulated ( i.e. , computer - generated ) data across a series of independent simulations . generated time series started on 1 january 2013 and ended on 30 june 2014 , with a perturbation beginning on 1 january 2014 . the data were simulated using a dynamic regression component with two covariates whose coefficients evolved according to independent random walks , @xmath165 , initialised at @xmath166 . the covariates themselves were simple sinusoids with wavelengths of 90 days and 360 days , respectively . the latent state underlying the observed data was generated using a local level that evolved according to a random walk , @xmath167 , initialised at @xmath168 . independent observation noise was sampled using @xmath169 . in summary , observations @xmath12 were generated using @xmath170 to simulate the effect of advertising , the post - intervention portion of the preceding series was multiplied by @xmath171 , where @xmath172 ( not to be confused with @xmath173 ) represented the true effect size specifying the ( uniform ) relative lift during the campaign period . an example is shown in figure [ fig : apps : synthetic : inference](a ) . prior . interval coverage . using an effect size of 10% , the plot shows the proportion of simulations in which the pointwise central 95% credible interval contained the true impact , as a function of campaign duration . intervals should contain ground truth in 95% of simulations , however much uncertainty its predictions may be associated with . error bars represent 95% credible intervals . ] to study the properties of our model , we began by considering under what circumstances we successfully detected a causal effect , that is , the statistical power or sensitivity of our approach . a related property is the probability of _ not _ detecting an absent impact , that is , specificity . we repeatedly generated data , as described above , under different true effect sizes . we then computed the posterior predictive distribution over the counterfactuals and recorded whether or not we would have concluded a causal effect . for each of the effect sizes 0% , 0.1% , 1% , 10% and 100% , a total of @xmath174 simulations were run . this number was chosen simply on the grounds that it provided reasonably tight intervals around the reported summary statistics without requiring excessive amounts of computation . in each simulation , we concluded that a causal effect was present if and only if the central 95% posterior probability interval of the cumulative effect excluded zero . the model used throughout this section comprised two structural blocks . the first one was a local level component . we placed an inverse - gamma prior on its diffusion variance with a prior estimate of @xmath175 and a prior sample size @xmath176 . the second structural block was a dynamic regression component . we placed a gamma prior with prior expectation @xmath177 on the diffusion variance of both regression coefficients . by construction , the outcome variable did not exhibit any local trends or seasonality other than the variation conveyed through the covariates . this obviated the need to include an explicit local linear trend or seasonality component in the model . in a first analysis , we considered the empirical proportion of simulations in which a causal effect had been detected . when taking into account only those simulations where the true effect size was greater than zero , these empirical proportions provide estimates of the sensitivity of the model w.r.t . the process by which the data were generated . conversely , those simulations where the campaign had no effect yield an estimate of the model s specificity . in this way , we obtained the power curve shown in figure [ fig : apps : synthetic : inference](b ) . the curve shows that , in data such as these , a market perturbation leading to a lift no larger than 1% is missed in about 90% of cases . by contrast , a perturbation that lifts market activity by 25% is correctly detected as such in most cases . in a second analysis , we assessed the coverage properties of the posterior probability intervals obtained through our model . it is desirable to use a diffuse prior on the local level component such that central 95% intervals contain ground truth in about 95% of the simulations . this coverage frequency should hold regardless of the length of the campaign period . in other words , a longer campaign should lead to posterior intervals that are appropriately widened to retain the same coverage probability as the narrower intervals obtained for shorter campaigns . this was approximately the case throughout the simulated campaign [ figure [ fig : apps : synthetic : inference](c ) ] . to study the accuracy of the point estimates supported by our approach , we repeated the preceding simulations with a fixed effect size of 10% while varying the length of the campaign . when given a quadratic loss function , the loss - minimizing point estimate is the posterior expectation of the predictive density over counterfactuals . thus , for each generated data set @xmath178 , we computed the expected causal effect for each time point , @xmath179 to quantify the discrepancy between estimated and true impact , we calculated the absolute percentage estimation error , @xmath180 this yielded an empirical distribution of absolute percentage estimation errors [ figure [ fig : apps : synthetic : acc](a ) , blue ] , showing that impact estimates become less and less accurate as the forecasting period increases . this is because , under the local linear trend model in ( [ eq : local - linear - trend ] ) , the true counterfactual activity becomes more and more likely to deviate from its expected trajectory . 2 s.e.m . ) at which predictions become less accurate as the length of the counterfactual forecasting period increases ( blue ) . the well - behaved decrease in estimation accuracy breaks down when the data are subject to a sudden structural change ( red ) , as simulated for 1 april 2014 . illustration of a structural break . the plot shows one example of the time series underlying the red curve in . on 1 april 2014 , the standard deviation of the generating random walk of the local level was tripled , causing the rapid decline in estimation accuracy seen in the red curve in . ] it is worth emphasising that all preceding results are based on the assumption that the model structure remains intact throughout the modelling period . in other words , even though the model is built around the idea of multiple ( nonstationary ) components ( i.e. , a time - varying local trend and , potentially , time - varying regression coefficients ) , this structure itself remains unchanged . if the model structure does change , estimation accuracy may suffer . we studied the impact of a changing model structure in a second simulation in which we repeated the procedure above in such a way that 90 days after the beginning of the campaign the standard deviation of the random walk governing the evolution of the regression coefficient was tripled ( now 0.03 instead of 0.01 ) . as a result , the observed data began to diverge much more quickly than before . accordingly , estimations became considerably less reliable [ figure [ fig : apps : synthetic : acc](a ) , red ] . an example of the underlying data is shown in figure [ fig : apps : synthetic : acc](b ) . the preceding simulations highlight the importance of a model that is sufficiently flexible to account for phenomena typically encountered in seasonal empirical data . this rules out entirely static models in particular ( such as multiple linear regression ) . to illustrate the practical utility of our approach , we analysed an advertising campaign run by one of google s advertisers in the united states . in particular , we inferred the campaign s causal effect on the number of times a user was directed to the advertiser s website from the google search results page . we provide a brief overview of the underlying data below [ see @xcite for additional details ] . the campaign analysed here was based on product - related ads to be displayed alongside google s search results for specific keywords . ads went live for a period of 6 consecutive weeks and were geo - targeted to a randomised set of 95 out of 190 designated market areas ( dmas ) . the most salient observable characteristic of dmas is offline sales . to produce balance in this characteristic , dmas were first rank - ordered by sales volume . pairs of regions were then randomly assigned to treatment / control . dmas provide units that can be easily supplied with distinct offerings , although this fine - grained split was not a requirement for the model . in fact , we carried out the analysis as if only one treatment region had been available ( formed by summing all treated dmas ) . this allowed us to evaluate whether our approach would yield the same results as more conventional treatment - control comparisons would have done . the outcome variable analysed here was search - related visits to the advertiser s website , consisting of organic clicks ( i.e. , clicks on a search result ) and paid clicks ( i.e. , clicks on an ad next to the search results , for which the advertiser was charged ) . since paid clicks were zero before the campaign , one might wonder why we could not simply count the number of paid clicks after the campaign had started . the reason is that paid clicks tend to cannibalise some organic clicks . since we were interested in the net effect , we worked with the total number of clicks . the first building block of the model used for the analyses in this section was a local level component . for the inverse - gamma prior on its diffusion variance we used a prior estimate of @xmath181 and a prior sample size @xmath176 . the second structural block was a static regression component . we used a spike - and - slab prior with an expected model size of @xmath182 , an expected explained variance of @xmath183 and 50 prior @xmath184 . we deliberately kept the model as simple as this . since the covariates came from a randomised experiment , we expected them to already account for any additional local linear trends and seasonal variation in the response variable . if one suspects that a more complex model might be more appropriate , one could optimise model design through bayesian model selection . here , we focus instead on comparing different sets of covariates , which is critical in counterfactual analyses regardless of the particular model structure used . model estimation was carried out using 10,000 mcmc samples . we began by applying the above model to infer the causal effect of the campaign on the time series of clicks in the treated regions . given that a set of unaffected regions was available in this analysis , the best possible set of controls was given by the untreated dmas themselves ( see below for a comparison with a purely observational alternative ) . as shown in figure [ fig : apps : empirical : forecast : original](a ) , the model provided an excellent fit on the pre - campaign trajectory of clicks ( including a spike in `` week @xmath185 '' and a dip at the end of `` week @xmath50 '' ) . following the onset of the campaign , observations quickly began to diverge from counterfactual predictions : the actual number of clicks was consistently higher than what would have been expected in the absence of the campaign . the curves did not reconvene until one week after the end of the campaign . subtracting observed from predicted data , as we did in figure [ fig : apps : empirical : forecast : original](b ) , resulted in a posterior estimate of the incremental lift caused by the campaign . it peaked after about three weeks into the campaign , and faded away after about one week after the end of the campaign . thus , as shown in figure [ fig : apps : empirical : forecast : original](c ) , the campaign led to a sustained cumulative increase in total clicks ( as opposed to a mere shift of future clicks into the present or a pure cannibalization of organic clicks by paid clicks ) . specifically , the overall effect amounted to 88,400 additional clicks in the targeted regions ( posterior expectation ; rounded to three significant digits ) , that is , an increase of @xmath186 , with a central 95% credible interval of @xmath187 $ ] . to validate this estimate , we returned to the original experimental data , on which a conventional treatment - control comparison had been carried out using a two - stage linear model [ @xcite ] . this analysis had led to an estimated lift of 84,700 clicks , with a 95% confidence interval for the relative expected lift of @xmath188 $ ] . thus , with a deviation of less than 5% , the counterfactual approach had led to almost precisely the same estimate as the randomised evaluation , except for its wider intervals . the latter is expected , given that our intervals represent prediction intervals , not confidence intervals . moreover , in addition to an interval for the sum over all time points , our approach yields a full time series of pointwise intervals , which allows analysts to examine the characteristics of the temporal evolution of attributable impact . the posterior predictive intervals in figure [ fig : apps : empirical : forecast : original](b ) widen more slowly than in the illustrative example in figure [ fig : intro : illustration ] . this is because the large number of controls available in this data set offers a much higher pre - campaign predictive strength than in the simulated data in figure [ fig : intro : illustration ] . this is not unexpected , given that controls came from a randomised experiment , and we will see that this also holds for a subsequent analysis ( see below ) that is based on yet another data source for predictors . a consequence of this is that there is little variation left to be captured by the random - walk component of the model . a reassuring finding is that the estimated counterfactual time series in figure [ fig : apps : empirical : forecast : original](a ) eventually almost exactly rejoins the observed series , only a few days after the end of the intervention . an important characteristic of counterfactual - forecasting approaches is that they do not require a setting in which a set of controls , selected at random , was exempt from the campaign . we therefore repeated the preceding analysis in the following way : we discarded the data from all control regions and , instead , used searches for keywords related to the advertiser s industry , grouped into a handful of verticals , as covariates . in the absence of a dedicated set of control regions , such industry - related time series can be very powerful controls , as they capture not only seasonal variations but also market - specific trends and events ( though not necessarily advertiser - specific trends ) . a major strength of the controls chosen here is that time series on web searches are publicly available through google trends ( http://www.google.com/trends/ ) . this makes the approach applicable to virtually any kind of intervention . at the same time , the industry as a whole is unlikely to be moved by a single actor s activities . this precludes a positive bias in estimating the effect of the campaign that would arise if a covariate was negatively affected by the campaign . as shown in figure [ fig : apps : empirical : forecast : observational ] , we found a cumulative lift of 85,900 clicks ( posterior expectation ) , or @xmath189 , with a @xmath190 $ ] interval . in other words , the analysis replicated almost perfectly the original analysis that had access to a randomised set of controls . one feature in the response variable which this second analysis failed to account for was a spike in clicks in the second week before the campaign onset ; this spike appeared both in treated and untreated regions and appears to be specific to this advertiser . in addition , the series of point - wise impact [ figure [ fig : apps : empirical : forecast : observational](b ) ] is slightly more volatile than in the original analysis ( figure [ fig : apps : empirical : forecast : original ] ) . on the other hand , the overall point estimate of 85,900 , in this case , was even closer to the randomised - design baseline ( 84,700 ; deviation ca . 1% ) than in our first analysis ( 88,400 ; deviation ca . 4% ) . in summary , the counterfactual approach effectively obviated the need for the original randomised experiment . using purely observational variables led to the same substantive conclusions . ) that had been based on a randomised design . ] to go one step further still , we analysed clicks in those regions that had been exempt from the advertising campaign . if the effect of the campaign was truly specific to treated regions , there should be no effect in the controls . to test this , we inferred the causal effect of the campaign on _ unaffected _ regions , which should _ not _ lead to a significant finding . in analogy with our second analysis , we discarded clicks in the treated regions and used searches for keywords related to the advertiser s industry as controls . . time series of clicks to the advertiser s website . pointwise ( daily ) incremental impact of the campaign on clicks . cumulative impact of the campaign on clicks . ] as summarised in figure [ fig : apps : empirical : forecast : null ] , no significant effect was found in unaffected regions , as expected . specifically , we obtained an overall nonsignificant lift of @xmath191 in clicks with a central 95% credible interval of @xmath192 $ ] . in summary , the empirical data considered in this section showed : ( i ) a clear effect of advertising on treated regions when using randomised control regions to form the regression component , replicating previous treatment - control comparisons ( figure [ fig : apps : empirical : forecast : original ] ) ; ( ii ) notably , an equivalent finding when discarding control regions and instead using observational searches for keywords related to the advertiser s industry as covariates ( figure [ fig : apps : empirical : forecast : observational ] ) ; ( iii ) reassuringly , the absence of an effect of advertising on regions that were not targeted ( figure [ fig : apps : empirical : forecast : null ] ) . the increasing interest in evaluating the incremental impact of market interventions has been reflected by a growing literature on applied causal inference . with the present paper we are hoping to contribute to this literature by proposing a bayesian state - space model for obtaining a counterfactual prediction of market activity . we discuss the main features of this model below . in contrast to most previous schemes , the approach described here is fully bayesian , with regularizing or empirical priors for all hyperparameters . posterior inference gives rise to complete - data ( smoothing ) predictions that are only conditioned on past data in the treatment market and both past and present data in the control markets . thus , our model embraces a dynamic evolution of states and , optionally , coefficients ( departing from classical linear regression models with a fixed number of static regressors ) , and enables us to flexibly summarise posterior inferences . because closed - form posteriors for our model do not exist , we suggest a stochastic approximation to inference using mcmc . one convenient consequence of this is that we can reuse the samples from the posterior to obtain credible intervals for all summary statistics of interest . such statistics include , for example , the average absolute and relative effect caused by the intervention as well as its cumulative effect . posterior inference was implemented in c@xmath193 and r and , for all empirical data sets presented in section [ sec : apps : empirical ] , took less than 30 seconds on a standard linux machine . if the computational burden of sampling - based inference ever became prohibitive , one option would be to replace it by a variational bayesian approximation [ see , e.g. , @xcite ] . another way of using the proposed model is for power analyses . in particular , given past time series of market activity , we can define a point in the past to represent a hypothetical intervention and apply the model in the usual fashion . as a result , we obtain a measure of uncertainty about the response in the treated market after the beginning of the hypothetical intervention . this provides an estimate of what incremental effect would have been required to be outside of the 95% central interval of what would have happened in the absence of treatment . the model presented here subsumes several simpler models which , in consequence , lack important characteristics , but which may serve as alternatives should the full model appear too complex for the data at hand . one example is classical multiple linear regression . in principle , classical regression models go beyond difference - in - differences schemes in that they account for the full counterfactual trajectory . however , they are not suited for predicting stochastic processes beyond a few steps . this is because ordinary least - squares estimators disregard serial autocorrelation ; the static model structure does not allow for temporal variation in the coefficients ; and predictions ignore our posterior uncertainty about the parameters . put differently : classical multiple linear regression is a special case of the state - space model described here in which ( i ) the gaussian random walk of the local level has zero variance ; ( ii ) there is no local linear trend ; ( iii ) regression coefficients are static rather than time - varying ; ( iv ) ordinary least squares estimators are used which disregard posterior uncertainty about the parameters and may easily overfit the data . another special case of the counterfactual approach discussed in this paper is given by synthetic control estimators that are restricted to the class of convex combinations of predictor variables and do not include time - series effects such as trends and seasonality [ @xcite ] . relaxing this restriction means we can utilise predictors regardless of their scale , even if they are negatively correlated with the outcome series of the treated unit . other special cases include autoregressive ( ar ) and moving - average ( ma ) models . these models define autocorrelation among observations rather than latent states , thus precluding the ability to distinguish between state noise and observation noise [ @xcite ] . in the scenarios we consider , advertising is a planned perturbation of the market . this generally makes it easier to obtain plausible causal inferences than in genuinely _ observational _ studies in which the experimenter had no control about treatment [ see discussions in @xcite , robinson , mcnulty and krasno ( @xcite ) , @xcite ] . the principal problem in observational studies is endogeneity : the possibility that the observed outcome might not be the result of the treatment but of other omitted , endogenous variables . in principle , propensity scores can be used to correct for the selection bias that arises when the treatment effect is correlated with the likelihood of being treated [ @xcite ] . however , the propensity - score approach requires that exposure can be measured at the individual level , and it , too , does not guarantee valid inferences , for example , in the presence of a specific type of selection bias recently termed `` activity bias '' [ @xcite ] . counterfactual modelling approaches avoid these issues when it can be assumed that the treatment market was chosen at random . overall , we expect inferences on the causal impact of designed market interventions to play an increasingly prominent role in providing quantitative accounts of return on investment [ @xcite ] . this is because marketing resources , specifically , can only be allocated to whichever campaign elements jointly provide the greatest return on ad spend ( roas ) if we understand the causal effects of spend on sales , product adoption or user engagement . at the same time , our approach could be used for many other applications involving causal inference . examples include problems found in economics , epidemiology , biology or the political and social sciences . with the release of the causalimpact r package we hope to provide a simple framework serving all of these areas . structural time - series models are being used in an increasing number of applications at google , and we anticipate that they will prove equally useful in many analysis efforts elsewhere . the authors wish to thank jon vaver for sharing the empirical data analysed in this paper .

inflation in the early universe @xcite is now an indispensable ingredient of modern cosmology not only to explain the global properties of homogeneous and isotropic space with a vanishingly small spatial curvature but also to account for the origin of the primordial curvature perturbation that seeded cosmic structure formation @xcite . at present , despite the significant progress in the state - of - the - art precise measurements of the cosmic microwave background radiation ( cmb ) by wmap @xcite and planck @xcite missions , there is no single observational result in conflict with the single - field inflation paradigm @xcite . in particular , the anti - correlation of the temperature and the e - mode polarization anisotropies on large scales observed by the wmap mission strongly supports the superhorizon perturbations suggested by inflation @xcite . in other words , once inflation sets in , virtually all the available cosmological observation data can be explained simultaneously irrespective of the initial condition of the universe this does not mean that we may be indifferent to the initial condition of the universe before inflation . on the contrary , in order to achieve complete understanding of the cosmic history , we must work out the very beginning of the universe that may smoothly evolve into the inflationary phase . as is well known , as long as the null energy condition ( nec ) is satisfied in the expanding phase , the hubble parameter and the energy density of the universe increase backward in cosmic time . so , it is often claimed that , if one tries to discuss what happened before inflation and/or how inflation started , one needs to know the information of very high energy physics , and challenge the initial singularity problem @xcite in terms of quantum gravity . but , this is not always the case . recently , it was recognized that , if an action includes higher derivative terms of a scalar field like the galileon terms , the nec can be violated without encountering ghost nor gradient instabilities . see , e.g. , ref . @xcite for a recent review and ref . @xcite for a subtle issue of nonlinear instabilities . if the nec is violated , the energy density can grow as time proceeds , contrary to the conventional wisdom . in the nec violating theories , the universe can therefore start from the static zero - energy state described by the minkowski spacetime from infinite past @xcite , and the universe starts expansion with the increase of the energy density . such a picture of the emergence of the universe was first proposed by creminelli _ et al . _ @xcite with the name galilean genesis . in their model , however , the hot big bang state was postulated to be realized after the effective field theory description breaks down as the energy density blows up beyond its realm of validity . therefore , the theory to describe the most important epoch of the early universe is lacking there . nevertheless , since their original idea is so interesting that a number of extension has been made in a wider class of scalar field theories @xcite and various aspects of the genesis scenario have been explored in the literature @xcite , such as avoidance of the superluminal propagation of perturbations and absence of primordial tensor perturbations . they have been unsuccessful , however , to realize transition from the genesis phase to the hot big bang state within their model lagrangians . in this paper , we take a different approach , namely , to make use of the galilean genesis to explain the initial condition of the universe before inflation and smoothly connect it to the inflationary phase , thereby solving the initial singularity problem @xcite and the trans - planckian problem @xcite ( see also @xcite ) in inflationary cosmology . in fact , such an approach has also been put forward by pirtskhalava _ et al . _ @xcite recently . their model lagrangian , however , gives rise to gradient instability as it is , although it has been argued there that higher - order structure of the effective field theory for perturbations possesses enough freedom to cure the gradient instability . discussion on termination of inflation and reheating is not presernted there , either . in the present paper , we construct a specific model free from any catastrophic instabilities and with subluminal velocities of primordial perturbations . in our setup the universe starts from the minkowski spacetime from infinite past and is smoothly connected to the inflationary phase followed by the graceful exit . for this purpose , we provide a generic lagrangian capable of describing the background and perturbation evolution in all the above phases instead of choosing the effective field theory approach because the latter can not capture the evolution of the background and perturbations from pre - inflationary genesis to the exit from inflation with the same single lagrangian . although we start with asymptotically minkowski space at the past infinity for aesthetic beauty , it has been shown that the galilean genesis solution is an attractor for a variety of initial conditions including those with a negative hubble parameter and/or finite curvature , provided that the time derivative of the scalar field has the right sign @xcite . the horndeski theory @xcite or the generalized galileon @xcite , whose mutual equivalence was first shown in @xcite , is known to be the most general scalar - tensor tensor theory with the second - order field equations , and thereby avoid ostrogradski instabilities in spite of having higher derivative terms in the action . the theory can be generalized to have second - order field equations only in a specific gauge while maintaining the number of propagating degrees of freedom . this possibility was realized recently by gleyzes _ et al . _ @xcite ( see also ref . @xcite ) and was extended further by gao @xcite . the number of propagating degrees of freedom in these theories is indeed shown to be the same as that of the horndeski theory @xcite . in this paper , we use the subclass of gao s framework as a concrete realization of the unified scenario starting from galilean genesis through inflation to the graceful exit . this paper is organized as follows . in the next section , we give a framework of our model and derive the background equations of motion and the quadratic actions of cosmological perturbations . in sec . iii , a concrete lagrangian is constructed to describe our scenario beginning from the genesis phase through the inflationary one to the graceful exit , and such a background dynamics is presented explicitly . in sec . iv , we discuss the stability during each phase based on the quadratic actions of cosmological perturbations . in sec . v , a concrete realization of our scenario is given . the final section is devoted to our conclusions and discussion . let us start with describing the general framework to construct and study our explicit realization of the early - time completion of inflation . we would like to consider theories composed of a metric @xmath0 and a single scalar field @xmath1 , and hence it will be appropriate to work in the horndeski theory . the lagrangian of the horndeski theory is of the form @xmath2,\label{horndeski - l}\end{aligned}\ ] ] where @xmath3 , @xmath4 is the four - dimensional ricci scalar , and @xmath5 is the four - dimensional einstein tensor . we have four arbitrary functions of @xmath1 and @xmath6 in the horndeski theory . this is the most general lagrangian having second - order field equations . nevertheless , it will turn out that this framework is insufficient for our purpose , and hence we have to go beyond the horndeski theory . one can generalize the horndeski theory to possess higher order field equations while maintaining the number of propagating degrees of freedom @xcite . the first step to do so is to perform an adm decomposition by taking @xmath7 const hypersurfaces as constant time hypersurfaces . in the adm language , the metric is written as @xmath8 by definition @xmath1 is a function of only @xmath9 , @xmath10 , and @xmath11 , where a dot denotes differentiation with respect to @xmath9 , so any function of @xmath1 and @xmath6 can be regarded as a function of @xmath9 and the lapse function @xmath12 , provided that @xmath13 and @xmath14 never vanish . then , the horndeski lagrangian ( [ horndeski - l ] ) can be written in terms of the adm variables as @xmath15 with @xmath16 where @xmath17 and @xmath18 are the extrinsic and intrinsic curvature tensors on the constant time hypersurfaces , and @xmath19 , @xmath20 , @xmath21 , and @xmath22 are subject to the relations @xmath23 variation of the above lagrangian with respect to @xmath12 gives a second - class constraint that eliminates only one degree of freedom , as opposed to general relativity . the key trick to generalize the horndeski theory is to notice that this property remains the same even if one liberates @xmath19 and @xmath20 from the restriction imposed by eq . ( [ ab - constraint ] ) @xcite . we thus arrive at the so called glpv theory that is more general than horndeski but has the same number of propagating degrees of freedom . one can move back to a covariant form of the lagrangian by introducing the unit normal to the constant time hypersurfaces as @xmath24 , writing the extrinsic curvature tensor in terms of @xmath25 , and using the gauss - codazzi equations . since there are six arbitrary functions of @xmath9 and @xmath12 in the adm form , the resultant covariant lagrangian has six arbitrary functions of @xmath1 and @xmath6 . the above idea has been pushed forward by gao @xcite , who proposed a unified framework to study single scalar - tensor theories beyond horndeski . one can write a general lagrangian in the adm form as @xmath26,\end{aligned}\ ] ] where the coefficients @xmath27 , @xmath28 , ... are arbitrary functions of @xmath9 and @xmath12 . the hamiltonian depends nonlinearly on @xmath12 as in the glpv theory , giving rise to a single scalar degree of freedom on top of the traceless and transverse gravitons @xcite . in this paper , we will employ the lagrangian @xmath29 with @xmath30 where @xmath31 , @xmath32 , and @xmath33 are constant parameters of the theory . this is a deformation of the glpv lagrangian and belongs to a subclass of gao s framework . the generalization to this level is sufficient for the purpose of the present paper . the glpv theory is recovered by taking @xmath34 . given the lagrangian ( [ ourlag ] ) in the adm form , one can restore the scalar degree of freedom @xmath1 to write its covariant expression in the same way as in the glpv theory . however , it will be more convenient for our purpose to use the explicitly time - dependent lagrangian , because by doing so one can easily design the lagrangian so as to admit the desired cosmological evolution . before specifying the suitable form of @xmath35 , @xmath36 , ... to construct our early universe model , let us derive the general equations governing the background and perturbation dynamics of cosmologies based on the lagrangian ( [ ourlag ] ) . the adm variables are given by @xmath37 where @xmath38 is the curvature perturbation in the unitary gauge and @xmath39 is the transverse and traceless tensor perturbation . a spatially flat background has been assumed and the spatial diffeomorphism invariance was used to write @xmath40 in the above form . in the following , the background value of the lapse function is denoted by @xmath12 where there is no worry about confusion . substituting eq . ( [ adm - cosmological ] ) to the lagrangian ( [ ourlag ] ) , we obtain the background part of the lagrangian as @xmath41 where @xmath42 , @xmath43 , and @xmath44 . at the background level , @xmath31 , @xmath32 , and @xmath33 just rescale @xmath19 and @xmath20 . in what follows we simply consider the case with @xmath45 . since we are considering a spatially flat universe , we have @xmath46 at zeroth order , and hence @xmath21 and @xmath22 play no role in the background dynamics . varying eq . ( [ zero - bg ] ) with respect to @xmath12 and @xmath47 , we obtain , respectively , @xmath48 where a prime represents differentiation with respect to @xmath12 . the background equations contain at most second derivatives of the scale factor and first derivatives of the lapse function . the quadratic lagrangian for the tensor perturbation is given by @xmath49,\end{aligned}\ ] ] where @xmath50 the equation of motion contains at most second derivatives both in time and space . the tensor perturbation is stable provided that @xmath51 and @xmath52 . the quadratic lagrangian for the scalar perturbations is given by @xmath53,\label{lags2}\end{aligned}\ ] ] where the coefficients are defined as @xmath54 and note the relation @xmath55 . one has @xmath56 in the horndeski and glpv theories , in which @xmath34 . therefore , the last term in the lagrangian ( [ lags2 ] ) is the novel consequence of theories beyond glpv . from @xmath57 and @xmath58 we obtain @xmath59,\label{eldeltan } \\ \frac{\partial^2\chi}{a^2 } & = & \frac{1}{\theta^2+\sigma{\cal c}}\left [ ( 3\theta^2+\sigma{\cal g}_b)\frac{\dot\zeta}{n } -\theta{\cal g}_b\frac{\partial^2\zeta}{a^2 } \right].\label{elchi}\end{aligned}\ ] ] substituting eqs . ( [ eldeltan ] ) and ( [ elchi ] ) into eq . ( [ lags2 ] ) , we obtain the reduced lagrangian for the curvature perturbation , @xmath60,\end{aligned}\ ] ] where @xmath61 thus , if @xmath62 , the equation of motion for @xmath38 has the fourth derivative in space , giving the dispersion relation @xmath63 we require that @xmath64 in order to avoid ghost instabilities . however , we allow for a negative sound speed squared , @xmath65 , for a short period of time . in the absence of the @xmath66 term ( @xmath56 ) , a negative sound speed squared would cause a rapid growth of instabilities for large @xmath67 modes . in this paper , we consider theories with @xmath62 , so that the curvature perturbation with large @xmath67 can be stabilized by requiring that @xmath68 . as will be seen in the rest of the paper , the sound speed squared becomes negative at the transition from one phase to another . such a behavior should not occur even for a tiny period because high wavenumber modes would grow exponentially rapidly . however , we could not avoid it not only within the horndesky theory but also the glpv theory despite we analyzed extensive models . on the other hand , we have not been successful in proving that this is an inevitable consequence . since our primary purpose is to show an existence proof of the model to realize our intended cosmic evolution without any instabilities , we construct a specific model by going beyond the glpv theory and invoking the @xmath66 term . the lagrangian we study in this paper is characterized by a single time - dependent function @xmath69 and four functions @xmath70 of @xmath12 : @xmath71 where @xmath72 is a constant parameter . we have introduced the mass scales @xmath73 ( and the planck mass @xmath74 ) , so that @xmath69 and @xmath75 are dimensionless . the other two functions , @xmath21 and @xmath22 , are arbitrary at this stage because they have no impacts on the background dynamics . note that @xmath76 is not a dynamical variable . specifying the functions @xmath77 and @xmath78 amounts to defining a concrete theory . the above forms of @xmath79 are chosen so that the theory admits an inflationary universe preceded by the generalized galilean genesis while retaining much of the generality . other choices could be possible and hence we do not claim that this is the most general description of such scenarios at all . instead , as we mentioned above , we would provide the existence proof of desired models by demonstrating that a sufficiently wide class of healthy models can indeed be constructed . we design @xmath69 so as to implement the ( generalized ) galilean genesis followed by inflation and a graceful exit from the prolonged inflationary phase . our choice is @xmath80 well before @xmath81 , and @xmath82 for @xmath83 . as our time variable starts at @xmath84 with asymptotically minkowski spacetime configuration , @xmath9 is large and negative in the beginning , so we find @xmath85 in eq . ( [ f : gen ] ) . as will be seen shortly , the initial stage described by eq . ( [ f : gen ] ) corresponds to the generalized galilean genesis , while the subsequent stage described by eq . ( [ f : ds ] ) to inflation . after a sufficiently long period of the inflationary stage , we assume that @xmath86 for @xmath87 , where @xmath88 is the time at the end of inflation . with this the universe exits from inflation . in what follows we will investigate the background evolution of each stage . assuming that @xmath89 in the first stage where @xmath76 is given by eq . ( [ f : gen ] ) , let us look for a consistent solution for large @xmath76 . the background field equations read @xmath90 it can be seen from eq . ( [ fr : gen ] ) that the lapse function @xmath12 is a constant , @xmath91 , satisfying @xmath92 then , @xmath93 is consistently determined from eq . ( [ pres : gen ] ) , which can be written as @xmath94 where @xmath95 is a constant . this leads to the generalized galilean genesis solution @xcite : @xmath96 it is required that @xmath97 to guarantee @xmath98 . we have thus arrived at the generalized galilean genesis solution starting from the lagrangian written in the adm form rather than in the covariant form . the original galilean genesis solution found in ref . @xcite corresponds to @xmath99 . in deriving the above solution , @xmath100 and @xmath101 are always subdominant due to the assumed scalings @xmath102 and @xmath103 . therefore , any choices of @xmath104 and @xmath105 will not spoil the above galilean genesis solution . as will be seen in the next section , those two terms are also irrelevant to the stability conditions during the genesis phase . the galilean genesis phase will end at @xmath106 since the function @xmath76 is constant for @xmath83 . in the subsequent phase we obtain the de sitter solution , @xmath107 const and @xmath108 const , satisfying @xmath109 ( note that @xmath79 is now a function of @xmath12 only and is independent of @xmath9 . ) a @xmath9-independent lagrangian in the adm form can be recast in a covariant lagrangian with the shift symmetry , @xmath110 . this implies that the above exact de sitter solution corresponds to kinetically driven g - inflation . if one invokes a weak time - dependence in @xmath76 , one obtains quasi - de sitter inflation instead . after the prolonged phase of inflation , @xmath76 is given by eq . ( [ f : exit ] ) . we assume that @xmath9 is sufficiently large , so that @xmath85 . then , we have a consistent solution with @xmath111 const and @xmath112 satisfying @xmath113 thus , one can implement a graceful exit from inflation . it follows from eq . ( [ exit : eq1 ] ) that @xmath114 it can be shown using eqs . ( [ exit : eq1 ] ) and ( [ exit : eq2 ] ) that , during this third stage , @xmath115 it is therefore necessary to impose @xmath116 . in the standard potential - driven inflation models @xcite inflation is followed by coherent field oscillation of the inflaton scalar field which decays to radiation to reheat the universe . in the present approach the scalar field @xmath1 is used to specify constant time hypersurfaces , so that @xmath13 may not vanish in order to preserve one - to - one correspondence between @xmath1 and the cosmic time @xmath9 . hence one must switch from the adm language we used to construct the action to the conventional `` @xmath1 language '' at this point in order to apply the standard reheating mechanism , which is all right but looks like sewing a fox s skin to the lion s . here instead we consider another reheating mechanism which can take place without breaking the one - to - one correspondence between @xmath1 and @xmath9 , namely , the gravitational reheating due to the change of geometry or the cosmic expansion law @xcite . during the transition from the de sitter inflation to a decelerated power - law expansion , conformally non - invariant particles are produced with the initial energy density @xmath117 where @xmath118 is a factor determined by the effective number of conformally noninvariant fields and the change of the geometry . for example , for @xmath119 or 4 , a single minimally coupled massless scalar field contributes to @xmath118 by @xmath120 respectively @xcite . here @xmath121 is the time required for the transition . in case it is nonminimally coupled with a coupling parameter @xmath122 , a factor @xmath123 is multiplied there . in order for the radiation thus created to dominate the universe , the energy density of the scalar field must dissipate more rapidly , namely , @xmath124 then , the reheating temperature at the radiation domination is given by @xmath125 where @xmath126 is the effective number of relativistic degrees of freedom and we have assumed the universe would evolve in the same way as in the einstein gravity after inflation . if long - lived massive particles are copiously produced at the gravitational particle production , the reheating temperature may be significantly higher then the above value . furthermore , the decay of quasi - flat direction may produce a large amount of entropy to reheat the universe efficiently and create matter particles @xcite . at this point one may wonder if standard particle production due to the decay of the inflaton might work in our scenario . in such a reheating process , the inflaton decay ( and the associated energy loss of the inflaton ) must be very efficient in order to avoid the dominance of the inflaton energy density in comparison to that of produced radiation . for this reason the inflaton must oscillate after inflation . however , in our construction of the action in the adm language , the field @xmath1 is used to specify constant time hypersurfaces and so @xmath1 and the cosmic time @xmath9 have a one - to - one correspondence . therefore , our approach can not describe the oscillatory phase of the inflaton and we need to resort to gravitational reheating instead of the standard reheating scenario due to the inflaton decay . we expect that a different action based on the @xmath1 language instead of the adm language can accommodate the oscillatory phase as well as the genesis and inflationary phases . minimum modification would be such that one moves to the @xmath1 language at the end of inflation to construct a @xmath1-lagrangian that admits the standard reheating scenario after inflation . however , this is beyond the scope of the present paper and is left for future work . having obtained the background evolution of our scenario , let us investigate the nature of primordial perturbations and stability , using the result of the generic analysis in sec . [ sec : cp ] . during the genesis phase , we have @xmath127 obviously , the kinetic term of the tensor perturbations has the right sign , @xmath51 . for large @xmath76 , we see @xmath128 ( as long as @xmath62 ) , and hence @xmath129 this implies that @xmath130 const , while @xmath131 . the kinetic term of the curvature perturbation has the right sign if @xmath132 thus , it is sufficient to impose @xmath133 ( we are considering only the case with @xmath134 . ) another stability condition , @xmath135 , is equivalent to requiring that @xmath136 since @xmath137 depends on @xmath138 and @xmath139 and these two functions are irrelevant to the background dynamics , the condition @xmath140 can easily be satisfied without spoiling the genesis background . suppose for simplicity that @xmath141 where @xmath142 is a constant . then , @xmath143 . for the scalar perturbations we have @xmath144 \nonumber\\ & = & { \rm const.}\end{aligned}\ ] ] this can also be made positive by an appropriate choice of @xmath145 . it should be noted that if @xmath146 then we inevitably have @xmath147 ; the @xmath148 term is crucial for the stable violation of the nec . note also that , if we take sufficiently small @xmath149 , the sound speed @xmath150 can be smaller than unity , which applies also to the other two phases discussed below . let us move to discuss the nature of the primordial fluctuations in the genesis phase . since @xmath151 const , the tensor perturbations behave in the same way as in the minkowski spacetime . therefore , no large tensor modes are generated during the first stage of our scenario . plane with @xmath152 decreasing toward the right . in the region below ( above ) the red broken curve , @xmath153 is dominated by the term proportional to @xmath66 @xmath154 . modes with @xmath155 experience the break down of the wkb approximation around the point crossing the blue solid curve beyond which @xmath38 is frozen , while modes with @xmath156 do not . , height=415 ] the behavior of the curvature perturbation turns out to be more nontrivial , as sketched in fig . [ fig : horizon.eps ] . recalling that @xmath157 const , @xmath158 const , and @xmath159 , the equation of motion for @xmath38 in the fourier space is of the form @xmath160 where @xmath161 and @xmath162 with @xmath163 and @xmath164 being some constants . for sufficiently large @xmath152 , we have @xmath165 . one may define the time at which this approximation breaks down as @xmath166 , and for @xmath167 we have @xmath168 . with some manipulation , it is found that @xmath169 where @xmath170 . this implies that for the modes with @xmath171 the wkb approximation is always good in the genesis phase , @xmath172 giving @xmath173 for @xmath167 . thus , the amplitude of those modes at late times in the genesis phase is given by @xmath174 for the modes with @xmath175 , the wkb approximation breaks down at some time and then the curvature perturbation freezes . this `` horizon crossing '' occurs at @xmath176 . it can be seen that @xmath177 for @xmath175 , for the @xmath178 mode . the genesis phase could end sufficiently early so that @xmath179 . if this is the case , we only need to care about the modes with @xmath175 . ] which allows us to study the freezing process by using the solution to eq . ( [ zetaeqgen ] ) with @xmath180 . the exact solution in this case that matches the positive frequency wkb solution for @xmath181 is given by @xmath182 where @xmath183 is the hankel function of the first kind . the frozen amplitude can thus be evaluated by taking the limit @xmath184 in the solution ( [ smallksol ] ) , leading to @xmath185 for @xmath186 , @xmath187 is dominated by the @xmath188 term where the solution ( [ smallksol ] ) is no longer exact . the frozen amplitude ( [ frozen ] ) , however , is still valid even in this regime since the solution to eq . ( [ zetaeqgen ] ) with the effective frequency ( [ omega ] ) does not oscillate any more and remains constant . hence , the expression of the power spectrum ( [ frozen ] ) is correct for the entire range of @xmath189 . to summarize , the power spectrum of the curvature perturbation generated during the genesis phase is blue and hence is suppressed on large scales . in the ( de sitter ) inflationary phase , @xmath190 , @xmath191 , @xmath192 , @xmath193 , and @xmath194 are time - independent . we require that all those coefficients are positive during inflation in order to avoid instabilities . since the quadratic action for the tensor perturbations is essentially the same as that of generalized g - inflation , the power spectrum of the primordial tensor perturbations is given by @xcite @xmath195 the equation of motion for the canonically normalized variable @xmath196 during inflation is of the form @xmath197 where @xmath198 with @xmath199 and @xmath200 being dimensionless constants . here , we have introduced the conformal time @xmath201 defined by @xmath202 . the dispersion relations of this form have been studied in the context of inflation , e.g. , in refs . the positive frequency modes are given by @xcite @xmath203 where @xmath204 is the whittaker function . taking the limit @xmath205 , the power spectrum of the curvature perturbation can be calculated as @xmath206 where @xmath207 even in the presence of the @xmath66 term in the dispersion relation , the power spectrum is scale - invariant in the case of exact de sitter inflation . since we have @xmath208 as @xmath209 , we recover the result of generalized g - inflation @xcite in the limit @xmath210 . for @xmath211 we have @xmath212 , so that one can take the limit @xmath213 smoothly to get @xmath214 we have approximated the inflationary phase as exact de sitter . if we consider a background slightly different from de sitter by incorporating weak time dependence in @xmath76 , we would be able to obtain a tilted spectrum of @xmath38 . after inflation , we have @xmath215 , @xmath216 , @xmath217 where to simplify the expression we introduced @xmath218 recalling that we have been imposing @xmath219 , all of these coefficients are positive provided that @xmath220 . this condition can be written equivalently as @xmath221 and ( b ) the lapse function @xmath12 around the genesis - de sitter transition . , height=377 ] , and ( b ) the coefficient of @xmath222 ( divided by @xmath223 ) around the genesis - de sitter transition . , height=377 ] let us provide a concrete lagrangian exhibiting the genesis - de sitter transition . the lagrangian is characterized by @xmath224 where @xmath225 and @xmath226 are constants . we take @xmath227 , @xmath228 , and @xmath229 . we also take @xmath230 to guarantee the stability . this corresponds to the ( @xmath219 generalization of the ) unitary gauge description of the lagrangian considered in ref . @xcite . in the genesis stage we have @xmath231<0.\end{aligned}\ ] ] since @xmath230 and @xmath232 , we see that @xmath233 and @xmath234 . we also see that @xmath235 -1,\end{aligned}\ ] ] and hence it is easy to satisfy @xmath236 during the genesis phase by choosing the parameters appropriately . a numerical example of the genesis - de sitter transition is illustrated in figs . [ fig : hn.eps ] and [ fig : stability.eps ] . our numerical calculation was performed as follows : we solve the evolution equations @xmath237 and @xmath238 with initial data @xmath239 satisfying @xmath240 , and confirm that the constraint @xmath240 is satisfied at each time step . in the numerical calculation , the parameters are given by @xmath241 , @xmath242 , @xmath243 , @xmath244 , and @xmath245 . the function @xmath69 is taken to be @xmath246+f_{1},\label{f : ex1}\end{aligned}\ ] ] with @xmath247 , @xmath248 , and @xmath249 . the background evolution is shown in fig . [ fig : hn.eps ] . the evolution of the sound speed squared , @xmath250 , and the coefficient of @xmath222 in the dispersion relation is shown in fig . [ fig : stability.eps ] . as pointed out in ref . @xcite , @xmath251 flips the sign at the transition . the sound speed squared is positive except in this finite period . during the genesis and subsequent de sitter phases we have @xmath64 and @xmath135 , and therefore we may conclude that this model is stable . and ( b ) the lapse function @xmath12 around the end of inflation . height=377 ] , and ( b ) the coefficient of @xmath222 ( divided by @xmath223 ) around the end of inflation . , height=377 ] although we have thus obtained the stable example of the genesis - de sitter transition , the simple example ( [ example ] ) is not completely satisfactory if one would want successful gravitational reheating . indeed , the condition ( [ exit : c1 ] ) implies that @xmath252 , but @xmath253 for such @xmath254 . this problem can be evaded easily by the following small deformation of @xmath255 : @xmath256 where @xmath257 is a parameter smaller than @xmath258 . the condition ( [ exit : c1 ] ) now reads @xmath259 , i.e. , @xmath260 , while @xmath261 is positive for @xmath262 . the stability condition further restricts the allowed ranges of @xmath254 and @xmath257 . the necessary condition for stability is @xmath263 [ see eq . ( [ ge : st ] ) ] . this translates to @xmath264 , leading to @xmath265 = 4.8 . note that the small deformation of @xmath255 with @xmath266 does not change the background and perturbation dynamics of the genesis and inflationary phases . to illustrate the final stage of inflation , let us take @xmath267\right\}^{1/(\alpha + 1)},\label{f : ex2}\end{aligned}\ ] ] where the origin of time is shifted so that the end of inflation is given by @xmath268 . in the numerical plots presented in figs . [ fig : hn - ds - r.eps ] and [ fig : stability - ds - r.eps ] , the parameters are given by @xmath269 , @xmath270 , and @xmath271 , while the other parameters are taken to be the same as the previous example of the genesis - de sitter transition . it is found that @xmath272 . again , we see that @xmath273 in the finite period around the transition . however , @xmath192 and @xmath194 remain positive all through the inflation and subsequent stages . in this paper , we have introduced a generic description of galilean genesis in terms of the adm lagrangian and constructed a concrete realization of inflation preceded by galilean genesis , _ i.e. _ , the scenario in which the universe starts from minkowski spacetime in the asymptotic past and is connected smoothly to the inflationary phase followed by the graceful exit . our model utilizes the recent extension of the horndeski theory , which has the same number of propagating degrees of freedom as the horndeski theory and thus can avoid ostrogradski instabilities . this approach allows us to cover the background and perturbation evolution in all the three phases with the same single lagrangian , as opposed to the effective field theory approach . in our scenario , the sound speed squared during the transition from the genesis phase to inflation becomes negative for a short period . however , thanks to the nonlinear dispersion relation arising from the fourth - order derivative term in the quadratic action , modes with higher momenta are stable and the growth rate of perturbations with smaller momenta is finite and under control . it should also be noted that the sound speed of the primordial perturbations can be smaller than unity by choosing the parameter of the model appropriately . although we have constructed our inflation model in order to resolve the initial singularity and possible trans - planckian problems by incorporating galilean genesis phase before inflation , we could make use of our model to realize the original galilean genesis scenario , which is an alternative to inflationary cosmology , simply by taking vanishingly short period of inflation there . as discussed in the appendix , the sound speed squared becomes negative at the transition also in this case , but the instabilities are relevant only for small @xmath67 modes thanks to the @xmath66 term in the dispersion relation . thus , the transition from the genesis phase to the reheating stage is described in a healthy and controllable manner . in fact , it would be fair to say that such a cosmology works quite well among the proposed alternatives to inflation , because , in contrast with the bouncing cosmology , in which all the would - be decaying modes in the expanding universe such as vector fluctuations and spatial anisotropy severely increase in an undesirable manner , the genesis solution is an attractor and generation of nearly scale - invariant curvature perturbation is also possible with an appropriate choice of model parameters @xcite . since no first - order tensor perturbation is generated in this type of scenarios , detection of tensor perturbation with its amplitude larger than @xmath274 would be a smoking gun of inflation . this work was supported in part by the jsps grant - in - aid for scientific research nos . 24740161 ( t.k . ) , 25287054 and 26610062 ( m.y . ) , 23340058 and 15h02082 ( j.y . ) . and ( b ) the lapse function @xmath12 around the genesis - reheating transition . , height=377 ] in the main text , we consider the scenario in which galilean genesis is followed by inflation . in this appendix , we will go back to the original motivation of galilean genesis and study how we can match smoothly the genesis phase to the reheating phase . our approach based on the adm lagrangian is quite useful in analyzing such a situation as well . it is now obvious that by taking @xmath275 and gluing the two functions smoothly at around @xmath276 , one can describe the genesis - reheating transition . as a concrete example , we glue @xmath277 and @xmath278 smoothly at around @xmath276 and perform a numerical calculation as shown in figs . [ fig : hn - g - r.eps ] and [ fig : stability - g - r.eps ] . the other parameters are the same as those taken in the main text . as is expected , the numerical result here is much the same as the case where a duration of the intermediate inflationary phase is taken to be very short . in particular , @xmath279 becomes negative at the genesis - reheating transition . the model is nevertheless stable since the conditions @xmath64 and @xmath135 remain satisfied . 99 a. a. starobinsky , `` a new type of isotropic cosmological models without singularity , '' phys . b * 91 * , 99 ( 1980 ) . k. sato , `` first order phase transition of a vacuum and expansion of the universe , '' mon . not . soc . * 195 * , 467 ( 1981 ) . a. h. guth , `` the inflationary universe : a possible solution to the horizon and flatness problems , '' phys . d * 23 * , 347 ( 1981 ) . a. d. linde , `` a new inflationary universe scenario : a possible solution of the horizon , flatness , homogeneity , isotropy and primordial monopole problems , '' phys . b * 108 * , 389 ( 1982 ) . for a review of inflation , see , e.g. , j. yokoyama , `` inflation : 1980 - 201x , '' ptep * 2014 * , no . 6 , 06b103 ( 2014 ) . v. f. mukhanov and g. v. chibisov , `` quantum fluctuation and nonsingular universe . 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the computational effort needed to deal with large combinatorial structures considerably varies with the task to be performed and the resolution procedure used@xcite . the worst case complexity of a task , more precisely an optimization or decision problem , is defined as the time required by the best algorithm to treat any possible inputs to the problem . for instance , the sorting problem of a list of @xmath0 numbers has worst - case complexity @xmath1 : there exists several algorithms that can order any list in at most @xmath2 elementary operations , and none with asymptotically less operations . unfortunately , the worst - case complexities of many important computational problems , called np - complete , is not known . partitioning a list of @xmath0 numbers in two sets with equal partial sums is one among hundreds of such np - complete problems . it is a fundamental conjecture of theoretical computer science that there exists no algorithm capable of partitioning any list of length @xmath0 , or of solving any other np - complete problem with inputs of size @xmath0 , in a time bounded by a polynomial of @xmath0 . therefore , when dealing with such a problem , one necessarily uses algorithms which may takes exponential times on some inputs . quantifying how ` frequent ' these hard inputs are for a given algorithm is the question answered by the analysis of algorithms . in this paper , we will present an overview of recent works done by physicists to address this point , and more precisely to characterize the average performances , called hereafter complexity , of a given algorithm over a distribution of inputs to an optimization problem . the history of algorithm analysis by physical methods / ideas is at least as old as the use of computers by physicists . one well - established chapter in this history is , for instance , the analysis of monte carlo sampling algorithms for statistical mechanics models . in this context , it is well known that phase transitions , _ i.e. _ abrupt changes in the physical properties of the model , can imply a dramatic increase in the time necessary to the sampling procedure . this phenomenon is commonly known as critical slowing down . the physicists insight in this problem comes mainly from the analogy between the dynamics of algorithms and the physical dynamics of the system . this analogy is quite natural : in fact many algorithms mimick the physical dynamics itself . a quite new idea is instead to abstract from physically motivated problems and use statistical mechanics ideas for analyzing the dynamics of algorithms . in effect there are many reasons which suggest that analysis of algorithms and statistical physics should be considered close relatives . in both cases one would like to understand the asymptotic behavior of dynamical processes acting on exponentially large ( in the size of the problem ) configuration spaces . the differences between the two disciplines mainly lie in the methods ( and , we are tempted to say , the style ) of investigation . theoretical computer science derives rigorous results based on probability theory . however these results are sometimes too weak for a complete characterization of the algorithm . physicists provide instead heuristic results based on intuitively sensible approximations . these approximations are eventually validated by a comparison with numerical experiments . in some lucky cases , approximations are asymptotically irrelevant : estimates are turned into conjectures left for future rigorous derivations . perhaps more interesting than stylistic differences is the _ point of view _ which physics brings with itself . let us highlight two consequences of this point of view . first , a particular importance is attributed to `` complexity phase transitions '' _ i.e. _ abrupt changes in the resolution complexity as some parameter defining the input distribution is varied@xcite . we shall consider two examples in the next sections : * random satisfiability of boolean constraints ( sat ) . in @xmath3-sat one is given an instance , that is , a set of @xmath4 logical constraints ( clauses ) among @xmath0 boolean variables , and wants to find a truth assignment for the variables which fulfill all the constraints . each clause is the logical or of @xmath3 literals , a literal being one of the @xmath0 variables or its negation e.g. @xmath5 for 3-sat . random @xmath3-sat is the @xmath3-sat problem supplied with a distribution of inputs uniform over all instances having fixed values of @xmath0 and @xmath4 . the limit of interest is @xmath6 at fixed ratio @xmath7 of clauses per variable@xcite . * vertex cover of random graphs ( vc ) . an input instance of the vc decision problem consists in a graph @xmath8 and an integer number @xmath9 . the problem consists in finding a way to distribute @xmath9 covering marks over the vertices in such a way that every edge of the graph is covered , that is , has at least one of its ending vertices marked . a possible distribution of inputs is provided by drawing random graphs @xmath8 _ la _ erds - reny _ i.e. _ with uniform probability among all the graphs having @xmath0 vertices and @xmath10 edges . the limit of interest is @xmath11 at fixed ratio @xmath12 of edges per vertex . the algorithms for random sat and vc we shall consider in the next sections undergo a complexity phase transition as the input parameter @xmath13 ( @xmath14 for sat , @xmath15 for vc ) crosses some critical threshold @xmath16 . typically resolution of a randomly drawn instance requires linear time below the threshold @xmath17 and exponential time above @xmath18 . the observation that most difficult instances are located near the phase boundary confirms the relevance of the phase - transition phenomenon . secondly , a key role is played by the intrinsic ( algorithm independent ) properties of the instance under study . the intuition is that , underlying the dramatic slowing down of a particular algorithm , there can be some _ qualitative _ change in some structural property of the problem e.g. the geometry of the space of solutions . while there is no general understanding of this question , we can further specify the above statements case - by - case . let us consider , for instance , a local search algorithm for a combinatorial optimization problem . if the algorithm never increases the value of the cost function @xmath19 where @xmath20 is the configuration ( assignment ) of variables to be optimized over , the number and geometry of the local minima of @xmath19 will be crucial for the understanding of the dynamics of the algorithm . this example is illustrated in sec . [ gradxor ] . while the `` dynamical '' behavior of a particular algorithm is not necessarily related to any `` static '' property of the instance , this approach is nevertheless of great interest because it could provide us with some ` universal ' results . some properties of the instance , for example , may imply the ineffectiveness of an entire class of algorithms . while we shall mainly study in this paper the performances of search algorithms applied to hard combinatorial problems as sat , vc , we will also consider easy , that is , polynomial problems as benchmarks for these algorithms . the reason is that we want to understand if the average hardness of resolution of solving np - complete problems with a given distribution of instances and a given algorithm truly reflects the intrinsic hardness of these combinatorial problems or is simply due to some lack of efficiency of the algorithm under study . the benchmark problem we shall consider is random xorsat . it is a version of a satisfiability problem , much simpler than sat from a computational complexity point of view@xcite . the only but essential difference with sat is that a clause is said to be satisfied if the exclusive , and not inclusive , disjunction of its literals is true . xorsat may be recast as a linear algebra problem , where a set of @xmath4 equations involving @xmath0 boolean variables must be satisfied modulo 2 , and is therefore solvable in polynomial time by various methods e.g. gaussian elimination . nevertheless , it is legitimate to ask what are the performances of general search algorithms for this kind of polynomial computational problem . in particular , we shall see that some algorithms requiring exponential times to solve random sat instances behave badly on random xorsat instances too . a related question we shall focus on in sec . [ codesection ] is decoding , which may also , in some cases , be expressed as the resolution of a set of boolean equations . the paper is organized as follows . in sec . [ dpllsection ] we shall review backtracking search algorithms , which , roughly speaking , work in the space of instances . we explain the general ideas and then illustrate them on random sat ( sec . [ dpllsatsection ] ) and vc ( sec . [ dpllvcsection ] ) . in sec . [ dpllflucsection ] we consider the fluctuations in running times of these algorithms and analyze the possibility of exploiting these fluctuations in random restart strategies . in sec . [ localsection ] we turn to local search algorithms , which work in the space of configurations . we review the analysis of such algorithms for decoding problems ( sec . [ codesection ] ) , random xorsat ( sec . [ gradxor ] ) , and sat ( sec . [ walksatsection ] ) . finally in the conclusion we suggest some possible future developments in the field . in this section , we briefly review the davis - putnam - loveland - logemann ( dpll ) procedure @xcite . a decision problem can be formulated as a constrained satisfaction problem , where a set of variables must be sought for to fulfill some given constraints . for simplicity , we suppose here that variables may take a finite set of values with cardinality @xmath21 e.g. @xmath22 for sat or vc . dpll is an exhaustive search procedure operating by trials and errors , the sequence of which can be graphically represented by a search tree ( fig . [ trees ] ) . the tree is defined as follows : * ( 1 ) * a node in the tree corresponds to a choice of a variable . * ( 2 ) * an outgoing branch ( edge ) codes for the value of the variable and the logical implications of this choice upon not yet assigned variables and clauses . obviously a node gives birth to @xmath21 branches at most . * ( 3 ) * implications can lead to : * ( 3.1 ) * a violated constraint , then the branch ends with @xmath20 ( contradiction ) , the last choice is modified ( backtracking of the tree ) and the procedure goes on along a new branch ( point 2 above ) ; * ( 3.2 ) * a solution when all constraints are satisfied , then the search process is over ; * ( 3.3 ) * otherwise , some constraints remain and further assumptions on the variables have to be done ( loop back to point 1 ) . a computer independent measure of computational complexity , that is , the amount of operations necessary to solve the instance , is given by the size @xmath23 of the search tree _ i.e. _ the number of nodes it contains . performances can be improved by designing sophisticated heuristic rules for choosing variables ( point 1 ) . the resolution time ( or complexity ) is a stochastic variable depending on the instance under consideration and on the choices done by the variable assignment procedure . its average value , @xmath24 , is a function of the input distribution parameters @xmath13 e.g. the ratio @xmath25 of clauses per variable for sat , or the average degree @xmath15 for the vc of random graphs , which can be measured experimentally and that we want to calculate theoretically . more precisely , our aim is to determine the values of the input parameters for which the complexity is linear , @xmath26 or exponential , @xmath27 , in the size @xmath0 of the instance and to calculate the coefficients @xmath28 as functions of @xmath13 . the dpll algorithm gives rise to a dynamical process . indeed , the initial instance is modified during the search through the assignment of some variables and the simplification of the constraints that contain these variables . therefore , the parameters of the input distribution are modified as the algorithm runs . this dynamical process has been rigorously studied and understood in the case of a search tree reducing to one branch ( tree a in figure [ trees])@xcite . study of trees with massive backtracking e.g. trees b and c in fig . [ trees ] is much more difficult . backtracking introduces strong correlations between nodes visited by dpll at very different times , but close in the tree . in addition , the process is non markovian since instances attached to each node are memorized to allow the search to resume after a backtracking step . the study of the operation of dpll is based on the following , elementary observation . since instances are modified when treated by dpll , description of their statistical properties generally requires additional parameters with respects to the defining parameters @xmath13 of the input distribution . our task therefore consists in 1 . identifying these extra parameters @xmath29@xcite ; 2 . deriving the phase diagram of this new , extended distribution @xmath30 to identify , in the @xmath30 space , the critical surface separating instances having solution with high probability ( satisfiable phase ) from instances having generally no solution ( unsatisfiable phase ) , see fig . [ schemoins ] . 3 . tracking the evolution of an instance under resolution with time @xmath31 ( number of steps of the algorithm ) , that is , the trajectory of its characteristic parameters @xmath32 in the phase diagram . whether this trajectory remains confined to one of the two phases or crosses the boundary inbetween has dramatic consequences on the resolution complexity . we find three average behaviours , schematized on fig . [ schemoins ] : * if the initial instance has a solution and the trajectory remains in the sat phase , resolution is typically linear , and almost no backtracking is present ( fig . [ trees]a ) . the coordinates of the trajectory @xmath32 of the instance in the course of the resolution obey a set of coupled ordinary differential equations accounting for the changes in the distribution parameters done by dpll . * if the initial instance has no solution , solving the instance , that is , finding a proof of unsatisfiability , takes exponentially large time and makes use of massive backtracking ( fig . [ trees]b ) . analysis of the search tree is much more complicated than in the linear regime , and requires a partial differential equation that gives information on the population of branches with parameters @xmath30 throughout the growth of the search tree . * in some intermediary regime , instances have solutions but finding one requires an exponentially large time ( fig . [ trees]c ) . this may be related to the crossing of the boundary between sat and unsat phases of the instance trajectory . we have therefore a mixed behaviour which can be understood through combination of the two above cases . we now explain how to apply concretly this approach to the cases of random sat and vc . and @xmath29 are scalar and not vectorial parameters . vertical axis is the instance distribution defining parameter @xmath13 . instances are almost always satisfiable if @xmath33 , unsatisfiable if @xmath34 . under the action of dpll , the distribution of instances is modified and requires another parameter @xmath29 to be characterized ( horizontal axis ) , equal to , say , zero prior to any action of dpll . for non zero values of @xmath29 , the critical value of the defining parameter @xmath13 obviously changes ; the line @xmath35 defines a boundary separating typically sat from unsat instances ( bold line ) . when the instance is unsat ( point u ) , dpll takes an exponential time to go through the tree trajectory . for satisfiable and easy instances , dpll goes along a branch trajectory in a linear time ( point s ) . the mixed case of hard sat instances ( point ms ) correspond to the branch trajectory crossing the boundary separating the two phases ( bold line ) , which leads to the exploration of unsat subtrees before a solution is finally found . ] clauses involving @xmath36 variables @xmath37 , which can be assigned to true ( t ) or false ( f ) . @xmath38 means ( not @xmath39 ) and @xmath40 denotes the logical or . the search tree is empty . dpll randomly selects a clause among the shortest ones , and assigns a variable in the clause to satisfy it , e.g. @xmath41 t ( splitting with the generalized unit clause guc heuristic @xcite ) . a node and an edge symbolizing respectively the variable chosen ( @xmath39 ) and its value ( t ) are added to the tree . the logical implications of the last choice are extracted : clauses containing @xmath39 are satisfied and eliminated , clauses including @xmath38 are simplified and the remaining ones are left unchanged . if no unitary clause ( _ i.e. _ with a single variable ) is present , a new choice of variable has to be made . * * splitting takes over . another node and another edge are added to the tree . * * same as step 2 but now unitary clauses are present . the variables they contain have to be fixed accordingly . the propagation of the unitary clauses results in a contradiction . the current branch dies out and gets marked with c. * 6 . * dpll backtracks to the last split variable ( @xmath42 ) , inverts it ( f ) and creates a new edge . * * same as step 4 . the propagation of the unitary clauses eliminates all the clauses . a solution s is found and the instance is satisfiable . for an unsatisfiable instance , unsatisfiability is proven when backtracking ( see step 6 ) is not possible anymore since all split variables have already been inverted . in this case , all the nodes in the final search tree have two descendent edges and all branches terminate by a contradiction c. ] the input distribution of 3-sat is characterized by a single parameter @xmath13 , the ratio @xmath25 of clauses per variable . the action of dpll on an instance of 3-sat , illustrated in fig . [ algo ] , causes the changes of the overall numbers of variables and clauses , and thus of @xmath25 . furthermore , dpll reduces some 3-clauses to 2-clauses . we use a mixed 2+p - sat distribution@xcite , where @xmath43 is the fraction of 3-clauses , to model what remains of the input instance at a node of the search tree . using experiments and methods from statistical mechanics@xcite and rigorous calculations@xcite , the threshold line @xmath44 , separating sat from unsat phases , may be estimated with the results shown in fig . [ sche ] . for @xmath45 , _ i.e. _ left to point t , the threshold line is given by @xmath46 , and saturates the upper bound for the satisfaction of 2-clauses . above @xmath47 , no exact value for @xmath44 is known . the phase diagram of 2+p - sat is the natural space in which the dpll dynamics takes place . an input 3-sat instance with ratio @xmath25 shows up on the right vertical boundary of fig . [ sche ] as a point of coordinates @xmath48 . under the action of dpll , the representative point moves aside from the 3-sat axis and follows a trajectory in the @xmath49 plane . in this section , we show that the location of this trajectory in the phase diagram allows a precise understanding of the search tree structure and of complexity as a function of the ratio @xmath25 of the instance to be solved ( inset of fig . [ sche ] ) . in addition , we shall present an approximate calculation of trajectories accounting for the case of massive backtracking , that is for unsat instances , and slightly below the threshold in the sat phase . our approach is based on a non rigorous extension of works by chao and franco who first studied the action of dpll ( without backtracking ) on easy , sat instances@xcite as a way to obtain lower bounds to the threshold @xmath50 , see @xcite for a recent review . let us emphasize that the idea of trajectory is made possible thanks to an important statistical property of the heuristics of split we consider @xcite , * unit - clause ( uc ) heuristic : pick up randomly a literal among a unit clause if any , or any unset variable otherwise . * generalized unit - clause ( guc ) heuristic : pick up randomly a literal among the shortest avalaible clauses . * short clause with majority ( sc@xmath51 ) heuristic : pick up randomly a literal among unit clauses if any , or pick up randomly an unset variable @xmath21 , count the numbers of occurences @xmath52 of @xmath21 , @xmath53 in 3-clauses , and choose @xmath21 ( respectively @xmath53 ) if @xmath54 ( resp . @xmath55 ) . when @xmath56 , @xmath21 and @xmath53 are equally likely to be chosen . these heuristics do not induce any bias nor correlation in the instances distribution@xcite . such a statistical `` invariance '' is required to ensure that the dynamical evolution generated by dpll remains confined to the phase diagram of fig . [ sche ] . in the following , the initial ratio of clauses per variable of the instance to be solved will be denoted by @xmath57 . let us consider the first descent of the algorithm , that is the action of dpll in the absence of backtracking . the search tree is a single branch ( fig . [ trees]a ) . the numbers of 2 and 3-clauses are initially equal to @xmath58 respectively . under the action of dpll , @xmath59 and @xmath60 follow a markovian stochastic evolution process , as the depth @xmath61 along the branch ( number of assigned variables ) increases . both @xmath59 and @xmath60 are concentrated around their average values , the densities @xmath62 $ ] ( @xmath63 ) of which obey a set of coupled ordinary differential equations ( ode)@xcite , @xmath64 where @xmath65 is the probability that dpll fixes a variable at depth @xmath31 through unit - propagation . function @xmath66 depends upon the heuristic : @xmath67 , @xmath68 ( if @xmath69 ) , @xmath70 with @xmath71 and @xmath72 denotes the @xmath73 modified bessel function . to obtain the single branch trajectory in the phase diagram of fig . [ sche ] , we solve the odes ( [ ode ] ) with initial conditions @xmath74 , and perform the change of variables @xmath75 results are shown for the guc heuristics and starting ratios @xmath76 and 2.8 in fig . [ sche ] . trajectories , indicated by light dashed lines , first head to the left and then reverse to the right until reaching a point on the 3-sat axis at a small ratio . further action of dpll leads to a rapid elimination of the remaining clauses and the trajectory ends up at the right lower corner s , where a solution is found . frieze and suen @xcite have shown that , for ratios @xmath77 ( for the guc heuristics ) , the full search tree essentially reduces to a single branch , and is thus entirely described by the odes ( [ ode ] ) . the number of backtrackings necessary to reach a solution is bounded from above by a power of @xmath78 . the average size @xmath24 of the branch then scales linearly with @xmath0 with a multiplicative factor @xmath79 that can be analytically computed @xcite . the boundary @xmath80 of this easy sat region can be defined as the largest initial ratio @xmath57 such that the branch trajectory @xmath81 issued from @xmath57 never leaves the sat phase in the course of dpll resolution . for ratios above threshold ( @xmath82 ) , instances almost never have a solution but a considerable amount of backtracking is necessary before proving that clauses are incompatible . as shown in fig . [ trees]b , a generic unsat tree includes many branches . the number of branches ( leaves ) , @xmath83 , or the number of nodes , @xmath84 , grow exponentially with @xmath0@xcite . it is convenient to define its logarithm @xmath85 through @xmath86 . contrary to the previous section , the sequence of points @xmath87 characterizing the evolution of the 2+p - sat instance solved by dpll does not define a line any longer , but rather a patch , or cloud of points with a finite extension in the phase diagram of fig . [ schemoins ] . we have analytically computed the logarithm @xmath85 of the size of these patches , as a function of @xmath88 , extending to the unsat region the probabilistic analysis of dpll . this is , _ a priori _ , a very difficult task since the search tree of fig . 1b is the output of a complex , sequential process : nodes and edges are added by dpll through successive descents and backtrackings . we have imagined a different building up , that results in the same complete tree but can be mathematically analyzed : the tree grows in parallel , layer after layer ( fig . [ struct ] ) . denotes the depth in the tree , that is the number of variables assigned by dpll along each ( living ) branch . at depth @xmath61 , one literal is chosen on each branch among 1-clauses ( unit - propagation , grey circles not represented on figure 1 ) , or 2,3-clauses ( split , black circles as in figure 1 ) . if a contradiction occurs as a result of unit - propagation , the branch gets marked with c and dies out . the growth of the tree proceeds until all branches carry c leaves . the resulting tree is identical to the one built through the usual , sequential operation of dpll . ] a new layer is added by assigning , according to dpll heuristic , one more variable along each living branch . as a result , a branch may split ( case 1 ) , keep growing ( case 2 ) or carry a contradiction and die out ( case 3 ) . cases 1,2 and 3 are stochastic events , the probabilities of which depend on the characteristic parameters @xmath89 defining the 2+p - sat instance carried by the branch , and on the depth ( fraction of assigned variables ) @xmath31 in the tree . we have taken into account the correlations between the parameters @xmath90 on each of the two branches issued from splitting ( case 1 ) , but have neglected any further correlation which appear between different branches at different levels in the tree@xcite . this markovian approximation permits to write an evolution equation for the logarithm @xmath91 of the average number of branches with parameters @xmath90 as the depth @xmath31 increases , @xmath92 \qquad . \label{croi}\ ] ] @xmath93 incorporates the details of the splitting heuristics . in terms of the partial derivatives @xmath94 , @xmath95 , we have for the uc and guc heuristics @xmath96 \nonumber \\ { h } _ { guc } % ( c_2 , c_3 , y_2 , y_3 , t ) & = & \log _ 2 \nu ( y_2 ) + \frac{1}{\ln2 } \left [ \frac { 3\ , c_3}{1-t}\ ; \left ( e^{y_3 } \frac{1+e^{-y_2}}{2 } -1 \right)+ \frac{c_2}{1-t } \ ; \left ( \nu(y_2 ) -2 \right ) \right ] \nonumber \\ \hbox{\rm where } & & \nu ( y_2 ) = \frac 12\ ; e^{y_2}\left ( 1 + \sqrt{1 + 4 e^{-y_2 } } \right)\qquad .\end{aligned}\ ] ] partial differential equation ( pde ) ( [ croi ] ) is analogous to growth processes encountered in statistical physics @xcite . the surface @xmath85 , growing with `` time '' @xmath31 above the plane @xmath97 , or equivalently from ( [ change ] ) , above the plane @xmath87 ( fig . [ dome ] ) , describes the whole distribution of branches . the average number of branches at depth @xmath31 in the tree equals @xmath98 , where @xmath99 is the maximum over @xmath100 of @xmath101 reached in @xmath102 . in other words , the exponentially dominant contribution to @xmath103 comes from branches carrying 2+p - sat instances with parameters @xmath102 , which define the tree trajectories on fig . [ sche ] . the hyperbolic line in fig . [ sche ] indicates the halt points , where contradictions prevent dominant branches from further growing . each time dpll assigns a variable through unit - propagation , an average number @xmath104 of new 1-clauses is produced , resulting in a net rate of @xmath105 additional 1-clauses . as long as @xmath106 , 1-clauses are quickly eliminated and do not accumulate . conversely , if @xmath107 , 1-clauses tend to accumulate . opposite 1-clauses @xmath42 and @xmath108 are likely to appear , leading to a contradiction @xcite . the halt line is defined through @xmath109 . as far as dominant branches are concerned , the equation of the halt line reads @xmath110\;\frac 1{1-p } \simeq \frac{1.256}{1-p}\qquad .\ ] ] along the tree trajectory , @xmath99 grows from 0 , on the right vertical axis , up to some final positive value , @xmath111 , on the halt line . @xmath112 is our theoretical prediction for the logarithm of the complexity ( divided by @xmath0 ) . values of @xmath113 obtained for @xmath114 by solving equation ( [ croi ] ) compare very well with numerical results @xcite . .5 cm we have plotted the surface @xmath85 above the @xmath87 plane , with the results shown in fig . it must be stressed that , though our calculation is not rigorous , it provides a very good quantitative estimate of the complexity . furthermore , complexity is found to scale asymptotically as @xmath115 ^ 2 \simeq \frac{0.292}{\alpha _ 0 } \qquad ( \alpha _ 0 \gg \alpha _ c ) .\ ] ] this result exhibits the expected scaling@xcite , and could indeed be exact . as @xmath57 increases , search trees become smaller and smaller , and correlations between branches , weaker and weaker . the interest of the trajectory approach proposed in this paper is best seen in the upper sat phase , that is ratios @xmath57 ranging from @xmath80 to @xmath116 . this intermediate region juxtaposes branch and tree behaviors , see fig . [ trees]c . the branch trajectory starts from the point @xmath117 corresponding to the initial 3-sat instance and hits the critical line @xmath118 at some point g with coordinates ( @xmath119 ) after @xmath120 variables have been assigned by dpll ( fig . [ sche ] ) . the algorithm then enters the unsat phase and generates 2+p - sat instances with no solution . a dense subtree , that dpll has to go through entirely , forms beyond g till the halt line ( fig . [ sche ] ) . the size of this subtree , @xmath121 , can be analytically predicted from our theory . g is the highest backtracking node in the tree ( fig . [ trees]c ) reached back by dpll , since nodes above g are located in the sat phase and carry 2+p - sat instances with solutions . dpll will eventually reach a solution . the corresponding branch ( rightmost path in fig . [ trees]c ) is highly non typical and does not contribute to the complexity , since almost all branches in the search tree are described by the tree trajectory issued from g ( fig . [ sche ] ) . we have checked experimentally this scenario for @xmath122 . the coordinates of the average highest backtracking node , @xmath123 ) , coincide with the analytically computed intersection of the single branch trajectory and the critical line @xmath118@xcite . as for complexity , experimental measures of @xmath85 from 3-sat instances at @xmath124 , and of @xmath125 from 2 + 0.78-sat instances at @xmath126 , obey the expected identity @xmath127 and are in very good agreement with theory@xcite . therefore , the structure of search trees for 3-sat reflects the existence of a critical line for 2+p - sat instances . we now consider the vc problem , where inputs are random graphs drawn from the @xmath128 ensemble@xcite . in other words , graphs have @xmath0 vertices and the probability that a pair of vertices are linked through an edge is @xmath129 , independently of other edges . when the number @xmath130 of covering marks is lowered , the model undergoes a cov / uncov transition at some critical density of covers @xmath131 for @xmath132 . for @xmath133 , vertex covers of size @xmath134 exist with probability one , for @xmath135 the available covering marks are not sufficient . the statistical mechanics analysis of ref . @xcite gave the result @xmath136 where @xmath137 solves the equation @xmath138 . this result is compatible with the bounds of refs . @xcite , and was later shown to be exact @xcite . for @xmath139 , eq . ( [ critical_vc ] ) only gives an approximate estimate of @xmath131 . more sophisticated calculations can be found in ref . @xcite . , high-@xmath15 uncov phase is separated by the dashed line , cf . ( [ critical_vc ] ) , from the high-@xmath42 , low-@xmath15 cov phase . the symbols ( numerics ) and continuous lines ( analytical prediction , cf . ( [ eqtrajvc ] ) ) refer to the simple search algorithm described in the text . the dotted line is the separatrix between two types of trajectories . ] let us consider a simple implementation of the dpll procedure for the present problem . during the computation , vertices can be _ covered _ , _ uncovered _ or just _ free _ , meaning that the algorithm has not yet assigned any value to that vertex . at the beginning all the vertices are set _ free_. at each step the algorithm chooses a vertex @xmath140 at random among those which are _ free_. if @xmath140 has neighboring vertices which are either _ free _ or _ uncovered _ , then the vertex @xmath140 is declared _ covered _ first . in case @xmath140 has only covered neighbors , the vertex is declared _ uncovered_. the process continues unless the number of covered vertices exceeds @xmath9 . in this case the algorithm backtracks and the opposite choice is taken for the vertex @xmath140 unless this corresponds to declaring _ uncovered _ a vertex that has one or more _ uncovered _ neighbors . the algorithm halts if it finds a solution ( and declares the graph to be cov ) or after exploring all the search tree ( in this case it declares the graph to be uncov ) . , is plotted versus the number of covering marks . here we consider random instances with average connectivity @xmath141 . the phase transition is at @xmath142 and corresponds to the peak in computational complexity . ] of course one can improve over this algorithm by using smarter heuristics @xcite . one remarkable example is the `` leaf - removal '' algorithm defined in ref . @xcite . instead of picking any vertex randomly , one chooses a connectivity - one vertex , declare it _ uncovered _ , and declare _ covered _ its neighbor . this procedure is repeated iteratively on the subgraph of _ free _ nodes , until no connectivity - one nodes are left . in the low - connectivity , cov region @xmath143 , it stops only when the graph is completely covered . as a consequence , this algorithm can solve vc in linear time with high probability in all this region . no equally good heuristics exists for higher connectivity , @xmath139 . under the action of one of the above algorithms , the instance is progressively modified and the number of variables is reduced . in fact , at each step a vertex is selected and can be eliminated from the graph regardless whether it is declared _ covered _ or _ uncovered_. the analysis of the first algorithm is greatly simplified by the remark that , as long as backtracking has not begun , the new vertex is selected randomly . this implies that the modified instance produced by the algorithm is still a random graph . its evolution can be effectively described by a trajectory in the @xmath144 space . if one starts from the parameters @xmath145 , @xmath146 , after @xmath147 steps of the algorithm , he will end up with a new instance of size @xmath148 and parameters @xcite @xmath149 some examples of the two types of trajectories ( the ones leading to a solution and the ones which eventually enter the uncov region ) are shown in fig . [ traj_vc ] . the separatrix is given by @xmath150 and corresponds to the dotted line in fig . [ traj_vc ] . above this line the algorithm solves the problem in linear time . for more general heuristics the analysis becomes less straightforward because the graph produced by the algorithm does not belong to the standard random - graph _ ensemble_. it may be necessary to augment the number of parameters which describe the evolution of the instance . as an example , the leaf - removal algorithm mentioned in the previous section is conveniently described by keeping track of three numbers which parametrize the degree profile ( i.e. the fraction of vertices @xmath151 having a given degree @xmath152 ) of the graph @xcite . below the critical line @xmath131 , cf . ( [ critical_vc ] ) , no solution exists to the typical random instance of vc . our algorithm must explore a large backtracking tree to prove it and this takes an exponential time . the size of the backtracking tree could be computed along the lines of sec . however a good result can be obtained with a simple `` static '' calculation @xcite . as explained in sec . ii.b.2 , we imagine the evolution of the backtracking tree as proceeding `` in parallel '' . at the level @xmath4 of the tree a set of @xmath4 vertices has been visited . call @xmath153 the subgraph induced by these vertices . since we always put a covering mark on a vertex which is surrounded by vertices declared _ uncovered _ , each node of the backtracking tree will carry a vertex cover of the associated subgraph @xmath153 . therefore the number of backtracking nodes is given by @xmath154 where @xmath155 is the number of vc s of @xmath153 using at most @xmath9 marks . a very crude estimate of the right - hand side of the above equation is : @xmath156 where we bounded the number of vc s of size @xmath157 on @xmath153 with the number of ways of placing @xmath157 marks on @xmath4 vertices . the authors of @xcite provided a refined estimate based on the _ annealed approximation _ of statistical mechanics . the results of this calculation are compared in fig . [ time_vc ] with the numerics . if the parameters which characterize an instance of vc lie in the region between @xmath131 , cf . ( [ critical_vc ] ) , and @xmath158 , cf . ( [ separatrix_vc ] ) , the problem is still soluble but our algorithm takes an exponential time to solve it . in practice , after a certain number of vertices has been visited and declared either _ covered _ or _ uncovered _ , the remaining subgraph @xmath159 can not be any longer covered with the leftover marks . this happens typically when the first descent trajectory ( [ eqtrajvc ] ) crosses the critical line ( [ critical_vc ] ) . it takes some time for the algorithm to realize this fact . more precisely , it takes exactly the time necessary to prove that @xmath159 is uncoverable . this time dominates the computational complexity in this region and can be calculated along the lines sketched in the previous section . the result is , once again , reported in fig . [ time_vc ] , which clearly shows a computational peak at the phase boundary . finally , let us notice that this mixed behavior disappears in the entire @xmath160 region if the leaf - removal heuristics is adopted for the first descent . up to now we have studied the typical resolution complexity . the study of fluctuations of resolution times is interesting too , particularly in the upper sat phase where solutions exist but are found at a price of a large computational effort . we may expect that there exist lucky but rare resolutions able to find a solution in a time much smaller than the typical one . due to the stochastic character of dpll complexity indeed fluctuates from run to run of the algorithm on the same instance . in fig . [ historun ] we show this run - to - run distribution of the logarithm @xmath85 of the resolution complexity for four instances of random 3-sat with the same ratio @xmath161 . the run to run distribution are qualitatively independent of the particular instances , and exhibit two bumps . the wide right one , located in @xmath162 , correspond to the major part of resolutions . it acquires more and more weight as @xmath0 increases and corresponds to the typical behavior analysed in section ii.b.3 . the left peak corresponds to much faster resolutions , taking place in linear time . the weight of this peak ( fraction of runs with complexities falling in the peak ) decreases exponentially fast with @xmath0 , and can be numerically estimated to @xmath163 with @xmath164 . therefore , instances at @xmath161 are typically solved in exponential time but a tiny ( exponentially small ) fraction of runs are able to find a solution in linear time only . a systematic stop - and - restart procedure may be introduced to take advantage of this fluctuation phenomenon and speed up resolution . if a solution is not found before @xmath0 splits , dpll is stopped and rerun after some random permutations of the variables and clauses . the expected number @xmath165 of restarts necessary to find a solution being equal to the inverse probability @xmath166 of linear resolutions , the resulting complexity scales as @xmath167 . to calculate @xmath168 we have analyzed , along the lines of the study of the growth of the search tree in the unsat phase , the whole distribution of the complexity for a given ratio @xmath25 in the upper sat phase . calculations can be found in @xcite . linear resolutions are found to correspond to branch trajectories that cross the unsat phase without being hit by a contradiction , see fig . results are reported in fig . [ histolin ] and compare very well with the experimentally measured number @xmath165 of restarts necessary to find a solution . in the whole upper sat phase , the use of restarts offers an exponential gain with respect to usual dpll resolution ( see fig . [ histolin ] for comparison between @xmath168 and @xmath85 ) , but the completeness of dpll is lost . a slightly more general restart strategy consists in stopping the backtracking procedure after a fixed number of nodes @xmath169 has been visited . a new ( and statistically independent ) dpll procedure is then started from the beginning . in this case one exploits lucky , but still exponential , stochastic runs . the tradeoff between the exponential gain of time and the exponential number of restarts , can be optimized by tuning the parameter @xmath170 . this approach has been analyzed in ref . @xcite taking vc as a working example . in fig . [ time_rvc ] we show the computatonal complexity of such a strategy as a function of the restart parameter @xmath170 . we compare the numerics with an approximate calculation @xcite . the instances were random graphs with average connectivity @xmath171 , and @xmath172 covering marks per vertex . the optimal choice of the parameter seems to be ( in this case ) @xmath173 , corresponding to polynomial runs . the analytical prediction reported in fig . [ time_rvc ] requires , as for 3-sat , an estimate of the execution - time fluctuations of the dpll procedure ( without restart ) . it turns out that one major source of fluctuations is , in the present case , the location in the @xmath144 plane of the highest node in the backtracking tree . in the typical run this coincides with the intersection @xmath174 between the first descent trajectory ( [ eqtrajvc ] ) and the critical line ( [ critical_vc ] ) . one can estimate the probability @xmath175 for this node to have coordinates @xmath144 ( obviously @xmath176 ) . when an upper bound @xmath170 on the backtracking time is fixed , the problem is solved in those lucky runs which are characterized by an atypical highest backtracking node . roughly speaking , this means that the algorithm has made some very good ( random ) choices in its first steps . in fig . [ root_rvc ] we plot the position of the highest backtracking point in the ( last ) successful runs for several values of @xmath170 . once again the numerics compare favourably with an approximate calculation . we now turn to the description and study of algorithms of another type , namely local search algorithms . as a common feature , these algorithms start from a configuration ( assignment ) of the variables , and then make successive improvements by changing at each step few of the variables in the configuration ( local move ) . for instance , in the sat problem , one variable is flipped from being true to false , or _ vice versa _ , at each step . whereas complete algorithms of the dpll type give a definitive answer to any instance of a decision problem , exhibiting either a solution or a proof of unsatisfiability , local search algorithms give a sure answer when a solution is found but can not prove unsatisfiability . however , these algorithms can sometimes be turned into one - sided probabilistic algorithms , with an upper bound on the probability that a solution exists and has not been found after @xmath61 steps of the algorithm , decreasing to zero when @xmath177@xcite . local search algorithms perform repeated changes of a configuration @xmath20 of variables ( values of the boolean variables for sat , status marked or unmarked of vertices for vc ) according to some criterion , usually based on the comparison of the cost function @xmath178 ( number of unsatisfied clauses for sat , of uncovered edges for vc ) evaluated at @xmath20 and over its neighborhood . it is therefore clear that the shape of the multidimensional surface @xmath179 , called cost function landscape , is of high importance . on intuitive grounds , if this landscape is relatively smooth with a unique minimum , local procedures as gradient descent should be very efficient , while the presence of many local minima could hinder the search process ( fig . [ landscape ] ) . the fundamental underlying question is whether the performances of the dynamical process ( ability to find the global minimum , time needed to reach it ) can be understood in terms of an analysis of the cost function landscape only . this question was intensively studied and answered for a limited class of cost functions , called mean field spin glass models , some years ago@xcite . the characterization of landscapes is indeed of huge importance in physical systems . there , the cost function is simply the physical energy , and local dynamics are usually low or zero temperature monte carlo dynamics , essentially equivalent to gradient descent . depending on the parameters of the input distribution , the minima of the cost functions may undergo structural changes , a phenomenon called clustering in physics . clustering has been rigorously shown to take place in the random 3-xorsat problem@xcite , and is likely to exist in many other random combinatorial problems as 3-sat@xcite . instances of the 3-xorsat problem with @xmath180 clauses and @xmath0 variables have almost surely solutions as long as @xmath181@xcite . the clustering phase transition takes place at @xmath182 and is related to a change in the geometric structure of the space of solutions , see fig . [ landscape ] : * when @xmath183 , the space of solutions is connected . given a pair of solutions @xmath184 , _ i.e. _ two assignments of the @xmath0 boolean variables that satisfy the clauses , there almost surely exists a sequence of solutions , @xmath185 , with @xmath186 , @xmath187 , @xmath188 , connecting the two solutions such that the hamming distance ( number of different variables ) between @xmath189 and @xmath190 is bounded from above by some finite constant when @xmath132 . * when @xmath191 , the space of solutions is not connected any longer . it is made of an exponential ( in @xmath0 ) number of connected components , called clusters , each containing an exponentially large number of solutions . clusters are separated by large voids : the hamming distance between two clusters , that is , the smallest hammming distance between pairs of solutions belonging to these clusters , is of the order of @xmath0 . from intuitive grounds , changes of the statistical properties of the cost function landscape e.g. of the structure of the solutions space may potentially affect the search dynamics . this connection between dynamics and static properties was established in numerous works in the context of mean field models of spin glasses @xcite , and subsequently also put forward in some studies of local search algorithms in combinatorial optimization problems@xcite . so far , there is no satisfying explanation to when and why features of _ a priori _ algorithm dependent dynamical phenomena should be related to , or predictable from some statistical properties of the cost function landscape . we shall see some examples in the following where such a connection indeed exist ( sec . iii.b ) and other ones where its presence is far less obvious ( sec . iiic , d ) . , while vertical axis is the associated cost @xmath19 . left : smooth cost function , with a single minimum easily reachable with local search procedures e.g. gradient descent . middle : rough cost function with a lot of local minima whose presence may damage the performances of local search algorithms . the various global minima are spread out homogeneously over the configuration space . right : rough cost function with global minima clustered in some portions of the configuration space only . , title="fig : " ] , while vertical axis is the associated cost @xmath19 . left : smooth cost function , with a single minimum easily reachable with local search procedures e.g. gradient descent . middle : rough cost function with a lot of local minima whose presence may damage the performances of local search algorithms . the various global minima are spread out homogeneously over the configuration space . right : rough cost function with global minima clustered in some portions of the configuration space only . , title="fig : " ] , while vertical axis is the associated cost @xmath19 . left : smooth cost function , with a single minimum easily reachable with local search procedures e.g. gradient descent . middle : rough cost function with a lot of local minima whose presence may damage the performances of local search algorithms . the various global minima are spread out homogeneously over the configuration space . right : rough cost function with global minima clustered in some portions of the configuration space only . , title="fig : " ] 0 00 coding theory is a rich source of computational problems ( and algorithms ) for which the average case analysis is relevant @xcite . let us focus , for sake of concreteness , on the decoding problem . codewords are sequences of symbols with some built - in redundancy . if we consider the case of linear codes on a binary alphabet , this redundancy can be implemented as a set of linear constraints . in practice , a codeword is a vector @xmath192 ( with @xmath193 ) which satisfies the equation @xmath194 where @xmath195 is an @xmath196 binary matrix ( _ parity check matrix _ ) . each one of the @xmath4 linear equations involved in eq . ( [ paritycheckmatrix ] ) is called a _ parity check_. this set of equation can be represented graphically by a _ tanner graph _ , cf . [ tanner ] . this is a bipartite graph highlighting the relations between the variables @xmath197 and the constraints ( parity checks ) acting on them . the decoding problem consists in finding , among the solutions of eq . ( [ paritycheckmatrix ] ) , the `` closest '' one @xmath198 to the output @xmath199 of some communication channel . this problem is , in general , np - hard @xcite . the precise meaning of `` closest '' depends upon the nature of the communication channel . let us make two examples : * the binary symmetric channel ( bsc ) . in this case the output of the communication channel @xmath199 is a codeword , i.e. a solution of ( [ paritycheckmatrix ] ) , in which a fraction @xmath200 of the entries has been flipped . `` closest '' has to be understood in the hamming - distance sense . @xmath198 is the solution of eq . ( [ paritycheckmatrix ] ) which minimizes the hamming distance from @xmath199 . * the binary erasure channel ( bec ) . the output @xmath199 is a codeword in which a fraction @xmath200 of the entries has been erased . one has to find a solution @xmath198 of eq . ( [ paritycheckmatrix ] ) which is compatible with the remaining entries . such a problem has a _ unique _ solution for small enough erasure probability @xmath200 . there are two sources of randomness in the decoding problem : @xmath201 the matrix @xmath195 which defines the code is usually drawn from some random _ ensemble _ ; @xmath202 the received message which is distributed according to some probabilistic model of the communication channel ( in the two examples above , the bits to be flipped / erased were chosen randomly ) . unlike many other combinatorial problems , there is therefore a `` natural '' probability distribution defined on the instances . average case analysis with respect to this distribution is of great practical relevance . recently , amazingly good performances have been obtained by using low - density parity check ( ldpc ) codes @xcite . ldpc codes are defined by parity check matrices @xmath195 which are large and sparse . as an example we can consider gallager _ regular _ codes @xcite . in this case @xmath195 is chosen with flat probability distribution within the family of matrices having @xmath203 ones per column , and @xmath204 ones per row . these are decoded using a suboptimal linear - time algorithm known as `` belief - propagation '' or `` sum - product '' algorithm @xcite . this is an iterative algorithm which takes advantage of the locally tree - like structure of the tanner graph , see fig . [ tanner ] , for ldpc codes . after @xmath205 iterations it incorporates the information conveyed by the variables up to distance @xmath205 from the one to be decoded . this can be done in a recursive fashion allowing for linear - time decoding . belief - propagation decoding shows a striking threshold phenomenon as the noise level @xmath200 crosses some critical ( code - dependent ) value @xmath206 . while for @xmath207 the transmitted codeword is recovered with high probability , for @xmath208 decoding will fail almost always . the threshold noise @xmath206 is , in general , smaller than the threshold @xmath209 for optimal decoding ( with unbounded computational resources ) . the rigorous analysis of ref . @xcite allows a precise determination of the critical noise @xmath206 under quite general circumstances . nevertheless some important theoretical questions remain open : can we find some smarter linear - time algorithm whose threshold is greater than @xmath206 ? is there any `` intrinsic '' ( i.e. algorithm independent ) characterization of the threshold phenomenon taking place at @xmath206 ? as a first step towards the answer to these questions , ref . @xcite explored the dynamics of local optimization algorithms by using statistical mechanics techniques . the interesting point is that `` belief propagation '' is by no means a local search algorithm . for sake of concreteness , we shall focus on the binary erasure channel . in this case we can treat decoding as a combinatorial optimization problem within the space of bit sequences of length @xmath210 ( the number of erased bits , the others being fixed by the received message ) . the function to be minimized is the _ energy density _ @xmath211 where we denote as @xmath212 the hamming distance between two vectors @xmath213 and @xmath214 , and we introduced the normalizing factor for future convenience . notice that both arguments of @xmath215 in eq . ( [ costfunction ] ) are vectors in @xmath216 . we can define the @xmath217-neighborhood of a given sequence @xmath218 as the set of sequences @xmath219 such that @xmath220 , and we call @xmath217-stable states the bit sequences which are optima of the decoding problem within their @xmath217-neighborhood . one can easily invent local search algorithms @xcite for the decoding problem which use the @xmath217-neighborhoods . the algorithm start from a random sequence and , at each step , optimize it within its @xmath217-neighborhood . this algorithm is clearly suboptimal and halts on @xmath217-stable states . let us consider , for instance , a @xmath221 regular code and decode it by local search in @xmath222-neighborhoods . in fig . [ glauber ] we report the resulting energy density @xmath223 after the local search algorithm halts , as a function of the erasure probability @xmath200 . we averaged over 100 different realizations of the noise and of the matrix @xmath195 . for sake of comparison we recall that the threshold for belief - propagation decoding is @xmath224 @xcite , while the threshold for optimal decoding is at @xmath225 @xcite . it is evident that local search by @xmath222-neighborhoods performs quite poorly . a natural question is whether ( and how much ) , these performances are improved by increasing @xmath217 . it is therefore quite natural to study _ metastable _ states . these are @xmath217-stable states for any @xmath226 if @xmath227 . ] . there exists no completely satisfying definition of such states : here we shall just suggest a possibility among others . the tricky point is that we do not know how to compare @xmath217-stable states for different values of @xmath0 . this forbids us to make use of the above asymptotic statement . one possibility is to count without really defining them . this can be done , at least in principle , by counting @xmath217-stable states , take the @xmath228 limit and , at the end , the @xmath229 limit@xcite . on physical grounds , we expect @xmath217-stable states to be exponentially numerous . in particular , if we call @xmath230 the number of @xmath217-stable states taking a value @xmath223 of the cost function ( [ costfunction ] ) , we have @xmath231 we can therefore define the so called ( physical ) complexity @xmath232 as follows , @xmath233 roughly speaking we can say that the number of metastable states is @xmath234 . of course there are several alternative ways of taking the limits @xmath229 , @xmath228 , and we do not yet have a proof that these procedures give the same result for @xmath232 nevertheless it is quite clear that the existence of an exponential number of metastable states should affect dramatically the behavior of local search algorithms . statistical mechanics methods @xcite allows to determine the complexity @xmath232 @xcite . in `` difficult '' cases ( such as for error - correcting codes ) , the actual computation may involve some approximation , e.g. the use of a variational ansatz . nevertheless the outcome is usually quite accurate . in fig . [ complexity ] we consider a @xmath235 regular code on the binary erasure channel . we report the resulting complexity for three different values of the erasure probability @xmath200 . the general picture is as follows . below @xmath206 there is no metastable state , except the one corresponding to the correct codeword . between @xmath206 and @xmath209 there is an exponential number of metastable states with energy density belonging to the interval @xmath236 ( @xmath232 is strictly positive in this interval ) . above @xmath209 , @xmath237 . the maximum of @xmath232 is always at @xmath238 . the above picture tell us that any local algorithm will run into difficulties above @xmath206 . in order to confirm this picture , the authors of ref . @xcite made some numerical computations using simulated annealing as decoding algorithm for quite large codes ( @xmath239 bits ) . for each value of @xmath200 , we start the simulation fixing a fraction @xmath240 of spins to @xmath241 ( this part will be kept fixed all along the run ) . the remaining @xmath242 spins are the dynamical variables we change during the annealing in order to try to satisfy all the parity checks . the energy of the system counts the number of unsatisfied parity checks . the cooling schedule has been chosen in the following way : @xmath243 monte carlo sweeps ( mcs ) proposed spin flips . each proposed spin flip is accepted or not accordingly to a standard metropolis test . ] at each of the 1000 equidistant temperatures between @xmath244 and @xmath245 . the highest temperature is such that the system very rapidly equilibrates . typical values for @xmath243 are from 1 to @xmath246 . notice that , for any fixed cooling schedule , the computational complexity of the simulated annealing method is linear in @xmath0 . then we expect it to be affected by metastable states of energy @xmath238 , which are present for @xmath208 : the energy relaxation should be strongly reduced around @xmath238 and eventually be completely blocked . some results are plotted in fig . [ annealing ] together with the theoretical prediction for @xmath238 . the good agreement confirm our picture : the algorithm gets stucked in metastable states , which have , in the great majority of cases , energy density @xmath238 . both `` belief propagation '' and local search algorithms fail to decode correctly between @xmath206 and @xmath209 . this leads naturally to the following conjecture : no linear time algorithm can decode in this regime of noise . the ( typical case ) computational complexity changes from being linear below @xmath206 to superlinear above @xmath206 . in the case of the binary erasure channel it remains polynomial between @xmath206 and @xmath209 ( since optimal decoding can be realized with linear algebra methods ) . however it is plausible that for a general channel it becomes non - polynomial . in this section the local procedure we consider is gradient descent ( gd ) . gd is defined as follows . * ( 1 ) * start from an initial randomly chosen configuration of the variables . call @xmath10 the number of unsatisfied clauses . * ( 2 ) * if @xmath247 then stop ( a solution is found ) . otherwise , pick randomly one variable , say @xmath248 , and compute the number @xmath249 of unsatisfied clauses when this variable is negated ; if @xmath250 then accept this change _ i.e. _ replace @xmath248 with @xmath251 and @xmath10 with @xmath249 ; if @xmath252 , do not do anything . then go to step 2 . the study of the performances of gd to find the minima of cost functions related to statistical physics models has recently motivated various studies@xcite . numerics indicate that gd is typically able to solve random 3-sat instances with ratios @xmath253 @xcite close to the onset of clustering @xcite . we shall rigorously show below that this is not so for 3-xorsat . let us apply gd to an instance of xorsat . the instance has a graph representation explained in fig . [ xorgr ] . vertices are in one to one correspondence with variables . a clause is fully represented by a plaquette joining three variables and a boolean label equal to the number of negated variables it contains modulo 2 ( not represented on fig . [ xorgr ] ) . once a configuration of the variables is chosen , each plaquette may be labelled by its status , s or u , whether the associated clause is respectively satisfied or unsatisfied . a fundamental property of xorsat is that each time a variable is changed , _ i.e. _ its value is negated , the clauses it belongs to change status too . this property makes the analysis of some properties of gd easy . consider the hypergraph made of 15 vertices and 7 plaquettes in fig . [ bi ] , and suppose the central plaquette is violated ( u ) while all other plaquettes are satisfied ( s ) . the number of unsatisfied clauses is @xmath254 . now run gd on this special instance of xorsat . two cases arise , symbolized in fig . [ bi ] , whether the vertex attached to the variable to be flipped belongs , or not , to the central plaquette . it is an easy check that , in both cases , @xmath255 and the change is not permitted by gd . the hypergraph of fig . [ bi ] will be called hereafter island . when the status of the plaquettes is u for the central one and s for the other ones , the island is called blocked . though the instance of the xorsat problem encoded by a blocked island is obviously satisfiable ( think of negating at the same time one variable attached to a vertex @xmath256 of the central plaquette and one variable in each of the two peripherical plaquettes joining the central plaquette at @xmath256 ) , gd will never be able to find a solution and will be blocked forever in the local minimum with height @xmath254 . the purpose of this section is to show that this situation typically happens for random instances of xorsat . more precisely , while almost all instances of xorsat with a ratio of clauses per variables smaller than @xmath257 have a lot of solutions , gd is almost never able to find one . even worse , the number of violated clauses reached by gd is bounded from below by @xmath258 where @xmath259 in other words , the number of clauses remaining unsatisfied at the end of a typical gd run is of the order of @xmath0 . our demonstration , inspired from @xcite , is based on the fact that , with high probability , a random instance of xorsat contains a large number of blocked islands of the type of fig . [ bi ] . to make the proof easier , we shall study the following fixed clause probability ensemble . instead of imposing the number of clauses to be equal to @xmath260 , any triplet @xmath243 of three vertices ( among @xmath0 ) is allowed to carry a plaquette with probability @xmath261 . notice that this probability ensures that , on the average , the number of plaquettes equals @xmath262 . let us now draw a hypergraph with this distribution . for each triplet @xmath243 of vertices , we define @xmath263 if @xmath243 is the center of a island , 0 otherwise . we shall show that the total number of islands , @xmath264 , is highly concentrated in the large @xmath0 limit , and calculate its average value . the expectation value of @xmath265 is equal to @xmath266 = \frac{(n-3)\times ( n-4 ) \times \ldots \times ( n-13)\times ( n-14)}{8\times 8\times 8 } \times \mu ^a \ , ( 1-\mu ) ^b \ , \ ] ] where @xmath267 is the number of plaquettes in the island , and @xmath268 is the number of triplets with at least one vertex among the set of 15 vertices of the island that do not carry plaquette . the significance of the terms in eq . ( [ expitau ] ) is transparent . the central triplet @xmath243 occupying three vertices , we choose 2 vertices among @xmath269 to draw the first peripherical plaquette of the island , then other 2 vertices among @xmath270 for the other peripherical plaquette having a common vertex with the latter . the order in which these two plaquettes are built does not matter and a factor @xmath271 permits us to avoid double counting . the other four peripherical plaquettes have multiplicities calculable in the same way ( with less and less available vertices ) . the terms in @xmath272 and @xmath273 correspond to the probability that such a 7 plaquettes configuration is drawn on the 15 vertices of the island , and is disconnected from the remaining @xmath274 vertices . the expectation value of the number @xmath275 of islands per vertex thus reads , @xmath276 = \lim _ { n\to\infty } \frac 1n \ , { n \choose 3}\ , e[i_{\tau } ] = \frac{729}{8 } \ , \alpha^7 \ , e^{-45\ , \alpha } \quad .\ ] ] chebyshev s inequality can be used to show that @xmath140 is concentrated around its above average value . let us calculate the second moment of the number of islands , @xmath277= \sum _ { \tau , \sigma } e [ i_{\tau } i_{\sigma } ] $ ] . clearly , @xmath278 $ ] depends only on the number @xmath279 of vertices common to triplets @xmath243 and @xmath280 . it is obvious that no two triplets of vertices can be centers of islands when they have @xmath281 or @xmath282 common vertices . if @xmath283 , @xmath284 and @xmath285=e [ i_{\tau}]$ ] has been calculated above . for @xmath286 , a similar calculation gives @xmath287 finally , we obtain @xmath288 = \frac 1{n^2 } \left [ { n \choose 3 } e_{\ell =3 } + { n \choose 3}{n -3\choose 3 } e_{\ell = 0 } \right ] = e[i]^2 + o\left ( \frac 1n \right ) \quad .\ ] ] therefore the variance of @xmath140 vanishes and @xmath140 is , with high probability , equal to its average value given by ( [ eq89 ] ) . to conclude , an island has a probability @xmath289 to be blocked by definition . therefore the number ( per vertex ) of blocked islands in a random xorsat instance with ratio @xmath25 is almost surely equal to @xmath290 given by eq . ( [ teh ] ) . since each blocked island has one unsatisfied clause , this is also a lower bound to the number of violated clauses per variable . notice however that @xmath291 is very small and bounded from above by @xmath292 over the range of interest , @xmath293 . therefore , one would in principle need to deal with billions of variables not to reach solutions and be in the true asymptotic regime of gd . the proof is easily generalizable to gradient descent with more than one look ahead . to extend the notion of blocked island to the case where gd is allowed to invert @xmath217 , and not only 1 , variables at a time , it is sufficient to have @xmath294 , and not 2 , peripherical plaquettes attached to each vertex of the central plaquette . the calculation of the lower bound @xmath295 to the number of violated clauses ( divided by @xmath0 ) reached by gd is straightforward and not reproduced here . as a consequence , gd , even with @xmath217 simultaneous flips permitting to overcome local barriers , remains almost surely trapped at an extensive ( in @xmath0 ) level of violated clauses for any finite @xmath217 . actually the lower bound @xmath296 tends to zero only if @xmath217 is of the order of @xmath78 . we stress that the statistical physics calculation of physical ` complexity ' @xmath297 ( see sec . [ codesection ] ) predicts there is no metastable states for @xmath298@xcite , while gd is almost surely trapped by the presence of blocked islands for any @xmath299 . this apparent discrepancy comes from the fact that gd is sensible to the presence of configurations blocked for finite @xmath217 , while the physical ` complexity ' considers states metastable in the limit @xmath229 only@xcite . and @xmath300 . each clause or equation is represented by a plaquette whose vertices are the attached variables . when the variables are assigned some values , the clauses can be satisfied ( s ) or unsatisfied ( u ) . ] the pure random walksat ( prwsat ) algorithm for solving @xmath3-sat is defined by the following rules@xcite . 1 . choose randomly a configuration of the boolean variables . 2 . if all clauses are satisfied , output `` satisfiable '' . 3 . if not , choose randomly one of the unsatisfied clauses , and one among the @xmath3 variables of this clause . flip ( invert ) the chosen variable . notice that the selected clause is now satisfied , but the flip operation may have violated other clauses which were previously satisfied . 4 . go to step 2 , until a limit on the number of flips fixed beforehand has been reached . then output `` do nt know '' . what is the output of the algorithm ? either `` satisfiable '' and a solution is exhibited , or `` do nt know '' and no certainty on the status of the formula is achieved . papadimitriou introduced this procedure for @xmath301 , and showed that it solves with high probability any satisfiable 2-sat instance in a number of steps ( flips ) of the order of @xmath302@xcite . recently schning was able to prove the following very interesting result for 3-sat@xcite . call ` trial ' a run of prwsat consisting of the random choice of an initial configuration followed by @xmath303 steps of the procedure . if none of @xmath61 successive trials on a given instance has been successful ( has provided a solution ) , then the probability that this instance is satisfiable is lower than @xmath304 . in other words , after @xmath305 trials of prwsat , most of the configuration space has been ` probed ' : if there were a solution , it would have been found . though this local search algorithm is not complete , the uncertainty on its output can be made as small as desired and it can be used to prove unsatisfiability ( in a probabilistic sense ) . schning s bound is true for any instance . restriction to special input distributions allows to strengthen this result . alekhnovich and ben - sasson showed that instances drawn from the random 3-satisfiability ensemble described above are solved in polynomial time with high probability when @xmath25 is smaller than @xmath306@xcite . in this section , we briefly sketch the behaviour of prwsat , as seen from numerical experiments @xcite and the analysis of @xcite . a dynamical threshold @xmath307 ( @xmath308 for 3-sat ) is found , which separates two regimes : * for @xmath309 , the algorithm finds a solution very quickly , namely with a number of flips growing linearly with the number of variables @xmath0 . figure [ wsat_phen]a shows the plot of the fraction @xmath310 of unsatisfied clauses as a function of time @xmath31 ( number of flips divided by @xmath4 ) for one instance with ratio @xmath311 and @xmath312 variables . the curve shows a fast decrease from the initial value ( @xmath313 in the large @xmath0 limit independently of @xmath25 ) down to zero on a time scale @xmath314 . fluctuations are smaller and smaller as @xmath0 grows . @xmath315 is an increasing function of @xmath25 . this _ relaxation _ regime corresponds to the one studied by alekhnovich and ben - sasson , and @xmath316 as expected@xcite . * for instances in the @xmath317 range , the initial relaxation phase taking place on @xmath318 time scale is not sufficient to reach a solution ( fig . [ wsat_phen]b ) . the fraction @xmath319 of unsat clauses then fluctuates around some plateau value for a very long time . on the plateau , the system is trapped in a _ metastable _ state . the life time of this metastable state ( trapping time ) is so huge that it is possible to define a ( quasi ) equilibrium probability distribution @xmath320 for the fraction @xmath319 of unsat clauses . ( inset of fig . [ wsat_phen]b ) . the distribution of fractions is well peaked around some average value ( height of the plateau ) , with left and right tails decreasing exponentially fast with @xmath0 , @xmath321 with @xmath322 . eventually a large negative fluctuation will bring the system to a solution ( @xmath323 ) . assuming that these fluctuations are independant random events occuring with probability @xmath324 on an interval of time of order @xmath222 , the resolution time is a stochastic variable with exponential distribution . its average is , to leading exponential order , the inverse of the probability of resolution on the @xmath325 time scale : @xmath326 \sim \exp ( n \zeta)$ ] with @xmath327 . escape from the metastable state therefore takes place on exponentially large in@xmath0 time scales , as confirmed by numerical simulations for different sizes . schning s result@xcite can be interpreted as a lower bound to the probability @xmath328 , true for any instance . the plateau energy , that is , the fraction of unsatisfied clauses reached by prwsat on the linear time scale is plotted on fig . [ wsat_plateau ] . notice that the `` dynamic '' critical value @xmath329 above which the plateau energy is positive ( prwsat stops finding a solution in linear time ) is strictly smaller than the `` static '' ratio @xmath330 , where formulas go from satisfiable with high probability to unsatisfiable with high probability . in the intermediate range @xmath331 , instances are almost surely satisfiable but prwsat needs an exponentially large time to prove so . interestingly , @xmath332 and @xmath330 coincides for 2-sat in agreement with papadimitriou s result@xcite . furthermore , the dynamical transition is apparently not related to the onset of clustering taking place at @xmath333 . a b .5 cm .5 cm when prwsat finds easily a solution , the number of steps it requires is of the order of @xmath0 , or equivalently , @xmath4 . let us call @xmath334 the average of this number divided by the number of clauses @xmath4 . by definition of the dynamic threshold , @xmath315 diverges when @xmath335 . assuming that @xmath336 can be expressed as a series of powers of @xmath25 , we find the following expansion@xcite @xmath337 around @xmath338 . as only a finite number of terms in this expansion have been computed , we do not control its radius of convergence , yet as shown in fig . [ wsat_fig_tres_q1 ] the numerical experiments provide convincing evidence in favour of its validity . the above calculation is based on two facts . first , for @xmath339 the instance under consideration splits into independent subinstances ( involving no common variable ) that contains a number of variables of the order of @xmath78 at most . moreover , the number of the connected components containing @xmath340 clauses , computed with probabilistic arguments very similar to those of section [ gradxor ] , contribute to a power expansion in @xmath25 only at order @xmath341 . secondly , the number of steps the algorithm needs to solve the instance is simply equal to the sum of the numbers of steps needed for each of its independent subinstances . this additivity remains true when one averages over the initial configuration and the choices done by the algorithm . one is then left with the enumeration of the different subinstances with a given size and the calculation of the average number of steps for their resolution . a detailed presentation of this method has been given in a general case in @xcite , and applied more specifically to this problem in @xcite ; the reader is referred to these previous works for more details . equation ( [ dev_cluster_tresk ] ) is the output of the enumeration of subinstances with up to three clauses . the above small @xmath25 expansion does not allow us to investigate the @xmath342 regime . we turn now to an approximate method more adapted to this situation . let us denote by @xmath20 an assignment of the boolean variables . prwsat defines a markov process on the space of the configurations @xmath20 , a discrete set of cardinality @xmath343 . it is a formidable task to follow the probabilities of all these configurations as a function of the number of steps @xmath61 of the algorithm so one can look for a simpler description of the state of the system during the evolution of the algorithm . the simplest , and crucial , quantity to follow is the number of clauses unsatisfied by the current assignment of the boolean variables , @xmath344 . indeed , as soon as this value vanishes , the algorithm has found a solution and stops . a crude approximation consists in assuming that , at each time step @xmath61 , all configurations with a given number of unsatisfied clauses are equiprobable , whereas the hamming distance between two configurations visited at step @xmath61 and @xmath345 of the algorithm is at most @xmath204 . however , the results obtained are much more sensible that one could fear . within this simplification , a markovian evolution equation for the probability that @xmath346 clauses are unsatisfied after @xmath61 steps can be written . using methods similar to the ones in section [ dpllsatsection ] , we obtain ( see @xcite for more details and @xcite for an alternative way of presenting the approximation ) : * the average fraction of unsatisfied clauses , @xmath347 , after @xmath348 steps of the algorithm . for ratios @xmath349 , @xmath319 remains positive at large times , which means that typically a large formula will not be solved by prwsat , and that the fraction of unsat clauses on the plateau is @xmath350 . the predicted value for @xmath351 , @xmath352 , is in good but not perfect agreement with the estimates from numerical simulations , around @xmath353 . the plateau height , @xmath354 , is compared to numerics in fig . [ wsat_plateau ] . * the probability @xmath321 that the fraction of unsatisfied clauses is @xmath310 . it has been argued above that the distribution of resolution times in the @xmath355 phase is expected to be , at leading order , an exponential distribution of average @xmath356 , with @xmath357 . predictions for @xmath358 are plotted and compared to experimental measures of @xmath168 in fig . [ wsat_fig_zeta ] . despite the roughness of our markovian approximation , theoretical predictions are in qualitative agreement with numerical experiments . a similar study of the behaviour of prwsat on xorsat problems has been also performed in @xcite , with qualitatively similar conclusions : there exists a dynamic threshold @xmath329 for the algorithm , smaller both than the satisfiability and clustering thresholds ( known exactly in this case @xcite ) . for low values of @xmath25 , the resolution time is linear in the size of the formula ; between @xmath329 and @xmath330 resolution occurs on exponentially large time scales , through fluctuations around a plateau value for the number of unsatisfied clauses . in the xorsat case , the agreement between numerical experiments and this approximate study ( which predicts @xmath359 ) is quantitatively better and seems to improve with growing @xmath3 . in this article , we have tried to give an overview of the studies that physicists have devoted to the analysis of algorithms . this presentation is certainly not exhaustive . let us mention that use of statistical physics ideas have permitted to obtain very interesting results on related issues as number partitioning@xcite , binary search trees @xcite , learning in neural networks @xcite , extremal optimization @xcite ... it may be objected that algorithms are mathematical and well defined objects and , as so , should be analysed with rigorous techniques only . though this point of view should ultimately prevail , the current state of available probabilistic or combinatorics techniques compared to the sophisticated nature of algorithms used in computer science make this goal unrealistic nowadays . we hope the reader is now convinced that statistical physics ideas , techniques , ... may be of help to acquire a quantitative intuition or even formulate conjectures on the average performances of search algorithms . a wealth of concepts and methods familiar to physicists e.g. phase transitions and diagrams , dynamical renormalization flow , out - of - equilibrium growth phenomena , metastability , perturbative approaches ... are found to be useful to understand the behaviour of algorithms . it is a simple bet that this list will get longer in next future and that more and more powerful techniques and ideas issued from modern theoretical physics will find their place in the field . open questions are numerous . variants of dpll with complex splitting heuristics , random backtrackings@xcite or applied to combinatorial problems with internal symmetries@xcite would be worth being studied . as for local search algorithms , it would be very interesting to study refined versions of the pure walksat procedure that alternate random and greedy steps @xcite to understand the observed existence and properties of optimal strategies . one of the main open questions in this context is to what extent performances are related to intrinsic features of the combinatorial problems and not to the details of the search algorithm@xcite . this raises the question of how the structure of the cost function landscape may induce some trapping or slowing down of search algorithms@xcite . last of all , the input distributions of instances we have focused on here are far from being realistic . real instances have a lot of structure which will strongly reflect on the performances of algorithms . going towards more realistic distributions or , even better , obtaining results true for any instance would be of great interest . d. mitchell , b. selman , h. levesque , hard and easy distributions of sat problems , _ proc . of the tenth natl . conf . on artificial intelligence ( aaai-92 ) _ , 440 - 446 , the aaai press / mit press , cambridge , ma ( 1992 ) . j. gu , p.w . purdom , j. franco , b.w . wah , algorithms for satisfiability ( sat ) problem : a survey . _ dimacs series on discrete mathematics and theoretical computer science _ * 35 * , 19 - 151 , american mathematical society ( 1997 ) . for a review on the analysis of search heuristics in the absence of backtracking , see : + d. achlioptas , lower bounds for random 3-sat via differential equations , _ theor . sci . _ * 265 * , 159 - 186 ( 2001 ) . kaporis , l.m . kirousis , e.g. lalas , the probabilistic analysis of a greedy satisfiability algorithm , _ proceedings of sat 2002 _ , pp 362376 ( 2002 ) , available at http://gauss.ececs.uc.edu/conferences/sat2002/abstracts/kaporis.ps a.c . kaporis , l.m . kirousis , y.c . stamatiou , how to prove conditional randomness using the principle of deferred decisions , technical report , computer technology institute , greece ( 2002 ) , available at http://www.ceid.upatras.fr/faculty/kirousis/kks-pdd02.ps r. monasson , r. zecchina , s. kirkpatrick , b. selman , l. troyansky , 2+p - sat : relation of typical - case complexity to the nature of the phase transition , _ random structure and algorithms _ * 15 * , 414 ( 1999 ) . s. cocco , r. monasson , trajectories in phase diagrams , growth processes and computational complexity : how search algorithms solve the 3-satisfiability problem , _ phys . lett . _ * 86 * , 1654 ( 2001 ) ; analysis of the computational complexity of solving random satisfiability problems using branch and bound search algorithms , _ eur . phys . j. b _ * 22 * , 505 ( 2001 ) . s. cocco , r. monasson , exponentially hard problems are sometimes polynomial , a large deviation analysis of search algorithms for the random satisfiability problem , and its application to stop - and - restart resolutions , _ phys . e _ * 66 * , 037101 ( 2002 ) ; restart method and exponential acceleration of random 3-sat instances resolutions : a large deviation analysis of the davis putnam loveland logemann algorithm , to appear in amai ( 2003 ) . s. boettcher , m. grigni , jamming model for the extremal optimization heuristic , _ j. phys . a _ * 35 * , 1109 - 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since its inception , it has been clear that general relativity has many striking similarities to gauge theories . both are based on the idea of local symmetry and therefore share a number of formal properties . nevertheless , their dynamical behavior can be quite different . while maxwell electrodynamics describes a long - range force similar to the situation with gravity , the non - abelian gauge theories used to describe the weak and strong nuclear forces have rather different behaviors . quantum chromodynamics , which describes the strong nuclear forces , for example , exhibits confinement of particles carrying the non - abelian gauge charges . certainly , there is no obvious corresponding property for gravity . moreover , consistent quantum gauge theories have existed for a half century , but as yet no satisfactory quantum field theory of gravity has been constructed ; indeed , there are good arguments suggesting that it is not possible to do so . the structures of the lagrangians are also rather different : the non - abelian yang - mills lagrangian contains only up to four - point interactions while the einstein - hilbert lagrangian contains infinitely many . despite these differences , string theory teaches us that gravity and gauge theories can , in fact , be unified . the maldacena conjecture @xcite , for example , relates the weak coupling limit of a gravity theory on an anti - de sitter background to a strong coupling limit of a special supersymmetric gauge field theory . there is also a long history of papers noting that gravity can be expressed as a gauging of lorentz symmetry @xcite , as well as examples of non - trivial similarities between classical solutions of gravity and non - abelian gauge theories @xcite . in this review a different , but very general , relationship between the weak coupling limits of both gravity and gauge theories will be described . this relationship allows gauge theories to be used directly as an aid for computations in perturbative quantum gravity . the relationship discussed here may be understood most easily from string perturbation theory . at the semi - classical or `` tree level , '' kawai , lewellen and tye ( klt ) @xcite derived a precise set of formulas expressing closed string amplitudes in terms of sums of products of open string amplitudes . in the low - energy limit ( _ i.e. _ anywhere well below the string scale of @xmath1 gev ) where string theory effectively reduces to field theory , the klt relations necessarily imply that similar relations must exist between amplitudes in gravity and gauge field theories : at tree - level in field theory , graviton scattering must be expressible as a sum of products of well defined pieces of non - abelian gauge theory scattering amplitudes . moreover , using string based rules , four - graviton amplitudes with one quantum loop in einstein gravity were obtained in a form in which the integrands appearing in the expressions were given as products of integrands appearing in gauge theory @xcite . these results may be interpreted heuristically as @xmath2 this remarkable property suggests a much stronger relationship between gravity and gauge theories than one might have anticipated by inspecting the respective lagrangians . the klt relations hold at the semi - classical level , _ i.e. _ with no quantum loops . in order to exploit the klt relations in quantum gravity , one needs to completely reformulate the quantization process ; the standard methods starting either from a hamiltonian or a lagrangian provide no obvious means of exploiting the klt relations . there is , however , an alternative approach based on obtaining the quantum loop contributions directly from the semi - classical tree - level amplitudes by using @xmath0-dimensional unitarity @xcite . these same methods have also been applied to non - trivial calculations in quantum chromodynamics ( see _ e.g. _ refs . @xcite ) and in supersymmetric gauge theories ( see _ e.g. _ refs . @xcite ) . in a sense , they provide a means for obtaining collections of quantum loop - level feynman diagrams without direct reference to the underlying lagrangian or hamiltonian . the only inputs with this method are the @xmath0-dimensional tree - level scattering amplitudes . this makes the unitarity method ideally suited for exploiting the klt relations . an interesting application of this method of perturbatively quantizing gravity is as a tool for investigating the ultra - violet behavior of gravity field theories . ultraviolet properties are one of the central issues of perturbative quantum gravity . the conventional wisdom that quantum field theories of gravity can not possibly be fundamental rests on the apparent non - renormalizability of these theories . simple power counting arguments strongly suggest that einstein gravity is not renormalizable and therefore can be viewed only as a low energy effective field theory . indeed , explicit calculations have established that non - supersymmetric theories of gravity with matter generically diverge at one loop @xcite , and pure gravity diverges at two loops @xcite . supersymmetric theories are better behaved with the first potential divergence occurring at three loops @xcite . however , no explicit calculations have as yet been performed to directly verify the existence of the three - loop supergravity divergences . the method described here for quantizing gravity is well suited for addressing the issue of the ultraviolet properties of gravity because it relates overwhelmingly complicated calculations in quantum gravity to much simpler ( though still complicated ) ones in gauge theories . the first application was for the case of maximally supersymmetric gravity , which is expected to have the best ultra - violet properties of any theory of gravity . this analysis led to the surprising result that maximally supersymmetric gravity is less divergent @xcite than previously believed based on power counting arguments @xcite . this lessening of the power counting degree of divergence may be interpreted as an additional symmetry unaccounted for in the original analysis @xcite . ( the results are inconsistent , however , with an earlier suggestion @xcite based on the speculated existence of an unconstrained covariant off - shell superspace for @xmath3 supergravity , which in @xmath4 implies finiteness up to seven loops . the non - existence of such a superspace was already noted a while ago @xcite . ) the method also led to the explicit construction of the two - loop divergence in eleven - dimensional supergravity @xcite . more recently , it aided the study of divergences in type i supergravity theories @xcite where it was noted that they factorize into products of gauge theory factors . other applications include the construction of infinite sequences of amplitudes in gravity theories . given the complexity of gravity perturbation theory , it is rather surprising that one can obtain compact expressions for an arbitrary number of external legs , even for restricted helicity or spin configurations of the particles . the key for this construction is to make use of previously known sequences in quantum chromodynamics . at tree - level , infinite sequences of maximally helicity violating amplitudes have been obtained by directly using the klt relations @xcite and analogous quantum chromodynamics sequences . at one loop , by combining the klt relations with the unitarity method , additional infinite sequences of gravity and super - gravity amplitudes have also been obtained @xcite . they are completely analogous to and rely on the previously obtained infinite sequences of one - loop gauge theory amplitudes @xcite . these amplitudes turn out to be also intimately connected to those of self - dual yang - mills @xcite and gravity @xcite . the method has also been used to explicitly compute two - loop supergravity amplitudes @xcite in dimension @xmath5 , that were then used to check m - theory dualities @xcite . although the klt relations have been exploited to obtain non - trivial results in quantum gravity theories , a derivation of these relations from the einstein - hilbert lagrangian is lacking . there has , however , been some progress in this regard . it turns out that with an appropriate choice of field variables one can separate the space - time indices appearing in the lagrangian into ` left ' and ` right ' classes @xcite , mimicking the similar separation that occurs in string theory . moreover , with further field redefinitions and a non - linear gauge choice , it is possible to arrange the off - shell three - graviton vertex so that it is expressible in terms of a sum of squares of yang - mills three - gluon vertices @xcite . it might be possible to extend this more generally starting from the formalism of siegel @xcite , which contains a complete gravity lagrangian with the required factorization of space - time indices . this review is organized as follows . in section [ section : traditional_approach ] the feynman diagram approach to perturbative quantum gravity is outlined . the kawai , lewellen and tye relations between open and closed string tree amplitudes and their field theory limit are described in section [ section : klt_relations ] . applications to understanding and constructing tree - level gravity amplitudes are also described in this section . in section [ section : einsteinhilbert ] the implications for the einstein - hilbert lagrangian are presented . the procedure for obtaining quantum loop amplitudes from gravity tree amplitudes is then given in section [ section : trees_to_loops ] . the application of this method to obtain quantum gravity loop amplitudes is described in section [ section : gravity_loops ] . in section [ section : divergence_properties ] the quantum divergence properties of maximally supersymmetric supergravity obtained from this method are described . the conclusions are found in section [ section : conclusions ] . there are a number of excellent sources for various subtopics described in this review . for a recent review of the status of quantum gravity the reader may consult the article by carlip @xcite . the conventional feynman diagram approach to quantum gravity can be found in the les houches lectures of veltman @xcite . a review article containing an early version of the method described here of using unitarity to construct complete loop amplitudes is ref . excellent reviews containing the quantum chromodynamics amplitudes used to obtain corresponding gravity amplitudes are the ones by mangano and parke @xcite and by lance dixon @xcite . these reviews also provide a good description of helicity techniques which are extremely useful for explicitly constructing scattering amplitude in gravity and gauge theories . broader textbooks describing quantum chromodynamics are refs . chapter 7 of _ superstring theory _ by green , schwarz , and witten @xcite contains an illuminating discussion of the relationship of closed and open string tree amplitudes , especially at the four - point level . a somewhat more modern description of string theory may be found in the book by polchinski @xcite . applications of string methods to quantum field theory are described in a recent review by schubert @xcite . scattering of gravitons in flat space may be described using feynman diagrams @xcite . the feynman rules for constructing the diagrams are obtained from the einstein - hilbert lagrangian coupled to matter using standard procedures of quantum field theory . ( the reader may consult any of the textbooks on quantum field theory @xcite for a derivation of the feynman rules starting from a given lagrangian . ) for a good source describing the feynman rules of gravity , the reader may consult the classic lectures of veltman @xcite . # 1#21.0#1 the momentum - space feynman rules are expressed in terms of vertices and propagators as depicted in fig . [ figure : gravityfeynman ] . in the figure , space - time indices are denoted by @xmath6 and @xmath7 while the momenta are denoted by @xmath8 or @xmath9 . in contrast to gauge theory , gravity has an infinite set of ever more complicated interaction vertices ; the three- and four - point ones are displayed in the figure . the diagrams for describing scattering of gravitons from each other are built out of these propagators and vertices . other particles can be included in this framework by adding new propagators and vertices associated with each particle type . ( for the case of fermions coupled to gravity the lagrangian needs to be expressed in terms of the vierbein instead of the metric before the feynman rules can be constructed . ) according to the feynman rules , each leg or vertex represents a specific algebraic expression depending on the choice of field variables and gauges . for example , the graviton feynman propagator in the commonly used de donder gauge is : @xmath10 { i\over k^2 + i\epsilon}\ , . \label{gravitypropagator}\ ] ] the three - vertex is much more complicated and the expressions may be found in dewitt s articles @xcite or in veltman s lectures @xcite . for simplicity , only a few of the terms of the three - vertex are displayed : 0.14 em @xmath11 where the indices associated with each graviton are depicted in the three - vertex of fig.[figure : gravityfeynman ] , _ i.e. _ , the two indices of graviton @xmath12 are @xmath13 . # 1#2.5#1 # 1#2.5#1 the loop expansion of feynman diagrams provide a systematic quantum mechanical expansion in planck s constant @xmath14 . the tree - level diagrams such as those in fig . [ figure : gravitytrees ] are interpreted as ( semi ) classical scattering processes while the diagrams with loops are the true quantum mechanical effects : each loop carries with it a power of @xmath14 . according to the feynman rules , each loop represents an integral over the momenta of the intermediate particles . the behavior of these loop integrals is the key for understanding the divergences of quantum gravity . in general , the loop momentum integrals in a quantum field theory will diverge in the ultraviolet where the momenta in the loops become arbitrarily large . unless these divergences are of the right form they indicate that a theory can not be interpreted as fundamental , but is instead valid only at low energies . gauge theories such as quantum chromodynamics are renormalizable : divergences from high energy scales can be absorbed into redefinitions of the original parameters appearing in the theory . in quantum gravity , on the other hand , it is not possible to re - absorb divergences in the original lagrangian for a very simple reason : the gravity coupling @xmath15 , where @xmath16 is newton s constant , carries dimensions of length ( in units where @xmath17 ) . by dimensional analysis , any divergence must be proportional to terms with extra derivatives compared to the original lagrangian and are thus of a different form . this may be contrasted to the gauge theory situation where the coupling constant is dimensionless , allowing for the theory to be renormalizable . the problem of non - renormalizability of quantum gravity does not mean that quantum mechanics is incompatible with gravity , only that quantum gravity should be treated as an effective field theory @xcite for energies well below the planck scale of @xmath1 gev ( which is , of course , many orders of magnitude beyond the reach of any conceivable experiment ) . in an effective field theory , as one computes higher loop orders new and usually unknown couplings need to be introduced to absorb the divergences . generally , these new couplings are suppressed at low energies by ratios of energy to the fundamental high energy scale , but at sufficiently high energies the theory loses its predictive power . in quantum gravity this happens at the planck scale . quantum gravity based on the feynman diagram expansion allows for a direct investigation of the non - renormalizability issue . for a theory of pure gravity with no matter , amazingly , the one - loop divergences cancel , as demonstrated by t hooft and veltman @xcite . unfortunately , this result is `` accidental , '' since it does not hold generically when matter is added to the theory or when the number of loops is increased . explicit calculations have shown that non - supersymmetric theories of gravity with matter generically diverge at one loop @xcite , and pure gravity diverges at two loops @xcite . the two - loop calculations were performed using various improvements to the feynman rules such as the background field method @xcite . supersymmetric theories of gravity are known to have less severe divergences . in particular , in any four - dimensional supergravity theory , supersymmetry ward identities @xcite forbid all possible one - loop @xcite and two - loop @xcite divergences . there is a candidate divergence at three loops for all supergravities including the maximally extended @xmath3 version @xcite . however , no explicit three - loop ( super ) gravity calculations have been performed to confirm the divergence . in principle it is possible that the coefficient of a potential divergence obtained by power counting can vanish , especially if the full symmetry of the theory is taken into account . as described in section [ section : divergence_properties ] , this is precisely what does appear to happen @xcite in the case of maximally supersymmetric supergravity . the reason no direct calculation of the three - loop supergravity divergences has been performed is the overwhelming technical difficulties associated with multi - loop gravity feynman diagrams . in multi - loop calculations the number of algebraic terms proliferates rapidly beyond the point where computations are practical . as a particularly striking example , consider the five - loop diagram in fig . [ figure : multiloop ] , which , as noted in section [ section : divergence_properties ] , is of interest for ultraviolet divergences in maximal @xmath3 supergravity in @xmath4 . in the standard de donder gauge this diagram contains twelve vertices , each of the order of a hundred terms , and sixteen graviton propagators , each with three terms , for a total of roughly @xmath18 terms , even before having evaluated any integrals . this is obviously well beyond what can be implemented on any computer . the standard methods for simplifying diagrams , such as background - field gauges and superspace , are unfortunately insufficient to reduce the problem to anything close to manageable levels . the alternative of using string- based methods that have proven to be useful at one loop and in certain two - loop calculations @xcite also does not as yet provide a practical means for performing multi - loop scattering amplitude calculations @xcite , especially in gravity theories . # 1#20.3#1 the heuristic relation ( [ heuristicformula ] ) suggests a possible way to deal with multi - loop diagrams such as the one in fig . [ figure : multiloop ] by somehow factorizing gravity amplitudes into products of gauge theory ones . since gauge theory feynman rules are inherently much simpler than gravity feynman rules , it clearly would be advantageous to re - express gravity perturbative expansions in terms of gauge theory ones . as a first step , one might , for example , attempt to express the three - graviton vertex as a product of two yang - mills vertices , as depicted in fig . [ figure : threevertex ] : @xmath19 where the two indices of each graviton labeled by @xmath12 are @xmath13 , _ i.e. _ @xmath20 . # 1#20.4#1 such relations , however , do not hold in any of the standard formulations of gravity . for example , the three vertex in the standard de donder gauge ( [ threededonder ] ) contains traces over gravitons , _ i.e. _ a contraction of indices of a single graviton . for physical gravitons the traces vanish , but for gravitons appearing inside feynman diagrams it is in general crucial to keep such terms . a necessary condition for obtaining a factorizing three - graviton vertex ( [ threefactorizing ] ) is that the `` left '' @xmath6 indices never contract with the `` right '' @xmath7 indices . this is clearly violated by the three - vertex in eq . ( [ threededonder ] ) . indeed , the standard formulations of quantum gravity generate a plethora of terms that violate the heuristic relation ( [ heuristicformula ] ) . in section [ section : einsteinhilbert ] the question of how one rearranges the einstein action to be compatible with string theory intuition is returned to . however , in order to give a precise meaning to the heuristic formula ( [ heuristicformula ] ) and to demonstrate that scattering amplitudes in gravity theories can indeed be obtained from standard gauge theory ones , a completely different approach from the standard lagrangian or hamiltonian ones is required . this different approach is described in the next section . our starting point for constructing perturbative quantum gravity is the kawai , lewellen , and tye ( klt ) relations @xcite between closed and open string tree - level amplitudes . since closed string theories are theories of gravity and open string theories include gauge bosons , in the low energy limit , where string theory reduces to field theory , these relations then necessarily imply relations between gravity and gauge theories . the realization that ordinary gauge and gravity field theories emerge from the low energy limit of string theories has been appreciated for nearly three decades . ( see , for example , refs . @xcite ) . the klt relations between open and closed string theory amplitudes can be motivated by the observation is that any closed string vertex operator for the emission of a closed string state ( such as a graviton ) is a product of open string vertex operators ( see _ e.g. _ ref . @xcite ) , @xmath21 this product structure is then reflected in the amplitudes . indeed , the celebrated koba - nielsen form of string amplitudes @xcite , which may be obtained by evaluating correlations of the vertex operators , factorize at the level of the integrands before world sheet integrations are performed . amazingly , kawai , lewellen , and tye were able to demonstrate a much stronger factorization : complete closed string amplitudes factorize into products of open string amplitudes , even _ after _ integration over the world sheet variables . ( a description of string theory scattering amplitudes and the history of their construction may be found in standard books on string theory @xcite . ) as a simple example of the factorization property of string theory amplitudes , the four - point partial amplitude of open superstring theory for scattering any of the massless modes is given by @xmath22 where @xmath23 is the open string regge slope proportional to the inverse string tension , @xmath24 is the gauge theory coupling , and @xmath25 is a gauge invariant kinematic coefficient depending on the momenta @xmath26 . explicit forms of @xmath25 may be found in ref . ( the metric is taken here to have signature @xmath27 . ) in this and subsequent expressions , @xmath28 , @xmath29 and @xmath30 . the indices can be either vector , spinor or group theory indices and the @xmath31 can be vector polarizations , spinors , or group theory matrices , depending on the particle type . these amplitudes are the open string partial amplitudes before they are dressed with chan - paton @xcite group theory factors and summed over non - cyclic permutations to form complete amplitudes . ( any group theory indices in eq . ( [ openstringfourpoint ] ) are associated with string world sheet charges arising from possible compactifications . ) for the case of a vector , @xmath31 is the usual polarization vector . similarly , the four - point amplitudes corresponding to a heterotic closed superstring @xcite are , 0.14 em @xmath32 where @xmath23 is the open string regge slope or equivalently twice the close string one . up to prefactors , the replacements @xmath33 and substituting @xmath34 , the closed string amplitude ( [ closedstringfourpoint ] ) is a product of the open string partial amplitudes ( [ openstringfourpoint ] ) . for the case of external gravitons the @xmath35 are ordinary graviton polarization tensors . for further reading , chapter 7 of _ superstring theory _ by green , schwarz , and witten @xcite provides an especially enlightening discussion of the four - point amplitudes in various string constructions . as demonstrated by klt , the property that closed string tree amplitudes can be expressed in terms of products of open string tree amplitudes is completely general for any string states and for any number of external legs . in general , it holds also for each of the huge number of possible string compactifications @xcite . an essential part of the factorization of the amplitudes is that any closed - string state is a direct product of two open - string states . this property directly follows from the factorization of the closed - string vertex operators ( [ closedvertex ] ) into products of open- string vertex operators . in general , for every closed - string state there is a fock space decomposition @xmath36 in the low energy limit this implies that states in a gravity field theory obey a similar factorization , @xmath37 for example , in four dimensions each of the two physical helicity states of the graviton are given by the direct product of two vector boson states of identical helicity . the cases where the vectors have opposite helicity correspond to the antisymmetric tensor and dilaton . similarly , a spin 3/2 gravitino state , for example , is a direct product of a spin 1 vector and spin 1/2 fermion . note that decompositions of this type are not especially profound for free field theory and amount to little more than decomposing higher spin states as direct products of lower spin ones . what is profound is that the factorization holds for the full non - linear theory of gravity . the fact that the klt relations hold for the extensive variety of compactified string models @xcite implies that they should also be generally true in field theories of gravity . for the cases of four- and five - particle scattering amplitudes , in the field theory limit the klt relations @xcite reduce to : @xmath38 0.14 em @xmath39 where the @xmath40 s are tree - level amplitudes in a gravity theory , the @xmath41 s are color - stripped tree - level amplitudes in a gauge theory and @xmath42 . in these equations the polarization and momentum labels are suppressed , but the label `` @xmath43 '' is kept to distinguish the external legs . the coupling constants have been removed from the amplitudes , but are reinserted below in eqs . ( [ fullgaugetheory ] ) and ( [ gravitycouplingamplitude ] ) . an explicit generalization to @xmath44-point field theory gravity amplitudes may be found in appendix a of ref . the klt relations before the field theory limit is taken may , of course , be found in the original paper @xcite . the klt equations generically hold for any closed string states , using their fock space factorization into pairs of open string states . although not obvious , the gravity amplitudes ( [ kltfourpoint ] ) and ( [ kltfivepoint ] ) have all the required symmetry under interchanges of identical particles . ( this is easiest to demonstrate in string theory by making use of an @xmath45 symmetry on the string world sheet . ) in the field theory limit the klt equations must hold in any dimension , because the gauge theory amplitudes appearing on the right - hand - side have no explicit dependence on the space - time dimension ; the only dependence is implicit in the number of components of momenta or polarizations . moreover , if the equations hold in , say , ten dimensions , they must also hold in all lower dimensions since one can truncate the theory to a lower dimensional subspace . the amplitudes on the left - hand side of eqs . ( [ kltfourpoint ] ) and ( [ kltfivepoint ] ) are exactly the scattering amplitudes that one obtains via standard gravity feynman rules @xcite . the gauge theory amplitudes on the right - hand - side may be computed via standard feynman rules available in any modern textbook on quantum field theory @xcite . after computing the full gauge theory amplitude , the color - stripped partial amplitudes @xmath41 appearing in the klt relations ( [ kltfourpoint ] ) and ( [ kltfivepoint ] ) , may then be obtained by expressing the full amplitudes in a color trace basis @xcite : @xmath46 where the sum runs over the set of all permutations , but with cyclic rotations removed and @xmath24 as the gauge theory coupling constant . the @xmath41 partial amplitudes that appear in the klt relations are defined as the coefficients of each of the independent color traces . in this formula , the @xmath47 are fundamental representation matrices for the yang - mills gauge group @xmath48 , normalized so that @xmath49 . note that the @xmath41 are completely independent of the color and are the same for any value of @xmath50 . ( [ fullgaugetheory ] ) is quite similar to the way full open string amplitudes are expressed in terms of the string partial amplitudes by dressing them with chan - paton color factors @xcite . instead , it is somewhat more convenient to use color - ordered feynman rules @xcite since they directly give the @xmath41 color - stripped gauge theory amplitudes appearing in the klt equations . these feynman rules are depicted in fig . [ figure : rules ] . when obtaining the partial amplitudes from these feynman rules it is essential to order the external legs following the order appearing in the corresponding color trace . one may view the color - ordered gauge theory rules as a new set of feynman rules for gravity theories at tree level , since the klt relations allow one to convert the obtained diagrams to tree - level gravity amplitudes @xcite as follows : # 1#20.8#1 to obtain the full amplitudes from the klt relations in eqs . ( [ kltfourpoint ] ) , ( [ kltfivepoint ] ) and their @xmath44-point generalization , the couplings need to be reinserted . in particular , when all states couple gravitationally , the full gravity amplitudes including the gravitational coupling constant are : @xmath51 where @xmath52 expresses the coupling @xmath53 in terms of newton s constant @xmath16 . in general , the precise combination of coupling constants depends on how many of the interactions are gauge or other interactions and how many are gravitational . for the case of four space - time dimensions , it is very convenient to use helicity representation for the physical states @xcite . with helicity amplitudes the scattering amplitudes in either gauge or gravity theories are , in general , remarkably compact , when compared with expressions where formal polarization vectors or tensors are used . for each helicity , the graviton polarization tensors satisfy a simple relation to gluon polarization vectors : @xmath54 the @xmath55 are essentially ordinary circular polarization vectors associated with , for example , circularly polarized light . the graviton polarization tensors defined in this way automatically are traceless , @xmath56 , because the gluon helicity polarization vectors satisfy @xmath57 . they are also transverse , @xmath58 , because the gluon polarization vectors satisfy @xmath59 , where @xmath60 is the four momentum of either the graviton or gluon . using a helicity representation @xcite , berends , giele , and kuijf ( bgk ) @xcite were the first to exploit the klt relations to obtain amplitudes in einstein gravity . in quantum chromodynamics ( qcd ) an infinite set of helicity amplitudes known as the parke - taylor amplitudes @xcite were already known . these maximally helicity violating ( mhv ) amplitudes describe the tree - level scattering of @xmath44 gluons when all gluons but two have the same helicity , treating all particles as outgoing . ( the tree amplitudes in which all or all but one of the helicities are identical vanish . ) bgk used the klt relations to directly obtain graviton amplitudes in pure einstein gravity , using the known qcd results as input . remarkably , they also were able to obtain a compact formula for @xmath44-graviton scattering with the special helicity configuration in which two legs are of opposite helicity from the remaining ones . as a particularly simple example , the color - stripped four - gluon tree amplitude with two minus helicities and two positive helicities in qcd is given by @xmath61 @xmath62 where the @xmath24 subscripts signify that the legs are gluons and the @xmath63 superscripts signify the helicities . with the conventions used here , helicities are assigned by treating all particles as outgoing . ( this differs from another common choice which is to keep track of which particles are incoming and which are outgoing . ) in these amplitudes , for simplicity , overall phases have been removed . the gauge theory partial amplitude in eq . ( [ fourgluonamplitude ] ) may be computed using the color - ordered feynman diagrams depicted in fig . [ figure : gluon ] . the diagrams for the partial amplitude in eq . ( [ fourgluonamplitudeb ] ) are similar except that the labels for legs 3 and 4 are interchanged . although qcd contains fermion quarks , they do not contribute to tree amplitudes with only external gluon legs because of fermion number conservation ; for these amplitudes qcd is entirely equivalent to pure yang - mills theory . # 1#20.9#1 the corresponding four - graviton amplitude follows from the klt equation ( [ kltfourpoint ] ) . after including the coupling from eq . ( [ gravitycouplingamplitude ] ) , the four - graviton amplitude is : @xmath64 where the subscript @xmath65 signifies that the particles are gravitons and , as with the gluon amplitudes , overall phases are removed . as for the case of gluons , the @xmath63 superscripts signify the helicity of the graviton . this amplitude necessarily must be identical to the result for pure einstein gravity with no other fields present , because any other states , such as an anti - symmetric tensor , dilaton , or fermion , do not contribute to @xmath44-graviton tree amplitudes . the reason is similar to the reason why the quarks do not contribute to pure glue tree amplitudes in qcd . these other physical states contribute only when they appear as an external state , because they couple only in pairs to the graviton . indeed , the amplitude ( [ gravityexample ] ) is in complete agreement with the result for this helicity amplitude obtained by direct diagrammatic calculation using the pure gravity einstein - hilbert action as the starting point @xcite ( and taking into account the different conventions for helicity ) . the klt relations are not limited to pure gravity amplitudes . cases of gauge theory coupled to gravity have also been discussed in ref . for example , by applying the feynman rules in fig . [ figure : rules ] , one can obtain amplitudes for gluon amplitudes dressed with gravitons . a sampling of these , to leading order in the graviton coupling , is : @xmath66 for the coefficients of the color traces @xmath67 $ ] following the ordering of the gluon legs . again , for simplicity , overall phases are eliminated from the amplitudes . ( in ref . @xcite mixed graviton matter amplitudes including the phases may be found . ) these formulae have been generalized to infinite sequences of maximally helicity - violating tree amplitudes for gluon amplitudes dressed by external gravitons . the first of these were obtained by selivanov using a generating function technique @xcite . another set was obtained using the klt relations to find the pattern for an arbitrary number of legs @xcite . in doing this , it is extremely helpful to make use of the analytic properties of amplitudes as the momenta of various external legs become soft ( _ i.e. _ @xmath68 ) or collinear ( _ i.e. _ @xmath9 parallel to @xmath69 ) , as discussed in the next subsection . cases involving fermions have not been systematically studied , but at least for the case with a single fermion pair the klt equations can be directly applied using the feynman rules in fig . [ figure : rules ] , without any modifications . for example , in a supergravity theory , the scattering of a gravitino by a graviton is @xmath70 where the subscript @xmath71 signifies a spin 3/2 gravitino and @xmath72 signifies a spin 1/2 gluino . as a more subtle example , the scattering of fundamental representation quarks by gluons via graviton exchange also has a klt factorization : @xmath73 where @xmath74 and @xmath75 are distinct massless fermions . in this equation , the gluons are factorized into products of fermions . on the right - hand side the group theory indices @xmath76 are interpreted as global flavor indices but on the left - hand side they should be interpreted as color indices of local gauge symmetry . as a check , in ref . @xcite , for both amplitudes ( [ gavitinoampl ] ) and ( [ qqggexchange ] ) , ordinary gravity feynman rules were used to explicitly verify that the expressions for the amplitudes are correct . cases with multiple fermion pairs are more involved . in particular , for the klt factorization to work in general , auxiliary rules for assigning global charges in the color - ordered amplitudes appear to be necessary . this is presumably related to the intricacies associated with fermions in string theory @xcite . when an underlying string theory does exist , such as for the case of maximal supergravity discussed in section [ section : divergence_properties ] , then the klt equations necessarily must hold for all amplitudes in the field theory limit . the above examples , however , demonstrate that the klt factorization of amplitudes is not restricted only to the cases where an underlying string theory exists . the analytic properties of gravity amplitudes as momenta become either soft @xmath77 or collinear ( @xmath69 parallel to @xmath69 ) are especially interesting because they supply a simple demonstration of the tight link between the two theories . moreover , these analytic properties are crucial for constructing and checking gravity amplitudes with an arbitrary number of external legs . the properties as gravitons become soft have been known for a long time @xcite but the collinear properties were first obtained using the known collinear properties of gauge theories together with the klt relations . helicity amplitudes in quantum chromodynamics have a well - known behavior as momenta of external legs become collinear or soft @xcite . for the collinear case , at tree - level in quantum chromodynamics when two nearest neighboring legs in the color - stripped amplitudes become collinear , _ e.g. , _ @xmath78 , @xmath79 , and @xmath80 , the amplitude behaves as @xcite : @xmath81 the function @xmath82 is a splitting amplitude , and @xmath83 is the helicity of the intermediate state @xmath84 . ( the other helicity labels are implicit . ) the contribution given in eq . ( [ ymcollinear ] ) is singular for @xmath85 parallel to @xmath86 ; other terms in the amplitude are suppressed by a power of @xmath87 , which vanishes in the collinear limit , compared to the ones in eq . ( [ ymcollinear ] ) . for the pure glue case , one such splitting amplitude is @xmath88 where @xmath89 = -\sqrt{s_{ij}}\ ; e^{-i \phi_{jl}}\ , , \label{spinorsdefs}\ ] ] are spinor inner products , and @xmath90 is a momentum - dependent phase that may be found in , for example , ref . @xcite . in general , it is convenient to express splitting amplitudes in terms of these spinor inner products . the ` @xmath91 ' and ` @xmath92 ' labels refer to the helicity of the outgoing gluons . since the spinor inner products behave as @xmath93 , the splitting amplitudes develop square - root singularities in the collinear limits . if the two collinear legs are not next to each other in the color ordering , then there is no singular contribution , _ e.g. _ no singularity develops in @xmath94 for @xmath85 collinear to @xmath95 . from the structure of the klt relations it is clear that a universal collinear behavior similar to eq . ( [ ymcollinear ] ) must hold for gravity since gravity amplitudes can be obtained from gauge theory ones . the klt relations give a simple way to determine the gravity splitting amplitudes , @xmath96 . the value of the splitting amplitude may be obtained by taking the collinear limit of two of the legs in , for example , the five - point amplitude . taking @xmath85 parallel to @xmath86 in the five - point relation ( [ kltfivepoint ] ) and using eq . ( [ gravitycouplingamplitude ] ) yields : @xmath97 where @xmath98 more explicitly , using eq . ( [ ymsplitexample ] ) then gives : @xmath99 \over \langle { 1}{2}\rangle } \ , . \label{gravtreecollexample}\ ] ] using the klt relations at @xmath44-points , it is not difficult to verify that the splitting behavior is universal for an arbitrary number of external legs , _ i.e. _ : @xmath100 ( since the klt relations are not manifestly crossing - symmetric , it is simpler to check this formula for some legs being collinear rather than others ; at the end all possible combinations of legs must give the same results , though . ) the general structure holds for any particle content of the theory because of the general applicability of the klt relations . in contrast to the gauge theory splitting amplitude ( [ ymsplitexample ] ) , the gravity splitting amplitude ( [ gravtreecollexample ] ) is not singular in the collinear limit . the @xmath101 factor in eq . ( [ gravtreecoll ] ) has canceled the pole . however , a phase singularity remains from the form of the spinor inner products given in eq . ( [ spinorsdefs ] ) , which distinguishes terms with the splitting amplitude from any others . in eq . ( [ spinorsdefs ] ) , the phase factor @xmath102 rotates by @xmath103 as @xmath104 and @xmath105 rotate once around their sum @xmath106 as shown in fig . [ figure : rotcol ] . the ratio of spinors in eq . ( [ gravtreecollexample ] ) then undergoes a @xmath107 rotation accounting for the angular - momentum mismatch of 2@xmath14 between the graviton @xmath108 and the pair of gravitons @xmath109 and @xmath110 . in the gauge theory case , the terms proportional to the splitting amplitudes ( [ ymcollinear ] ) dominate the collinear limit . in the gravitational formula ( [ gravcollinear ] ) , there are other terms of the same magnitude as @xmath111/ \langle{1}{2}\rangle$ ] as @xmath112 . however , these non - universal terms do not acquire any additional phase as the collinear vectors @xmath104 and @xmath105 are rotated around each other . thus , they can be separated from the universal terms . the collinear limit of any gravity tree amplitude must contain the universal terms given in eq . ( [ gravcollinear ] ) thereby putting a severe restriction on the analytic structure of the amplitudes . # 1#20.6#1 even for the well - studied case of momenta becoming soft one may again use the klt relation to extract the behavior and to rewrite it in terms of the soft behavior of gauge theory amplitudes . gravity tree amplitudes have the well known behavior @xcite , @xmath113 as the momentum of graviton @xmath44 , becomes soft . ( [ gravtreesoft ] ) the soft graviton is taken to carry positive helicity ; parity can be used to obtain the other helicity case . one can obtain the explicit form of the soft factors directly from the klt relations , but a more symmetric looking soft factor can be obtained by first expressing the three - graviton vertex in terms of a yang - mills three vertex @xcite . ( see eq . ( [ berngrantvertex ] ) . ) this three - vertex can then be used to directly construct the soft factor . the result is a simple formula expressing the universal function describing soft gravitons in terms of the universal functions describing soft gluons @xcite : @xmath114 where @xmath115 is the eikonal factor for a positive helicity soft gluon in qcd labeled by @xmath44 , and @xmath116 and @xmath117 are labels for legs neighboring the soft gluon . in eq . ( [ finalsoftgrav ] ) the momenta @xmath118 and @xmath119 are arbitrary null `` reference '' momenta . although not manifest , the soft factor ( [ finalsoftgrav ] ) is independent of the choices of these reference momenta . by choosing @xmath120 and @xmath121 the form of the soft graviton factor for @xmath122 used in , for example , refs . @xcite is recovered . the important point is that in the form ( [ finalsoftgrav ] ) , the graviton soft factor is expressed directly in terms of the qcd gluon soft factor . since the soft amplitudes for gravity are expressed in terms of gauge theory ones , the probability of emitting a soft graviton can also be expressed in terms of the probability of emitting a soft gluon . one interesting feature of the gravitational soft and collinear functions is that , unlike the gauge theory case , they do not suffer any quantum corrections @xcite . this is due to the dimensionful nature of the gravity coupling @xmath53 , which causes any quantum corrections to be suppressed by powers of a vanishing kinematic invariant . the dimensions of the coupling constant must be absorbed by additional powers of the kinematic invariants appearing in the problem , which all vanish in the collinear or soft limits . this observation is helpful because it can be used to put severe constraints on the analytic structure of gravity amplitudes at any loop order . consider the einstein - hilbert and yang - mills lagrangians , @xmath123 where @xmath124 is the usual scalar curvature and @xmath125 is the yang - mills field strength . an inspection of these two lagrangians does not reveal any obvious factorization property that might explain the klt relations . indeed , one might be tempted to conclude that the klt equations could not possibly hold in pure einstein gravity . however , although somewhat obscure , the einstein - hilbert lagrangian can in fact be rearranged into a form that is compatible with the klt relations ( as argued in this section ) . of course , there should be such a rearrangement , given that in the low energy limit pure graviton tree amplitudes in string theory should match those of einstein gravity . all other string states either decouple or can not enter as intermediate states in pure graviton amplitudes because of conservation laws . indeed , explicit calculations using ordinary gravity feynman rules confirm this to be true @xcite . ( in loops , any state of the string that survives in the low energy limit will in fact contribute , but in this section only tree amplitudes are being considered . ) one of the key properties exhibited by the klt relations ( [ kltfourpoint ] ) and ( [ kltfivepoint ] ) is the separation of graviton space - time indices into ` left ' and ` right ' sets . this is a direct consequence of the factorization properties of closed strings into open strings . consider the graviton field , @xmath126 . we define the @xmath127 index be a `` left '' index and the @xmath128 index to be a `` right '' one . in string theory , the `` left '' space - time indices would arise from the world - sheet left - mover oscillator and the `` right '' ones from the right - mover oscillators . of course , since @xmath126 is a symmetric tensor it does not matter which index is assigned to the left or to the right . in the klt relations each of the two indices of a graviton are associated with two distinct gauge theories . for convenience , we similarly call one of the gauge theories the `` left '' one and the other the `` right '' one . since the indices from each gauge theory can never contract with the indices of the other gauge theory , it must be possible to separate all the indices appearing in a gravity amplitude into left and right classes such that the ones in the left class only contract with left ones and the ones in the right class , only with right ones . this was first noted by siegel , who observed that it should be possible to construct a complete field theory formalism that naturally reflects the left - right string theory factorization of space - time indices . in a set of remarkable papers @xcite , he constructed exactly such a formalism . with appropriate gauge choices , indices separate exactly into `` right '' and `` left '' categories , which do not contract with each other . this does not provide a complete explanation of the klt relations , since one would still need to demonstrate that the gravity amplitudes can be expressed directly in terms of gauge theory ones . nevertheless , this formalism is clearly a sensible starting point for trying to derive the klt relations directly from einstein gravity . hopefully , this will be the subject of future studies , since it may lead to a deeper understanding of the relationship of gravity to gauge theory . a lagrangian with the desired properties could , for example , lead to more general relations between gravity and gauge theory classical solutions . here we outline a more straightforward order - by - order rearrangement of the einstein - hilbert lagrangian , making it compatible with the klt relations @xcite . a useful side - benefit is that this provides a direct verification of the klt relations up to five points starting from the einstein - hilbert lagrangian in its usual form . this is a rather non - trivial direct verification of the klt relations , given the algebraic complexity of the gravity feynman rules . in conventional gauges , the difficulty of factorizing the einstein - hilbert lagrangian into left and right parts is already apparent in the kinetic terms . in de donder gauge , for example , the quadratic part of the lagrangian is @xmath129 so that the propagator is the one given in eq . ( [ gravitypropagator ] ) . although the first term is acceptable since left and right indices do not contract into each other , the appearance of the trace @xmath130 in eq . ( [ quadraticlagrang ] ) is problematic since it contracts a left graviton index with a right one . ( the indices are raised and lowered using the flat space metric @xmath131 and its inverse . ) in order for the kinematic term ( [ quadraticlagrang ] ) to be consistent with the klt equations , all terms which contract a `` left '' space - time index with a `` right '' one need to be eliminated . a useful trick for doing so is to introduce a `` dilaton '' scalar field that can be used to remove the graviton trace from the quadratic terms in the lagrangian . the appearance of the dilaton as an auxiliary field to help rearrange the lagrangian is motivated by string theory , which requires the presence of such a field . following the discussion of refs . @xcite , consider instead a lagrangian for gravity coupled to a scalar : @xmath132 since the auxiliary field @xmath133 is quadratic in the lagrangian , it does not appear in any tree diagrams involving only external gravitons @xcite . it therefore does not alter the tree @xmath134-matrix of purely external gravitons . ( for theories containing dilatons one can allow the dilaton to be an external physical state . ) in de donder gauge , for example , taking @xmath135 , the quadratic part of the lagrangian including the dilaton is : @xmath136 the term involving @xmath130 can be eliminated with the field redefinitions , @xmath137 and @xmath138 yielding @xmath139 one might be concerned that the field redefinition might alter gravity scattering amplitudes . however , because this field redefinition does not alter the trace - free part of the graviton field it can not change the scattering amplitudes of traceless gravitons @xcite . of course , the rearrangement of the quadratic terms is only the first step . in order to make the einstein - hilbert lagrangian consistent with the klt factorization , a set of field variables should exist where all space - time indices can be separated into `` left '' and `` right '' classes . to do so , all terms of the form @xmath140 need to be eliminated since they contract left indices with right ones . a field redefinition that accomplishes this is @xcite : @xmath141 this field redefinition was explicitly checked in ref . @xcite through @xmath142 , to eliminate all terms of the type in eq . ( [ badterms ] ) , before gauge fixing . however , currently there is no formal understanding of why this field variable choice eliminates terms that necessarily contract left and right indices . it turns out that one can do better by performing further field redefinitions and choosing a particular non - linear gauge . the explicit forms of these are a bit complicated and may be found in ref . @xcite . with a particular gauge choice it is possible to express the off - shell three - graviton vertex in terms of yang - mills three vertices : @xmath143\ , , \label{berngrantvertex}\end{aligned}\ ] ] where @xmath144 is the color - ordered gervais - neveu @xcite gauge yang - mills three- vertex , from which the color factor has been stripped . this is not the only possible reorganization of the three - vertex that respects the klt factorization . it just happens to be a particularly simple form of the vertex . for example , another gauge that has a three - vertex that factorizes into products of color - stripped yang - mills three - vertices is the background - field @xcite version of de donder gauge for gravity and feynman gauge for qcd . ( however , background field gauges are meant for loop effective actions and not for tree - level @xmath134-matrix elements . ) interestingly , these gauge choices have a close connection to string theory @xcite . the above ideas represent some initial steps in reorganizing the einstein - hilbert lagrangian so that it respects the klt relations . an important missing ingredient is a derivation of the klt equations starting from the einstein - hilbert lagrangian ( and also when matter fields are present ) . in this section , the above discussion is extended to quantum loops through use of @xmath0-dimensional unitarity @xcite . the klt relations provide gravity amplitudes only at tree level ; @xmath0-dimensional unitarity then provides a means of obtaining quantum loop amplitudes . in perturbation theory this is tantamount to quantizing the theory since the complete scattering matrix can , at least in principle , be systematically constructed this way . amusingly , this bypasses the usual formal apparatus @xcite associated with quantizing constrained systems . more generally , the unitarity method provides a way to systematically obtain the complete set of quantum loop corrections order - by - order in the perturbative expansion whenever the full analytic behavior of tree amplitudes as a function of @xmath0 is known . it always works when the particles in the theory are all massless . the method is well tested in explicit calculations and has , for example , recently been applied to state - of - the - art perturbative qcd loop computations @xcite . in quantum field theory the @xmath134-matrix links initial and final states . a basic physical property is that the @xmath134 matrix must be unitary @xcite : @xmath145 . in perturbation theory the feynman diagrams describe a transition matrix @xmath146 defined by @xmath147 , so that the unitarity condition reads @xmath148 where @xmath149 and @xmath150 are initial and final states , and the `` sum '' is over intermediate states @xmath151 ( and includes an integral over intermediate on - mass - shell momenta ) . perturbative unitarity consists of expanding both sides of eq . ( [ basicunitarity ] ) in terms of coupling constants , @xmath24 for gauge theory and @xmath53 for gravity , and collecting terms of the same order . for example , the imaginary ( or absorptive ) parts of one - loop four - point amplitudes , which is order @xmath152 in gravity , are given in terms of the product of two four - point tree amplitudes , each carrying a power of @xmath153 . this is then summed over all two - particle states that can appear and integrated over the intermediate phase space . ( see fig . [ figure : twoparticle ] . ) # 1#20.9#1 this provides a means of obtaining loop amplitudes from tree amplitudes . however , if one were to directly apply eq . ( [ basicunitarity ] ) in integer dimensions one would encounter a difficulty with fully reconstructing the loop scattering amplitudes . since eq . ( [ basicunitarity ] ) gives only the imaginary part one then needs to reconstruct the real part . this is traditionally done via dispersion relations , which are based on the analytic properties of the @xmath134 matrix @xcite . however , the dispersion integrals do not generally converge . this leads to a set of subtraction ambiguities in the real part . these ambiguities are related to the appearance of rational functions with vanishing imaginary parts , @xmath154 , where the @xmath155 are the kinematic variables for the amplitude . a convenient way to deal with this problem @xcite is to consider unitarity in the context of dimensional regularization @xcite . by considering the amplitudes as an analytic function of dimension , at least for a massless theory , these ambiguities are not present , and the only remaining ambiguities are the usual ones associated with renormalization in quantum field theory . the reason there can be no ambiguity relative to feynman diagrams follows from simple dimensional analysis for amplitudes in dimension @xmath156 . with dimensional regularization , amplitudes for massless particles necessarily acquire a factor of @xmath157 for each loop , from the measure @xmath158 . for small @xmath159 , @xmath160 , so every term has an imaginary part ( for some @xmath161 ) , though not necessarily in terms which survive as @xmath162 . thus , the unitarity cuts evaluated to @xmath163 provide sufficient information for the complete reconstruction of an amplitude . furthermore , by adjusting the specific rules for the analytic continuation of the tree amplitudes to @xmath0-dimensions one can obtain results in the different varieties of dimensional regularization , such as the conventional one @xcite , the t hooft - veltman scheme @xcite , dimensional reduction @xcite , and the four - dimensional helicity scheme @xcite . it is useful to view the unitarity - based technique as an alternate way of evaluating sets of ordinary feynman diagrams by collecting together gauge - invariant sets of terms containing residues of poles in the integrands corresponding to those of the propagators of the cut lines . this gives a region of loop - momentum integration where the cut loop momenta go on shell and the corresponding internal lines become intermediate states in a unitarity relation . from this point of view , even more restricted regions of loop momentum integration may be considered , where additional internal lines go on mass shell . this amounts to imposing cut conditions on additional internal lines . in constructing the full amplitude from the cuts it is convenient to use unrestricted integrations over loop momenta , instead of phase space integrals , because in this way one can obtain simultaneously both the real and imaginary parts . the generalized cuts then allow one to obtain multi - loop amplitudes directly from combinations of tree amplitudes . as a first example , the generalized cut for a one - loop four - point amplitude in the channel carrying momentum @xmath164 , as shown in fig . [ figure : twoparticle ] , is given by : @xmath165 where @xmath166 , and the sum runs over all physical states of the theory crossing the cut . in this generalized cut , the on - shell conditions @xmath167 are applied even though the loop momentum is unrestricted . in addition , any physical state conditions on the intermediate particles should also be included . the real and imaginary parts of the integral functions that do have cuts in this channel are reliably computed in this way . however , the use of the on - shell conditions inside the unrestricted loop momentum integrals does introduce an arbitrariness in functions that do not have cuts in this channel . such integral functions should instead be obtained from cuts in the other two channels . # 1#20.9#1 a less trivial two - loop example of a generalized `` double '' two particle cut is illustrated in fig . [ figure : doubledoubleampl](a ) . the product of tree amplitudes appearing in this cut is : @xmath168 where the loop integrals and cut propagators have been suppressed for convenience . in this expression the on - shell conditions @xmath169 are imposed on the @xmath170 , @xmath171 appearing on the right - hand side . this double cut may seem a bit odd from the traditional viewpoint in which each cut can be interpreted as the imaginary part of the integral . it should instead be understood as a means to obtain part of the information on the structure of the integrand of the two - loop amplitude . namely , it contains the information on all integral functions where the cut propagators are not cancelled . there are , of course , other generalized cuts at two loops . for example , in fig . [ figure : doubledoubleampl](b ) a different arrangement of the cut trees is shown . complete amplitudes are found by combining the various cuts into a single function with the correct cuts in all channels . this method works for any theory where the particles can be taken to be massless and where the tree amplitudes are known as an analytic function of dimension . the restriction to massless amplitudes is irrelevant for the application of studying the ultra - violet divergences of gravity theories . in any case , gravitons and their associated superpartners in a supersymmetric theory are massless . ( for the case with masses present the extra technical complication has to do with the appearance of functions such as @xmath172 which have no cuts in any channel . see ref . @xcite for a description and partial solution of this problem . ) this method has been extensively applied to the case of one- and two - loop gauge theory amplitudes @xcite and has been carefully cross - checked with feynman diagram calculations . here , the method is used to obtain loop amplitudes directly from the gravity tree amplitudes given by the klt equations . in the next section an example of how the method works in practice for the case of gravity is provided . the unitarity method provides a natural means for applying the klt formula to obtain loop amplitudes in quantum gravity , since the only required inputs are tree - level amplitudes valid for @xmath0-dimensions ; this is precisely what the klt relations provide . although einstein gravity is almost certainly not a fundamental theory , there is no difficulty in using it as an effective field theory @xcite , in order to calculate quantum loop corrections . the particular examples discussed in this section are completely finite and therefore do not depend on a cutoff or on unknown coefficients of higher curvature terms in the low energy effective action . they are therefore a definite low energy prediction of _ any _ fundamental theory of gravity whose low energy limit is einstein gravity . ( although they are definite predictions , there is , of course , no practical means to experimentally verify them . ) the issue of divergences is deferred to section [ section : divergence_properties ] . as a simple example of how the unitarity method gives loop amplitudes , consider the one - loop amplitude with four identical helicity gravitons and a scalar in the loop @xcite . the product of tree amplitudes appearing in the @xmath101 channel unitarity cut depicted in fig . [ figure : twoparticle ] is : @xmath173 where the superscript @xmath174 indicates that the cut lines are scalars . the @xmath65 subscripts on legs @xmath175 indicate that these are gravitons , while the `` @xmath91 '' superscripts indicate that they are of plus helicity . from the klt expressions ( [ kltfourpoint ] ) the gravity tree amplitudes appearing in the cuts may be replaced with products of gauge theory amplitudes . the required gauge theory tree amplitudes , with two external scalar legs and two gluons , may be obtained using color - ordered feynman diagrams and are : @xmath176 @xmath177 \,.\ ] ] the external gluon momenta are four dimensional , but the scalar momenta @xmath178 and @xmath179 are @xmath0 dimensional since they will form the loop momenta . in general , loop momenta will have a non - vanishing @xmath180-dimensional component @xmath181 , with @xmath182 . the factors of @xmath183 appearing in the numerators of these tree amplitudes causes them to vanish as the scalar momenta are taken to be four dimensional , though they are non - vanishing away from four dimensions . for simplicity , overall phases have been removed from the amplitudes . after inserting these gauge theory amplitudes in the klt relation ( [ kltfourpoint ] ) , one of the propagators cancels , leaving @xmath184 \,.\ ] ] for this cut , one then obtains a sum of box integrals that can be expressed as : @xmath185 \nonumber \\ & & \hskip 2 cm \times { i \over ( l_1 - k_1 - k_2)^2 } \biggl[{i \over ( l_1 + k_3)^2 } + { i \over ( l_1 + k_4)^2 } \biggr ] \,.\end{aligned}\ ] ] by symmetry , since the helicities of all the external gravitons are identical , the other two cuts also give the same combinations of box integrals , but with the legs permuted . the three cuts can then be combined into a single function that has the correct cuts in all channels yielding @xmath186(s_{12},s_{23 } ) + { \cal i}_4^{\rm 1\ , loop}[\mu^8](s_{12},s_{13 } ) \nonumber \\ & & \null + { \cal i}_4^{\rm 1\ , loop}[\mu^8](s_{23},s_{13 } ) \bigr)\ , , \label{fourgravallplus}\end{aligned}\ ] ] and where @xmath187(s_{12},s_{23 } ) = \int { d^d l \over ( 2\pi)^d } \ , { { \cal p } \over l^2 ( l - k_1)^2 ( l - k_1 - k_2)^2 ( l + k_4)^2 } \ , , \label{oneloopintegral}\ ] ] is the box integral depicted in fig . [ figure : oneloopintegral ] with the external legs arranged in the order 1234 . in eq . ( [ fourgravallplus ] ) @xmath188 is @xmath189 . the two other integrals that appear correspond to the two other distinct orderings of the four external legs . the overall factor of 2 in eq . ( [ fourgravallplus ] ) is a combinatoric factor due to taking the scalars to be complex with two physical states . since the factor of @xmath189 is of @xmath163 , the only non - vanishing contributions come where the @xmath159 from the @xmath189 interferes with a divergence in the loop integral . these divergent contributions are relatively simple to obtain . after extracting this contribution from the integral , the final @xmath4 result for a complex scalar loop , after reinserting the gravitational coupling , is @xmath190 in agreement with a calculation done by a different method relying directly on string theory @xcite . ( as for the previous expressions , the overall phase has been suppressed . ) 1#20.5#1 this result generalizes very simply to the case of any particles in the loop . for any theory of gravity , with an arbitrary matter content one finds : @xmath191 where @xmath192 is the number of physical bosonic states circulating in the loop minus the number of fermionic states . the simplest way to demonstrate this is by making use of supersymmetry ward identities @xcite , which provide a set of simple linear relations between the various contributions showing that they must be proportional to each other . surprisingly , the above four - point results can be extended to an arbitrary number of external legs . using the unitarity methods , the five- and six - point amplitudes with all identical helicity have also been obtained by direct calculation @xcite . then by demanding that the amplitudes have the properties described in section [ subsection : soft_collinear ] for momenta becoming either soft @xcite or collinear @xcite , an ansatz for the one - loop maximally helicity - violating amplitudes for an arbitrary number of external legs has also been obtained . these amplitudes were constructed from a set of building blocks called `` half - soft - function , '' which have `` half '' of the proper behavior as gravitons become soft . the details of this construction and the explicit forms of the amplitudes may be found in refs . @xcite . the all - plus helicity amplitudes turn out to be very closely related to the infinite sequence of one - loop maximally helicity - violating amplitudes in @xmath3 supergravity . the two sequences are related by a curious `` dimension shifting formula . '' in ref . @xcite , a known dimension shifting formula @xcite between identical helicity qcd and @xmath193 super - yang - mills amplitudes was used to obtain the four- , five- , and six - point @xmath3 amplitudes from the identical helicity gravity amplitudes using the klt relations in the unitarity cuts . armed with these explicit results , the soft and collinear properties were then used to obtain an ansatz valid for an arbitrary number of external legs @xcite . this provides a rather non - trivial illustration of how the klt relations can be used to identify properties of gravity amplitudes using known properties of gauge theory amplitudes . interestingly , the all - plus helicity amplitudes are also connected to self - dual gravity @xcite and self - dual yang - mills @xcite , _ i.e. _ gravity and gauge theory restricted to self - dual configurations of the respective field strengths , @xmath194 and @xmath195 , with @xmath196 . this connection is simple to see at the linearized ( free field theory ) level since a superposition of plane waves of identical helicity satisfies the self - duality condition . the self - dual currents and amplitudes have been studied at tree and one - loop levels @xcite . in particular , chalmers and siegel @xcite have presented self - dual actions for gauge theory ( and gravity ) , which reproduce the all - plus helicity scattering amplitudes at both tree and one - loop levels . the ability to obtain exact expressions for gravity loop amplitudes demonstrates the utility of this approach for investigating quantum properties of gravity theories . the next section describes how this can be used to study high energy divergence properties in quantum gravity . in general , the larger the number of supersymmetries , the tamer the ultraviolet divergences because of the tendency for these to cancel between bosons and fermions in a supersymmetric theory . in four - dimensions maximal @xmath3 supergravity may therefore be expected to be the least divergent of all possible supergravity theories . moreover , the maximally supersymmetric gauge theory , @xmath193 super - yang - mills , is completely finite @xcite , leading one to suspect that the superb ultraviolet properties of @xmath193 super - yang - mills would then feed into improved ultra - violet properties for @xmath3 supergravity via its relation to gauge theory . this makes the ultraviolet properties of @xmath3 supergravity the ideal case to investigate first via the perturbative relationship to gauge theory . the maximal @xmath3 supergravity amplitudes can be obtained by applying the klt equations to express them in terms of maximally supersymmetric @xmath193 gauge theory amplitudes . for @xmath3 supergravity , each of the states of the multiplet factorizes into a tensor product of @xmath193 super - yang - mills states , as illustrated in eq . ( [ statefactorization ] ) . applying the klt equation ( [ kltfourpoint ] ) to the product of tree amplitudes appearing in the @xmath101 channel two - particle cuts yields : @xmath197 where the sum on the left - hand side runs over all 256 states in the @xmath3 supergravity multiplet . on the right - hand side the two sums run over the 16 states ( ignoring color degrees of freedom ) of the @xmath193 super - yang - mills multiplet : a gluon , four weyl fermions and six real scalars . the @xmath193 super - yang - mills tree amplitudes turn out to have a particularly simple sewing formula @xcite , @xmath198 which holds in _ any _ dimension ( though some care is required to maintain the total number of physical states at their four - dimensional values so as to preserve the supersymmetric cancellations ) . the simplicity of this result is due to the high degree of supersymmetry . using the gauge theory result ( [ yangmillssewing ] ) , it is a simple matter to evaluate eq . ( [ gravitysewingstart ] ) . this yields : @xmath199 \nonumber \\ & & \hskip 5.03 cm \times \biggl[{1\over ( l_3 - k_3)^2 } + { 1\over ( l_3 - k_4)^2 } \biggr]\ , . \label{basicgravcutting}\end{aligned}\ ] ] the sewing equations for the @xmath200 and @xmath201 kinematic channels are similar to that of the @xmath101 channel . applying eq . ( [ basicgravcutting ] ) at one loop to each of the three kinematic channels yields the one - loop four graviton amplitude of @xmath3 supergravity , @xmath202 in agreement with previous results @xcite . the gravitational coupling @xmath53 has been reinserted into this expression . the scalar integrals are defined in eq . ( [ oneloopintegral ] ) , inserting @xmath203 . this is a standard integral appearing in massless field theories ; the explicit value of this integral may be found in many articles , including refs . this result actually holds for any of the states of @xmath3 supergravity , not just external gravitons . it is also completely equivalent to the result one obtains with covariant feynman diagrams including fadeev - popov @xcite ghosts and using regularization by dimensional reduction @xcite . the simplicity of this result is due to the high degree of supersymmetry . a generic one - loop four - point gravity amplitude can have up to eight powers of loop momenta in the numerator of the integrand ; the supersymmetry cancellations have reduced it to no powers . at two loops , the two - particle cuts are obtained easily by iterating the one - loop calculation , since eq . ( [ basicgravcutting ] ) returns a tree amplitude multiplied by some scalar factors . the three - particle cuts are more difficult to obtain , but again one can `` recycle '' the corresponding cuts used to obtain the two - loop @xmath193 super - yang - mills amplitudes @xcite . it turns out that the three - particle cuts introduce no other functions than those already detected in the two - particle cuts . after all the cuts are combined into a single function with the correct cuts in all channels , the @xmath3 supergravity two - loop amplitude @xcite is : @xmath204 where `` @xmath91 cyclic '' instructs one to add the two cyclic permutations of legs ( 2,3,4 ) . the scalar planar and non - planar loop momentum integrals , @xmath205 and @xmath206 , are depicted in fig . [ figure : planarnonplanar ] . in this expression , all powers of loop momentum have cancelled from the numerator of each integrand in much the same way as at one loop , leaving behind only the feynman propagator denominators . the explicit values of the two - loop scalar integrals in terms of polylogarithms may be found in refs . @xcite . # 1#20.5#1 the two - loop amplitude ( [ twoloopamplitude ] ) has been used by green , kwon , and vanhove @xcite to provide an explicit demonstration of the non - trivial m - theory duality between @xmath5 supergravity and type ii string theory . in this case , the finite parts of the supergravity amplitudes are important , particularly the way they depend on the radii of compactified dimensions . a remarkable feature of the two - particle cutting equation ( [ basicgravcutting ] ) is that it can be iterated to _ all _ loop orders because the tree amplitude ( times some scalar denominators ) reappears on the right - hand - side . although this iteration is insufficient to determine the complete multi - loop four - point amplitudes , it does provide a wealth of information . in particular , for planar integrals it leads to the simple insertion rule depicted in fig . [ figure : insertline ] for obtaining the higher loop contributions from lower loop ones @xcite . this class includes the contribution in fig . [ figure : multiloop ] , because it can be assembled entirely from two - particle cuts . according to the insertion rule , the contribution corresponding to fig . [ figure : multiloop ] is given by loop integrals containing the propagators corresponding to all the internal lines multiplied by a numerator factor containing 8 powers of loop momentum . this is to be contrasted with the 24 powers of loop momentum in the numerator expected when there are no supersymmetric cancellations . this reduction in powers of loop momenta leads to improved divergence properties described in the next subsection . # 1#20.5#1 since the two - loop @xmath3 supergravity amplitude ( [ twoloopamplitude ] ) has been expressed in terms of scalar integrals , it is straightforward to extract the divergence properties . the scalar integrals diverge only for dimension @xmath207 ; hence the two - loop @xmath3 amplitude is manifestly finite in @xmath208 and @xmath209 , contrary to earlier expectations based on superspace power counting @xcite . the discrepancy between the above explicit results and the earlier superspace power counting arguments may be understood in terms of an unaccounted higher dimensional gauge symmetry @xcite . once this symmetry is accounted for , superspace power counting gives the same degree of divergence as the explicit calculation . the cutting methods provide much more than just an indication of divergence ; one can extract the explicit numerical coefficients of the divergences . for example , near @xmath210 the divergence of the amplitude ( [ twoloopamplitude ] ) is : @xmath211 which clearly diverges when the dimensional regularization parameter @xmath212 . in all cases the linearized divergences take the form of derivatives acting on a particular contraction of riemann tensors , which in four dimensions is equivalent to the square of the bel - robinson tensor @xcite . this operator appears in the first set of corrections to the @xmath3 supergravity lagrangian , in the inverse string - tension expansion of the effective field theory for the type ii superstring @xcite . therefore , it has a completion into an @xmath3 supersymmetric multiplet of operators , even at the non - linear level . it also appears in the m - theory one - loop and two - loop effective actions @xcite . interestingly , the manifest @xmath0-independence of the cutting algebra allows the calculation to be extended to @xmath5 , even though there is no corresponding @xmath5 super - yang - mills theory . the result ( [ twoloopamplitude ] ) then explicitly demonstrates that @xmath213 @xmath5 supergravity diverges . in dimensional regularization there are no one - loop divergences so the first potential divergence is at two loops . ( in a momentum cutoff scheme the divergences actually begin at one loop @xcite . ) further work on the structure of the @xmath5 two - loop divergences in dimensional regularization has been carried out in ref . the explicit form of the linearized @xmath213 , @xmath5 counterterm expressed as derivatives acting on riemann tensors along with a more general discussion of supergravity divergences may be found in ref . @xcite . using the insertion rule of fig [ figure : insertline ] , and counting the powers of loop momenta in these contributions leads to the simple finiteness condition : @xmath214 ( with @xmath215 ) , where @xmath216 is the number of loops . this formula indicates that @xmath3 supergravity is finite in some other cases where the previous superspace bounds suggest divergences @xcite , _ @xmath4 , @xmath217 : the first @xmath4 counterterm detected via the two - particle cuts of four - point amplitudes occurs at five , not three loops . further evidence that the finiteness formula is correct stems from the maximally helicity violating contributions to @xmath218-particle cuts , in which the same supersymmetry cancellations occur as for the two - particle cuts @xcite . moreover , a recent improved superspace power count @xcite , taking into account a higher dimensional gauge symmetry , is in agreement with the finiteness formula ( [ finitenessformuala ] ) . further work would be required to prove that other contributions do not alter the two - particle cut power counting . a related open question is whether one can prove that the five - loop @xmath4 divergence encountered in the two - particle cuts does not somehow cancel against other contributions @xcite because of some additional symmetry . it would also be interesting to explicitly demonstrate the non - existence of divergences after including all contributions to the three - loop amplitude . in any case , the explicit calculations using cutting methods do establish that at two loops maximally supersymmetric supergravity does not diverge in @xmath208 @xcite , contrary to earlier expectations from superspace power counting @xcite . this review described how the notion that gravity @xmath219 ( gauge theory ) @xmath220 ( gauge theory ) can be exploited to develop a better understanding of perturbative quantum gravity . the kawai - lewellen - tye ( klt ) string theory relations @xcite gives this notion a precise meaning at the semi - classical or tree level . quantum loop effects may then be obtained by using @xmath0-dimensional unitarity @xcite . in a sense , this provides an alternative method for quantizing gravity , at least in the context of perturbative expansions around flat space . with this method , gauge theory tree amplitudes are converted into gravity tree amplitudes which are then used to obtain gravity loop amplitudes . the ability to carry this out implies that gravity and gauge theory are much more closely related than one might have deduced by an inspection of the respective lagrangians . some concrete applications were also described , including the computation of the two - loop four - point amplitude in maximally supersymmetric supergravity . the result of this and related computations is that maximal supergravity is less divergent in the ultraviolet than had previously been deduced from superspace power counting arguments @xcite . for the case of @xmath4 , maximal supergravity appears to diverge at five instead of three loops . another example for which the relation is useful is for understanding the behavior of gravitons as their momenta become either soft or collinear with the momenta of other gravitons . the soft behavior was known long ago @xcite , but the collinear behavior is new . the klt relations provide a means for expressing the graviton soft and collinear functions directly in terms of the corresponding ones for gluons in quantum chromodynamics . using the soft and collinear properties of gravitons , infinite sequences of maximally helicity violating gravity amplitudes with a single quantum loop were obtained by bootstrapping @xcite from the four- , five- , and six - point amplitudes obtained by direct calculation using the unitarity method together with the klt relations . interestingly , for the case of identical helicity , the sequences of amplitudes turn out to be the same as one gets from self - dual gravity @xcite . there are a number of interesting open questions . using the relationship of gravity to gauge theory one should be able to systematically re - examine the divergence structure of non - maximal theories . some salient work in this direction may be found in ref . @xcite , where the divergences of type i supergravity in @xmath221 were shown to split into products of gauge theory factors . more generally , it should be possible to systematically re - examine finiteness conditions order - by - order in the loop expansion to more thoroughly understand the divergences and associated non - renormalizability of quantum gravity . an important outstanding problem is the lack of a direct derivation of the klt relations between gravity and gauge theory tree amplitudes starting from their respective lagrangians . as yet , there is only a partial understanding in terms of a `` left - right '' factorization of space - time indices @xcite , which is a necessary condition for the klt relations to hold . a more complete understanding may lead to a useful reformulation of gravity where properties of gauge theories can be used to systematically understand properties of gravity theories and vice versa . connected with this is the question of whether the heuristic notion that gravity is a product of gauge theories can be given meaning outside of perturbation theory . in summary , the perturbative relations between gravity and gauge theory provide a new tool for understanding non - trivial properties of quantum gravity . however , further work will be required to unravel fully the intriguing relationship between the two theories . the author thanks abilio de freitas , aaron grant , david dunbar , david kosower , maxim perelstein , joel rozowsky , henry wong , and especially lance dixon for collaboration on work described here and for sharing their insight into quantum gravity . the author also thanks eduardo guendelman for a number of interesting discussions on the einstein - hilbert action and its relation to gauge theory . this work was supported by the us department of energy under grant de - fg03 - 91er40662 . f. w. hehl , j. d. mccrea , e. w. mielke and y. neeman , `` metric affine gauge theory of gravity : field equations , noether identities , world spinors , and breaking of dilation invariance , '' _ phys . _ * 258 * , 1 ( 1995 ) [ arxiv : gr - qc/9402012 ] . z. bern , l. j. dixon , d. c. dunbar , m. perelstein and j. s. rozowsky , `` on the relationship between yang - mills theory and gravity and its implication for ultraviolet divergences , '' _ nucl . _ b * 530 * , 401 ( 1998 ) [ hep - th/9802162 ] . z. bern , l. j. dixon , d. dunbar , b. julia , m. perelstein , j. rozowsky , d. seminara , and m. trigiante `` counterterms in supergravity , '' in _ proceeedings of 4th annual european tmr conference on integrability , nonperturbative effects and symmetry in quantum field theory _ , paris , france , 7 - 13 sep 2000 [ hep - th/0012230 ] . a. n. leznov , `` on equivalence of four - dimensional selfduality equations to continual analog of the main chiral field problem , '' _ theor . math . _ * 73 * ( 1988 ) 1233 [ _ teor . mat . fiz . _ * 73 * ( 1988 ) 302 ] . m. t. grisaru , p. van nieuwenhuizen and j. a. vermaseren , `` one loop renormalizability of pure supergravity and of maxwell - einstein theory in extended supergravity , '' _ phys . lett . _ * 37 * , 1662 ( 1976 ) . k. s. stelle , `` extended supercurrents and the ultraviolet finiteness of n=4 supersymmetric yang - mills theory , '' in _ c81 - 08 - 20.4 _ lptens 81/24 , in _ proc . of 1981 nuffield quantum gravity workshop _ , london , england , aug 20 , 1981

over the past two decades , the utility of type ia supernovae ( sne ia ) as standardizable candles to trace the expansion history of the universe has been underscored by the increasing resources dedicated to optical / near - ir discovery and follow - up campaigns ( @xcite ; @xcite ) . at the same time , the nature of their progenitor system(s ) has remained elusive , despite aggressive studies to unveil them ( see e.g. @xcite ) . the second nearest ia sn discovered in the digital era , sn 2011fe @xcite located at @xmath6 @xcite , represents a natural test bed for a detailed sn ia progenitor study @xcite . ] . the best studied type ia sn at early times before sn 2011fe , sn 2009ig , demonstrated how single events can provide significant insight into the properties of this class of explosions @xcite . the fundamental component of sn ia progenitor models is an accreting white dwarf ( wd ) in a binary system . currently , the most popular models include ( i ) a single - degenerate ( hereafter , sd ) scenario in which a massive wd accretes material from a h - rich or he - rich companion , potentially a giant , subgiant or main - sequence star , ( @xcite ; @xcite ) . mass is transferred either via roche - lobe overflow ( rlof ) or through stellar winds . alternatively , ( ii ) models invoke a double sub-@xmath7 wd binary system that eventually merges ( double degenerate model , dd ; @xcite , @xcite ) . in sd models , the circumbinary environment may be enriched by the stellar wind of the donor star or through non - conservative mass transfer in which a small amount of material is lost to the surroundings . winds from the donor star shape the local density profile as @xmath8 over a @xmath9 parsec region encompassing the binary system . theoretical considerations indicate that the wind - driven mass loss rate must be low , since an accretion rate of just @xmath10 is ideal for the wd to grow slowly up to @xmath7 and still avoid mass - losing nova eruptions ( steady burning regime , @xcite ) . strong evidence for the _ lack _ of a wind - stratified medium and/or the detection of a constant local density ( with a typical interstellar medium density of @xmath11 ) may instead point to a dd model . arising from the interaction of the sn shock blast wave with the circumbinary material , radio and x - ray observations can potentially discriminate between the two scenarios by shedding light on the properties of the environment , shaped by the evolution of the progenitor system ( see e.g. @xcite , @xcite ) . motivated thus , several dozen sne ia at distances @xmath12 mpc have been observed with the very large array ( vla ; @xcite ; @xcite ; soderberg in prep . ) , the chandra x - ray observatory @xcite , and the swift x - ray telescope ( @xcite ; russel & immler , in press ) revealing no detections to date . these limits were used to constrain the density of the circumbinary material , and in turn the mass loss rate of the progenitor system . however these data poorly constrain the wd companion , due in part to the limited sensitivity of the observations ( and the distance of the sne ) . the improved sensitivity of the expanded very large array ( evla ) coupled with a more detailed approach regarding the relevant radio and x - ray emission ( and absorption ) processes in type ia supernovae , has enabled the deepest constraints to date on a circumbinary progenitor as discussed in our companion paper on the recent type ia sn2011fe/ ptf11kly ( @xcite . see also @xcite ) . here we report a detailed panchromatic study of sn2011fe bridging optical / uv and gamma - ray observations . drawing from observations with the _ swift _ and chandra satellites as well as the interplanetary network ( ipn ; @xcite ) , we constrain the properties of the bulk ejecta and circumbinary environment through a self - consistent characterization of the dynamical evolution of the shockwave . first we present optical / uv light - curves for the sn , indicating that the object appears consistent with a `` normal '' sn ia . next we discuss deep limits on the x - ray emission in the month following explosion . we furthermore report gamma - ray limits ( 25 - 150 kev ) for the shock breakout pulse . in the appendix we present an analytic generalization for the the inverse compton ( ic ) x - ray luminosity expected from hydrogen poor sne that builds upon previous work by @xcite and @xcite but is broadly applicable for a wide range of shock properties , metallicity , photon temperatures , and circumstellar density profiles ( stellar wind or ism ; see appendix [ sec : iclum ] ) . we apply this analytic model to sn2011fe to constrain the density of the circumbinary environment , and find that our limits are a factor of @xmath13 10 deeper than the results recently reported by @xcite . observations are described in sec . [ sec : obs ] ; limits to the sn progenitor system from x - ray observations are derived and discussed in sec . [ sec : xray ] using the ic formalism from appendix [ sec : iclum ] . we combine our radio @xcite and x - ray limits to constrain the post - shock energy density in magnetic fields in sec . [ sec : epsilonb ] , while the results from the search of a burst of gamma - ray radiation from the sn shock break - out is presented in sec . [ sec : gammaray ] . conclusions are drawn in sec . [ sec : conc ] . -0.0 true cm region around the sn is marked with a white box . _ inset : _ _ chandra _ 0.5 - 8 kev deep observation of the same region obtained at day 4 since the explosion . no source is detected at the sn position ( white circle ) . , title="fig : " ] sn 2011fe was discovered by the palomar transient factory ( ptf ) on 2011 august 24.167 ut and soon identified as a very young type ia explosion in the pinwheel galaxy ( m101 ) ( @xcite ) . from early time optical observations @xcite were able to constrain the sn explosion date to august 23 , @xmath14 ( ut ) . the sn site was fortuitously observed both by the _ hubble space telescope _ ( hst ) and by _ chandra _ on several occasions prior to the explosion in the optical and x - ray band , giving the possibility to constrain the progenitor system ( @xcite ; @xcite ) . very early optical and uv photometry has been used by @xcite and @xcite to infer the progenitor and companion radius and nature , while multi - epoch high - resolution spectroscopy taken during the evolution of the sn has been employed as a probe of the circumstellar environment @xcite . limits to the circumstellar density have been derived from deep radio observations in our companion paper @xcite , where we consistently treat the shock parameters and evolution . here we study sn 2011fe from a complementary perspective , bridging optical / uv , x - ray and gamma - ray observations . _ swift _ observations were acquired starting from august 24 , @xmath15 days since the onset of the explosion . _ swift_-xrt data have been analyzed using the latest release of the heasoft package at the time of writing ( v11 ) . standard filtering and screening criteria have been applied . no x - ray source consistent with the sn position is detected in the 0.3 - 10 kev band either in promptly available data ( @xcite ; @xcite ) or in the combined @xmath16 ks exposure covering the time interval @xmath17 days ( see fig . [ fig : x - rays ] ) . in particular , using the first 4.5 ks obtained on august 24th , we find a psf ( point spread function ) and exposure map corrected @xmath18 count - rate limit on the undetected sn @xmath19 . for a simple power - law spectrum with photon index @xmath20 and galactic neutral hydrogen column density @xmath21 @xcite this translates into an unabsorbed 0.3 - 10 kev flux @xmath22 corresponding to a luminosity @xmath23 at a distance of 6.4 mpc @xcite . collecting data between 1 and 65 days after the explosion ( total exposure of @xmath16 ks ) we obtain a @xmath18 upper limit of @xmath24 ( @xmath25 , @xmath26 ) . finally , extracting data around maximum light ( the time interval 8 - 38 days ) , the x - rays are found to contribute less than @xmath27 ( @xmath18 limit , total exposure of 61 ks ) corresponding to @xmath28 , @xmath29 . we observed sn 2011fe with the _ chandra _ x - ray observatory on aug 27.44 ut ( day 4 since the explosion ) under an approved ddt proposal ( pi hughes ) . data have been reduced with the ciao software package ( version 4.3 ) , with calibration database caldb ( version 4.4.2 ) . we applied standard filtering using ciao threads for acis data . no x - ray source is detected at the sn position during the 50 ks exposure @xcite , with a @xmath18 upper limit of @xmath30 in the 0.5 - 8 kev band , from which we derive a flux limit of @xmath31 corresponding to @xmath32 ( assuming a simple power - law model with spectral photon index @xmath33 ) . @xmath18 upper limits from _ swift _ and _ chandra _ observations are shown in fig . [ fig : ic ] . the sn was clearly detected in _ swift_-uvot observations . photometry was extracted from a @xmath34 aperture , closely following the prescriptions by @xcite ( see fig . [ fig : ic ] ) . pre - explosion images of the host galaxy acquired by uvot in 2007 were used to estimate and subtract the host galaxy light contribution . our photometry agrees ( within the uncertainties ) with the results of @xcite . with respect to @xcite we extend the uvot photometry of sn 2011fe to day @xmath35 since the explosion . due to the brightness of sn 2011fe , u , b and v observations strongly suffer from coincidence losses @xcite around maximum light ( see @xcite for details ) : supernova templates from @xcite were used to fit the u and b light - curves and infer the sn luminosity during those time intervals in the u and b bands . for the v - band , it was possible to ( partially ) recover the original light - curve applying standard coincidence losses corrections : however , due to the extreme coincidence losses , our v - band light - curve may still provide a lower limit to the real sn luminosity in the time interval @xmath36 days since explosion . in fig . [ fig : ic ] we present the _ swift_-uvot 6-filter light - curves , and note that the re - constructed v - band is broadly consistent with the nugent template . we adopted a galactic reddening of @xmath37 @xcite . in the case of the `` golden standard '' ia sn 2005cf ( which is among the best studied ia sne ) , the v band is found to contribute @xmath38 to the bolometric luminosity @xcite , with limited variation over time . for sn 2011fe , we measure at day 4 a v - band luminosity @xmath39 , corresponding to @xmath40 and note that at this time the luminosity in the v , b , u , w1 and w2 bands account for @xmath41 . we therefore assumed that the v , b , u , w1 and w2 bands represent @xmath42 . in the following we explicitly provide the dependence of our density limits on @xmath43 , so that it is easily possible to re - scale our limits to any @xmath43 value . given that the optical properties point to a normal sn ia ( parrent at al . in prep . ) we adopt fiducial parameters @xmath44 and @xmath45 for the ejecta mass and sn energy , respectively , throughout this paper . -0.0 true cm -0.0 true cm upper limit as a function of the power - law index of the electron distribution @xmath46 assuming @xmath47 . upper limit contours in the cases @xmath48 k and @xmath49 k are also shown for comparison ( black dashed lines ) . yellow bullets : upper limit to the csm density as derived from radio observations for @xmath50 in the range @xmath51 . @xmath52 gives the tightest constraint @xcite . we assume @xmath45 , @xmath53 , @xmath54 . , title="fig : " ] upper limit as a function of the power - law index of the electron distribution @xmath46 assuming @xmath47 . upper limit contours in the cases @xmath48 k and @xmath49 k are also shown for comparison ( black dashed lines ) . yellow bullets : upper limit to the csm density as derived from radio observations for @xmath50 in the range @xmath51 . @xmath52 gives the tightest constraint @xcite . we assume @xmath45 , @xmath53 , @xmath54 . , title="fig : " ] x - ray emission from sne may be attributed to a number of emission processes including ( i ) synchrotron , ( ii ) thermal , ( iii ) inverse compton ( ic ) , or ( iv ) a long - lived central engine ( see @xcite for a review ) . it has been shown that the x - ray emission from stripped supernovae exploding into low density environments is dominated by ic on a timescale of weeks to a month since explosion , corresponding to the peak of the optical emission ( @xcite , @xcite ) . in specific cases , this has been shown to be largely correct ( e.g. , sn 2008d @xcite , sn 2011dh @xcite ) . in this framework the x - ray emission is originated by up - scattering of optical photons from the sn photosphere by a population of relativistic electrons ( e.g. @xcite ) . the ic x - ray luminosity depends on the density structure of the sn ejecta , the structure of the circumstellar medium ( csm ) and the details of the relativistic electron distribution responsible for the up - scattering . here we assume the sn outer density structure @xmath55 with @xmath56 @xcite , as found for sne arising from compact progenitors ( as a comparison , @xcite found the outermost profile of the ejecta to scale as @xmath57 . see @xcite , soderberg in prep . for a discussion ) ; the sn shock propagates into the circumstellar medium and is assumed to accelerate the electrons in a power - law distribution @xmath58 for @xmath59 . radio observations of type ib / c sne indicate @xmath60 @xcite . however , no radio detection has ever been obtained for a type ia sn so that the value of @xmath46 is currently unconstrained : this motivates us to explore a wider parameter space @xmath61 ( fig . [ fig : icwind ] ) as seen for mildly relativistic and relativistic explosions ( e.g. , gamma - ray bursts , @xcite ; @xcite ; @xcite ) . finally , differently from the thermal or synchrotron mechanisms , the ic luminosity is directly related to the bolometric luminosity of the sn ( @xmath62 ) : the environment directly determines the _ ratio _ of the optical to the x - ray luminosity , so that possible uncertainties on the distance of the sn do not affect the ic computation ; it furthermore does _ not _ require any assumption on magnetic field related parameters . for a population of optical photons with effective temperature @xmath63 , the ic luminosity at frequency @xmath64 reads ( see appendix [ sec : iclum ] ) : @xmath65 where @xmath66 is the extension of the region containing fast electrons ; @xmath67 is the circumstellar medium density the sn shock is impacting on , which we parametrize as a power - law in shock radius @xmath68 ; together with @xmath69 , @xmath67 determines the shock dynamics , directly regulating the evolution of the shock velocity @xmath70 , shock radius @xmath71 and @xmath72 as derived in appendix [ sec : iclum ] . for the special case @xmath73 , @xmath74 , its dependence on @xmath63 cancels out and it is straightforward to verify that eq . [ eq : icgeneral ] matches the predictions from @xcite , their eq . ( 31 ) for @xmath75 ( wind medium ) . in the following we use eq . [ eq : icgeneral ] and the @xmath76 evolution calculated from _ swift_-uvot observations of sn 2011fe ( sec . [ sec : obs ] ) to derive limits on the sn environment assuming different density profiles . we assume @xmath54 , as indicated by well studied sn shocks @xcite . each limit on the environment density we report below has to be re - scaled of a multiplicative factor @xmath77 for other @xmath78 values . a star which has been losing material at constant rate @xmath79 gives rise to a `` wind medium '' : @xmath80 . [ eq : icwind ] and the _ chandra _ non - detection constrain the wind density to @xmath81 ( where @xmath82 is the wind velocity ) . this is a @xmath18 limit obtained integrating eq . [ eq : icwind ] over the 0.5 - 8 kev _ pass band and assuming @xmath73 , @xmath54 , @xmath45 and @xmath53 . the observation was performed on day 4 after the explosion : at this time @xmath83 while the shock wave probes the environment density at a radius @xmath84 ( eq . [ eq : vshockrshock ] and [ eq : vshockwind ] ) for @xmath85 ( see fig . [ fig : ic ] ) . for the wind scenario @xmath86 ( see appendix [ sec : iclum ] ) . while giving less deep constraints , _ swift _ observations have the advantage of being spread over a long time interval giving us the possibility to probe the csm density over a wide range of radii . integrating eq . [ eq : icwind ] in the time interval 1 - 65 days to match the _ swift _ coverage ( and using the 0.3 - 10 kev band ) leads to @xmath87 for @xmath88 from the progenitor site , eq . [ eq : vshockwind ] ) , these values are accurate within a factor 10 of @xmath89 variation . ] . a similar value is obtained using the x - ray limit around maximum optical light , when the x - ray emission from ic is also expected to peak ( fig . [ fig : ic ] the _ swift _ limits are arbitrarily assigned to the linear midpoint of the temporal intervals . the limit on the ambient density is however calculated integrated the model over the entire time interval so that the arbitrary assignment of the `` central '' bin time has no impact on our conclusions . ] ) . sn 2011fe might have exploded in a uniform density environment ( ism , @xmath90 ) . in this case , integrating eq . [ eq : icism ] over the 0.5 - 8 kev energy range , the _ chandra _ limit implies a csm density @xmath91 at @xmath18 confidence level for fiducial parameter values @xmath73 , @xmath54 , @xmath45 and @xmath53 . this limit applies to day 4 after the explosion ( or , alternatively to a distance @xmath92 , see fig . [ fig : ic ] ) . integrating eq . [ eq : icism ] over the time interval 1 - 65 days ( and in the energy window 0.3 - 10 kev ) the _ swift _ upper limit implies @xmath93 ( @xmath18 level ) , over a distance range @xmath94 from the progenitor site has a very gentle ( @xmath95 , see eq . [ eq : vshockism ] ) dependence on the environment density . the @xmath96 values we list are representative of an ism medium with a wide range of density values : @xmath97 . ] . around maximum light ( days 8 - 38 ) , we constrain @xmath98 for distances @xmath99 . for an ism scenario our constraints on the particle density @xmath100 ( see appendix [ sec : iclum ] ) . figure [ fig : icwind ] ( lower panel ) shows how our _ chandra _ limit compares to deep radio observations of sn 2011fe . we explore a wide parameter space to understand how a different photon effective temperature and/or electron power - law index @xmath46 would affect the inferred density limit : we find @xmath101 for @xmath102 k and @xmath103 . x - ray observations are less constraining than radio observations in the ism case when compared to the wind case : this basically reflects the higher sensitivity of the synchrotron radio emission to the blastwave velocity , which is faster for an ism - like ambient ( for the same density at a given radius ) . from the _ chandra _ non detection we derive @xmath81 . this is the deepest limit obtained from x - ray observations to date and directly follows from ( i ) unprecedented deep _ chandra _ observations , ( ii ) proximity of sn 2011fe coupled to ( iii ) a consistent treatment of the dynamics of the sn shock interaction with the environment ( appendix [ sec : iclum ] ) . before sn 2011fe , the deepest x - ray non - detection was reported for type ia sn 2002bo at a level of @xmath104 ( distance of 22 mpc ) : using 20 ks of _ chandra _ observations obtained @xmath105 after explosion , @xcite constrained @xmath106 . this limit was computed conservatively assuming thermal emission as the leading radiative mechanism in the x - rays . using a less conservative approach , other studies were able to constrain the x - ray luminosity from type ia sne observed by _ swift _ to be @xmath107 @xcite , leading to @xmath108 ( a factor @xmath109 above our result ) . our limit on sn 2011fe strongly argues against a symbiotic binary progenitor for _ this _ supernova . according to this scenario the wd accretes material from the wind of a giant star carrying away material at a level of @xmath110 for @xmath111 ( see e.g. @xcite ; @xcite ; @xcite ) . we reached the same conclusion in our companion paper @xcite starting from deep radio observations of sn 2011fe . the radio limit is shown in fig . [ fig : icwind ] for the range of values @xmath112 , with @xmath52 leading to the most constraining limit ( where @xmath50 is the post shock energy density fraction in magnetic fields ) . historical imaging at the sn site rules out red - giant stars and the majority of the parameter space associated with he star companions ( @xcite , their fig . 2 ) : however , pre - explosion images could _ not _ constrain the roche - lobe overflow ( rlof ) scenario , where the wd accretes material either from a subgiant or a main - sequence star . in this case , winds or transferred material lost at the outer lagrangian points of the system are expected to contribute at a level @xmath113 _ if _ a fraction @xmath114 of the transferred mass is lost at the lagrangian points and the wd is steadily burning ( see e.g. @xcite et al and references therein ) . the real fraction value is however highly uncertain , so that it seems premature to rule out the entire class of models based on the present evidence . x - ray limits would be compatible with rlof scenarios where the fraction of lost material is @xmath115 ( for any @xmath116 and @xmath117 k , fig . [ fig : icwind ] ) . however , from the analysis of early uv / optical data , @xcite found the companion radius to be @xmath118 , thus excluding roche - lobe overflowing red - giants and main sequence secondary stars ( see also @xcite ) . x - ray non - detections are instead consistent with ( but can hardly be considered a proof of ) the class of double degenerate ( dd ) models for type ia sne , where two wds in a close binary system eventually merge due to the emission of gravitational waves . no x - ray emission is predicted ( apart from the shock break out at @xmath119 , see sec . [ sec : gammaray ] ) and sn 2011fe might be embedded in a constant and low - density environment ( at least for @xmath120 ) . pre - explosion radio hi imaging indicates an ambient density of @xmath121 @xcite ( on scales @xmath122 ) , while our tightest limits in the case of an ism environment are @xmath123 . our observations can not however constrain the presence of material at distances in the range @xmath124 from the sn explosion : recent studies suggest that significant material from the secondary ( disrupted ) wd may indeed reside at those distances either as a direct result of the dd - merger @xcite or as an outcome of the subsequent evolution of the system @xcite . whatever the density profile of the environment , our findings are suggestive of a _ clean _ environment around sn 2011fe for distances @xmath125 . the presence of significant material at larger distances ( @xmath126 ) can not be excluded , so that our observations can not constrain models that predict a large delay ( @xmath127 yr ) between mass loss and the sn explosion ( see e.g. @xcite , @xcite and references therein ) . finally , it is interesting to note that the high - resolution spectroscopy study by @xcite lead to a similar , _ environment conclusion : at variance with sn 2006x @xcite , sn 1999cl @xcite and sn 2007le @xcite , sn 2011fe shows no evidence for variable sodium absorption in the time period @xmath128 days since explosion . in this context , a recent study by @xcite found evidence for gas outflows from type ia progenitor systems in at least @xmath129 of cases . independent constraints on the circumstellar medium density around type ia sne come from galactic type ia supernova remnants ( snr ) : the study of tycho s snr in the x - rays lead @xcite to determine a pre - shock ambient density of less than @xmath130 ; the ambient density is likely @xmath131 both in the case of kepler s snr @xcite and in the case of snr 0509 - 67.5 @xcite . we emphasize that different type ia sne might have different progenitor systems as suggested by the increasing evidence of diversity among this class : we know that 30% of local sne ia have peculiar optical properties ( @xcite , @xcite ) . the above discussion directly addresses the progenitor system of sn 2011fe : our conclusions can not be extended to the entire class of type ia sne . -0.0 true cm has been used in the case of a wind medium . the horizontal dashed line marks equipartition ( @xmath132 ) for the assumed @xmath54 . things stands for the hi nearby galaxy survey " @xcite.,title="fig : " ] has been used in the case of a wind medium . the horizontal dashed line marks equipartition ( @xmath132 ) for the assumed @xmath54 . things stands for the hi nearby galaxy survey " @xcite.,title="fig : " ] while the ic emission model discussed here is primarily sensitive to csm density , the associated radio synchrotron emission is sensitive to both the csm density and @xmath50 ( post shock energy density in magnetic fields ) . as a consequence , when combined with radio observations of synchrotron self - absorbed sne , deep x - ray limits can be used to constrain the @xmath50 vs. ambient density parameter space ( @xcite ; @xcite ) . this is shown in fig . [ fig : epsilonb ] for a wind ( upper panel ) and ism ( lower panel ) environment around sn 2011fe : the use of the same formalism ( and assumptions ) allows us to directly combine the radio limits from @xcite with our results . we exclude the values of @xmath133 coupled to @xmath134 for a wind medium , while @xmath135 for any @xmath136 . in the case of an ism profile , x - ray limits rule out the @xmath137 @xmath138 parameter space . the exact value of the microphysical parameters @xmath50 and @xmath78 is highly debated both in the case of non - relativistic ( e.g. sne ) and relativistic ( e.g. gamma - ray bursts , grbs ) shocks : equipartition ( @xmath139 ) was obtained for sn 2002ap from a detailed modeling of the x - ray and radio emission @xcite while significant departure from equipartition ( @xmath140 ) has recently been suggested by @xcite to model sn 2011dh . the same is true for sn 1993j , for which @xmath141 @xcite . in the context of relativistic shocks , grb afterglows seem to exhibit a large range of @xmath50 and @xmath78 values ( e.g. @xcite ) ; furthermore , values as low as @xmath142 have recently been be suggested by @xcite from accurate multi - wavelength modeling of grbs with gev emission . it is at the moment unclear if this is to be extended to the entire population of grbs . on purely theoretical grounds , starting from relativistic mhd simulations @xcite concluded @xmath143 : this result applies to grb internal shocks , the late stage of grb afterglows , transrelativistic sn explosions ( like sn 1998bw , @xcite ) and shock breakout from type ibc supernova ( e.g. sn 2008d , @xcite ) . it is not clear how different the magnetic field generation and particle acceleration might be between relativistic and non - relativistic shocks . figure [ fig : epsilonb ] constitutes the first attempt to infer the @xmath50 value combining deep radio and x - ray observations of a type ia sn : better constraints on the parameters could in principle be obtained _ if _ x - ray observations are acquired at the sn optical maximum light . in the case of sn 2011fe we estimate that a factor @xmath144 improvement on the density limits would have been obtained with a _ observation at maximum light . shock break out from wd explosions is expected to produce a short ( @xmath145 ) pulse with typical @xmath146 photon energy , luminosity @xmath147 and energy in the range @xmath148 @xcite . such an emission episode would be easily detected if it were to happen close by ( either in the milky way or in the magellanic clouds ) , while sn 2011fe exploded @xmath149 mpc away @xcite . given the exceptional proximity of sn 2011fe we nevertheless searched for evidence of high - energy emission from the shock break - out using data collected by the nine spacecrafts of the interplanetary network ( ipn mars odyssey , konus - wind , rhessi , integral ( spi - acs ) , _ swift_-bat , suzaku , agile , messenger , and fermi - gbm ) . the ipn is full sky with temporal duty cycle @xmath150 and is sensitive to radiation in the range @xmath151 kev @xcite . within a 2-day window centered on aug 23rd a total of 3 bursts were detected and localized by multiple instruments of the ipn . out of these 3 confirmed bursts , one has localization consistent with sn 2011fe . interestingly , this burst was detected by konus , suzaku and integral ( spi - acs ) on august 23rd 13:28:25 ut : for comparison , the inferred explosion time of sn 2011fe is @xmath152 minutes , @xcite . the ipn error box area for this burst is @xmath153 sr . the poor localization of this event does not allow us to firmly associate this burst with sn 2011fe : from poissonian statistics we calculate a @xmath154 chance probability for this burst to be spatially consistent with sn 2011fe . a more detailed analysis reveals that sn 2011fe lies inside the konus - integral triangulation annulus but outside the konus - suzaku triangulation annulus . furthermore , at the inferred time of explosion , sn 2011fe was slightly above the fermi - gbm horizon , but no burst was detected ( in spite of the stable gbm background around this time ) . we therefore conclude that there is no statistically significant evidence for a sn - associated burst down to the fermi - gbm threshold ( fluence @xmath155 in the 8 - 1000 kev band ) for a sn - associated burst with fluence above the _ swift _ threshold and below the fermi - gbm one to occur without being detected . ] . the early photometry of sn 2011fe constrains the progenitor radius to be @xmath156 @xcite . using the fiducial values @xmath45 , @xmath53 , the shock break out associated with sn 2011fe is therefore expected to have released @xmath157 over a time - scale @xmath158 with luminosity @xmath159 at typical @xmath160 ( see @xcite , their eq . 29 ) . at the distance of sn 2011fe , the expected fluence is as low as @xmath161 which is below the threshold of all gamma - ray observatories currently on orbit ( the weakest burst observed by bat had a 15 - 150 kev fluence of @xmath162 ) . for comparison , the konus - suzaku - integral burst formally consistent with the position of sn 2011fe was detected with fluence @xmath163 and duration of a few seconds ( peak flux of @xmath164 ) . if it were to be connected with the sn , the associated @xmath165sec peak luminosity would be @xmath166 and total energy @xmath167 ( quantities computed in the 20 - 1400 kev energy band ) which are orders of magnitudes above expectations . for @xmath168 , the temperature and luminosity drop quickly ( see @xcite for details ) : in particular , for @xmath169 the emitting shell enters the newtonian phase . for sn 2011fe we estimate @xmath170 ( @xcite , their eq . 30 ) ; for @xmath156 the luminosity at @xmath171 is @xmath172 with typical emission in the soft x - rays : @xmath173 . at later times @xmath174 @xcite while @xmath175 rapidly drops below the _ swift_-xrt energy band ( 0.3 - 10 kev ) . _ swift_-xrt observations were unfortunately not acquired early enough to constrain the shock break out emission from sn 2011fe . uv observations were not acquired early enough either : after @xmath176 hr the uv emission connected with the shock break out is expected to be strongly suppressed due to the deviation from pure radiation domination ( e.g. @xcite ) . it is however interesting to note the presence of a `` shoulder '' in the uv light - curve @xcite particularly prominent in the uvm2 filter for @xmath177 days ( see @xcite , their fig . 2 ) whose origin is still unclear ( see however @xcite ) . a detailed modeling is required to disentangle the contribution of different physical processes to the early uv emission ( and understand which is the role of the `` red leak '' -see e.g. @xcite- of the uvm2 filter in shaping the observed light - curve ) . the collision of the sn ejecta with the companion star is also expected to produce x - ray emission with typical release of energy @xmath178 in the hours following the explosion ( a mechanism which has been referred to as the analog of shock break out emission in core collapse sne , @xcite ) . according to @xcite , in the most favorable scenario of a red - giant companion of @xmath179 at separation distance @xmath180 , the interaction time - scale is @xmath181 after the sn explosion and the burst of x - ray radiation lasts @xmath182 ( with a typical luminosity @xmath183 ) : too short to be caught by our _ swift_-xrt re - pointing 1.25 days after the explosion . we furthermore estimate the high energy tail of the longer lasting thermal optical / uv emission associated to the collision with the companion star to be too faint to be detected either : at @xmath184 , the emission has @xmath185 and peaks at frequency @xmath186 ( eq . 25 from @xcite ) . non - thermal particle acceleration might be a source of x - rays at these times , a scenario for which we still lack clear predictions : future studies will help understand the role of non - thermal emission in the case of the collision of a sn with its companion star . ic emission provides solid limits to the environment density which are _ not _ dependent on assumptions about the poorly constrained magnetic field energy density ( i. e. the @xmath50 parameter ; see also @xcite and @xcite ) . this is different from the synchrotron emission , which was used in our companion paper @xcite to constrain the environment of the same event from the deepest radio observations ever obtained for a sn ia . the two perspectives are complementary : the use of the same assumptions and of a consistent formalism furthermore allows us to constrain the post - shock energy density in magnetic fields vs. ambient density parameter space ( see fig . [ fig : epsilonb ] ) . this plot shows how deep and contemporaneous radio and x - rays observations of sne might be used to infer the shock parameters . the ic luminosity is however strongly dependent on the sn bolometric luminosity : @xmath62 . here we presented the deepest limit on the ambient density around a type ia sn obtained from x - ray observations . our results directly benefit from : ( i ) unprecedented deep _ chandra _ observations of one of the nearest type ia sne , coupled to ( ii ) a consistent treatment of the dynamics of the sn shock interaction with the environment ( appendix [ sec : iclum ] and @xcite ) , together with ( iii ) the direct computation of the sn bolometric luminosity from _ swift_/uvot data . in particular we showed that : * assuming a wind profile the x - ray non - detections imply a mass loss @xmath187 for @xmath188 . this is a factor of @xmath189 deeper than the limit reported by @xcite . this rules out symbiotic binary progenitors for sn 2011fe and argues against roche - lobe overflowing subgiants and main sequence secondary stars _ if _ a fraction @xmath114 of the transferred mass is lost at the lagrangian points and the wd is steadily burning . * were sn 2011fe to be embedded in an ism environment , our calculations constrain the density to @xmath190 . whatever the density profile , the x - ray non - detections are suggestive of a _ clean _ environment around sn 2011fe , for distances in the range @xmath191 . this is either consistent with the bulk of material ( transferred from the donor star to the accreting wd or resulting from the merging of the two wds ) to be confined within the binary system or with a significant delay @xmath127 yr between mass loss and sn explosion ( e.g. @xcite , @xcite ) . note that in the context of dd mergers , the presence of material on distances @xmath124 ( as recently suggested by e.g. @xcite and @xcite ) has been excluded by @xcite based on the lack of bright , early uv / optical emission . we furthermore looked for bursts of gamma - rays associated with the shock break out from sn 2011fe . we find no statistically significant evidence for a sn - associated burst for fluences @xmath192 . however , with progenitor radius @xmath193 the expected sn 2011fe shock break out fluence is @xmath194 , below the sensitivity of gamma - ray detectors currently on orbit . the proximity of sn 2011fe coupled to the sensitivity of _ chandra _ observations , make the limits presented in this paper difficult to be surpassed in the near future for type ia sne . however , the generalized ic formalism of appendix [ sec : iclum ] is applicable to the entire class of hydrogen poor sne , and will provide the tightest constraints to the explosion environment _ if _ x - ray observations are acquired around maximum light ( see fig . [ fig : ic ] ) for type i supernovae ( ia , ib and ic ) . we thank harvey tananbaum and neil gehrels for making _ chandra _ and _ swift _ observations possible . we thank reem sari , bob kirshner , sayan chakraborti , stephan immler , brosk russel and rodolfo barniol duran for helpful discussions . l.c . is a jansky fellow of the national radio astronomy observatory . is supported by a clay fellowship . kh is grateful for ipn support under the following nasa grants : nnx10ar12 g ( suzaku ) , nnx12ad68 g ( swift ) , nnx07ar71 g ( messenger ) , and nnx10au34 g ( fermi ) . the konus - wind experiment is supported by a russian space agency contract and rfbr grant 11 - 02 - 12082-ofi_m . pos acknowledges partial support from nasa contract nas8 - 03060 . 80 natexlab#1#1 ? ? ? ? \08 . 1 , c .- i . , & fransson , c. 2004 , , 605 , 823 , s. , prieto , j. l. , patat , f. , challis , p. , hicken , m. , kirshner , r. p. , matheson , t. , & modjaz , m. 2009 , , 693 , 207 , j. s. , et al . 2011 , arxiv e - 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ray frequencies via inverse compton ( ic ) , see e.g. @xcite , @xcite . here we generalize eq . ( 31 ) from @xcite for a population of relativistic electrons with arbitrary distribution @xmath195 for @xmath196 , both for an ism ( eq . [ eq : icism ] ) and a wind ( eq . [ eq : icwind ] ) scenario . using the ic emissivity given by @xcite , their eq . 27 , the ic luminosity reads : @xmath197 where @xmath198 is the energy density of photons of effective temperature @xmath63 which are upscattered to @xmath199 ; @xmath66 is the extension of the region containing fast electrons while @xmath96 is the ( forward ) shock radius . the emission is expected to originate from a shell of shocked gas between the reverse and the forward shock which are separated by the contact discontinuity at @xmath200 @xcite . for @xmath55 with @xmath201 the forward shock is at @xmath202 ( @xmath203 ) while the reverse shock is at @xmath204 ( @xmath205 ) in the case of a wind ( ism ) environment @xcite . the fraction of the volume within the forward shock with shocked gas is @xmath206 ( @xmath207 ) corresponding to a sphere of radius @xmath208 ( @xmath209 ) for an assumed wind ( ism ) density profile . if a fraction @xmath78 of the post - shock energy density goes into non thermal relativistic electrons , from @xmath210 we have : @xmath211 for @xmath212 . combining eq . [ eq : one ] with eq . [ eq : two ] , we obtain eq . [ eq : icgeneral ] . the temporal evolution of @xmath213 directly depends on @xmath76 ; @xmath214 ; @xmath215 ; @xmath216 and @xmath217 . the properties of the sn and of its progenitor determine @xmath76 , @xmath214 and the profile of the outer ejecta @xmath55 . we assume @xmath218 through out the paper ( e.g. @xcite ) . the environment sets the @xmath67 profile , which we parametrize as @xmath219 . both the sn explosion properties _ and _ the environment determine the shock dynamics : evolution of the shock radius @xmath216 , shock velocity @xmath215 and , as a consequence @xmath220 . under those conditions the shock interaction region can be described by a self - similar solution @xcite with the shock radius evolving as @xmath221 which implies : @xmath222 the shock velocity directly determines @xmath223 . from @xcite , assuming that _ all _ electrons go into a power - law spectrum with spectral index @xmath46 : @xmath224 where @xmath225 is the shock compression parameter , @xmath226 ( @xmath227 ) is the electron ( ion ) number density and @xmath228 is the average number of nucleons per atom . we furthermore define @xmath229 . for solar metallicity @xmath230 . in the following we assume @xmath231 @xcite , @xmath232 . the self - similar solutions for the interaction of the sn ejecta with an ism - like circumstellar medium ( @xmath90 , @xmath233 ) lead to ( @xcite , soderberg et al . , in prep ) : @xmath234 where @xmath235 is the mass of the ejected material and @xmath236 is the energy of the supernova explosion . [ eq : two ] , [ eq : vshockrshock ] , [ eq : gammamin ] and [ eq : vshockism ] , together with eq . [ eq : one ] , predict an ic luminosity : @xmath237 with @xmath238 . in the body of the paper @xmath239 will be reported in ( hydrogen ) particles per @xmath240 . for @xmath75 ( @xmath241 ) the self - similar solutions lead to ( @xcite , soderberg et al . , in prep ) : @xmath242 combining eq . [ eq : two ] , [ eq : vshockrshock ] , [ eq : gammamin ] and [ eq : vshockwind ] with eq . [ eq : one ] we obtain : @xmath243 with @xmath244 . + note that @xmath245 , so that @xmath246 , where @xmath79 and @xmath82 are the mass loss rate and the wind velocity of the sn progenitor , respectively . in the body of the paper , for the wind scenario , we refer to @xmath239 in terms of mass loss rate for a given wind velocity so that it is easier to connect our results to known physical systems .

the recent experimental developments in ultracold fermi gases and in particular the experimental realization of the bec - bcs crossover through feshbach resonance @xcite have arised in a field where theoretical investigations started a long time ago . indeed fairly soon after the publication of the bardeen - cooper - shrieffer ( bcs ) theory @xcite , the extension to the neutral fermi liquid @xmath4he was considered @xcite . although the case of liquid @xmath4he is more complex than the one of ultracold gases , it is nevertheless analogous to the situation found on the bcs side of the crossover , in the strong interaction regime . similarly the possibility of bose - einstein condensation ( bec ) of composite bosons was considered quite early for excitons in semiconductors @xcite , where the composite nature of these bosons is expected to be an essential physical feature . this is the situation met in the bec range of the bec - bcs crossover . deep theoretical investigation of this last situation was made not long afterwards by keldysh and kozlov @xcite , following a seminal work by popov @xcite which considered a physical model with short range potentials , much closer to the ultracold gas case . in both cases , due to the formal similarity between the bogoliubov and the bcs frameworks , the theoretical treatment relied on the fact that in the dilute limit the formalism leads directly to the schrdinger equation for the molecular state corresponding to the composite boson . this provided an anticipation of the study of the whole bec - bcs crossover . independently eagles @xcite , in the course of a study of superconductivity in doped semiconductors ( where the electron gas has a low concentration compared to standard metals ) , was led to investigate the extrapolation of the simple bcs formalism toward the dilute regime where pair formation would take place , due to the attractive interaction , and bose condensation of these pairs should occur at lower temperatures . in the context of cold gases , the physics of the crossover was considered by leggett @xcite , who stressed the physical interest of considering cooper pairs as giant molecules , emphasized the change of physical regime when the chemical potential goes through zero , and introduced the scattering length in the bcs formulation . the matter was taken up by nozires and schmitt - rink @xcite who studied the smooth evolution of the critical temperature in the crossover within the simplest @xmath5matrix approximation , showing in particular how the critical temperature of the ideal bose gas is recovered in the strong coupling limit . s de melo , randeria and engelbrecht @xcite addressed the crossover with the functional integral formalism . as it has often been stressed , the physical situation found in this bec - bcs crossover is extremely interesting since ( at least for wide feshbach resonances @xcite ) the scattering length @xmath6 of the different fermions with mass @xmath7 is enough to fully describe their interaction , while one has still to deal with the full complexity of a strongly interacting system . as a result the theoretical problem has not been solved exactly and one has to rely on various approximate schemes @xcite , the simplest one being the straight bcs formalism ( often called bcs - mf ) which is known to give a correct description at zero temperature in both the extreme bec and bcs limits @xmath8 . the imperfections of the approximate treatments are clearer in the bec regime . indeed if one wants to describe properly the departure from the ideal bose gas of dimers , one has to provide a correct description of the dimer - dimer interaction , in order to find the expected physics of a weakly interacting bose gas . as stressed by keldysh and kozlov @xcite the composite nature of the dimers has already to be properly taken into account at this stage in order to obtain correct results . a first step in this direction for the specific case of short range interactions , relevant for cold gases , was made by haussmann @xcite who obtained , both in the normal and in the superfluid state ( where this can be found naturally out of the bcs ansatz ) , that the dimer - dimer scattering length @xmath1 is related to the fermion scattering length @xmath6 by @xmath9 . this result was also obtained in ref.@xcite in the superfluid state by other methods . the level of approximation corresponding to this result turns out to be the born approximation for the dimer - dimer scattering . at the same level of approximation pieri and strinati @xcite derived quite recently the gross - pitaevskii equation from the bogoliubov - de gennes equations . however this level of approximation was markedly improved by pieri and strinati @xcite who performed the calculation at the level of a @xmath5matrix approximation and found @xmath10 . nevertheless their calculation does not include all the processes resulting from the existence of the fermions making up these composite bosons . the exact numerical result was obtained recently by petrov , salomon and shlyapnikov @xcite who basically solved the relevant schrdinger equation for the four - body problem and found for this dimer - dimer scattering length @xmath11 . thereafter it was shown @xcite how this same problem could be exactly formulated and solved within the methods of quantum field theory , with naturally exactly the same result . this last work is quite relevant to the present paper , and represents the basis out of which we will work . this is naturally linked to the fact that field theoretic methods are a convenient , and probably necessary , framework for an exact formulation and solution of this superfluid strong coupling problem raised by the bec - bcs crossover . in this paper , we provide an exact approach to the bec - bcs crossover problem from the bec side by presenting an exact fermionic theory of a bec superfluid of composite bosons in the low density range . the two interacting fermion species we deal with are actually the two lowest energy hyperfine states used experimentally @xcite in ultracold gases of @xmath12li or @xmath13k , and as it is usually done we will for convenience refer to them as `` spin up '' and `` spin down '' states . specifically working in the low density range means that we proceed basically to an expansion in powers of the gas density . otherwise our framework is completely general . in the present paper we deal with the lower orders in this expansion , but nothing in principle prevents our approach to be extended to higher orders , although admittedly this will require much more work . similarly we will restrict ourselves to the @xmath14 situation , which brings strong simplifications as we will see because in this case there are no fermions at all in the normal state . however it is quite possible to extend our approach to non zero temperature . finally we deal here with thermodynamics , but our framework allows us naturally to calculate dynamical quantities . as we mentionned our approach is in principle a low density expansion . however the density @xmath0 is not a convenient basic parameter since it rather appears in the formalism as the result of a calculation . on the other hand it is known ( and we will see it explicitely below ) that in the simple bcs approximation for the bec limit , which as mentionned above corresponds to a born approximation at the level of the mean field term , the density @xmath0 is proportional to @xmath15 , where @xmath16 is the gap parameter of the bcs theory . hence a convenient way to proceed effectively to a low density expansion is rather to expand in powers of @xmath16 . however this quantity is specific of the bcs approximation , which has no general validity . but it is directly related to the anomalous self - energy @xmath17 ( where @xmath18 is an energy - momentum four - vector ) which is a completely general microscopic quantity characteristic of a superfluid system . hence our method will specifically be an expansion of the general equations in powers of the anomalous self - energy @xmath17 . here we will calculate the equation of state , or more precisely the dependence of the chemical potential @xmath19 on the density @xmath0 ( a short account of our calculation has already been published @xcite ) . actually this is rather the reciprocal of this function which comes naturally out of the calculation , since @xmath19 enters the green s functions while @xmath0 is obtained at the end of the calculation , after elimination of @xmath17 which comes in the intermediate steps . the lowest order term in the expression of @xmath19 , beyond the trivial term of half the molecular binding energy @xmath20 , is the mean - field term . one has to note that , even at this stage , the composite nature of the bosons we deal with is entering as noted by keldysh and kozlov @xcite . however in our case it is lumped into the full dimer - dimer scattering length @xmath1 . this result is physically natural since , beyond the zeroth order term @xmath20 in the chemical potential @xcite corresponding to an isolated molecule , we expect in a density expansion to find the physics of two coexisting molecules , which should be fully describable by @xmath1 for its static properties , corresponding to the zero energy scattering properties . nevertheless , as far as we know , this quite reasonable result has always been in the literature more or less taken for granted or cursorily obtained , but not fully explicitely derived from a microscopic theory for composite bosons . hence our first step will be to obtain this fully explicit derivation . however we are naturally essentially interested by the next order term , at which we will also limit our present calculation . from the theory of weakly interacting elementary bosons this term is proportional to @xmath2 , as first obtained by lee and yang @xcite . we refer to this term as the lhy term . this term has then been rederived by beliaev @xcite making use of field theoretic methods . naturally it is reasonable to expect that a similar term arises for composite bosons . what is however less clear is that the coefficient is just the same as for elementary bosons , i.e. that it can be simply expressed in the same way in terms of @xmath1 . indeed it is clear that , at some stage in the expansion , the composite nature of the bosons will enter by other quantities than those describing the physics of elementary bosons . this is quite obvious physically since , by going to higher densities ( which corresponds to take into account higher order terms in the expansion ) , we will go toward unitarity and find physical properties linked to the existence of fermi seas . obviously this can not be obtained from the physics of elementary bosons . hence it is conceivable that additional processes contribute to the @xmath2 term for composite bosons . nevertheless we will find at the end of our paper that the @xmath2 term is indeed identical to the lhy term provided the dimer - dimer scattering length is used . the fact that this is not obvious is immediately realized from the number and the complexity of our steps , which do not map systematically on the elementary boson derivation . on the other hand this result makes sense physically when it is realized that the lhy term is directly linked to the existence of the gapless collective mode . a standard perturbation expansion should produce a @xmath3 term as the next term after the mean - field one . the existence of the lhy term is linked to the presence of these gapless elementary excitations which produce a singularity in the expansion , leading formally to a divergence in the coefficient of the @xmath3 term . obtaining the proper result is equivalent to the resummation of a partial series to all order in perturbations in order to get rid of the singularity . since these steps involve only low energies , they do not test the internal structure of the bosons which comes into play when energies of order of the binding energy @xmath21 are involved . in such a case it is reasonable that no difference appears between elementary and composite bosons . on the other hand the calculation of the coefficient of the regular @xmath3 term in the expansion involves the consideration of energies of order of the binding energy , as we will see explicitely , making explicit the composite nature of the bosons . this implies that at the level of this term one finds a result different from the case of elementary bosons . note that this result has been previously assumed to be correct in calculations of collective mode frequencies @xcite . on one hand this was in agreement with monte - carlo calculations @xcite , and on the other hand this has been supported by very recent experiments @xcite . at the level of handling the collective mode , very strong analogies with a purely bosonic approach , in particular with beliaev s work @xcite will appear for obvious physical reasons , although it can be seen that our treatment bypasses a good deal of beliaev s detailed diagrammatic analysis . we will not try to put these analogies away , but rather make use of these similarities and correspondances to make easier the physical and technical understanding of our calculation . however we will never make an actual use of a bosonic approach and all our theory is a purely fermionic theory , the essential physical feature being that we deal with fermion physics but with negative chemical potential @xmath22 . in particular we often use the word dimer to describe two fermions with spin up and spin down . but since the theory contains , in the corresponding propagator , not only the bound state of these fermions but also the scattering states , it should be taken as a simple convenient wording , although it is naturally physically suggestive . more generally and formally the anomalous self - energies @xmath23 and @xmath24 play clearly a role analogous to @xmath25 and @xmath26 for bosonic superfluidity . our paper is organized according to the above introduction . after introducing briefly the general formalism in the next section [ general ] , we proceed to the calculation of the normal and anomalous self - energies in section [ deltexp ] to lowest order in @xmath17 , which leads us to the mean field term in the expression of the chemical potential . then in section [ collmod ] we proceed to discuss the collective mode which plays an essential role in the calculation of the lhy term . then the calculation of the contribution of this collective mode to the normal and anomalous self - energies is performed in section [ sigcollmod ] , at the end of which the lhy contribution to the chemical potential is obtained . a number of technical details are treated in appendices , to try to make more palatable the main part of this paper which is already fairly heavy . our work deals only with the @xmath14 situation . hence it would be most natural to use the standard @xmath14 green s function formalism and naturally all the content of our paper can be reproduced with this formalism . however one has in this case to be careful in dealing with real frequencies , and specify precisely the location , with respect to the real axis , of singularities in the corresponding complex plane . this is often left implicit . a simple way to clarify this question is to deal with complex frequencies . but this is actually what one has to do naturally when one works at @xmath27 , where the frequencies run on the imaginary frequency axis . in the @xmath28 limit the discrete sums over matsubara frequencies turn into integrals over frequencies running on the imaginary axis ( the link with the @xmath14 formalism is merely obtained by continuing analytically the frequencies toward the real frequency axis , that is performing a so - called wick rotation on the frequency ) . this leads to somewhat clearer results . hence , despite its somewhat artificial character , we will work quite specifically with this @xmath14 limit of the @xmath27 formalism . actually this will not appear in many our expressions because of their formal nature and for this reason it does not make any real problem . the essential point is that , in this way , we will always have perfectly regular expressions , with an effective handling which is quite clear and easy . we deal only with the formalism of fermionic superfluidity , as it is known in superconductivity @xcite and superfluid @xmath4he @xcite . this entails the use of both the normal fermionic green s functions @xmath29 , where we make use of the four - vector notation @xmath30 , and their anomalous counterpart @xmath31 and @xmath32 , where we make use of standard notations @xcite . this could be written in a compact nambu formalism , but for clarity we prefer to write the normal and anomalous parts explicitely , with slightly different sign conventions . in practice we will always consider in the following , when we calculate @xmath33 , that we deal with @xmath34 , without writing it explicitely for simplicity . the existence ( i.e. the fact that they are non zero ) of these anomalous green s function is the basic ingredient of fermionic superfluidity in our formalism . this is quite clear physically since they are related to the condensate . , [ eqdysona]),width=377 ] we start by writing dyson s equations for this superfluid : @xmath35 their corresponding diagrammatic representation being given in fig . [ figdyson ] . here @xmath36^{-1}$ ] is the free atom green s function , with @xmath37 the atom kinetic energy @xmath38 measured from the chemical potential @xmath19 . the quantities @xmath39 and @xmath17 are respectively the normal and the anomalous self - energies . by definition @xmath39 is the sum of all diagrams , with one entering @xmath18 line and one outgoing @xmath18 line , which can not separated into two disconnected parts by cutting a single @xmath40 line ( or @xmath41 line ) . @xmath17 has the same definition except that it has two entering lines , one ( @xmath42 ) line and one ( @xmath43 ) line . we note that @xmath17 depends naturally in the general case on momentum and frequency . in the second equation , for @xmath32 which begins with an outgoing ( @xmath43 ) line and finishes with an outgoing ( @xmath42 ) line , it would look more symmetrical to introduce a corresponding anomalous self - energy @xmath44 , analogous to @xmath17 , but with two outgoing lines , one ( @xmath45 ) line and one ( @xmath46 ) line . however it can be related directly to @xmath47 by time reversal , as we have written explicitely . all formulae in the following turn out to be simpler to interpret when this substitution is done , and we will follow this standard habit . we note also that , since we deal with an equilibrium situation , where the gas is homogeneous , we do not need to introduce green s functions with different entering and outgoing momenta , as it would be required in the case of an inhomogeneous system . dyson s equations eq . ( [ eqdyson ] ) can be understood as a way of ordering the full perturbation expansion , by gathering terms into @xmath39 and @xmath17 . however it is interesting to note that this can be done in slightly different ways . for example we can gather in eq . ( [ eqdyson ] ) all the terms where only the free green s function @xmath40 and the normal self - energy @xmath39 appear , that is the terms where no anomalous terms are present . this leads to introduce @xmath48 , that is @xmath49 . in this way , by making use of @xmath50 instead of @xmath40 , we are only left to write in dyson s equations the anomalous terms , that is to rewrite these equations as : @xmath51 it is indeed checked easily that the algebraic solution of eq . ( [ eqdyson],[eqdysona ] ) , as well as the one of eq . ( [ eqdyson1 ] , [ eqdyson1a ] ) , is indeed : @xmath52 making use of the expression of @xmath53 this can be rewritten as : @xmath54\end{aligned}\ ] ] this result could have been obtained directly in the following way . in the definition we have taken above for the self - energy , we had excluded diagrams made of two blocks linked by a single propagator @xmath40 ( corresponding to an atom propagating to the right , as for the definition of the normal self - energy ) or @xmath41 ( corresponding to an atom propagating to the left , as it is produced by the anomalous self - energies ) . however we could take another definition where we would exclude only blocks linked by a rightward propagator , just as in the normal state . in this case the formal expression of @xmath33 in terms of the self - energy , i.e. @xmath55 , is just the same as in the normal state , as it is indeed the case in eq . ( [ eqdysonsol1 ] ) . but the expression of the self - energy has been naturally modified from @xmath39 to @xmath56 . indeed we have to include now terms containing leftward propagators @xmath41 . but it is easily realized that these new terms contain at most a product @xmath57 , corresponding to the presence of anomalous self - energies of opposite types at the extremities of the block . these two anomalous self - energies can be separated by any number of leftward propagators @xmath41 , with normal self - energies @xmath58 . the sum of all these produces a factor @xmath59 , which leads to the relation : @xmath60 in agreeement with eq . ( [ eqdysonsol1 ] ) . while they are completely general , the considerations presented in the preceding section are not of much practical use since we have no expressions for the normal and anomalous self - energies . in order to obtain explicit expressions , we will now begin to proceed along the lines outlined in the introduction , namely to perform a systematic expansion in powers of @xmath16 . in the present section , as an introduction and as a first step , we will only consider the lowest order in this expansion . the next order will then be addressed in the next section . in the first subsection we will consider the expansion of the normal self - energy , which turns out to be the easier one . then the anomalous self - energy will be handled along the same lines . as we have indicated in the introduction , and as we will check explicitely below , the expansion in powers of @xmath16 has the character of an expansion in powers of the density @xmath0 since we will find @xmath62 . hence the zeroth order in powers of @xmath16 corresponds to an infinitely dilute gas for which the normal self - energy is naturally zero . since by its definition this normal self - energy contains only processes which conserve globally the number of particles , the lowest order term @xmath63 in the normal self - energy must contain both @xmath16 and @xmath64 , because @xmath16 implies from its definition the annihilation of two atoms ( going physically in the condensate ) , while @xmath64 corresponds to the creation of two atoms ( coming out of the condensate ) . hence this lowest order term in the normal self - energy is the sum of all diagrams which contain @xmath65 and @xmath66 , and which have one entering line @xmath18 and one outgoing line @xmath18 , the rest of the diagrams containing only ( since we proceed to an expansion ) @xmath67 quantities , that is normal state quantities ( see fig . [ figt3 ] ) . we call the sum of all these normal state diagrams the normal part of @xmath63 . this is just @xmath63 , except for the @xmath65 and @xmath66 terms . , width=226 ] explicitely this means that this normal part of the diagrams contains only any number of bare propagators @xmath40 and of interactions lines . moreover the two atom lines coming out of @xmath66(with opposite spins ) are not allowed to interact immediately after entering the normal part of the diagram , since such an interaction is already included in @xmath66 ( as we will see explicitely below ) and allowing it would amount to double counting of diagrams . similarly the two atom lines entering @xmath65 are not allowed to interact immediately before . as it happens , such a normal state set of diagrams has already been essentially considered in ref.@xcite . indeed in this paper the scattering of a single atom by a dimer ( made of atoms with opposite spins ) has been calculated . the fact that a dimer is entering the diagram implied that the two corresponding atom lines can not interact immediately in the diagram , since such an interaction is already taken into account in the dimer propagator and this would be again double counting . there is a similar requirement for the two lines corresponding to the outgoing dimer . we see that the requirements on the normal part of @xmath63 are just the same as in the above paper . in this ref.@xcite the sum of all these diagrams was denoted @xmath68 where @xmath69 ( respectively @xmath70 ) is the four - vector of the entering ( respectively outgoing ) atom , and @xmath71 is the total four - vector of the single atom and the dimer . in our statement we apparently overlook the fact that @xmath68 contains only three atom lines , which go from the beginning to the end of the diagram , with various interactions between them in the diagram , since we deal with the vacuum scattering of an atom and a dimer . in contrast we could think that , in addition to these lines , the normal part of @xmath63 may contain any number of loops of atom lines . however it is clear physically that these loops are not allowed . indeed if we think of this normal part as representing , in a time representation , a succession of processes due to interaction , the appearance of a loop corresponds to the creation of a particle - hole pair . while such processes can always occur at non zero temperature , in our @xmath14 situation they are possible only if the chemical potential @xmath19 is positive , thereby allowing the existence of a fermi sea in which the hole can be created . but in our dilute regime we have @xmath72 and these processes are forbidden . this argument is developped more technically in appendix [ appnoloop ] . hence it is correct to identify the normal part of @xmath63 with @xmath73 since both contain only three atom lines , subject to the same restrictions indicated above . nevertheless there is still a small difference in the general case between the normal part we want and @xmath73 . in the definition of @xmath73 one has to sum over all possible four - vectors for the two lines making up the entering dimer , with the only restriction that their sum is fixed at @xmath74 . a similar summation applies for the two lines making up the outgoing dimer . on the other hand in our normal part the two lines ( p,@xmath75 ) and ( -p,@xmath76 ) entering in @xmath65 have a total zero four - vector , and because @xmath65 depends on @xmath77 , when we sum over @xmath77 , there is a weight @xmath65 which depends on @xmath77 , in contrast with the case of @xmath73 . nevertheless , as shown in appendix [ appt3 ] , it is possible to express in the general case our self - energy in terms of @xmath73 . but in our specific problem , this is an unnecessary complication since we will see that , at the order of the expansion we are considering , the dependence of @xmath65 on @xmath77 can be neglected , and @xmath65 can be considered as a constant which we denote merely as @xmath16 . hence we can factor @xmath61 out . similar considerations apply to @xmath78 and the entering dimer lines . there is still a very important point to consider before writing our expression for @xmath63 . among all the diagrams we have considered contributing to @xmath63 , there is necessarily a diagram which gives a contribution @xmath79 because it satisfies all our requirements . indeed this is just what is coming from the lowest order diagram in @xmath73 , which gives a contribution @xmath80 to @xmath73 and corresponds merely to the born approximation . this diagram is necessarily present because we considered it already in the preceding section [ general ] , when we were dealing with the general expansion of @xmath33 in powers of @xmath81 and @xmath17 . this is just the last term in eq.([eqdysonsol1 ] ) , when we replace @xmath59 by @xmath41 ( since we want only normal state diagrams ) and take @xmath17 independent of @xmath18 . however , since it is already explicitely present in eq.([eqdysonsol1 ] ) , this implies automatically that it is reducible ( and indeed it is obviously reducible ) and for this reason should not be included in @xmath63 ( otherwise we would double count this diagram ) . on the other hand this is clearly the only reducible contribution coming from @xmath73 , as seen directly from the diagrammatic expansion @xcite of @xmath73 , or because any such reducible contribution has to appear explicitely from eq.([eqdysonsol1 ] ) as explained above . accordingly we have to subtract this contribution out and we obtain finally : @xmath82 we can also rewrite this result in terms of our self - energy @xmath83 as : @xmath84 this result is also directly quite natural since , when we evaluate @xmath83 , the diagrams with @xmath41 are not considered as reducible , as we have seen , and accordingly @xmath79 should not be subtracted . we are now in position to calculate the single spin atomic density @xmath0 , which is given quite generally by : @xmath85 if @xmath86 is replaced by @xmath87 , one checks easily that @xmath88 , which is physically obvious since in this case we deal with a non interacting fermion gas , at @xmath14 , and chemical potential @xmath22 . there is no fermi sea ( which arises only for @xmath89 ) and there are no fermions . hence we can replace completely generally @xmath33 by @xmath90 in eq . ( [ eqdefn ] ) . since we work in this section at the lowest order in @xmath16 , which means at order @xmath91 in the present case , we can replace @xmath92 by @xmath93 , and @xmath33 by @xmath40 . this leads to : @xmath94 ^ 2 \label{eqdefn1}\end{aligned}\ ] ] where , as in eq . ( [ eqdefn ] ) , @xmath18 is for @xmath95 . at this stage it seems that , in order to calculate the density , we are required to have a deep knowledge of @xmath73 . this seems physically reasonable since , in eq . ( [ eqdefn1 ] ) , @xmath73 describes the scattering of an atom with the condensate which has clearly to be taken into account . however the actual situation turns out to be much simpler . in order to calculate the integral over @xmath96 in eq . ( [ eqdefn1 ] ) , it is convenient to close the contour in the upper complex half - plane @xmath97 ( which corresponds to @xmath98 ) , since the pole of @xmath99^{-1}$ ] is found for @xmath100 ( since @xmath101 ) . however it is shown in appendix [ appt3anal ] that , except for the born contribution , @xmath102 is analytical in the upper complex half - plane and does not contribute to the integral . the result comes entirely from the born contribution . this means that surprisingly the contribution of @xmath63 to @xmath0 is zero , and that the only non zero contribution comes from the explicit bcs term @xmath103 in eq . ( [ eqsigm ] ) . in other words the approximate bcs formalism gives the exact result at the lowest order in our @xmath16 expansion . it reads : @xmath104 ^ 2=\frac{m^2\,|\delta |^2}{8\pi ( 2m|\mu |)^{1/2}}\end{aligned}\ ] ] if we make use in this result of the zeroth order value of the fermion chemical potential @xmath105 , we end up with : @xmath106 actually this relation @xmath107 is precisely obtained from the dimer propagator @xmath108 ( see section [ collmod ] ) when it is evaluated near its pole and with the purely bosonic propagator factor merely replaced by @xmath0 . we proceed now in a somewhat similar way for the anomalous self - energy . the resulting equation is the generalization of the standard gap equation of the conventional bcs theory and it has the same character of being finally a self - consistent equation for @xmath17 . in writing an expression for @xmath17 , we can in full generality separate all contributing diagrams in two sets . in the first set , for both of the two entering fermionic lines , the first interaction is between these lines themselves . the second set corresponds to the opposite case , where for at least one of these two entering lines the first interaction is with a line which is not the other entering fermionic line ( see fig.[figxxa ] ) . this separation of the diagrams in two classes is completely analogous to the separation of the diagrams , coming in the standard self - energy @xcite , into an hartree - fock like set and another set involving the full vertex . the set of diagrams where the two fermionic lines interact can be immediately written in terms of @xmath31 , which is analogous ( and actually directly related ) to @xmath32 , except that it has , instead of outgoing lines , an ingoing ( @xmath46 ) line and an ingoing ( @xmath45 ) line . it satisfies a dyson s equation analogous to eq . ( [ eqdyson],[eqdysona ] ) , namely : @xmath109 introducing the fourier transform @xmath110 of the bare instantaneous interaction between @xmath75-atom and @xmath76-atom , this first contribution to the anomalous self - energy is : @xmath111 which has naturally just the form of the standard bcs term . let us now switch to the other set of diagrams , for which we proceed to an expansion similar to the one carried out in the preceding subsection . because of particle conservation this expansion involves now odd powers in @xmath112 ) . we consider first the term proportional to @xmath113 . the normal part of this diagram contains only one @xmath114 line and one @xmath76 line ( with any number of interactions ) , since additional lines are not allowed as discussed above . moreover , because the interaction is instantaneous , there is no possibility of crossing of interactions when the diagram is drawn in time representation . finally our lines are not allowed to go backward in time since this would correspond to the creation of a particle - hole pair , which is again not allowed in our case where we have a negative chemical potential at @xmath14 ( by contrast , for @xmath89 , such processes could naturally occur ) . in summary the only possible diagrams are those of repeated interactions between the @xmath114 and the @xmath115 atom . this corresponds merely to the propagation of a normal dimer . however these diagrams are already generated by the iteration of the term ( [ bcs ] ) when @xmath116 is written as @xmath117 from eq.([eqdysond ] ) at lowest order , and they should not be double - counted . hence there is no term of order @xmath16 in this set , and our expansion starts with terms containing @xmath118 . the analysis is now somewhat similar to the one made in the preceding subsection [ normself ] . to obtain the above third order term in the equation for @xmath17 , we have to write , together with the factor @xmath118 , all the normal state diagrams which have , in addition to the two entering lines ( @xmath46 ) and ( @xmath119 , another couple of entering lines coming from @xmath120 . moreover , we have two couples of outgoing lines , going into @xmath65 and @xmath121 . just as in the preceding subsection , we have a no first interaction restriction for all these couples of lines , including the ( @xmath46 ) and ( @xmath119 ones . as discussed above , in addition to these ingoing and outgoing ones with their continuations , no other lines can appear in these diagrams since they form loops , which are forbidden since we are at @xmath14 and @xmath22 . we find that such normal state diagrams have already been essentially encountered in ref.@xcite , where their sum has been called @xmath122 . here @xmath123 and @xmath124 are for the entering fermionic lines ( corresponding to @xmath18 and @xmath125 in our case ) , the total four - vector for the ingoing and outgoing lines being @xmath126 . finally there is , in the definition of @xmath127 , one entering @xmath128 dimer , while there are two outgoing dimers @xmath129 and @xmath130 . just as in the preceding section , we should take into account that , in the general case , we have the weight @xmath118 which depends on the four - vectors of the atom propagators making up the dimers , while in the definition of @xmath127 the summation over these variables is already performed , with only their sums @xmath128 , @xmath129 and @xmath130 being fixed . this problem could nevertheless be circumvented and an expression for the normal part obtained in terms of @xmath127 , in a way analogous to appendix [ appt3 ] . however , just as in subsection [ normself ] , this is not necessary . indeed from the lowest order term @xmath131 given by eq.([bcs ] ) , we see that @xmath17 does not depend at lowest order on the frequency variable @xmath132 . moreover , since we deal with a bare interaction potential @xmath133 which has a very short range , its fourier transform @xmath110 is a constant , except for very large wave vectors . but , as we will see , these wave vectors will not come into play because all the integrals we deal with converge at a much shorter wave vector . hence we find that , at lowest order , @xmath17 turns out to be independent of @xmath18 , just as in standard bcs theory . now , when we want to calculate the third order term , we may insert in it the expression of @xmath17 to lowest order , which is just a constant which we denote again as @xmath16 . accordingly the above problem disappears and the normal part of our diagrams can be expressed directly in terms of @xmath122 . nevertheless we have to take into account , just as in subsection [ normself ] , that some of the diagrams appearing in @xmath134 are actually not irreducible , and should not be included in our expression for @xmath17 . hence we have to subtract them out of @xmath135 . we denote @xmath136 what remains of @xmath127 after this subtraction . this will be taken care of just below . gathering our results , we end up with the following ( self - consistent ) equation : @xmath137 the factor 1/2 in front of @xmath136 is a general topological factor ( it is present even if we do not take @xmath16 as constant ) . indeed exchanging @xmath65 and @xmath121 corresponds to a mere change of variable and not to a different diagram . on the other hand , by making use of @xmath138 , any given such diagram would appear twice ( even if @xmath127 is actually unchanged when we exchange the bosonic variables of the two outgoing dimers , which amounts to change @xmath70 into @xmath139 ) . hence the factor 1/2 is required to avoid this double counting . we note that clearly @xmath17 depends indeed on @xmath18 in general , since this is already explicitely the case in eq.([eqdelta ] ) where the third order term depends on @xmath18 through @xmath140 . let us now be specific about the difference between @xmath127 and @xmath141 , due to the reducible diagrams contained in @xmath135 . one can obtain them by looking at the diagrammatic expansion @xcite of @xmath127 . however it is easier to notice , as indicated above , that they appear automatically explicitely from eq.([eqdysonc],[eqdysond ] ) , when their solution are expanded in order to obtain the complete perturbative series expansion of @xmath33 and @xmath31 in powers of @xmath142 and @xmath16 , because this expansion provides precisely all the reducible contributions in terms of the irreducible self - energies @xmath142 and @xmath16 . when we expand the solution for @xmath31 : @xmath143[g^{-1}_0(-p)-\sigma ( -p)]+\delta ( p ) \delta^ { * } ( p)}\end{aligned}\ ] ] to third order , we find : @xmath144g_0(-p)\end{aligned}\ ] ] where for consistency we have to take only the second order contribution @xmath63 ( given by eq.([eqsigmn2 ] ) ) to @xmath81 . the third order terms in the bracket are clearly the reducible contributions to @xmath145 , as it can be seen directly . they are displayed diagrammatically in fig.[figxxxx ] . in particular the last term of the bracket comes from the born term of @xmath146 . hence we have explicitely : @xmath147\end{aligned}\ ] ] . in the first two terms @xmath148 is obtained from @xmath73 by excluding the born term , since it is taken into account explicitely by the third term.,width=377 ] however it is now more convenient to work with @xmath127 rather than @xmath141 . hence we add to both members of eq.([eqdelta ] ) the reducible contributions . after multiplication by @xmath149 and taking into account eq.([eqf3 ] ) , we end up with : @xmath150 finally we obtain a closed equation by eliminating @xmath31 in favor of @xmath17 through eq.([eqdelta ] ) , or more simply through the definition eq.([bcs ] ) . multiplying eq.([eqf ] ) by @xmath151 and summing over @xmath18 we find : @xmath152 if we had only the first term in the right - hand side , we would merely have the linearized bcs gap equation . accordingly we proceed in the standard way @xcite to eliminate the bare potential in favor of the scattering amplitude . this is more easily understood by proceeding in a formal way . introducing the operator @xmath153 , with matrix elements @xmath154 , and the diagonal operator @xmath155 with matrix elements @xmath156 , the @xmath157-matrix for two fermions ( in the center of mass frame ) in vacuum satisfies @xmath158 , where the operator product implies naturally the summation on the intermediate variables with @xmath159 . this can just be written @xmath160 . if we note @xmath161 the second term in the right - hand side of eq.([eqf ] ) , we can rewrite eq.([eqgap1 ] ) as @xmath162 , or @xmath163 . inserting the above value of @xmath164 , this leads to @xmath165 , or @xmath166 . this is explicitely : @xmath167 where @xmath168 is the energy at which the @xmath157-matrix is calculated . in the present case , since @xmath169^{-1}$ ] , the total energy for the two fermions is @xmath170 ( @xmath19 for each fermion ) , as it can be seen directly by integrating the first term of eq.([eqgap1 ] ) over @xmath132 . in standard scattering theory the scattering amplitude @xmath171 at energy @xmath172 ( the reduced mass is @xmath173 ) is obtained from the evaluation of the @xmath157-matrix `` on the shell '' , i.e. taking @xmath174 with @xmath175 . in the present case we deal with a potential with a short range of order @xmath176 . this implies that @xmath177 and @xmath178 have to vary on a typical scale @xmath179 in order to have a significant variation of @xmath174 . since in the case of ultracold atoms we are dealing with much smaller wavevectors , and we may replace @xmath174 by @xmath180 . hence the scattering amplitude is merely given by @xmath181 . on the other hand , in this same regime , the scattering amplitude is given by its s - wave component @xmath182^{-1}$ ] . this gives @xmath183^{-1}$ ] . now we evaluate eq.([eqgap2 ] ) for @xmath184 small compared to @xmath179 . we see that in this range @xmath185 is essentially a constant , which we call simply @xmath186 . on the other hand @xmath161 is a rapidly decreasing function @xcite of @xmath187 , on a typical scale @xmath188 . hence we may replace in eq.([eqgap2 ] ) @xmath189 by @xmath190 , which can be expressed in terms of the scattering amplitude . this leads explicitely to : @xmath191\,\delta_1=\frac{1}{2 } \,\delta\ , |\delta |^2 \sum_{p}g_0(p)g_0(-p ) \phi ( p ,- p;0 , 0)\end{aligned}\ ] ] finally we take into account the fact that the sum over @xmath18 appearing in the right - hand side of eq.([eqgap3 ] ) is directly related to the dimer - dimer scattering amplitude @xmath192 in vacuum @xcite by : @xmath193 whereas : @xmath194 we note that , in eq.([eqt4 ] ) , we end up with @xmath192 being evaluated at a dimer energy @xmath195 , while the dimer - dimer scattering in eq.([eqam ] ) is related to @xmath192 evaluated at the binding energy @xmath196 . however at this stage the small difference between @xmath195 and @xmath197 is unimportant in @xmath192 , since it appears in the corrective term which is on the right - hand side of eq.([eqgap3 ] ) . in this term we may also , within the same level of accuracy , use @xmath198 . this leads finally to : @xmath199 when we substitute in this expression the relation @xmath200 found in section [ deltexp ] a , we obtain at the same level of accuracy : @xmath201 which is the standard mean field correction to the chemical potential , with the proper dimer - dimer scattering length @xmath1 . since the lhy terms come physically from the low energy collective mode , we need first to derive the expression for the collective mode propagator , as well as its anomalous counterpart . these propagators can be seen as the extension to the superfluid state of the dimer propagator in the normal state . for the normal component of this propagator , this is just the full vertex with two entering and two outgoing fermion lines , the two fermions having opposite spins . in the normal state , in the dilute limit , this vertex is just the single dimer propagator @xmath202 already used in ref.@xcite . it depends only on the total momentum @xmath203 and the total energy @xmath204 of the two fermions . in contrast with the above reference we work in the grand canonical ensemble and the energies are taken with respect to the chemical potential @xmath19 of single fermions . hence for all the energies @xmath205 used in ref.@xcite , we have to make the substitution @xmath206 . for the single dimer propagator we have to make @xmath207 . however we keep for simplicity the same notation as in ref.@xcite for the various quantities after having made this substitution . also for coherence , and in contrast with the preceding sections , we work in the present section with real frequencies , rather than with matsubara frequencies . this is also more natural physically since we are interested in particular in the ( real ) frequency of the collective mode . making use of the quadrivector notation @xmath208 we have thus , for this single dimer propagator in the dilute limit : @xmath209 where @xmath210 is the dimer binding energy , and we have made explicit the fact that @xmath211 . note that physically this propagator describes the bound state of the two fermions , through its pole at @xmath212 , as well as the continuum of scattering states where the dimer is broken , through the continuum in the spectral function starting at @xmath213 . in the superfluid we want to obtain the correction to this normal state result to lowest order in @xmath214 . more precisely we want to correct what corresponds to the self - energy if we had a single particle propagator . in the present case this is the irreducible vertex in the particle - particle channel , which by definition can not be separated in two disconnected parts by cutting two fermion propagators . more precisely , since the bare interaction between fermions is already taken into account in @xmath108 , this is the part of this irreducible vertex beyond the bare interaction we are interested in . this part @xmath215 of the irreducible vertex depends on the quadrivectors @xmath123 and @xmath124 of the entering particles , as well as on those @xmath216 and @xmath217 of the outgoing particles . assuming for the moment quite generally that the full vertex @xmath218 depends also on these four quadrivectors , we can write a bethe - salpeter equation , which is shown diagrammatically in fig . [ figgammaab]a : @xmath219 @xmath220 where @xmath221 . we have made explicit the fact that the normal state dimer propagator @xmath108 depends only on the sum @xmath222 of the incoming and outgoing fermions quadrivectors . in eq.([betsal ] ) the first sum is completely analogous to what should be written in the normal state . however we had also to write the second sum , which describes the anomalous processes where a dimer can be annihilated into the condensate or can be created from the condensate . this leads to introduce , in correspondance with the first sum , an anomalous irreducible vertex @xmath223 where four fermions are entering ( annihilated into the condensate ) and none is going out . correspondingly we need to introduce an anomalous full vertex @xmath224 with only outgoing fermion lines . naturally in order to close our set of equations we have to write for this vertex the corresponding bethe - salpeter equation , which is shown diagrammatically in fig . [ figgammaab]b , the major difference with the preceding one being that there is no normal state term : @xmath225 @xmath226 where we have introduced the irreducible vertex @xmath227 where there are only outgoing fermions , physically created from the condensate . now it is explicit in eq.([betsal ] ) that @xmath218 depends only on @xmath228 . by time reversal symmetry it depends also only on @xmath229 , this last equality resulting from momentum - energy conservation . hence , just as @xmath108 , it depends only on the total momentum and energy @xmath71 , and we merely denote this function as @xmath230 . similarly @xmath231 depends only on @xmath232 and we call it merely @xmath233 . in this way eq.([betsal ] ) becomes simply : @xmath234 and similarly : @xmath235 where we have introduced : @xmath236 these equations are naturally quite similar to the ones we had for the single fermion green s function . their solutions are : @xmath237 \left[t_2^{-1}(-p)-\gamma_{\mathrm irr}(-p)\right ] -\gamma^a_{\mathrm irr}(p){\bar \gamma}^a_{\mathrm irr}(-p)}\end{aligned}\ ] ] and @xmath238 \left[t_2^{-1}(-p)-\gamma_{\mathrm irr}(-p)\right ] -\gamma^a_{\mathrm irr}(p){\bar \gamma}^a_{\mathrm irr}(-p)}\end{aligned}\ ] ] as it could be expected physically , these equations are formally identical to those obtained for pure bosons @xcite . in the same spirit we note , for later use , that we could also introduce an anomalous full vertex @xmath239 with only ingoing fermion lines . it is easily shown , by writing the equivalent of eq.([betsalp ] ) with this vertex , to be related to the preceding ones by @xmath240 . this gives : @xmath241 \left[t_2^{-1}(-p)-\gamma_{\mathrm irr}(-p)\right ] -\gamma^a_{\mathrm irr}(p){\bar \gamma}^a_{\mathrm irr}(-p)}\end{aligned}\ ] ] we have now to find the expressions of @xmath242 , @xmath243 and @xmath227 within our expansion in powers of @xmath16 . since @xmath16 corresponds to diagrams with two ingoing fermions and @xmath64 to two outgoing fermions , particle conservation implies that the expansions start at second order with @xmath242 proportional to @xmath244 , @xmath243 proportional to @xmath91 and @xmath227 proportional to @xmath245 . since it can be seen that it is not required to go to higher order , the expressions of our irreducible vertices will be given by these second order expressions . since we have written explicitely the various factors @xmath16 and @xmath246 , the coefficients in the expressions of @xmath242,@xmath243 and @xmath227 correspond to diagrams containing only normal state propagators . for @xmath242 ( see fig . [ figxxabc]a ) , we have to write the expression for all the normal state diagrams with one ingoing dimer propagator and one outgoing dimer ( from the very definition of @xmath242 ) . moreover there are in addition two lines ingoing from @xmath64 and two lines outgoing to @xmath16 . these last two couples of lines have to be treated as dimer lines , which means that they are not allowed to interact immediately in the normal state diagrams we are drawing , because such interactions are already taken into account in @xmath16 and @xmath247 , and allowing them would amount to double counting . in summary we have to write all possible normal state diagrams with two ingoing dimer lines and two outgoing dimer lines ( these diagrams are automatically irreducible in the sense of the definition of @xmath242 , that is with respect to a single dimer propagator , since the four ingoing normal state lines can not disappear ) . this is depicted diagrammatically in fig . [ figxxabc]a . but this set of diagrams is just by definition the normal state dimer - dimer scattering vertex @xmath192 introduced in @xcite . more precisely , since the total momentum - energy linked to @xmath16 and @xmath247 is zero and the total momentum - energy of the ingoing lines is @xmath71 , this vertex is @xmath248 with the notations of ref.@xcite which gives for @xmath242 the explicit expression : @xmath249 with completely analogous arguments for @xmath243 , we obtain : @xmath250 and similarly : @xmath251 corresponding to the diagrams found in fig . [ figxxabc]b and fig . [ figxxabc]c . in eq.([gammirra ] ) and ( [ gammirrba ] ) we had to put a 1/2 topological factor corresponding to the fact that , for example , exchanging the two factors @xmath16 in eq.([gammirra ] ) ( that is precisely the set of diagrams they correspond to ) does not produce a different diagram , whereas this diagram would appear twice if we had just written @xmath252 . . a ) normal irreducible vertex @xmath253 . b ) and c ) anomalous irreducible vertices @xmath254 and @xmath255.,width=453 ] finally it is worth pointing out that @xmath192 satisfies @xmath256 . this is physically obvious since it corresponds to state that exchanging the arguments of the two ingoing dimer lines does not change the value of the vertex , since we exchange two bosons . but this property can also be proved directly from the integral equation @xcite satisfied by @xmath192 . similarly we have @xmath257 . let us consider now more specifically the collective mode dispersion relation . it is given by the pole of the collective mode propagator , that is : @xmath258 \left[t_2^{-1}(-p)-\gamma_{\mathrm irr}(-p)\right ] -\gamma^a_{\mathrm irr}(p){\bar \gamma}^a_{\mathrm irr}(-p)}=0\end{aligned}\ ] ] naturally we have first to check that for zero wave vector @xmath259 , the frequency @xmath260 is zero , that is @xmath261 is solution of this dispersion relation . for this case we have from eq . ( [ 2vertex ] ) for the bare dimer propagator : @xmath262 on the other hand we have @xmath263 where the last equality comes from ref.@xcite , and we have used the fact that , in @xmath264 , we may , at the order we are working , replace @xmath265 by its zeroth order value , namely @xmath266 ( we recall that we have to shift all the frequencies of ref.@xcite by @xmath195 ) . since we have already found that @xmath267 , we can easily check that eq.([moddisp ] ) is indeed satisfied in this case . now , as we will see , we are more specifically interested in the collective mode dispersion relation for wave vectors @xmath268 and frequencies @xmath269 . since @xmath270 and @xmath271 are of the same order of magnitude , we can expand the bare dimer propagator to first order as : @xmath272 on the other hand the typical scale of variation of @xmath192 on its arguments is @xmath188 for the wave vectors and @xmath270 for the energies , since we deal with a normal state quantity and there are accordingly no other scales in the corresponding problem . hence , for the wave vectors and frequencies we are considering , we can still take : @xmath273 on the other hand the above result at @xmath261 implies also that we have @xmath274 , and eq . ( [ moddisp ] ) for the collective mode frequency @xmath275 can be rewritten as : @xmath276 where we have set @xmath277 and in the last step we have used the lowest order expressions for @xmath278 and @xmath214 to calculate this quantity . naturally eq.([bog ] ) is just the bogoliubov dispersion relation , with the proper scattering dimer - dimer scattering length . as we have already indicated in the introduction , the lhy term arises from the collective mode contributions to the self - energies . if we were to continue naturally our above procedure and go to next order in @xmath16 , i.e. to order @xmath279 for example in the calculation of the normal self - energy , we would find singular results ( as it can be checked by proceeding to such a formal expansion in the explicit calculations performed below ) . this is due to the fact that the collective mode is gapless , and the lack of gap in the excitation spectrum leads to a singularity in the perturbative expansion . hence , instead of the next order terms , we include the whole contribution coming from the collective mode . avoiding in this way to proceed immediately to a @xmath16 expansion is equivalent to a series resummation , which allows to take care of singularities . writing the collective mode contributions to the self - energy is formally easy , when one notices for example the analogy of the normal part of the collective mode propagator with the product of @xmath64 and @xmath16 . indeed the term @xmath64 acts as a source of fermions in our diagrams , coming physically out of the condensate , while @xmath16 acts as a sink of fermions going into the condensate . these source and sink related to the condensate are required since , at @xmath280 , no fermions are present except coming from the superfluid . however when we allow to insert the normal part of a collective mode propagator , the starting part of the propagator acts as a sink for the rest of the diagram while the end part acts as a source . hence we have just to substitute @xmath244 by the normal part of a collective mode propagator . we will proceed in a similar way for the anomalous parts . naturally we will have to take care that the self - energy contributions already calculated above are not counted twice , by removing the corresponding terms . hence the collective mode contribution @xmath281 to the normal self - energy is obtained by replacing , in our lowest order expression , the product @xmath244 by the collective mode propagator , more specifically its diagonal component @xmath282 given by eq.([gamn ] ) , as depicted diagrammatically in fig.[figxab ] . we have naturally to sum over the variable @xmath71 of the collective mode . corresponding to eq.([eqsigmn2 ] ) , this leads to : @xmath283 . the diagonal component @xmath282 of the collective mode propagator is depicted by the heavy line , width=226 ] in principle this expression is improper because it contains , when we perform its expansion in powers of @xmath214 , contributions of zeroth and first order in @xmath214 which have already been taken explicitely into account in our above expansion . however in the present case these two contributions are zero . for the zeroth order term , this is fairly obvious physically since it corresponds to a normal state diagram with a normal state dimer propagator , and at @xmath14 no dimers are present which makes all diagrams of this kind equal to zero . mathematically this would happen because , in this diagram , the presence of a normal state dimer would necessarily imply , since there is no down - spin entering line , the presence of a down - spin loop ( if we calculate the up - spin self - energy ) which can not be present as explained in appendix [ appnoloop ] . in the collective mode contribution to the normal self - energy eq.([eqsigmod ] ) depicted in fig.[figxab],width=226 ] the conclusion is the same for the first order term , but the argument is slightly more complicated . the factor of @xmath214 in this term is @xmath284 ^ 2t_4(p/2,p/2;p/2)$ ] , which is a normal state diagram ( see fig.[figaaaa ] ) . it is actually a part of the diagrams belonging to @xmath102 since it has one entering fermion line and one entering dimer line , with the same for the outgoing lines . however we have seen , as it is detailed in appendix [ appt3anal ] , that only the born term in @xmath73 gives a non zero contribution to the particle number . on the other hand the term @xmath284 ^ 2t_4(p/2,p/2;p/2)$ ] does obviously not contain the born term of @xmath73 since this last one reduces to a simple propagator , whereas the presence of the factor @xmath108 implies at least two lines , with interaction between them . actually one can see more precisely that our term is equal to zero , since it contains necessarily propagators going backward in time ( see appendix [ appnoloop ] and [ appt3anal ] ) . taking the born contribution for @xmath73 and @xmath192 in @xmath284 ^ 2t_4(p/2,p/2;p/2)$ ] gives a good example of this . finally , after these considerations , we see that we can still use eq.([eqsigmod ] ) for @xmath285 . the conclusion will be different for the mode contribution to the anomalous self - energy discussed below . now when we look for the contribution @xmath286 to the density corresponding to eq.([eqsigmod ] ) , we have to replace @xmath33 in eq.([eqdefn ] ) by @xmath287 , just as it is done below eq.([eqdefn ] ) . quite similarly it will be enough to replace here @xmath33 by @xmath40 to obtain the lowest order contribution of the collective mode we are interested in . this gives : @xmath288 ^ 2= \sum_{p , p}\,t_3(p , p;p+p ) \,\gamma(p)\left[g_0(p)\right]^2\end{aligned}\ ] ] where again @xmath289 and @xmath290 . now the argument runs just as below eq.([eqdefn1 ] ) . we integrate first on the frequency variable @xmath96 of the quadrivector @xmath18 . as shown in appendix [ appt3anal ] , it turns out that , except for the born contribution , @xmath291 is analytical in the upper complex half - plane of the variable @xmath96 . since @xmath292 ^ 2=[i\nu -\xi_{\bf p } ] ^{-2}$ ] is also analytical in this half - plane , one obtains by closing the integration contour at infinity in this half - plane that only the born part @xmath293 of @xmath291 contributes to the result . hence for the integration over @xmath96 , we have just to calculate : @xmath294^{2}}\,\frac{1}{i\omega - i\nu- |\mu |-\epsilon _ { { \bf p}-{\bf p}}}= -\frac{1}{[i\omega -2|\mu |-\epsilon _ { \bf p}-\epsilon _ { { \bf p}-{\bf p } } ] ^{2}}\end{aligned}\ ] ] and we are left with : @xmath295^{2}}\end{aligned}\ ] ] the factor of @xmath282 is analytical in the upper complex half - plane of @xmath260 ( which corresponds to the negative frequency half - plane when we go back to the standard frequency language ) . hence by closing the @xmath204 contour in this half - plane , we are left to consider the singularities of @xmath282 . physically they correspond to the two - fermions excitations of the system , that is the collective mode and the broken molecule excited states . in the standard frequency language , the corresponding singularities are on the real frequency axis ( the imaginary axis for @xmath204 ) , on the negative as well as on the positive side . let us first consider the low frequency singularities . for a fixed value of @xmath296 , we have just as singularity a single pole corresponding to the collective mode frequency . let us call @xmath275 the positive solution of eq.([bog ] ) . our pole is located at @xmath297 . in eq.([gamn ] ) , which gives @xmath282 , for low frequency and wavevector , the denominator becomes @xmath298 , while from eq.([2vertex1 ] ) the numerator is @xmath299 $ ] . after going to imaginary frequency @xmath300 , this gives in @xmath282 , for the pole at @xmath301 , a residue : @xmath302\end{aligned}\ ] ] on the other hand , for small @xmath204 and @xmath296 , the @xmath187 integration in eq.([eqnmoda ] ) reduces to : @xmath303^{2}}=\frac{m^{3/2}}{8\pi \sqrt{2 |\mu |}}\end{aligned}\ ] ] this leads to : @xmath304\end{aligned}\ ] ] where in the last step we have made the change of variables @xmath305 . the numerical integral is @xmath306 and our final result is : @xmath307 where in the last equality we have made use of the lowest order result for @xmath214 , given by eq.([eqdefn3 ] ) . naturally , once we have taken care of @xmath73 , we are essentially back to the elementary boson theory @xcite and one recognizes in eq.([ncmfin ] ) the standard `` depletion of the condensate '' for the dilute bose gas , with @xmath308 being its coherence length . we have first considered the contribution of the low frequency singularities . we have now to finish up by showing that the higher frequency ones give a negligible contribution . first we see that , in eq.([eqnmoda ] ) , the denominator plays in this respect an unimportant role , since its modulus is larger than @xmath309 . hence to have an upper bound for the contributions we are investigating , we can replace this denominator by this lower bound , which gives after integration over @xmath187 an unimportant factor @xmath310 . hence we are again left with the evaluation of @xmath311 . it is now more convenient to go back to real frequencies ( equivalent to the change of variables @xmath312 ) . the integration is now over the imaginary @xmath313 axis and is closed in the negative @xmath313 half - plane . the contour can be deformed to merely enclose the negative @xmath313 axis , and corresponding to the singularities on this axis , @xmath314 has a jump across this axis . hence we have : @xmath315 now , from eq.([gamn ] ) , @xmath316 contains a first term @xmath317 which has all its singularities on the real positive @xmath313 axis , corresponding to the continuous spectrum of the broken dimer ( for @xmath108 ) and of the broken pair of dimers ( for @xmath318 ) . hence it is real on the negative @xmath313 axis and does not contribute to @xmath319 in eq.([eqcorr1 ] ) . from eq.([gamn ] ) the second term of @xmath320 is @xmath321 , which is directly proportional to @xmath322 from the product @xmath323 . accordingly we have proved basically our point since our result , being proportional to @xmath324 , is indeed negligible compared to the result eq.([ncmfin ] ) , which is proportional to @xmath2 . actually , to complete our proof , we have to be more careful and evaluate the coefficient of @xmath322 . this is quite obvious since eq.([ncmfin ] ) is in fact a contribution contained in eq.([eqcorr1 ] ) ( we have made no approximation to reach eq.([eqcorr1 ] ) ) . this detailed analysis is performed in appendix [ normselfa ] . we will now proceed in a similar way to obtain the mode contribution @xmath325 to the anomalous self - energy @xmath326 . this contribution is obtained by replacing , in the diagrams which lead to the second term of eq.([eqdelta ] ) , any second order product @xmath244 , @xmath327 or @xmath328 , by the corresponding collective mode propagator . accordingly we replace specifically @xmath244 by the diagonal component @xmath282 , and @xmath327 by the off - diagonal one @xmath329 . this is depicted diagrammatically in fig.[figxabcd ] . just as in eq.([eqdelta ] ) this last term comes with a topological factor @xmath330 to avoid double counting . on the other hand , since the two dimer outgoing lines in the first term play inequivalent roles , such a topological factor does not apply to this first term . it is worth noticing that , since the collective mode propagator depends only on the total momentum - energy four - vector , we do not meet at this stage the problems linked to the @xmath177-dependence of @xmath326 , discussed in appendix [ appt3 ] . . the diagonal component @xmath282 of the collective mode propagator is depicted by the heavy line with a single arrow and the off - diagonal one @xmath329 by the heavy line with two arrows . for the two diagrams the slash in the heavy lines means that the terms which are first order in @xmath214 have been removed.,width=377 ] just as in section [ anomself ] , we should not include the terms of @xmath122 leading to reducible contributions to the self - energy . these are easily obtained from the diagrams displayed in fig.[figxxxx ] by making the above substitution of @xmath244 by @xmath282 . this implies in the same way the replacement of @xmath122 by @xmath331 . however , as in section [ anomself ] , it is more convenient to add these reducible terms to both members of the equation for @xmath16 . in the left - hand side one obtains @xmath332^{-1}f(p)[g_0(-p)]^{-1}$ ] , while in the right - hand side one recovers @xmath127 instead of @xmath141 . finally we restrict ourselves to the contribution of the low frequency collective mode , the higher frequency contributions being expected to be of order @xmath333 and thus negligible , just as in the preceding section [ modnormal ] . accordingly the variable @xmath71 in @xmath282 and @xmath329 is small . this implies that @xmath334 , which comes as a factor of @xmath282 , can be replaced by @xmath335 at the order we are working . similarly @xmath336 , which comes as a factor of @xmath329 , can also be replaced by @xmath335 . after multiplication by @xmath149 this leads to the equation : @xmath337\end{aligned}\ ] ] which comes in place of eq.([eqf3 ] ) . in the preceding section [ modnormal ] we have already calculated @xmath338 at the level of eq.([eqintx ] ) and found @xmath339 , where @xmath340 is defined by eq.([ncmfin ] ) . in principle the calculation of @xmath341 follows the same line as the one of @xmath338 . one performs first the frequency integration by noticing that , for fixed @xmath296 , @xmath329 has a pole at @xmath297 . the corresponding residue @xmath342 is obtained from the residue @xmath343 for @xmath282 by comparing eq.([gama ] ) with eq.([gamn ] ) . one has to multiply @xmath343 by @xmath344 $ ] , evaluated at the pole , to obtain @xmath342 . this gives merely : @xmath345 now if we proceed as in eq.([eqintx ] ) we come to an integral with an ultraviolet divergence , which would invalidate our procedure . the reason for this problem is that we have not yet consistently removed in our calculation the terms of order zero and one in @xmath214 , as we have done precedingly . that we should do it was already appearent in eq.([eqfapp ] ) by the notation @xmath346 instead of @xmath347 ( which implies also a multiplication by @xmath348 ) . we recall that these terms have to be subtracted out because they have already been taken into account in our preceding lowest order calculation . however in the above preceding cases , it turned out that these terms were equal to zero , which made the subtraction transparent . here this is not the case . the most efficient way to perform this subtraction is to note that , from eq.([gamabar ] ) , @xmath329 contains already an explicit factor @xmath254 in the numerator , proportional to @xmath214 , which gives directly the factor @xmath349 in eq.([barza ] ) . hence the term of order @xmath214 to be subtracted is merely obtained by taking the other factors to order zero in @xmath214 , that is by replacing @xmath275 in eq.([barza ] ) by its zeroth order expression @xmath350 . actually this argument is a bit too fast . it misses the fact that we may at the same time evaluate @xmath351 in eq.([eqfapp ] ) with the chemical potential taken as @xmath352 , even for the @xmath353 term , as we had done in eq.([eqt4 ] ) . hence this whole step is taken up in more details in appendix [ muzero ] . we find in this way : @xmath354\end{aligned}\ ] ] making again the change of variables @xmath305 , this gives : @xmath355=\frac{32}{\pi } \mu _ b^{3/2}|\mu |^{1/2 } = \frac{24\pi \sqrt{2 |\mu |}}{m^{-3/2}}\,n_{cm}=3\sum_{p}{\gamma}(p)\end{aligned}\ ] ] hence the last bracket in eq.([eqfapp ] ) can be written @xmath356 . the rest of the argument proceeds just as in section [ anomself ] , provided @xmath353 is replaced by the above bracket . this leads to the replacement of eq.([mean ] ) by @xmath357\end{aligned}\ ] ] we may eliminate @xmath214 by making use of the number equation , which reads now from eq.([eqdefn2 ] ) : @xmath358 this gives : @xmath359=\pi a\ , a_m\left[n+4n_{cm}\right]\end{aligned}\ ] ] where in the last equality our replacement of @xmath360 by @xmath361 introduces only errors of order @xmath3 in the equation of state . to the same order this leads us to : @xmath362\end{aligned}\ ] ] in the last term we may replace @xmath286 by its expression eq.([ncmfin ] ) @xmath363 . this gives the familiar lee - huang - yang result for the chemical potential of a dimer , taking its binding energy as reference : @xmath364\end{aligned}\ ] ] in this paper we have presented a general theoretical framework to deal with fermionic superfluidity on the bec side of the bec - bcs crossover . our approach involves no approximation in principle , but in order to obtain explicit results we have to proceed to an expansion in powers ot the anomalous self - energy , which is basicallly equivalent to an expansion in powers of the gas parameter @xmath365 . we have applied our method to the @xmath14 thermodynamics and calculated the first terms in the expansion of the chemical potential in powers of the density . we have in this way shown explicitely that the mean - field contribution is indeed given by the standard expression in terms of the dimer - dimer scattering length @xmath1 . we have then proved that the lee - huang - yang contribution to the chemical potential retains also its standard expression from elementary boson superfluid theory provided @xmath1 is used in its expression . departures from elementary boson theory appear at the order @xmath3 . naturally our approach can be used to obtain other physical quantities and extended for example to finite temperature and dynamical properties . this will be considered elsewhere . `` laboratoire de physique statistique de lecole normale suprieure '' is `` associ au centre national de la recherche scientifique et aux universits pierre et marie curie - paris 6 et paris 7 '' . if we note @xmath367 the sum of all normal diagrams , with one single atom @xmath18 line going in and out , two entering atom lines @xmath368 ) and @xmath369 ) corresponding to an ingoing dimer , and two outgoing atom lines @xmath370 ) and @xmath371 ) corresponding to an outgoing dimer , we have : @xmath372 in order to obtain an expression for @xmath367 , one can find in the spirit of ref.@xcite integral equations for this quantity , and then identify in them a quantity which is precisely @xmath68 . another slightly more direct way is to separate , in @xmath373 , all the diagrams where 0 or 1 dimer lines are present , right at the beginning or right at the end of @xmath374 . we do not enter into the details but only give the result , on which the above mentionned separation is clear : @xmath375 ^ 2\,g(p')\,t_2(p - p')\,\delta_{p',p '' } \nonumber \\ + g(p')g(-p ' ) g(p'')g(-p'')t_2(p - p')t_2(p - p'')t_3(p',p'';p)\end{aligned}\ ] ] where @xmath376 is the kronecker symbol , and @xmath377 is the normal state dimer propagator . one can check , by making use of eq . ( [ eq2appt3 ] ) and of the integral equation @xcite satisfied by @xmath73 , that eq . ( [ eq1appt3 ] ) reduces to eq . ( [ eqsigmn2 ] ) when @xmath17 does not depend on @xmath18 . since we are at @xmath14 with a negative chemical potential , there is no fermi sea and no possibility of fermionic holes . when we consider the propagation of a fermionic atom , it has to be created before being destroyed . this corresponds to the fact that the free fermion propagator is retarded . in ( imaginary ) time and wavevector representation we have for the free atom green s function : @xmath378 where @xmath379 is the heaviside function . we consider now the contribution coming from any loop in a diagram . for clarity we do not write the wave vectors , which do not play any role in the argument . the loop starts for example at a vertex with time @xmath380 , with successive interactions at times @xmath381 . this implies in the term corresponding to this diagram the presence of the product @xmath382 . in order to have each term being non zero in this product of retarded green s function , we need @xmath383 , which is impossible to satisfy . hence one of the green s function in this product will necessarily be zero , and the contribution of any diagram with a loop will also be zero . note that the argument applies also if there is a single green s function in the loop @xcite . if we were to carry out calculations in frequency , rather than time , representation the corresponding free atom green s function is analytical in the upper complex half - plane , and the above result would appear through integration over frequencies with contours closed in the upper complex half - plane . we have considered for consistency the imaginary time formalism . naturally exactly the same arguments apply if we consider the real time representation . we give first an explicit proof of the analytical property of @xmath73 indicated in the text . we start from the integral equation satisfied by @xmath73 with at first essentially the notations of ref.@xcite : @xmath385 we follow ref.@xcite and perform the frequency integration in the second term , making use of the fact that the only singularity in the lower half complex plane comes from @xmath386 and is located at @xmath387 . we write then explicitely the resulting equation for @xmath388 , @xmath389 , while @xmath390 is replaced by @xmath391 , keeping in mind that the frequencies @xmath69 and @xmath70 have to be shifted by @xmath19 ( and @xmath391 by @xmath392 ) and that we work now with imaginary frequencies . this gives : @xmath393 @xmath394 when we write this equation for @xmath395 , we see that the first term , which is the born contribution , has a pole in the upper complex half - plane , at @xmath396 . on the other hand , in the integral , the first factor does not depend on frequency @xmath313 . the singularities coming from the dimer term will be in the lower complex half - plane with @xmath397 ( since we will have @xmath398 ) . regarding the @xmath399 term , if we write eq . ( [ 3vertex1 ] ) for the variable @xmath400 evaluated on the shell , i.e. @xmath401 , we obtain an integral equation for @xmath399 . in this equation the born term does not depend on frequency . the first term in the integral has its poles satisfying @xmath402 and the dimer term has again its singularities in the lower complex half - plane . hence by a recursive argument ( or equivalently by iteration of the equation ) , we conclude that the singularities of @xmath403 are also in the lower complex half - plane . this shows that indeed , except for the born contribution , @xmath384 is analytical in the upper complex half - plane of the frequency variable @xmath313 . involved in the calculation of the particle number @xmath0 . we use the standard representation where the propagator starts at the destruction time and ends at the creation time . b ) born term in the diagrammatic expansion of the propagator represented in a ) . c ) example of a diagram which gives a zero contribution due to the presence of a loop.,width=566 ] naturally we do not expect such a result to come by chance and we discuss now its physical origin . it is more easily seen in real time , rather than frequency , representation . the particle number eq.([eqdefn ] ) is merely given by : @xmath404 with @xmath405 . the average corresponds to a full propagator @xmath406 in time representation , where the particle is created at time @xmath407 , later than the time @xmath408 at which it is destroyed . in standard @xcite diagrammatic representation ( see fig.[figappc ] a ) ) , with time going from left to right and the propagator starting at the destruction time and ending at the creation time @xcite , we have to draw diagrams where the @xmath114 particle propagator starts from @xmath409 and goes globally forward ( that is to the right in fig.[figappc ] ) in time , ending up at @xmath407 . however when we write the perturbation expansion of this propagator , we have to use free particle propagators @xmath87 which are retarded and go only forward in time ( i.e. the arrows go to the left in fig.[figappc ] when we use the standard green s function representation @xcite ) . hence the zeroth order in our perturbation expansion is obviously zero . when we go to second order , corresponding to eq.([eqdefn1 ] ) , we can use @xmath64 which creates , at some given time , a pair of particles @xmath410 ( this process is instantaneous in time because , at lowest order , @xmath16 does not depend on frequency ) . similarly @xmath16 annihilates a pair of particles . the simplest combination of these ingredients , giving a non zero result for @xmath0 , is shown in fig.[figappc ] b ) : @xmath64 is located at positive time , and @xmath16 at negative time , allowing to have @xmath87 s going only to the left . this is just the born term of @xmath73 . however one sees easily that it is not possible to draw any other second order diagrams having the same property shown in fig.[figappc ] b ) . they would necessarily imply @xmath87 s going at some stage backward in time ( i.e. with arrows pointing to the right in fig.[figappc ] ) , which is not allowed . indeed we have to dispose of the @xmath114 propagator starting at @xmath409 , and this can only be done by annihilating it with @xmath16 at some negative time . similarly the @xmath114 propagator ending up at @xmath407 has to come from @xmath64 , acting at some positive time . then we are left with a @xmath115 propagator which can only go from @xmath64 to @xmath61 . any interaction between the @xmath114 and @xmath115 propagators is forbidden since this would merely amount to double count processes which are already taken into account in @xmath64 and @xmath61 . in this way we end up with the conclusion that only the born term leads to a non zero contribution to @xmath0 . in calculating the coefficient of @xmath322 in eq.([eqcorr1 ] ) , it is clear that we can evaluate all the other factors to zeroth order in @xmath16 , since we have already an explicit factor @xmath322 . this implies in particular to make @xmath411 in these terms . this leaves us with : @xmath412 @xmath413 . naturally this result could have been obtained more directly , by a straight expansion of @xmath282 in powers of @xmath16 , instead of resumming a whole perturbation series , as we have done in section [ collmod ] , and then expanding the result . we may first apply this formula for the collective mode contribution . in the present case it comes from the factor @xmath414 , which has a pole for @xmath415 . this is indeed the frequency of the bogoliubov mode , once the frequency is high enough for the zeroth order approximation in @xmath16 to be valid . correspondingly we have @xmath416 . on the other hand , we have for this value of the frequency @xmath417^{-1}$ ] . this has to be inserted in eq.([eqgam ] ) and the integration over @xmath296 has still to be performed . however if we concentrate on the low @xmath296 range ( where @xmath418 ) , we obtain a divergent integral @xmath419 . this is due to the fact that , in this range , our zeroth order handling in @xmath420 is not careful enough . from eq.([bog ] ) , we see that the dispersion relation has to be modified for @xmath421 . if we evaluate the low frequency contribution by putting a lower cut - off @xmath422 in the integral , we obtain @xmath423 and the overall result is of order @xmath2 . this is just the contribution eq.([ncmfin ] ) , where we have obtained the precise coefficient by our careful handling of this low frequency domain . now , if we define the low momentum region as @xmath424 , where @xmath425 is large compared to @xmath422 , but small compared to @xmath188 , our above treatment of the low frequency region is fully valid in this domain . on the other hand the integral in eq.([eqintx ] ) has already converged at the upper bound @xmath426 , and we may replace it by @xmath427 , just as we have done in eq.([eqintx ] ) . hence the contribution from the domain @xmath424 is exactly given by eq.([ncmfin ] ) . on the other hand , there are no divergences in the domain @xmath428 and its contribution is of order @xmath322 , and accordingly negligible compared to eq.([ncmfin ] ) . to be complete we have also to verify that there are no other sources of divergence in eq.([eqgam ] ) , analogous to what we have found at low momentum . since there is no singularity arising at finite frequency or momentum , the only possibility comes from the large frequency @xmath429 , large momentum @xmath430 region . in order to evaluate the contribution in this domain , it is more convenient to go back to the imaginary axis contour by performing backward the first step of eq.([eqcorr1 ] ) and write this overall contribution as : @xmath431 the scattering amplitudes @xmath432 and @xmath433 , introduced by @xmath254 and @xmath434 , may be shown to be equal and are dominated as expected in this range by the born contribution . they are real on this imaginary frequency axis . the born term has been obtained explicitely in @xcite , and @xmath432 is proportional to @xmath435 for large @xmath436 and @xmath437 . from eq.([2vertex ] ) there is in the modulus of the integrand a factor @xmath438^{3/4}$ ] coming from @xmath439 . then the overall integral is easily seen to be convergent for large @xmath436 and @xmath437 , which completes our proof . since there is no zeroth order term in powers of @xmath420 in @xmath329 , we have only to remove the term proportional to @xmath214 to avoid double counting , and go in this way from @xmath329 to @xmath440 . the corresponding diagram in @xmath31 is shown in fig.[figdiagsub ] . this leads to : @xmath441 where @xmath254 is given by eq.([gammirra ] ) . let us first replace the dimer propagator @xmath108 , given by eq.([2vertex ] ) , by its value @xmath442 where @xmath443 is replaced by its zeroth order value @xmath352 . when we consider the contribution from the low frequency and momentum contribution , we may replace @xmath254 by @xmath444 , and the pole of @xmath445 at @xmath446 gives : @xmath447= \frac{4\pi \mu _ b\sqrt{e_b}}{m^{3/2 } } \frac{4m}{{\bf p}^2}\end{aligned}\ ] ] this is indeed the second term in the right - hand side of eq.([eqgamaslash ] ) since in this last equation , which turns out to be of order @xmath448 , we have to make consistently @xmath449 ( and similarly we have to take @xmath450 , with @xmath451 ) . we are now left with the evaluation of the difference between our original expression and the one we have just considered , namely : @xmath452\end{aligned}\ ] ] which gives at the level of eq.([eqgap3 ] ) , after insertion at the level of eq.([eqfapp ] ) and summation over @xmath18 and @xmath71 : @xmath453 t_4(p,0;0 ) \end{aligned}\ ] ] where we have used eq.([gammirra ] ) for @xmath254 and the relation ( see ref . @xcite ) between @xmath192 and @xmath127 , namely @xmath454 . we show now that this difference is just what is required to be allowed to replace @xmath455 in eq.([eqt4 ] ) by @xmath456 , which is directly related to the dimer - dimer scattering scattering length @xmath1 by eq.([eqam ] ) . indeed it has been shown in @xcite that scattering amplitude @xmath457 satisfies quite generally an integral equation analogous to the standard integral equation for elementary bosons , namely : @xmath458 where @xmath459 is the sum of all irreducible diagrams with respect to cutting two dimer propagators , which is itself obtained from an integral equation with explicit kernel @xcite . formally this equation can be rewritten in terms of operators as : @xmath460 where we have introduced the diagonal operator @xmath461 ( the factor @xmath330 is again topological ) . we consider now this equation for @xmath462 and @xmath259 . with respect to the energy variable corresponding to @xmath71 , we take the two values @xmath195 and @xmath463 ( we note the operators with this last value with a superscript @xmath464 ) . the crucial point is that the difference between @xmath465 and @xmath466 is of order @xmath467 , that is the expansion of @xmath465 in powers of @xmath467 is regular . such a difference is negligible at the order we are working at . the regularity of the expansion can be seen from the integral equation ( eq.(b2 ) of ref.@xcite ) satisfied by @xmath465 , which is completely analogous to the one satisfied by @xmath127 ( eq.(9 ) of ref.@xcite ) , except that the last term ( with two dimer propagators ) in the equation for @xmath127 is not present for @xmath465 . this integral equation can be transformed in the same way , leading to more explicit expressions ( see eq.(11 ) and eq.(13 ) of ref.@xcite ) . on these expressions one checks easily that , for @xmath19 in the vicinity of @xmath352 , the propagator @xmath202 is evaluated for values of its argument which are never in the vicinity of its pole . hence one can proceed to a regular expansion . by contrast one sees also on the equations for @xmath127 that , for the term containing two dimer propagators the argument does reach the pole value , which makes the expansion singular . this explains why we can not argue that the difference between @xmath468 and @xmath469 is of order @xmath467 and negligible . the equation for @xmath470 reads : @xmath471^{-1}=\left[t_4 ^ 0\right]^{-1}+d^0\end{aligned}\ ] ] writing the difference with eq.([eqform ] ) and taking into account @xmath472 , we have : @xmath473^{-1}-t_4^{-1}=d - d_0\end{aligned}\ ] ] which can be rewritten explicitely , after premultiplication by @xmath474 and postmultiplication by @xmath475 , as : @xmath476t_4(q,0;\{{\bf 0},-e_b\})\end{aligned}\ ] ] where in the last term we have been allowed to replace @xmath195 by @xmath270 at the order we are working . the same substitution is allowed in eq.([eqabc ] ) , which completes our proof . let us finally note that , with respect to this part , our procedure has obvious strong analogies with the work of beliaev @xcite . indeed , as soon as we have 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this paper describes our computer program speden that reconstructs the density from the diffraction patterns of individual particles . speden is of interest for three reasons . diffractive imaging promises to improve the resolution , sensitivity , and practical wavelength range in x - ray microscopy , for three - dimensional objects that are tolerant to x - rays . a few examples are defects in semiconductor structures , phase separation in alloys , nano - scale machines and laser fusion targets . a long - term vision is the possibility of high - resolution reconstruction of diffraction patterns of single bio - molecules . of broad theoretical interest is speden s unique approach to the reconstruction of scatterers - a difficult mathematical problem . in the rest of this section we expand on these three topics . reconstruction of the electron density from non - uniformly sampled , three - dimensional diffraction patterns is of wide interest and applicability with present - day sources . in radiation - tolerant samples , x - ray diffraction and diffraction tomography are capable of higher resolution than ( straight- or cone - beam ) tomography alone . in tomography , resolution is limited by the quality of the incident beam and by the spatial resolution of the detector ; in diffraction the resolution can be as fine as the wavelength of the incident radiation . experimentally , diffraction imaging has already produced x - ray images at higher resolution than possible with available x - ray optics ( miao et al . , 1999 , and he et al . , 2002 ) . the price to be paid for these benefits is the intrinsic difficulty of the reconstruction . nevertheless , several successful reconstructions from experimental x - ray data , using the iterative hybrid input - output version of the gerchberg - saxton - fienup ( gsf ) algorithm , have been reported recently ( miao et al . , 2002 , and marchesini et al . , 2003 ) . the first successful application of this algorithm to electron diffraction data was reported in 2001 ( weierstall et al . , 2002 ) , and it has been used more recently to produce the first atomic - resolution image of a single carbon nanotube ( zuo et al . , 2003 ) . in biology , the use of the gsf has recently been shown to dramatically reduce the number of images needed for tomographic cryoelectronmicroscopy of protein monolayer crystals , so that phasing can be based mainly on the three - dimensional diffraction data ( spence et al . , 2003 ) . the development of speden was also prompted by the promise of new ways to image bio - molecules . free - electron lasers can , in principle , provide x - ray pulses of tens to hundreds of femtoseconds in length and brightness up to ten orders of magnitude greater than synchrotron radiation . it was predicted that , under such circumstances , it should be possible to dispense with crystals and reconstruct the electron density of single biological particles from their diffraction pattern ( neutze et al . , 2000 ) . in proposed experiments , a large number of single particles will be injected into the x - ray beam in random orientation and their diffraction patterns will be recorded , each in a single shot of the free - electron laser . such diffraction patterns will be very noisy and their resolution will be limited by the signal to noise ( s / n ) ratio . the measured diffraction patterns that correspond to different orientations of the particle will be classified into a number of mutually exclusive classes . the images within each class will then be averaged and the class averages assembled into a three - dimensional diffraction pattern by finding their mutual orientation relationships . finally , the three - dimensional diffraction pattern will be reconstructed to yield the electron density of the molecule . we have worked on the analysis of all three steps of such an experiment . the essence of the first analysis is that the maximum x - ray intensity at a given pulse length is limited by the requirement that the molecule stay intact during the pulse , even though it eventually disintegrates ( hau - riege et al . , the second analysis discusses the division of noisy diffraction patterns into a number of distinct classes . if the images are divided into too few classes , the available resolution is not realized . if the patterns are divided into too many classes , the class averages will be poor and the pattern quality suffers . the individual class averages , each corresponding to a well - defined orientation of the particle , will be assembled into a three - dimensional diffraction pattern . the result will be a three - dimensional diffraction pattern that is measured at a limited number of orientations . it will be , therefore , sparse , irregular and will have a limited signal to noise ratio ( huldt et al . , 2003 ) . the program speden , described in this paper , provides a way to optimally determine the electron density from such a three - dimensional ensemble of continuous diffraction patterns . we first give an analysis of their properties and discuss the methods and the expected difficulties of reconstructing a `` sensible '' electron density from them . we then describe how speden adapts the holographic method ( szoke et al . , 1997a ) in crystallography to deal with continuous diffraction patterns as opposed to discrete bragg spots ; this will be discussed in the next section . we then report quantitative results of preliminary tests for verifying the correctness of our method . these tests use computed and measured diffraction patterns from samples of inorganic particles . the reconstruction of the density of scatterers from its diffraction pattern is an `` inverse problem '' . other , well - studied inverse problems are those of computed tomography , image deblurring , phase recovery in astronomy , and crystallography . in tomography , for example , an inversion algorithm ( e.g. filtered back projection ) is used to recover the density of scatterers from the measured tomograms . it is widely recognized that the reconstructed density is very sensitive to inaccuracies in the measurement . small errors in the diffraction pattern cause large errors in the reconstruction . this property is called ill - posedness or ill conditioning . the reconstruction of the electron density from x - ray diffraction patterns is indeed ill conditioned . it also has two _ additional _ difficulties . first , in contrast to tomography , there are no direct inversion algorithms not even approximate ones . second , the reconstructed electron density at any sample point is influenced strongly by the electron densities of all sample points , as opposed to a limited number of them . therefore , errors in density are non - local and `` propagate '' far . fortunately , very good fundamental discussions of these subjects are provided in the books of daubechies ( daubechies , 1992 ) , bertero and boccacci ( bertero et al . , 1998 , bertero , 1989 ) and natterer ( natterer , 1996 , natterer and wbbeling , 2001 ) . in somewhat simplified terms , the reconstruction of the electron density is similar to finding the inverse of an ill - conditioned non - square matrix , a subject thoroughly discussed in golub and van loan ( golub et al . , 1996 ) . we consider these mathematical properties to be essential for understanding the successes and limitations of reconstruction algorithms ; we will try to be fully cognizant of them in the discussion that follows . the crystallographic phase problem is a good starting point for further discussion . it was first realized by sayre ( sayre , 1952 ) that the number of observable complex structure factors , limited by the bragg condition , is equivalent to a critical sampling of the electron density in the unit cell of the crystal . the sampling theorem of whittaker and shannon teaches us that , if the amplitudes and phases of all the diffraction peaks were accurately measured , the electron density could , in principle , be reconstructed everywhere ( bricogne , 1992 ) . unfortunately , only the amplitudes of the bragg reflections are measured , not their phases . therefore there is not enough information in the diffraction pattern for a unique reconstruction of the electron density . sayre proposed ( sayre , 1980 ) that if we could measure the diffraction amplitudes `` in between '' the bragg peaks , we should have enough information to reconstruct the electron density , or to `` phase '' the diffraction pattern . this is exactly the situation in diffraction from a single particle . nevertheless , it is still difficult to reconstruct the electron density , even from a well `` oversampled '' diffraction pattern . one corollary of critical sampling is that the amplitudes and phases of the bragg reflections of a crystal are independent of one other , but any structure factor in between them depends on the surrounding ones to some extent . therefore , too much oversampling does not help to obtain independent data , although it does improve the s / n ratio by reducing the noise . ill posedness is still with us , although with oversampling , the error propagates less . an additional difficulty with diffraction patterns from a set of discrete orientations of a particle is that at low resolution the diffraction pattern is well oversampled while at high resolution the sampling is sparse . a fundamental property of diffraction is that the position and the handedness of the electron density are undetermined , resulting sometimes in stagnation of the algorithm and drift in the position of the results ( stark , 1987 ) . there are two well - known , necessary remedies for the lack of information and for the ill posedness of the reconstruction problem . the more important one is the need for more information . for example , one way to include _ a priori _ knowledge is to accept reconstructed electron densities only if they are `` reasonable '' . the second remedy is to use `` stabilized '' or pseudo - inversion algorithms . in the next section we introduce our version of a real space reconstruction algorithm ; we will argue that our algorithm deals with all these problems optimally , at least in some sense . we return to the comparison of our algorithm with other methods for phase - recovery in section 2.3 . subsectionspeden , a real - space algorithm in this section we outline the workings of our reconstruction program , speden . speden uses a real - space method for reconstruction ; its acronym stands for single particle electron density . for computational efficiency the particle to be recovered is put into a _ fictitious _ unit cell that is several times larger than the particle itself . all reconstruction algorithms use this artifice in order to be able to calculate structure factors by fast fourier transform techniques . the resulting similarity with crystallography enables the use of many crystallographic concepts . in fact , the recognition of this similarity enabled us to write a program , speden , based on our crystallographic program , eden , with relatively small modifications . the most significant difference between the two programs is that in crystallography the bragg condition restricts the reciprocal lattice vectors to integer values , while the continuous diffraction pattern can be - and usually is - measured at arbitrary , non - integer values of the reciprocal lattice vectors . in speden , in common with eden , the ( unknown ) electron density is represented by a set of gaussian basis functions , with unknown amplitudes , that fill the fictitious unit cell uniformly . this way the recovery is reduced to the solution of a large set of quadratic equations . the program `` solves '' these equations by finding the number of electrons in each basis function so as to agree optimally with the measured diffraction intensities as well as with other `` prior knowledge '' . prior knowledge includes the emptiness of the unit cell outside the molecule , the positivity of the electron density , possibly some low - resolution image of the object , etc . each one of those conditions is described by a cost function that measures the deviation of the calculated data from the observed data . one of the cost functions describes the deviation of the calculated diffraction pattern from the measured one ; others depend on the deviation of the recovered density from prior knowledge . measured data are weighted by their certainty ( inverse uncertainty ) , other prior knowledge is weighted by its `` reliability '' . the mathematical method used is ( constrained ) conjugate gradient optimization of the sum of cost functions . at each step of the optimization , there is a set of amplitudes available that describe the current electron density in the full unit cell . a full set of structure factors is calculated by fourier transforming the current electron density . when the unit cell is larger than the particle , the structure factors can be stably interpolated to compare them with measured structure factor amplitudes . we refer to the cited literature that shows that the procedure we outlined is equivalent to a stabilized ( quasi ) solution of the inverse problem ( daubechies , 1992 ; bertero et al . , 1998 ; and natterer et al . , 2001 ) . as such , it is optimally suited for sparse , irregular , incomplete and noisy data . in the following subsections we describe very briefly the common features of eden and speden as well as their differences . a more complete description of eden can be found in previous papers ( szoke et al . , 1997a , and szoke , 1998 ) . the electron density is represented as a sum of basis functions , adapted to the resolution of the data . specifically , we take little gaussian `` blobs '' of width comparable to the resolution , and put their centers on a regular grid that fills the `` unit cell '' and whose grid spacing is comparable to the resolution . the amplitudes of the gaussians are proportional to the local electron density . in fact , the number of electrons contained in each gaussian constitutes the set of our basic unknowns . the above is identical to the representation of the electron density in eden . some mathematical details follow . the actual formula for the representation of the electron density as a sum of gaussians is @xmath0.\ ] ] the centers of the gaussians are positioned at grid points , @xmath1 , where @xmath2 is a counting index . in our fictitious unit cell , the grid is orthogonal , the grid spacing is @xmath3 and the centers of the gaussians are usually on two intercalating ( body centered ) grids for best representation of the electron density . the number @xmath4 of the order unity , determines the width of the gaussians relative to their spacing , @xmath3 . finally and most importantly , @xmath5 is the _ unknown _ number of electrons in the vicinity of the grid point @xmath6 . the values of @xmath5 are real , and in future may also be complex - valued to allow for photo - absorption in addition to scattering . ( the latter can be significant when diffraction measurements are made at longer x - ray wavelengths . ) given @xmath5 the structure factors can be calculated by @xmath7 } \sum\limits_{p = 1}^{p } n(p)\exp \left [ 2\pi i \bm{h } \cdot { \cal f}\bm{r}(p ) \right ] \,,\ ] ] using a fast fourier transform . the vector @xmath8 , a triplet of integers , denotes the reciprocal lattice vector , the operator f transforms from real space ( cartesian ) coordinates to fractional coordinates , and @xmath9 denotes the dual transformation . the constants appearing in eqs . ( [ eq1 ] , [ eq2 ] ) were discussed in some detail previously ( szoke et al . , 1997b ) . for completeness , we define them here . the crystallographic @xmath10 factor is given by @xmath11 . the `` crystallographic resolution '' , @xmath12 , determines the grid spacing , @xmath3 by the relation , @xmath13 , where @xmath14 is a constant of the order unity . for a body - centered lattice we set @xmath14 = 0.7 and @xmath15 = 0.6 . for a simple lattice , we use @xmath14 = 0.6 and @xmath15 = 0.8 . note that the gaussian basis functions are not used in a one - to - one correspondence with single atoms , but are simply used to describe the 3d electron density at the resolution that is appropriate to the data resolution . in the special case that the resolution was about the size of an atom and an atom happened to be sitting exactly on a grid point , that atom would be represented by a single basis function . if the atom is not on a grid point , or if the atom happens to be fat , because of thermal motion , that same atom would be represented by many basis functions . similarly , at lower resolution , a single basis function represents assemblies of atoms . the measured diffraction pattern of the molecule is proportional to the absolute square of the structure factors . in speden we do account for the curvature of the ewald sphere . there are two subtle points : the diffraction pattern is measured only in a finite number of directions , @xmath16 ; and as a rule , those directions are not along the reciprocal lattice vectors of the ( fictitious ) unit cell for a single particle . in other words , the measurement directions , @xmath16 , are usually not integers and they are not uniformly distributed in reciprocal space . this is the main difference between crystallography and single particle diffraction and , therefore , between eden and speden . the essence of _ any _ reconstruction algorithm is to try to find an electron density distribution such that the calculated diffraction pattern matches the observed one . in our representation , we try to find a set of @xmath5 , such that @xmath17 for each measurement direction , @xmath18 . let us assume for a moment that @xmath18 are integers . when the representation of the unknown density is substituted from ( [ eq2 ] ) , for each measured value of @xmath18 , equation ( [ eq3 ] ) becomes a quadratic equation in the unknowns , @xmath5 . the number of equations is the number of measured diffraction intensities . it is usually not equal to the number of independent unknowns that are the number of grid points in the unit cell . the equations usually contain inconsistent information , due to experimental errors . the equations are also ill conditioned and therefore their solutions are extremely sensitive to noise in the data . under these conditions the equations may have many solutions or , more usually , no solution at all . our way of circumventing these problems is to obtain a `` quasi solution '' of ( [ eq3 ] ) by minimizing the discrepancy , or cost function ( see e.g. stark , 1987 , and bertero et al . , 1998 ) @xmath19 ^ 2\ ] ] the weights , @xmath20 , are usually set to be proportional to the inverse square of the uncertainty of the measured structure factors , @xmath21 . as discussed by szoke ( szoke , 1999 ) , this is equivalent to a maximum likelihood solution of the equations . let us now discuss the first , previously ignored difficulty in the reconstruction . when we try to reconstruct the electron density from real experimental data , we have to compare the set of measured @xmath22 , where @xmath18 are not necessarily integers , with the calculated structure factor amplitudes , @xmath23 , that are on a regular grid , i.e. have integer @xmath24 . in principle , given an electron density of the molecule , one could calculate the structure factors in the experimental directions . nevertheless , for computational efficiency , we put the ( unknown ) molecule or particle into a fictitious unit cell that is larger than the molecule . we will also demand that the gaussians outside the molecular envelope be empty . ( in practice , sizes of molecules are known from their composition ; particle sizes and shapes may be known from lower - resolution imaging . ) as long as the distances of the gaussian basis functions are kept to be the experimental resolution , the number of `` independent '' unknowns neither increases nor decreases , in principle , by this computational device . the structure factors are calculated on an integer grid in the large unit cell , so they are essentially oversampled in each dimension by the same factor of the size of the large cell to the size of the molecule . the oversampling allows stable interpolation of the calculated structure factor amplitudes from integer @xmath24 to the fractional @xmath16 everywhere , independent of the density of the actual measurements . note that interpolation from fractional @xmath16 to integer @xmath24 is not always a stable procedure ! in the present implementation of speden , we get sufficient accuracy with the simplest , tri - linear interpolation in the amplitudes of @xmath25 if we choose the fictitious unit cell to be three times larger than the molecule in each dimension . now , some mathematical details : the reciprocal space vector * h*(@xmath26 is within a cube , bounded by eight corners @xmath27 with integer values . let us denote the fractional parts of the components of @xmath18 as @xmath28 . we define weights for the eight corners , @xmath29 , by taking the products of the fractional parts of @xmath30 , or @xmath31 with those of @xmath32 or @xmath33 and @xmath34 or @xmath35 . the cost function to be minimized now becomes @xmath36 ^ 2\,.\end{aligned}\ ] ] a similar approach of applying crystallographic algorithms to continuous diffraction data has been done with direct methods [ spence et al . , 2003 ] . in this case , however , the ewald sphere was approximated by a plane . let us assume that we have some , possibly uncertain , knowledge of the electron density in parts of the unit cell from an independent source , i.e. one that does not come from the x - ray measurement itself . this is the kind of knowledge present when the unknown molecule is placed into a larger unit cell and we demand that the unit cell be empty outside the molecule . this kind of knowledge was also referred to as a `` sensible '' electron density in the introduction . we represent this knowledge by a target electron density @xmath37 and by a real - space weight function @xmath38 . it will be desirable that the actual electron density of the molecule , @xmath39 , as represented by @xmath5 , be close to the target electron density ; the weight function @xmath38 expresses the strength of our belief in the suggested value of the electron density . note that target densities can be assigned in any region of the unit cell independently of those in any other region . the simplest way to express the above statement mathematically is to minimize the value of the cost function @xmath40 ^ 2 \,,\ ] ] where @xmath41 is a scale factor and @xmath42 is a normalizing constant , described in somoza et al . , 1995 . ( in the _ absence _ of information at and around the molecule , weights are generally unity where it is known that there is no molecule and zero elsewhere . ) the same procedure is used in eden . the knowledge of the electron density at low resolution can be expressed by a low - resolution spatial target . crystallographers call this phase extension . the essence is that , during the process of the search for an optimal electron density , we try to keep its low resolution component as close to the known density as possible . a convenient way to accomplish this is the following . given @xmath5 , we smear out its gaussian representation and compare it to the equally smeared out target . actually , it is easier to carry out the computation in reciprocal space . we define @xmath43 where the current `` smeared '' structure factors are calculated using the low resolution , @xmath44 , @xmath45\sum\limits_{p = 1}^{p } { n(p)\exp [ 2\pi i{\rm { \bf h } } \cdot { \cal f}{\rm { \bf r}}(p ) ] } .\ ] ] the original low - resolution target is prepared analogously from the ( presumably ) known electron density , @xmath46 . the same procedure is used in eden . in the presence of a target density , the actual cost function used in the computer program is the sum of @xmath47 ( [ eq5 ] ) , @xmath48 ( [ eq6 ] ) and @xmath49 ( [ eq7 ] ) @xmath50 the fast algorithm described in somoza et al . , 1995 , and szoke et al . , 1997b , is always applicable to the calculation of the full cost function , ( [ eq9 ] ) . there is a clear possibility of defining more target functions . they are all added together to form @xmath51 that is minimized to find the optimum electron density . the minimization of the cost function ( [ eq9 ] ) is carried out in speden ( as in eden ) by d. goodman s conjugate gradient algorithm ( goodman , 1991 ) . it has proven to be very robust and efficient in years of use in eden . the essential properties of the algorithm that make it so advantageous for our application is that the positivity of the electron density , @xmath52 , is always enforced and that the gradient vector in real space can be calculated by fast fourier transform . the gradient calculation needed only a very simple modification for the interpolation in reciprocal space , eq . ( [ eq5 ] ) . the line search algorithm does not use the hessian , so matrices are never calculated . as with any local minimization , global convergence is not achieved . we discussed this problem in our previous papers and came to the conclusion that , usually , the minimum surface of the cost function ( [ eq7 ] ) is so complicated that finding a global minimum would take more computer time than the existence of the universe . reconstruction of the scatterer from a continuous diffraction pattern has a tangled history replete with repeated discoveries . some of the present authors are also guilty of ignorance of prior work . we referred to the pioneering insights in section 2.1.2 . the `` recent '' period of algorithms started with the work of miao , sayre and chapman ( miao et al . 1998 ) who pointed out that the fraction of the unit cell where the density is known is an important parameter for convergence . in somewhat later work , with oversampled structure factors calculated on a regular grid , the crystallographic program eden successfully demonstrated the recovery of the electron density using a simulated data set from the photoactive yellow protein ( szoke , 1999 ) . the protein was put into a fictitious unit cell , twice the size of the original one , and a target with zero density was used outside the original unit cell of the protein . similarly , miao and sayre ( miao et al . , 2000 ) have studied empirically how much oversampling is required in two- and three - dimensional reconstructions of a simulated data set ; using a version of the gerchberg saxton fienup ( gsf ) algorithm . among recent articles we mention robinson et al ( 2001 ) , williams et al ( 2003 ) , marchesini et al . ( 2003 ) and references therein , in addition to those mentioned in the introduction . all reconstruction algorithms of oversampled diffraction patterns use _ a priori _ information on the shape and size of the particle . in our previous studies in crystals we found that such information is very valuable . for example , eden converges surprisingly well for proteins at low resolution where the only information used is that the molecule is a single `` blob '' . eden also converges for synthetic problems with a good knowledge of the solvent volume , which is greater than 50% ( beran et al . , 1995 ) or 60% ( eden ) . a similar conclusion was reached in miao et al . ( 1998 ) . in comparison , when a molecule is embedded in a 3-fold larger fictitious unit cell , the empty `` solvent '' occupies @xmath5396% of the cell volume . as discussed previously , the reconstruction of scatterers from their diffraction pattern is a difficult mathematical problem . in many cases the indeterminacy of the absolute position of the object and of its handedness causes difficulties in convergence . that is definitely the case with speden so , in that sense , speden is not a good algorithm . empirically , the gsf algorithm has a larger radius of convergence and deals better with stagnation ( marchesini et al . 2003 ) another family of difficulties arises when there is _ a priori _ information available , but there is only incomplete and noisy data . under such conditions the main questions are how to find a solution that optimally takes into account the available information and that is the best `` sensible '' one that reproduces the noisy and incomplete data to its limited accuracy . it is this second set of conditions for which speden was written . although , in this paper , we show only its performance for artificial and `` easy '' but incomplete data , speden s older sister , eden has been shown to have those properties on a large range of crystallographic data sets , ranging from cuo@xmath54 to the ribosome . we expect that such properties of eden will be inherited by speden , considering that their fundamental mathematical properties are sufficiently similar . the best known , and successful class of algorithms is the group of iterative transform algorithms that we refer to as gerchberg - saxton ( gs ) ( gerchberg et al . , 1972 ) , and its development , in which support constraints and feedback are added , the gs - fienup ( gsf ) or hybrid input - output algorithm ( fienup et al . , 1982 ; aldroubi et al . , 2001 ; bauschke et al . , 2002 ; bauschke at al . , 2003 ) . the essence of the gsf algorithms is that they iterate the n - pixel data between real and reciprocal spaces via ffts and enforce the known constraints in each of these spaces . depending on the degree of noise in the data , these algorithms usually converge in about a hundred to a thousand iterations . the weak convergence ( non divergence ) of the gs algorithm has been proven in the absence of noise ( fienup et al . , 1982 ) . there is no mathematical proof that these algorithms will converge in general , but it is reasonable that by sequentially projecting onto the set that satisfies the real space constraints and the set that satisfies the reciprocal space constraints , the intersection ( corresponding to a valid solution ) should be approached . this is definitely true for projections onto convex sets , but unfortunately these sets are not convex ( stark , 1987 ) . in practice , despite the fact that the modulus constraint is non - convex , the algorithms often converge even in the presence of noise in about several hundred iterations . the main difference between the gsf algorithms and speden is that speden does not iteratively project onto the sets of solutions that satisfy the real - space or reciprocal space constraints separately , but rather it minimizes a cost function that includes all the constraints of both spaces . it does this by varying quantities in real space only ( the @xmath5 s ) and the cost evaluation only requires a forward transform from real space to reciprocal space . as the cost function never increases , speden reaches only a local minimum . in spite of its `` small '' radius of convergence , there are some expected advantages to speden s algorithm .. in speden we compare the @xmath55 to the @xmath56 by interpolating from the samples of @xmath55(*h * ) , calculated on a regular grid , to the measured sample vectors * h*. since the gridded @xmath57 are a complete set ( due to the fact they are sampled above the nyquist frequency ) the interpolation is stable and performed with little error . since an inverse transform is required in the gsf algorithms , the measured diffraction data @xmath58 ( recorded on ewald spheres in reciprocal space ) must be interpolated to the gridded data points * h*. the observed data might not be a complete set , especially at high resolution where the density of samples is sparser : this may lead to error . an additional possible difficulty with the gsf algorithms is that the effective number of unknowns may increase with the size of the fictitious unit cell , while in speden the effective number of unknowns is constant . finally , in speden , weightings can be properly applied to all data and knowledge . the measured data is inversely weighted by its uncertainty ; it is a procedure equivalent to maximum likelihood methods and it should be optimal , at least in theory . as a side effect , when the constraints are inconsistent , speden still converges to a well - defined and correct solution . ( note that for noisy data the constraints are almost always inconsistent . ) weightings are also applied to reflect our confidence in real - space constraints , and these weightings are consistently used in real space . * tests * speden has certain built - in limitations . in particular , of course , reconstruction is only as good as the diffraction measurements and derived structure factor amplitudes are reliable . there are also other less obvious limitations . for example , there are inherent inaccuracies due to the gaussian representation of the electron density in real space . ( szoke et al . , 1997a ) also , the trilinear interpolation for representing integer ( hkl ) structure factor amplitudes on a non - integer grid is only approximate . finally , as a fundamental limitation of _ any _ reconstruction method , both the absolute position and the handedness of the molecule are undefined . we performed preliminary tests to verify the capabilities and limitations of our reconstruction method using computed and experimentally obtained diffraction patterns . in this section , we first describe the reconstruction of simple `` molecules '' from synthetic diffraction patterns with speden . specifically , we discuss how the convergence of speden is affected by the errors due to interpolation in reciprocal space , by the quantity of observed structure factors , @xmath59 , by the extent of the `` known '' starting model , and by the uniformity of sampling in reciprocal space . we then describe the reconstruction of simple two - dimensional objects from synthetic and measured diffraction patterns with speden . we created synthetic `` molecules '' in the format of the protein data bank ( pdb ) files ( berman et al . , 2000 ) . each molecule was composed of 15 point - like carbon atoms , placed at random positions within a cube measuring 16.8 in each dimension and `` measured '' to 4 resolution ; these values correspond to crystallographic @xmath10 factors of 185.7 @xmath60 . the molecule was then shifted so that its center - of - mass was at the center of the cube . all our simulations were repeated using molecules with several different random arrangements . initially , the `` unit cell '' coincided with the dimensions of the cube in which the atoms were placed ; later , larger cells were used and the atoms were positioned in their center . we also generated `` starting models '' by removing atoms from the full molecule . both full and partial molecules served to generate structure factors ( @xmath61 ) , using the atomic positions and the b factors . starting models were generated from the @xmath55 , using speden s preprocessor , back , which finds the optimal real - space representation for an input @xmath61 . initially , sets of `` measurements '' ( @xmath62 ) were generated by deleting the phases of the calculated structure factors of the full or partial molecule . the @xmath63 files so generated had all integer @xmath16 . the uncertainty of the measured structure factors , @xmath64 , were chosen to be @xmath65 with @xmath66 for @xmath67 , and @xmath68 for @xmath69 . we used two alternate methods to generate @xmath63 files for non - integer @xmath18 : in the first method , we used a unit cell whose dimensions were incommensurate with one another and with the edge of the cube , but whose volume equaled that of the cube . the resulting @xmath63 file , generated again from @xmath61 files by deleting the phases , then had its indices scaled back appropriately , yielding fractional @xmath18 . a different second method was used to sample the reciprocal @xmath18 space non - uniformly : @xmath70 at fractional @xmath18 were calculated using tri - linear interpolation from @xmath62 data on a regular grid that , in turn , was at four times the regular resolution in reciprocal space . for each of these @xmath63 files , some constraining information is required in order to find the atom positions . we used two types of constraints : one of them identified the ( approximate ) empty region ; the other one used @xmath61 at a considerably lower resolution . we call them the empty target and the low - resolution target , respectively . both are based on the assumption that at a considerably lower resolution , the general position of atoms as one or more `` blobs '' in empty space is known . the low - resolution @xmath71 was prepared by smearing the full @xmath72 file to 10 . the empty target identified the empty points in terms of a mask . then , throughout speden s iteration process , using the solver solve , the program attempted to match the current electron / voxel values in masked - in regions to empty ( @xmath73 ) values . the phase - extension target used the same @xmath71 file ; during the iteration process , at each step , the current real - space solution was smeared to that low resolution in reciprocal space and restrained to agree with that target . besides inspecting the reconstructed electron density visually , we used four quantitative measures to compare the reconstructed image with the electron density from the full molecule at 4 : 1 . real - space rms error : we calculated the real - space electron density from the electron / voxel files , a process we call regridding . we then calculated the root - mean - square ( rms ) error of the electron densities , @xmath74 and @xmath75 , defined as @xmath76 we permitted one file to be shifted with respect to the other file , in order to minimize the distance . phase difference : we calculated the average ( amplitude - weighted ) phase difference between the @xmath61 at the end of the run and the corresponding @xmath61 generated from the full molecule . final @xmath77 factor : we calculated the crystallographic r factor ( giacovazzo et al . , 2002 ) at the end of the run . count error : we compared the integrated real - space electron density generated from a run result with the true number of electron as identified in the pdb file , on an atom - by - atom basis . the integration was performed around each atom over a sphere with a radius that was 1.5 times the grid spacing . the figure - of - merit is the rms error . of all these measures , the final @xmath77 factor was the least useful to assess the quality of the reconstructed image . the @xmath77 factor tends to be lower for a small number of entries in the observation file since there are fewer equations to satisfy during the reconstruction . in such a case , a visual inspection shows that the reconstructed electron density may have little resemblance to the 15 carbon atoms . however , there was a good correlation among the other three measures . the solutions looked correct when the phase difference between solution and full @xmath61 was less than 20 , the count error between solution and pdb model was 0.2 or lower ( out of 6 ) , and the rms distance measure was less than 0.2 . for computational purposes , we placed the ( unknown ) molecule or particle into a fictitious unit cell that is larger than the molecule , and calculated the structure factors on an integer grid in the large unit cell , oversampled by the same ratio : the size of the large cell to the size of the molecule . we then calculated the structure factor at fractional * h * from the structure factor at integer * h * using tri - linear interpolation . the interpolation error becomes smaller when larger unit cells are used ( at the expense of computation time ) . we studied the effect of the unit cell size on convergence of speden . in the initial tests , the unit cell was the same size as the original molecule , and the starting ( known ) part of the molecule consisted of the full molecule . when the molecule consisted of atoms on grid points and the @xmath63 files had integer @xmath78 , unsurprisingly speden converged , as did eden on the same data set however , we found that speden did not converge to a unique solution if either the atoms were not on grid points or when the @xmath63 file had non - integer @xmath18 , or both , since the solution meandered in real space . in subsequent tests , we generated larger unit cells and applied a target over the empty part of the unit cell , in an attempt to restrain the meandering problem . we embedded the molecule in a cell that was 2 or 3 times greater in each dimension . an empty target was used that essentially covered the empty 7/8-th or 26/27-th of the unit cell , respectively . we applied a high relative weight for this empty target , and we still used the full molecule as a starting position . we found that both the larger cell and the empty target are of critical value in enabling speden to converge to the correct solution . comparing the 2-fold larger unit cell versus the 3-fold unit cell , there was a great improvement in the latter case . these results show that for tri - linear interpolation , it is adequate to use a unit cell that is 3 times greater in each dimension . we expect that more sophisticated interpolation algorithms should allow using smaller unit cells . a three - fold enlarged unit cell increases the number of unknown amplitudes of the gaussians , n(p ) , by a factor of 27 . in principle , the emptiness of the volume around the molecule restrains the effective number of independent unknowns . nevertheless , if the number of equations , which is given by the number of entries in the fobs file , is not increased , it is easy for the solver to `` hide '' electrons among the large number of unknowns in the system , even when the empty target constraint is used . in fact we found that when we compared the final fcalc from solve against the starting fcalc , on the one hand , and the correct fcalc on the other , solve s final fcalc was closer to the starting one than to the correct one . in other words , the cost function in reciprocal space was not a sufficiently strong constraint in speden s algorithm , for this synthetic problem . in a similar real case , more experimental diffraction patterns need to be collected in order that speden would be able to find the corresponding image without other information . in the next set of synthetic tests , we attempted to recover missing information by starting from a partial model that contained less than the full complement of 15 atoms . in these simulations , we used @xmath63 files with non - integer @xmath18 , a three - fold enlarged unit cell , an empty target or a phase extension target , and randomly positioned atoms . we found that a low - resolution spatial target significantly helps speden to converge . figure 1 ( a ) shows the results of the comparison of the reconstructed image with the electron density from the full pdb file when a phase extension target is used . the phase extension target was calculated at a resolution of 10 . we found that it was generally possible to recover 5 , 10 , or even all 15 of the atoms . please note that the amount of information in such a phase extension target is ( 4/10)@xmath79 6% of the information in the perfect solution . it was more difficult to reconstruct the original electron density when we used an empty target , as shown in figure 1 ( b ) . speden was able to recover 5 out of the 15 atoms , but did not converge when 10 atoms were unknown . perhaps surprisingly , the case where there was no starting model at all ( 0 atoms known ) did consistently better than those cases where a partial model was given as a starting condition . in this set of synthetic tests , we addressed the question of how difficult it is to recover the molecule from a non - uniform set of samples in reciprocal space , similar to real data sets , and how the results compare with the reconstruction from a uniformly - sampled data set . we generated two - dimensional diffraction patterns of the synthetic carbon molecule for different particle orientations , corresponding to recorded diffraction patterns in a `` real '' experiment . the two - dimensional diffraction patterns were linearly interpolated from a three - dimensional diffraction pattern ; the latter was calculated on an additionally double - fine grid over the already triple - sized unit cells , i.e. using a unit cell that was a total of six times larger than the molecule in each direction . further refining the grid of the three - dimensional diffraction pattern did not alter the results significantly . the three - dimensional diffraction pattern , in turn , was the fourier transform of the gridded electron density of the synthetic molecule . two - dimensional patterns do not sample the diffraction space uniformly . the sampling density near the center of the diffraction space is much larger than the sampling density further away . we used a completeness measure to characterize the sampling uniformity . the reciprocal space is divided into cells that are 4@xmath80/@xmath81 by 4@xmath80/@xmath82 by 4@xmath80/@xmath83 in size , where @xmath81 , @xmath82 , and @xmath83 are the molecule sizes in each dimension . the completeness then is the ratio of cells in reciprocal space that contain at least one measurement over the total number of cells . figure 2 shows the completeness of the input observation files as a function of the number of diffraction patterns . also shown in figure 2 is the number of calculated diffraction intensities . we then used speden to recover 7 out of the 15 atoms . the molecule was embedded in a unit cell that was three times larger in each dimension , and we used an empty solvent target . figure 3 shows the errors of the reconstructed electron density as a function of the number of two - dimensional diffraction patterns . the orientations of the diffraction patterns were chosen at random , and the calculations were repeated for four different molecules . for comparison , also shown in figure 3 are the errors of the electron density of the 8 known atoms ( `` partial model '' ) , and the errors of the reconstructed electron densities when a three - dimensional diffraction pattern is used which was oversampled three times ( `` integer hkl '' ) or six times ( `` fractional hkl '' ) . as discussed above , the @xmath77 factor is not a useful measure to assess the quality of the reconstructed image , but the rms and count errors are better measures for the reconstruction quality . surprisingly , we found that _ four _ two - dimensional patterns are sufficient to reconstruct the electron density as well as in the case the full three - dimensional diffraction pattern is given . four two - dimensional patterns have a remarkably low sampling completeness of only 14% . further increasing the completeness or the number of observations does not improve the quality of the reconstruction . we would like to note , however , that these results could be dependent on the choice of the test model , and that for different test models the number of required two - dimensional patterns may be larger . in the final set of tests , we demonstrate spedens ability to recover missing information for a two - dimensional configuration of 37 au balls in a plane . we reconstructed the au balls using ( i ) a synthetic diffraction pattern and ( ii ) an experimentally obtained diffraction pattern as discussed by he et al . , 2003 . in the following we will discuss both cases , starting with the synthetic diffraction data . the 37 au balls are arranged in a plane as shown in figure 4 . the arrangement of the balls is similar to the experimental case discussed by he et al . , 2003 . the au balls were 50 nm in diameter . we generated an artificial set of `` measurements '' ( @xmath84 ) by calculating the structure factors ( @xmath85 to 30 nm resolution and deleting the phases . the uncertainty of the measured structure factors , @xmath64 , were chosen according to equation ( [ eq10 ] ) . we generated an initial model by smearing the full @xmath72 file to 90 nm , and running back on it . the initial model is shown in figure 5 . in figures 5 7 , we only show one plane . we also used this smeared @xmath61 to generate a low - resolution spatial target as well as an empty target outside the molecule . the corresponding weight function is shown in figure 6 . we then used speden to reconstruct the au balls . as shown in figure 7 , speden reconstructed the electron density successfully . however , we further found that if we use an empty starting model , speden has difficulties converging to the correct electron density . there are two reasons for this behavior . first , without an initial model , the symmetry of the system is not broken , and speden stagnates since the support does not distinguish between the object and its centrosymmetric copy . second , the mask and the reconstructed electron density are possibly shifted with respect to each other . if the initial model is empty , the position of the reconstructed electron density is mostly determined in the early iteration of the solve algorithm and can be partially cut off by the solvent . the algorithm has difficulty shifting the result . it is necessary to provide information about the location of the electron density to a certain degree , for example in the form of a smeared model . note that the gsf algorithms are designed to overcome these problems when there is abundant and accurate data available . we will now discuss the reconstruction of the au balls using experimental data . to generate a starting model , we took the experimental @xmath59 data along with the phases obtained by he at al . , 2003 using a version of the gsf algorithm , and smeared this data to 80 nm . the starting model is shown in figure 8 . we used the same data to generate a real space target with a target fraction of 99.7% , shown in figure 9 . similar to the case of the synthetic test data , we chose @xmath86(*h * ) according to equation ( [ eq10 ] ) . we then used speden to reconstruct the au balls . figure 10 ( a ) shows the reconstructed electron density , and figure 10 ( b ) shows a scanning electron microscope ( sem ) picture of the sample . we found that speden reconstructed the electron density from the experimental data successfully . in this paper we presented speden , a method to reconstruct the electron density of single particles from their x - ray diffraction patterns , using an adaptation of the holographic method in crystallography . unlike existing gsf algorithms , speden minimizes a cost function that includes all the constraints of both real space and reciprocal space , by varying quantities in real space only , so that the cost evaluation requires only a forward transform from real space to reciprocal space . speden finds a local minimum of the cost function using the conjugate gradient algorithm . we implemented speden as a computer program , and tested it on synthetic and experimental data . our initial results indicate that speden works well . this work was performed under the auspices of the u.s . department of energy by university of california , lawrence livermore national laboratory under contract w-7405-eng-48 and doe contract de - ac03 - 76sf00098 ( lbl ) . sm acknowledges funding from the national science foundation . the center for biophotonics , an nsf science and technology center , is managed by the university of california , davis , under cooperative agreement no .

the rsp game @xcite is a game where players take their move simultaneously , each choosing a hand from the rock ( r ) , the scissors ( s ) and the paper ( p ) . the cyclic strength relation of the three hands determines the win and the loss ; the rock crushes the scissors , the scissors cut the paper and the paper wraps up the rock . the cyclic competition of the rsp game can mimic various relations in reality , particularly in the population dynamics ; _ e.g. _ a colony of three competing mutations of _ e. coli _ @xcite and a three - morph mating system of a lizard @xcite . the present study proposes a model of the rsp game on lattices . each player on a lattice point chooses the next hand from the hand of the neighboring player with the maximum point . ( we refer to such a player as a copy player . ) we found interesting spatial patterns , such as vortices and sinks , appearing particularly on the triangular lattice . the spatial pattern with vortices and sinks appears as a coexisting steady state on the triangular lattice . as far as we know , the previous studies considered the rsp game either on the square lattice @xcite or on various networks @xcite . it is , however , easy to imagine that triangles appear in the population dynamics in reality , _ e.g. _ clusters in complex networks . it is also known that elementary properties can be very different in many - body systems on non - bipartite lattices and on bipartite lattices . the vortex pattern that we observe in the present study is , in fact , due to the frustration of the triangular lattice ( fig . [ fig1 ] ) , a three - sided situation where each of the three players around a triangle chooses the rock , the scissors and the paper , respectively . ( hereafter , we refer to the three hands simply as the hands 2 , 1 and 0 , respectively . ) the existence of vortices was pointed out by some studies in the past @xcite . we here stress the importance of the frustration as the cause of the stationary vortex pattern . we show that the stationary vortex pattern does not appear on the square lattice nor on the honeycomb lattice . the paper is organized as follows . in sec . [ sec2 ] , we introduce the new model and discuss its elementary properties . we show that pairs of vortices and sinks can appear as spatial patterns . we also argue that players close a vortex core scores a high point while players close to a sink scores a low point . we report the results of our simulation on the triangular lattice in sec . [ sec3 ] and on the square and honeycomb lattices in sec . we confirm that the spatial pattern with vortices and sinks is stationary on the triangular lattice , while it is not on the square nor honeycomb lattices . in sec . [ sec5 ] , we introduce a random player , who chooses its hand randomly . we show that a random player can be a source in the spatial pattern . we propose a model where players residing on lattice points repeatedly play the rsp game with the nearest neighbors . we hereafter consider the triangular lattice , the square lattice and the honeycomb lattice , with an emphasis on the triangular lattice , whose frustration generates stationary vortices in the course of the rsp game . all players on the lattice points make their moves all at once , which constitutes one time step . a move is either 0 , 1 or 2 . the hand 1 wins over the hand 0 , the hand 2 wins over the hand 1 , and the hand 0 wins over the hand 2 . a win , a draw or a loss are determined between each pair of the nearest neighbors of the lattice . each player scores one point for a win , zero point for a draw and minus one point for a loss . hence , a player can score @xmath0 points at most and minus @xmath0 points at least in each time step , where @xmath0 is the number of the nearest neighbors on the lattice ( @xmath1 for the triangular lattice , @xmath2 for the square lattice and @xmath3 for the honeycomb lattice ) . this is a zero - sum game ; that is , the sum of the scores of all the players is always zero . particularly on the triangular lattice , we define the frustration and its sign ( fig . [ fig1 ] ) . we refer as a positive frustration to the situation where the hands 2 , 1 , and 0 appear in this order when we circle around a triangle counterclockwise . on the other hand , a negative frustration is the situation where the hands 0 , 1 , and 2 appear in this order when we circle around a triangle counterclockwise . we will argue in the next subsection that a positive frustration generates a counterclockwise vortex , whereas a negative frustration generates a clockwise vortex . all the players choose their hands at random in the initial time step with an equal probability . each player adopts the copy strategy or the random strategy afterwards . the copy strategy is to choose a hand of the player who , of all the nearest neighbors and the player itself , marked the highest score in the last time step . we refer to a player adopting the copy strategy as a copy player hereafter . if there are more than a player of the highest score with different hands , a copy player chooses a hand from their hands randomly . the random strategy is to choose a hand at random . we refer to a player adopting the random strategy as a random player . in the present study , we consider only the case where each player is either a copy player or a random player all through the game . we mostly consider copy players hereafter . we show that copy players on the triangular lattice exhibit vortex structure . in sec . [ sec5 ] , we discuss an impact of random players on the structure as impurities . before showing the simulation results , let us argue that two spatial patterns typically appear . one is a vortex and the other is a sink . they are logical consequences of the combination of the rsp game and the copy strategy . note first that copy players tend to form domains of the same hands . a copy player well inside a domain of , say , the hand 0 , will keep the hand 0 in the next time step because all its neighbors are of the hand 0 and their scores are all zero ; hence the bulk of the domain is stable . copy players on the boundary of a domain , on the other hand , may change their hands in the next time step , and hence the boundary moves . let us argue how the boundary moves in the following two cases . the three domains of the hands 0 , 1 and 2 can have either the topology of fig . [ fig2 ] ( a ) or ( b ) . ( a ) ( b ) + ( c ) in the topology of fig . [ fig2 ] ( a ) , there is a negative frustration around the point a and a positive frustration around the point b. a copy player of , say , the hand 0 , located just outside the domain of the hand 1 , tends to choose the hand 1 in the next time step because the neighbors with the hand 1 get high scores . thus the boundary between the domains of the hand 0 and the hand 1 moves onto the the domain of the hand 0 , so that the domain of the hand 1 expands . likewise , the boundary between the domains of the hand 1 and the hand 2 moves onto the domain of the hand 1 and the boundary between the domains of the hand 2 and the hand 0 moves on to the domain of the hand 2 . hence the boundaries rotate clockwise around the negative frustration at the point a and counterclockwise around the positive frustration at the point b. we will indeed show below in section [ sec3 - 2 ] that the boundaries take a configuration schematically illustrated in fig . [ fig2 ] ( c ) . that is , the topology of fig . [ fig2 ] ( a ) generates a pair of vortices of moving boundaries . ( a similar argument for a different model can be found in refs . @xcite . ) we refer to the counterclockwise vortex as a positive vortex and the clockwise vortex as a negative vortex . in short , a positive frustration of a configuration generates a positive vortex of moving boundaries and a negative frustration generates a negative vortex . in the topology of fig . [ fig2 ] ( b ) , on the other hand , the circular boundaries shrink toward the center ; the players with the hand 0 just inside the boundary mimic the players with the hand 1 just outside the boundary . the central domain of the hand 0 collapses eventually . then the domain of the hand 1 becomes the central domain and will collapse after a while . thus the topology of fig . [ fig2 ] ( b ) generates a sink . finally , a source does not appear when there are only copy players , because a new domain is never generated inside a domain . it is spontaneously generated only when some players adopt strategies other than the copy strategy . specifically , a random player can be a source as is shown in sec . [ sec5 ] . we here argue that the scores of the players near a vortex core are high , while those near a sink are low . both in fig . [ fig2 ] ( b ) and ( c ) , the boundaries are not straight . near the vortex cores in fig . [ fig2 ] ( c ) , the boundary is convex from the viewpoint of the winners ( the hand 2 in fig . [ fig3 ] ( a ) ) and concave from the viewpoint of the losers ( the hand 1 fig . [ fig3 ] ( a ) ) . near the sink in fig . [ fig2 ] ( b ) , on the other hand , the boundary is convex from the viewpoint of the losers ( the hand 1 in fig . [ fig3 ] ( b ) ) and concave from the viewpoint of the winners ( the hand 2 in fig . [ fig3 ] ( b ) ) . ( c ) and ( b ) : ( a ) a bend around the frustration a of fig . [ fig2 ] ( c ) ; ( b ) a bend around the sink of fig . [ fig2 ] ( b ) . ( c ) a simple case of two subsequent steps.,title="fig : " ] ( a ) ( c ) and ( b ) : ( a ) a bend around the frustration a of fig . [ fig2 ] ( c ) ; ( b ) a bend around the sink of fig . [ fig2 ] ( b ) . ( c ) a simple case of two subsequent steps.,title="fig : " ] ( b ) + ( c ) and ( b ) : ( a ) a bend around the frustration a of fig . [ fig2 ] ( c ) ; ( b ) a bend around the sink of fig . [ fig2 ] ( b ) . ( c ) a simple case of two subsequent steps.,title="fig : " ] + ( c ) around the bend of the boundary in fig . [ fig3 ] ( a ) , the number of the winners ( the hand 2 ) is one less than the number of the losers ( the hand 1 ) . since this is a zero - sum game even locally , the total of the positive scores of the winners is equal to the total of the negative scores of the losers . therefore , the time - averaged positive score of a winner is greater than the time - averaged negative score of a loser . for example , the player with the hand 2 at the corner of the boundary scores @xmath4 , whereas the player with the hand 1 at the corner of the boundary scores @xmath5 . in other words , each player wins a high score when it is a winner and loses a low score when it is a loser . since each player spends about equal time as a winner and as a loser over a long time , the time - averaged score is positive . in short , the time - averaged score of a player around a vortex is positive . the closer a player is to the vortex core , the more often the bend appears in the boundary , and the higher the time - averaged score of the player is . the situation is the opposite near a sink . around the bend of the boundary in fig . [ fig3 ] ( b ) , the number of the winners ( the hand 2 ) is one greater than the number of the losers ( the hand 1 ) . each player , as a winner , share the total positive score with more players and , as a loser , share the total negative score with less players . hence the time - averaged score of a player around a sink is negative . the closer a player is to the center of the sink , the lower the time - averaged score of the player is . for example , let us calculate the time - averaged score over the two steps of fig . [ fig3 ] ( c ) . the score of the central player is zero in the first step and @xmath6 in the next step . the time - averaged score over the two steps is @xmath7 for the central player . the score of the player next to the central player is @xmath7 in the first step while @xmath8 in the next step . the time - averaged score over the two steps is @xmath5 for the player next to the central player . in this section , we show results of our simulations on the triangular lattice . we simulated the society of copy players on a triangular lattice with @xmath9 players . we imposed periodic boundary conditions . we demonstrate that the stationary vortex structure appear . we first show the convergence to a steady pattern , presenting snapshots of the simulations . the initial configuration fig . [ fig4 ] ( a ) was chosen randomly . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( a ) players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( b ) + players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( c ) players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( d ) the domains of the three hands are quickly formed in the first few iterations as shown in fig . [ fig4 ] ( b)(d ) . a typical pattern consisting of vortices and sinks emerge by the 20th step as shown in fig . [ fig5 ] . players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( a ) players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( b ) + players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( c ) players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( d ) + players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( e ) players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( f ) in the 17th step ( fig . [ fig5 ] ( a ) ) , we have , for example , a pair of a positive vortex around @xmath10 and a negative vortex around @xmath11 , which is indicated by a red circle . these vortices have cancelled each other by the 22nd step ( fig . [ fig5 ] ( f ) ) . after such cancellations , the pattern settles into a fairly steady state by the 1 000th step as shown in fig . [ fig6 ] . players the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( a ) players the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( b ) + players the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( c ) players the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( d ) there is a vortex pair , for example , around @xmath12 and @xmath13 , which is indicated by a red circle . there is also a sink , for example , around @xmath14 , which is indicated by a blue circle . in order to look into details of the convergence , we plot the time dependence of the total number of frustrations per player in fig . [ fig7 ] . we can see that the number of frustrations becomes almost constant after the 3 000th step . we now discuss the structure of the steady pattern . the snapshots in fig . [ fig6 ] indicate that most of the domains in the steady pattern consist of three layers of the players , where a layer means a straight line on the triangular lattice ; see fig . [ fig8 ] ( a ) . ( a ) ( b ) the situation is the same for lattices of different sizes . the three - layer domains are generated at a vortex ; see fig . [ fig9 ] . players with the initial configuration shown in ( a ) . the con figurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the 45th step ( f ) . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( a ) players with the initial configuration shown in ( a ) . the con figurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the 45th step ( f ) . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( b ) players with the initial configuration shown in ( a ) . the con figurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the 45th step ( f ) . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( c ) + players with the initial configuration shown in ( a ) . the con figurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the 45th step ( f ) . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( d ) players with the initial configuration shown in ( a ) . the con figurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the 45th step ( f ) . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( e ) players with the initial configuration shown in ( a ) . the con figurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the 45th step ( f ) . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( f ) here we started a simulation from a configuration of the form in fig . [ fig2 ] ( a ) , specifically the configuration in fig . [ fig9 ] ( a ) . we can concretely see the vortex structure schematically shown in fig . [ fig2 ] ( c ) . furthermore , we can see that a vortex spontaneously takes the core structure of fig . [ fig10 ] with tails of three - layer domains . vortices thus generate domains with three layers . a domain with three layers , once generated , is stabilized because it has a layer to be beaten by a stronger hand , a layer to remain unchanged and a layer to win over a weaker hand . in fig . [ fig8 ] ( a ) , * every player in the layer of the domain 2 scores the point @xmath15 , * every player in the uppermost layer of the domain 1 scores the point @xmath16 , * every player in the mid layer of the domain 1 scores the point @xmath17 , * every player in the lowermost layer of the domain 1 scores the point @xmath15 , and * every player in the layer of the domain 0 scores the point @xmath16 , under the assumption that the layer above the shown area in fig . [ fig8 ] ( a ) belongs to the domain 2 and the layer below it belongs to the domain 0 . for every player in the uppermost layer of the domain 1 , a neighbor with the highest score is one in the layer of the domain 2 , and hence will change the hand to 2 in the next step as a copy player . for every player in the mid layer of the domain 1 , a neighbor with the highest score is one in the lowermost layer of the domain 1 , and hence hence will remain unchanged in the next step . for every player in the lowermost layer of the domain 1 , a neighbor with the highest score is in the same layer , and hence will remain unchanged in the next step . for every player in the layer of the domain 0 , a neighbor with the highest score is one in the layer of the domain 1 , and hence will change the hand to 1 in the next step as a copy player . therefore , the domain 1 will shift one layer below in the next step , remaining to be three layers . a domain with only two layers can grow to a domain with three layers . in fig . [ fig8 ] ( b ) , * every player in the layer of the domain 2 scores the point @xmath15 , * every player in the upper layer of the domain 1 scores the point @xmath16 , * every player in the lower layer of the domain 1 scores the point @xmath15 , and * every player in the layer of the domain 0 scores the point @xmath16 , under the assumption that the layer above the shown area in fig . [ fig8 ] ( b ) belongs to the domain 2 and the layer below it belongs to the domain 0 . for each player in the upper layer of the domain 1 , a neighbor with the highest score is _ either _ one in the layer of the domain 2 or one in the lower layer of the domain 1 . each player will choose either the hand 2 or the hand 1 randomly in the next step ; _ i.e. _ only half of the players will turn into the hand 2 . for every player in the lower layer of the domain 1 , a neighbor with the highest score is in the same layer , and hence will remain unchanged in the next step . for every player in the layer of the domain 0 , a neighbor with the highest score is one in the layer of the domain 1 , and hence will change the hand to 1 in the next step as a copy player . therefore , the upper boundary of the domain 1 shifts downward only halfway , whereas the lower boundary certainly shifts one layer below . thus the domain 1 grows gradually to a domain with three layers . we do not have any arguments for the fact that domains with more than three layers are rare . we speculate that a vortex , as in fig . [ fig10 ] , tends to generate the minimum stable domain , which is a domain with three layers . once its generated with just three layers , there is no mechanism that makes the domain grow to more than three layers . in this section , we show results of our simulations on the square lattice and the honeycomb lattice . we simulated the society of copy players on a square lattice with @xmath9 players and on a honeycomb lattice with @xmath18 players . we imposed periodic boundary conditions on both lattices . we do not repeat discussion on the convergence to the steady pattern here for the square and honeycomb lattices ; it is basically the same as the triangular lattice . we demonstrate that the stationary vortex structure does _ not _ appear in these lattices , thereby emphasizing that the frustration is essential to the stationary vortex structure . figure [ fig11 ] shows snapshots of a simulation on the triangular lattice with @xmath9 players . players the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( a ) players the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( b ) + players the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( c ) players the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( d ) there is obviously no vortex structure on the square lattice . the boundaries run diagonally in each snapshot . this means that each boundary runs between the two sublattices of the square lattice ; on the boundary indicated by the red circle in fig . [ fig11 ] ( a ) , for example , the players with the hand 0 on the immediately lower left side of boundary is on the different sublattice from the players with the hand 1 on the immediately upper right side of the boundary . the boundaries move either upward , downward or sideways in the next step the upward and downward movements do not interfere with the sideway movements . we demonstrate in fig . [ fig12 ] that an initial configuration of the type in fig . [ fig2 ] ( a ) never generate vortices . players with the initial configuration shown in ( a ) . the configurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the fifth step ( f ) . the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( a ) players with the initial configuration shown in ( a ) . the configurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the fifth step ( f ) . the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( b ) + players with the initial configuration shown in ( a ) . the configurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the fifth step ( f ) . the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( c ) players with the initial configuration shown in ( a ) . the configurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the fifth step ( f ) . the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( d ) + players with the initial configuration shown in ( a ) . the configurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the fifth step ( f ) . the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( e ) players with the initial configuration shown in ( a ) . the configurations at the first step ( b ) , the second step ( c ) , the third step ( d ) , the fourth step ( e ) , and the fifth step ( f ) . the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( f ) here we started a simulation from the configuration in fig . [ fig12 ] ( a ) , which mimics fig . [ fig2 ] ( a ) . we do not see any structure of the form schematically shown in fig . [ fig2 ] ( c ) . figure [ fig13 ] shows snapshots of a simulation on the honeycomb lattice with @xmath18 players . players the black triangles denote the player with the hand 0 , the gray triangles the hand 1 , and the white triangles the hand 2.,title="fig : " ] ( a ) players the black triangles denote the player with the hand 0 , the gray triangles the hand 1 , and the white triangles the hand 2.,title="fig : " ] ( b ) + players the black triangles denote the player with the hand 0 , the gray triangles the hand 1 , and the white triangles the hand 2.,title="fig : " ] ( c ) players the black triangles denote the player with the hand 0 , the gray triangles the hand 1 , and the white triangles the hand 2.,title="fig : " ] ( d ) the pattern may appear to have a vortex structure . we hereafter argue that the seemingly vortex structure on the honeycomb lattice is not stationary and hence is essentially different from the stationary vortex structure on the triangular lattice . as a piece of evidence for the essential difference , we first show the spatial distribution of the time - averaged frustration . figure [ fig14 ] ( a ) shows the time - averaged frustration on the _ triangular _ lattice with @xmath9 players . players . the red circles indicate positive frustrations and the blue circles indicate negative frustrations . ( c ) and ( d ) the square lattice with @xmath9 players . ( e ) and ( f ) the honeycomb lattice with @xmath18 players . in the panels ( a ) , ( c ) and ( e ) , white symbols indicate the time - averaged frustration more than @xmath19 , black symbols indicate the time - averaged frustration less than @xmath20 and gray symbols with gradation indicate the time - averaged frustration in between . in the panels ( b ) , ( d ) and ( f ) , white symbols indicate the time - averaged score more than @xmath21 , black symbols indicate the time - averaged score less than @xmath22 and gray symbols with gradation indicate the time - averaged score in between.,title="fig : " ] ( a ) players . the red circles indicate positive frustrations and the blue circles indicate negative frustrations . ( c ) and ( d ) the square lattice with @xmath9 players . ( e ) and ( f ) the honeycomb lattice with @xmath18 players . in the panels ( a ) , ( c ) and ( e ) , white symbols indicate the time - averaged frustration more than @xmath19 , black symbols indicate the time - averaged frustration less than @xmath20 and gray symbols with gradation indicate the time - averaged frustration in between . in the panels ( b ) , ( d ) and ( f ) , white symbols indicate the time - averaged score more than @xmath21 , black symbols indicate the time - averaged score less than @xmath22 and gray symbols with gradation indicate the time - averaged score in between.,title="fig : " ] ( b ) + players . the red circles indicate positive frustrations and the blue circles indicate negative frustrations . ( c ) and ( d ) the square lattice with @xmath9 players . ( e ) and ( f ) the honeycomb lattice with @xmath18 players . in the panels ( a ) , ( c ) and ( e ) , white symbols indicate the time - averaged frustration more than @xmath19 , black symbols indicate the time - averaged frustration less than @xmath20 and gray symbols with gradation indicate the time - averaged frustration in between . in the panels ( b ) , ( d ) and ( f ) , white symbols indicate the time - averaged score more than @xmath21 , black symbols indicate the time - averaged score less than @xmath22 and gray symbols with gradation indicate the time - averaged score in between.,title="fig : " ] ( c ) players . the red circles indicate positive frustrations and the blue circles indicate negative frustrations . ( c ) and ( d ) the square lattice with @xmath9 players . ( e ) and ( f ) the honeycomb lattice with @xmath18 players . in the panels ( a ) , ( c ) and ( e ) , white symbols indicate the time - averaged frustration more than @xmath19 , black symbols indicate the time - averaged frustration less than @xmath20 and gray symbols with gradation indicate the time - averaged frustration in between . in the panels ( b ) , ( d ) and ( f ) , white symbols indicate the time - averaged score more than @xmath21 , black symbols indicate the time - averaged score less than @xmath22 and gray symbols with gradation indicate the time - averaged score in between.,title="fig : " ] ( d ) + players . the red circles indicate positive frustrations and the blue circles indicate negative frustrations . ( c ) and ( d ) the square lattice with @xmath9 players . ( e ) and ( f ) the honeycomb lattice with @xmath18 players . in the panels ( a ) , ( c ) and ( e ) , white symbols indicate the time - averaged frustration more than @xmath19 , black symbols indicate the time - averaged frustration less than @xmath20 and gray symbols with gradation indicate the time - averaged frustration in between . in the panels ( b ) , ( d ) and ( f ) , white symbols indicate the time - averaged score more than @xmath21 , black symbols indicate the time - averaged score less than @xmath22 and gray symbols with gradation indicate the time - averaged score in between.,title="fig : " ] ( e ) players . the red circles indicate positive frustrations and the blue circles indicate negative frustrations . ( c ) and ( d ) the square lattice with @xmath9 players . ( e ) and ( f ) the honeycomb lattice with @xmath18 players . in the panels ( a ) , ( c ) and ( e ) , white symbols indicate the time - averaged frustration more than @xmath19 , black symbols indicate the time - averaged frustration less than @xmath20 and gray symbols with gradation indicate the time - averaged frustration in between . in the panels ( b ) , ( d ) and ( f ) , white symbols indicate the time - averaged score more than @xmath21 , black symbols indicate the time - averaged score less than @xmath22 and gray symbols with gradation indicate the time - averaged score in between.,title="fig : " ] ( f ) the time average was taken over 200 steps after the 5 000th step . we can see that the frustrations on the triangular lattice ( the red and blue circles ) remain mostly at the same positions over the 200 steps . the spatial distribution of the time - averaged score ( fig . [ fig14 ] ( b ) ) shows a corresponding structure , where players closer to the vortex cores get higher scores . we do not see such structures on the square and honeycomb lattices ( fig . [ fig14 ] ( c)(f ) ) . ( we define the frustration on the square and honeycomb lattices similarly to the definition for the triangular lattice as in sec . [ sec2 - 1 ] . on the honeycomb lattice , we can have from a @xmath16 frustration to a @xmath23 frustration on a honeycomb plaquette . ) in fig . [ fig14 ] ( c ) and ( e ) , we can vaguely see vortex cores crawl around over the 200 steps . they do not stay at the same positions . the corresponding spatial distributions of the time - averaged score do not show steady patterns . in order to show further the difference between the non - bipartite triangular lattice and the bipartite lattices , we plot in figure [ fig15 ] the auto - correlations of the score and the frustration on the three lattices . players . squares on a solid ( dotted ) line denotes the auto - correlation of the score ( the frustration ) of a simulation on the square lattice with @xmath9 players . hexagons on a solid ( dotted ) line denotes the auto - correlation of the score ( the frustration ) of a simulation on the honeycomb lattice with @xmath18 players . each data point represents the spatial average as well as the time average over the 10 000 steps after the 3 000th step.,scaledwidth=55.0% ] we can clearly see that the auto - correlations of the triangular lattice are one - order magnitude greater than the auto - correlations of the square lattice and the honeycomb lattice . we thereby conclude that vortices seen in fig . [ fig13 ] on the honeycomb lattice are not stationary in time and does not grow spatially . we now introduce random players among the copy players on the triangular lattice . we show that a random player can be a source , which was not present in the system with copy players only . we demonstrate in fig . [ fig16 ] that a random player can be a source . players . there is only one random player at the position indicated by the red circle ; the rest are copy players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( a ) players . there is only one random player at the position indicated by the red circle ; the rest are copy players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( b ) + players . there is only one random player at the position indicated by the red circle ; the rest are copy players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( c ) players . there is only one random player at the position indicated by the red circle ; the rest are copy players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( d ) + players . there is only one random player at the position indicated by the red circle ; the rest are copy players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( e ) players . there is only one random player at the position indicated by the red circle ; the rest are copy players . the black hexagons denote the player with the hand 0 , the gray hexagons the hand 1 , and the white hexagons the hand 2.,title="fig : " ] ( f ) figure [ fig16 ] shows snapshots of a simulation on the triangular lattice with one random player and @xmath24 copy players . the random player chooses its hand randomly at every step . when its hand happens to be stronger than the hand of the copy players around the random player ( such as in fig . [ fig16 ] ( a ) when the random player chooses the hand 2 among the copy players of the hand 1 ) , the copy players neighboring the random player will mimic the random player s hand in the next step . this may propagate as demonstrated in fig . [ fig16 ] . thus the random player can be a source with the probability of about @xmath25 . we argued in sec . [ sec2 - 3 ] that the players near a sink get lower scores . because of the same reason working in the opposite direction , the players around a random player , or a possible source , get higher scores than the average . ( the random player itself obviously gets the average . ) figure [ fig17 ] shows that the players around the random player have higher scores than the average . copy players . the average was take over 10 000 steps after the 3 000th step . white hexagons indicate the time - averaged score more than @xmath26 and black hexagons indicate the time - averaged score less than @xmath27 . gray hexagons with gradation indicate the time - averaged score in between @xmath27 and @xmath26 . , scaledwidth=45.0% ] in this subsection , we randomly scatter many random players over the triangular lattice . figure [ fig18 ] shows the population density distribution of the time - averaged scores of the copy players and the random players of a simulation on the triangular lattice with @xmath9 players . players in total . in every panel , the solid circles indicate the distribution of the copy players while the solid lines indicate the distribution of the random players . ( a ) no random players and @xmath9 copy players . ( b ) 400 random players ( @xmath28 ) . ( c ) 1 600 random players ( @xmath29 ) . ( d ) 3 600 random players ( @xmath30 ) . ( e ) 8 100 random players ( @xmath31 ) . note that a parabola on a semi - logarithmic plot is a gaussian distribution.,title="fig : " ] ( a ) players in total . in every panel , the solid circles indicate the distribution of the copy players while the solid lines indicate the distribution of the random players . ( a ) no random players and @xmath9 copy players . ( b ) 400 random players ( @xmath28 ) . ( c ) 1 600 random players ( @xmath29 ) . ( d ) 3 600 random players ( @xmath30 ) . ( e ) 8 100 random players ( @xmath31 ) . note that a parabola on a semi - logarithmic plot is a gaussian distribution.,title="fig : " ] ( b ) + players in total . in every panel , the solid circles indicate the distribution of the copy players while the solid lines indicate the distribution of the random players . ( a ) no random players and @xmath9 copy players . ( b ) 400 random players ( @xmath28 ) . ( c ) 1 600 random players ( @xmath29 ) . ( d ) 3 600 random players ( @xmath30 ) . ( e ) 8 100 random players ( @xmath31 ) . note that a parabola on a semi - logarithmic plot is a gaussian distribution.,title="fig : " ] ( c ) players in total . in every panel , the solid circles indicate the distribution of the copy players while the solid lines indicate the distribution of the random players . ( a ) no random players and @xmath9 copy players . ( b ) 400 random players ( @xmath28 ) . ( c ) 1 600 random players ( @xmath29 ) . ( d ) 3 600 random players ( @xmath30 ) . ( e ) 8 100 random players ( @xmath31 ) . note that a parabola on a semi - logarithmic plot is a gaussian distribution.,title="fig : " ] ( d ) + players in total . in every panel , the solid circles indicate the distribution of the copy players while the solid lines indicate the distribution of the random players . ( a ) no random players and @xmath9 copy players . ( b ) 400 random players ( @xmath28 ) . ( c ) 1 600 random players ( @xmath29 ) . ( d ) 3 600 random players ( @xmath30 ) . ( e ) 8 100 random players ( @xmath31 ) . note that a parabola on a semi - logarithmic plot is a gaussian distribution.,scaledwidth=45.0% ] ( e ) when there is no random players , the distribution of the score is almost gaussian ( fig . [ fig18 ] ( a ) ) . the fluctuation of the score is due to the fact that each player is occasionally close to a vortex , getting high scores , and occasionally close to a sink , getting low scores . as we introduce a few random players , the copy players just around the random players get scores higher than the average owing to the same reason described in the previous subsection . this generates the additional peak on the right of the highest peak in fig . [ fig18 ] ( b ) . as we increase the number of random players , the vortex structure disappears from the steady pattern ( fig . [ fig19 ] ) . ) and @xmath32 copy players the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( a ) ) and @xmath32 copy players the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( b ) + ) and @xmath32 copy players the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( c ) ) and @xmath32 copy players the black squares denote the player with the hand 0 , the gray squares the hand 1 , and the white squares the hand 2.,title="fig : " ] ( d ) when the number of random players is 1 600 ( fig , [ fig18 ] ( c ) ) , random players are scattered in the system every three lattice points on average . this is enough to destroy a vortex which generates the three - layer structure explained in sec . [ sec3 - 2 ] . the copy players not neighboring the random players can not get high scores generated by vortices and keep losing scores because of sinks . hence the highest peak in fig . [ fig18 ] ( b ) shifts in the direction of the lower score in fig . [ fig18 ] ( c ) . the copy players just around the random players , on the other hand , keep getting scores higher than the average . the number of such players is increased and hence the side peak in fig . [ fig18 ] ( b ) has grown in fig . [ fig18 ] ( c ) . as we further increase the number of random players , the peak on the side of the lower score keeps shrinking and the other peak keeps growing until the latter dominates as in fig . [ fig18 ] ( e ) . the width of the distribution is the greatest when random players are about @xmath33 of all players , or in the case fig . [ fig18 ] ( c ) . figure [ fig20 ] shows how the standard deviation of the distribution of the time - averaged score depends on the concentration of random players . players in total.,scaledwidth=55.0% ] we introduced a new lattice model of the rsp game with copy players , who mimic the hand of the player with the maximum score . the key feature is the existence of the frustration , which is the three - sided situation where the hands of the three players on a triangle are all different ; then the hand 1 wins over the hand 0 , the hand 2 wins over the hand 1 and the hand 0 wins over the hand 2 . we showed that the frustration generates a stationary vortex on the triangular lattice . we argued that the structure which consists of vortex pairs , sinks and domains of three layers is stable on the triangular lattice . the structure does not appear on the square lattice nor on the honeycomb lattice . finally , we introduced random players , each of which chooses the hand randomly at every step . a random player can be a source , which was not existent in the copy society . random players of about @xmath33 destroy the structure of vortex pairs . the authors express their sincere gratitude to dr . naoki masuda for many valuable comments to the earlier version of the paper .

gauge theories in three spacetime dimensions typically possess solitonic disorder operators that strongly affect the physics at long distances . a classic example is polyakov s proof that monopoles in a pure @xmath2 gauge theory lead to confinement even at weak coupling @xcite . that emit magnetic flux . we will give a more careful definition in section 2 , where we will also explain what we mean by monopole states in three dimensions . for a general review of basic properties of monopoles and their relation to other solitons in various dimensions , see e.g. @xcite . ] effects of gauge theory disorder operators on the phase structure of many other three - dimensional models have been analyzed in e.g. @xcite . the primary question concerning these operators is whether they cause confinement ; if not , as is often the case , the next order of business is to understand the ir fixed point . identifying disorder operators in such a nontrivial cft is often a daunting task . using various classical limits , their quantum numbers have been calculated in the ir of qed with many fermion flavors @xcite , in the critical @xmath3 model at large @xmath4 @xcite , in the ir of qcd with many fermion flavors and with various gauge groups @xcite , and in the infinite - level limit of conformal chern - simons - matter theories @xcite . disorder operators also feature prominently in supersymmetric theories , where their dimensions and intricate duality mappings can often be understood using holomorphy and localization @xcite . understanding disorder operators in the ir of non - abelian theories without supersymmetry is generally difficult . even in the planar limit , no results at finite t hooft coupling were available unless the number of matter flavors was much larger than the number of colors . however , recent progress in studying the t hooft limit of chern - simons ( cs ) theories coupled to matter @xcite makes it possible to elegantly compute the scaling dimensions of an interesting class of disorder operators in conformal cs - fermion theories with only one fermion flavor . we perform this computation in this brief note . our main result is that the lowest - dimension disorder operators with @xmath5 units of magnetic flux have dimension where @xmath6 is the yang - mills - regulated cs level . this result is valid at all t hooft couplings . our presentation starts with a technical definition of disorder operators in section 2 . in section 3 we obtain the advertised result by computing the thermal partition function of all cs - fermion states with a given magnetic flux on a two - sphere , and along the way we clarify certain aspects of the path integral computation given in @xcite . in section 4 we conclude by discussing implications of our findings for the bosonization duality between cs - fermion and cs - boson theories . we are interested in local operators @xmath7 that create magnetic flux in a theory with gauge group @xmath8 . a state created by a such a disorder operator at @xmath9 has flux that can be gauge - fixed into the form where @xmath10 is any two - sphere enclosing @xmath11 , @xmath12 is the gauge field strength in the theory , @xmath13 are the anti - hermitian cartans of @xmath8 , and @xmath14 are the gauge - invariant gno charges that index different monopole states in four spacetime dimensions @xcite . the quantization of gno charges is most easily demonstrated in the path integral language , where configurations with a disorder operator at the origin can be depicted ( in a suitable gauge ) as fluctuations around wu - yang gauge field configurations @xcite ( note that the field strength associated to the above connection gives @xmath15 when integrated over an @xmath16 centered at the origin . ) this is an allowed configuration if the gauge fields on the south and north hemispheres differ by a gauge transformation . fixing @xmath17 for convenience , the difference between north and south at @xmath18 is @xmath19 , and so the appropriate gauge transformation is enacted by the group element @xmath20 . this object must be well - defined as we circle the equator of the @xmath16 by letting @xmath21 . this condition , @xmath22 , is only satisfied for a discrete set of @xmath14 s . there are reasons to believe that not all gno charges correspond to different disorder operators . first , even if the wu - yang backgrounds are classical saddles , they need not all be stable in the quantum theory @xcite . second , the operators defined via eq . need not be eigenstates of the dilatation operator in the ir cft , and so they might not all be independent . third , the gno monopoles are not all topologically protected . topological charges exist only if the gauge group has a nontrivial fundamental group @xmath23 , e.g. for @xmath24 but not for @xmath25 @xcite . however , even though the gno classification may be too refined , we will find disorder operators with small amounts of flux in the planar limit that appear to have well - defined scaling dimensions with no indication of instabilities . this means that the situation is similar to the one found in many - flavor qcd @xcite , where large classes of gno charges were shown to correspond to stable monopoles even though there was no a priori reason for them to do so . so far we did not assume that the theory was conformal , and the operators @xmath7 were instantonic in character . these instantons represent transitions to / from states with magnetic flux that we will call monopoles . analyzing the spectrum of such states in general qfts is hard , but progress was made in the supersymmetric context in @xcite . things are simpler in a conformal theory , where each disorder operator corresponds to a monopole state @xmath26 on @xmath27 , a two - sphere of radius @xmath28 . in the path integral these states correspond to configurations with background fields @xmath29 on @xmath30 . in a yang - mills theory coupled to matter , disorder operators are defined in a gauge - invariant way using eq . . their scaling dimensions are determined , via the state - operator correspondence , as casimir energies of the fluctuations around wu - yang monopoles on @xmath31 , when these are good saddles . is given by a constant times the area form . if there is a reason to break the rotational invariance in the theory , it may happen that the true saddles are large deformations away from eq . for which the magnetic flux is not uniformly distributed across the sphere . in that case , all comments we make apply to fluctuations around these new backgrounds . spherical symmetry will never be broken in our examples , so we will work with fluctuations around uniform flux backgrounds . ] the fluctuations in question must be gauge - invariant themselves . in the case of yang - mills theory this means that no operators with electric charge may be present . this picture changes if a chern - simons term at level @xmath6 is present in the action . the wu - yang configuration in this case has electric charge @xmath32 , and the gauss law requires that matter fields compensate for this charge @xcite . this means that gauge - invariant disorder operators in cs - matter theories do not merely create magnetic flux ; they also create enough matter to dress themselves into an electric singlet . this dressing may introduce large corrections to the casimir energy due to matter self - interactions @xcite . for instance , in a cs - fermion theory at @xmath33 ( or at zero t hooft coupling in the planar limit ) , the matter fields can be heuristically treated as noninteracting fermions in a background magnetic field that fill up a fermi sea until they reach @xmath34 units of charge ; the landau levels are separated by gaps of order @xmath35 , giving a total fermi energy of @xmath36 @xcite . ( see also @xcite . ) we will show that this holds at all t hooft couplings . our approach to calculating disorder operator dimensions is the following . we focus on the conformal cs - fermion theory , radially quantized with a wu - yang background on the spatial two - sphere , and we compute the path integral over all gauge field and matter fluctuations around this background . if the euclidean spacetime is taken to be @xmath37 , the result is a thermal partition function of all monopole states with the given set of gno charges ( with matter dressing automatically taken care of ) . in the low temperature ( @xmath38 ) limit , the result must take the form @xmath39 , with @xmath40 being the desired energy ( or flat - space scaling dimension ) of the lightest disorder operator in this gno class . we will calculate this @xmath40 in the t hooft limit by using several subtle tricks that we carefully explain in the following section . consider the conformal @xmath0 chern - simons theory coupled to dirac fermions in the fundamental representation . the thermal partition function is given by the path integral on @xmath41 , with the two - sphere volume @xmath42 and temperature @xmath43 . the euclidean action is we normalize @xmath44 , and in particular we choose @xmath45 to be a matrix whose only nonzero entry is @xmath46 at the @xmath47th place on the diagonal . ( this choice means that gno charges have values @xmath48 for @xmath49 . ) we are interested in the planar limit , where @xmath4 and @xmath6 are taken to infinity with the t hooft coupling @xmath50 fixed . in dimensional regularization , the theory is unitary only for @xmath51 @xcite . the cs level @xmath6 is quantized by demanding invariance under large gauge transformations ; in order to offset the parity anomaly of the fermions , @xmath52 should take on values in @xmath53 @xcite , but in the t hooft limit this is immaterial and we may think of @xmath6 as being a ( large ) integer . at @xmath54 we recover the singlet fermion model @xcite , and as @xmath55 approaches unity the theory approaches pure cs theory @xcite . a subtlety ( pointed out by o. aharony ) arises here because we regulate the above theory using dimensional reduction , but the preceding discussion of disorder operators in cs theory assumed regularization using a yang - mills term . the cs levels in the two regularizations are related by @xmath56 . in dimensional reduction this shift arises because we must regulate the monopoles by giving them a core of nonzero size . this uv regularization effectively acts as an edge of the system , and the electric charge of the monopoles gets shifted by the @xmath4 edge states of cs theory . the operator content is simple to describe : cs has no dynamical degrees of freedom and the only operators with @xmath57 energies are products of @xmath58 `` single - trace '' singlets built out of two matter fields and covariant derivatives . these single - trace operators are the scalar operator @xmath59 and conserved currents @xmath60 ; they have protected ( @xmath61-independent ) scaling dimensions in the planar limit , and they are conjectured to be holographically dual to higher - spin fields of the vasiliev system in ads@xmath62 @xcite . if the gauge group is @xmath63 , the theory also contains baryons @xmath64 whose energies are @xmath65 @xcite . for any gauge group , there are also disorder operators defined in section 2 , and if the group is not simply connected ( e.g. for @xmath58 ) these operators may have nontrivial topological charge . the phase structure of planar cs - fermion theory at finite temperature is very rich ( see fig . [ fig phases ] ) . at any finite t hooft coupling @xmath66 there are two phase transitions ( which overlap at a critical @xmath61 ) . both transitions happen at temperatures of order @xmath67 . these different phases are observed in the density of eigenvalues of the matrix model obtained by integrating out matter fields from in the high temperature limit @xmath68 . interpreting the transitions in terms of the original variables is still an open question , but ref . @xcite has shown suggestive evidence that baryon - antibaryon pairs introduce relations between singlet states at high energies and thereby cause a qualitative change in the density of states an effective `` deconfinement '' that allows one set of singlets to recombine its constituents into a different set of singlets . this transition is accompanied by going from a singlet free energy @xmath69 to a deconfined free energy @xmath70 . it has also been speculated that the second phase transition is related to the proliferation of monopoles @xcite . ( -2 , 2 ) ( -2 , -2 ) ( 2 , -2 ) ( 2 , 2 ) ; ( 2 , -2 ) .. controls ( 0.5 , -1.95 ) and ( -2 , -1 ) .. ( -1.95 , 2 ) ; ( -2 , -1 ) .. controls ( -1 , -1 ) and ( 2 , 0.5 ) .. ( 1.95 , 2 ) ; ( 0 , -2.05 ) node[anchor = north ] @xmath71 ; ( -2 , -2.05 ) node[anchor = north ] @xmath72 ; ( 2 , -2.05 ) node[anchor = north ] @xmath73 ; ( -2.05 , 0 ) node[anchor = east ] @xmath74 ; ( -2.05 , -1.8 ) node[anchor = east ] @xmath75 ; ( -2.05 , 2 ) node[anchor = east ] @xmath76 ; a fascinating property of this planar cs - fermion theory is that it appears dual to the bosonic theory of cs gauge fields coupled to a wilson - fisher fixed point @xcite . ( similarly , the cs - gross - neveu model is related to the cs theory coupled to the ordinary free scalar ; our calculation holds for monopoles in this other cs - fermion theory , as well . ) this `` bosonization '' duality relates bosonic and fermionic theories with groups @xmath77 and @xmath78 and with couplings that satisfy @xmath79 . this is a strong - weak duality that can be viewed both as a nonsupersymmetric giveon - kutasov duality @xcite and as a level - rank duality of pure cs theory extended to theories with matter @xcite . the bosonization conjecture holds at the level of matching partition functions , correlators , s - matrices , and single - trace operators , but it is not known how nonperturbative operators map across the duality . since our result gives us dimensions of disorder operators at strong coupling , we will be able to tie this to the known facts about nonperturbative operators at weak coupling , and we will conjecture two possible patterns for how these nonperturbative operators may map to each other . let us now review the methodology of solving the cs - fermion theory in the planar limit and without monopole backgrounds . we present a streamlined version of the procedure that has been explained in great detail in ref . @xcite . for convenience , we will set the radius of the two - sphere to be @xmath80 ; this turns the temperature @xmath81 into a tunable dimensionless parameter . at finite temperature , matter fields develop thermal masses proportional to @xmath81 , so at @xmath82 the fermions become very massive , propagating only on scales @xmath83 much smaller than the size of the spatial @xmath16 . as for the gauge fields , the only gauge - invariant degrees of freedom are contained in the polyakov loop , i.e. in the holonomy of the gauge field around the thermal circle . this object is defined as the path - ordered exponential the polyakov loop can be viewed as a map from @xmath16 to @xmath58 . as such , it is not gauge - invariant it can still change under time - independent gauge transformations . its eigenvalues are gauge - invariant , however , and below we will describe the gauge fixing needed to reduce @xmath84 to its abelianized ( diagonal ) form @xmath85 with @xmath86 . the cs action produces a @xmath87-function potential that sharply suppresses spatial variations of the polyakov loop eigenvalues , and the large thermal mass of the fermions ensures that integrating out matter will not change this potential on length scales greater than @xmath88 . thus we may separate the fields in into `` high - energy '' modes ( all matter fields and the large - momentum gauge fields that glue them into singlets ) and `` low - energy '' modes ( the polyakov loop and the small - momentum gauge fields ) . the high - energy degrees of freedom are all fields at momenta above @xmath81 ( in particular , this includes all matter fields ) . since the curvature of @xmath16 is much smaller than @xmath81 , these fields should be thought of as living in a flat space with a constant polyakov loop in the background @xcite . the gauge fields can be dealt with by fixing the light - cone gauge , @xmath89 @xcite . this choice gives a trivial faddeev - popov determinant . integrating out the remaining gauge fields @xmath90 and @xmath91 gives a nonlocal potential for the matter fields , and matter can in turn be integrated out by the usual large-@xmath4 procedure for @xmath58 models @xcite . after all the high - energy modes are integrated out , we are left with an effective potential for the polyakov loop that takes the form of a derivative expansion : the potentials @xmath92 above are all of order @xmath4 and are formed by traces of arbitrarily many powers of @xmath84 and @xmath93 powers of @xmath94 . at large @xmath81 the derivative pieces can be ignored and we can replace the polyakov loop @xmath84 with a constant matrix @xmath95 governed by the eigenvalue potential exact forms of this potential for various theories at all @xmath61 have been given in ref . @xcite . integrating low - energy modes now proceeds by fixing the maximal torus gauge @xcite . this is done in three steps @xcite : 1 . fix the temporal gauge @xmath96 ; 2 . use time - independent gauge transformations to abelianize the polyakov loop and impose @xmath97 at each @xmath98 on the two - sphere ; 3 . use the remaining time - independent transformations to impose the coulomb gauge for the cartan components of spatial gauge fields @xmath99 . fixing this gauge and integrating all modes other than @xmath84 ( with @xmath81 as the uv regulator ) results in a matrix model for the polyakov loop . as we have mentioned , the gauge - fixed integral will impose a @xmath87-function on non - constant abelianized configurations @xmath84 , so once we go to the maximal torus gauge we may just talk about a constant matrix @xmath100 . the faddeev - popov determinants and the integration over the remaining modes ( the off - diagonal spatial components of gauge fields ) combine to give the vandermonde determinant of the @xmath58 matrix model , which is , modulo constant prefactors that we ignore , given by a subtle and crucially important point arises when abelianizing the polyakov loop : matrices @xmath84 with equal eigenvalues represent degenerate points in the set of all polyakov loop configurations . ( we retain position dependence because the gauge must be fully fixed before we can meaningfully say that the integration of gauge fields fixes the @xmath101 s to be constant . ) we can not perform the second gauge - fixing step outright , but instead we must break up the path integral over all @xmath84 s into a sum over domains in which there are no eigenvalue overlaps anywhere on the sphere , and then abelianize in each domain separately . remarkably , these domains are indexed by gno charges . more precisely , it is possible to show that time - independent gauge transformations @xmath102 needed to abelianize a generic nondegenerate @xmath84 at all @xmath103 need to be defined patch - wise on the two - sphere , just like in the wu - yang construction @xcite . these transformations @xmath102 are then classified by the winding profile around the equator , and in turn this means that nondegenerate @xmath84 s are classified by winding numbers of the gauge transformations needed to abelianize them . the winding numbers ( a.k.a . `` flux sectors '' ) are given by @xmath4 numbers @xmath104 , one for each cartan @xmath13 so that @xmath105 . performing a gauge transformation with nontrivial winding diagonalizes the holonomy @xmath84 but simultaneously turns a smooth profile of the spatial gauge fields into a monopole configuration of form . when evaluated on a configuration with a nontrivial winding of @xmath84 and after the @xmath106 s are forced to be constant , the cs term gives the @xmath107-dependent potential @xcite the integral over all polyakov loops in the maximal torus gauge thus reduces to summing over flux sectors of spatial fields and integrating over all spatially constant configurations of eigenvalues @xmath101 . the sum over flux sectors gives in other words , this sum discretizes eigenvalues in units of @xmath108 , meaning that instead of working with an @xmath4-fold integral over all @xmath109 we must work with a sum over @xmath110 for @xmath111 . in the planar limit , these eigenvalues are essentially continuous , although their discretization must be taken into account at large @xmath61 @xcite . it is important to emphasize that polyakov loops associated to nontrivial flux sectors do _ not _ correspond to the monopole backgrounds discussed in the previous section . we are talking about _ different _ gauge - invariant field configurations here : disorder operators correspond to nontrivial profiles of the spatial components of the gauge field , while flux sectors label domains defined by eigenvalues of @xmath84 , the integral of the temporal component of the gauge field . it is only in the maximal torus gauge that some polyakov loop configurations @xmath84 manifest themselves as monopole backgrounds . in the next subsection we will explain how bona fide monopoles are to be included in the computation . finally , once the gauge is fixed and all other degrees of freedom have been integrated out , the partition function is given by the matrix model with discretized eigenvalues this model is solved using the usual large @xmath4 methods , e.g. @xcite , by replacing @xmath101 with an eigenvalue density @xmath112 and carrying out a saddle - point calculation . this was done for all @xmath61 in refs . the four phases on fig . [ fig phases ] are seen when @xmath113 and they correspond to different qualitative behaviors of @xmath112 : crossing the red critical line causes @xmath112 to lose support on the entire circle ( the gww transition ) , and crossing the blue line causes @xmath112 to cap off at @xmath114 , saturating the maximal possible value it can attain given the quantization of eigenvalues due to flux sectors ( the dk transition ) . previous calculations of cs - matter partition functions using the above high-@xmath81 method did not sum over states with magnetic flux . the resulting partition function obeyed the dualities expected of the full partition function with monopoles included , however . this situation is similar to the calculation of the partition function for singlet vector models in @xcite . this partition function knew about the phase transition caused by baryons even though it was constructed just by counting single - trace operators , and in fact the @xmath58 and @xmath63 results were the same even though the @xmath58 theory did not even have baryons ; this indicates that the contribution of baryon states to the partition function was negligible . the same is expected to happen for monopole states . we now wish to take the sum over monopoles into account . we focus on those terms in this sum that correspond to states created by a single disorder operator with given gno charges @xmath115 . the calculation we need differs from the ones in earlier works in three ways : 1 . some of the gauge symmetry needs to be spent on fixing the monopole background to the maximal torus form at each spacetime point , as done in @xcite . for each set of @xmath93 equal gno charges , a @xmath116 factor contributes to the remaining gauge symmetry . thus , for instance , if @xmath117 and all other charges are zero , the symmetry left to be fixed is @xmath118 . if e.g. @xmath119 and all other charges are @xmath120 , the gauge symmetry is @xmath121 . the faddeev - popov determinant from this gauge - fixing depends on the saddle - point configurations , not on the dynamical fields . the space of single - monopole backgrounds is discrete , and we focus on one specific background at a time ; therefore the fp determinant can not influence the physics of fluctuations around the saddle that we wish to study , and we may ignore it just as we ignore other constant prefactors . the reduced gauge symmetry influences the vandermonde determinant @xmath122 and the abelianization of the polyakov loop . for instance , on backgrounds with all different gno charges , the remaining gauge symmetry is @xmath123 , the vandermonde is trivial , the contribution of the polyakov loop to the action is just as if it was abelian to begin with , and there are no flux sectors to sum over . 2 . high - energy modes will be insensitive to the flux background if the flux through a `` thermal cell '' of size @xmath124 is much less than unity . in other words , as long as each gno charge satisfies @xmath125 , the integral over high - energy modes proceeds just as in the case without monopoles . it is currently not known how to perform this integral at high gno charges . 3 . the computation we wish to perform is reliable at high temperatures , but we are ultimately interested in extracting the casimir energy at @xmath126 . we proceed by taking @xmath127 and then working in the double scaling limit @xmath128 , @xmath129 . this is not as outlandish as it may seem : the theory has no phase transitions below a critical @xmath130 , so any computations in the regime @xmath131 will be analytically connected to the @xmath132 region . indeed , @xcite has shown that the @xmath129 limit of the free energy on a monopole - free background always reproduces the @xmath133 result obtained by other means . in a monopole background the situation is only a bit more complicated . in the monopole - free sector , the dominant term of the partition function in the lowest - temperature phase is @xmath134 . in the case of interest for us , however , we are interested in the very small quantity @xmath135 , and so in our double scaling limit we should expect to reproduce the @xmath133 result up to corrections that have the same order in @xmath4 . in other words , we expect to find the partition function of the form @xmath136 with @xmath137 . this is the form of the partition function we will recover , and we will be able to indicate which states contributed to the @xmath138 correction . computing the partition function in the single - monopole background is now straightforward , and we can borrow most of the technical results from earlier work . in particular , the first two points above show that the integral over high - energy modes does not depend on the monopole background as long as the gno charges are much smaller than @xmath81 . thus , when setting up the matrix model with monopoles , in eq . we may use the result for the @xmath129 limit of the eigenvalue potential of cs - fermion theories obtained in eq . ( 6.33 ) of ref . @xcite , the partition function in which this potential figures is a modification of that takes into account the monopole background according to the three precepts above : in writing the above we arrange the gno charges such that @xmath139 with @xmath49 . for a general @xmath140 , there are @xmath141 segments of @xmath142 equal charges ( with @xmath143 ) . the primed sum over flux sectors @xmath144 leaves out all the @xmath145 s that belong to a segment of length one , i.e. all @xmath144 such that @xmath14 is not equal to any other gno charge . the remaining fluxes are summed just like in eq . . the vandermonde is the one appropriate for the @xmath146 group and is given by the product of determinants for subsets @xmath147 of eigenvalues that are conjugate to the same gno charge , eq . shows that monopoles with equal nonzero gno charges are special : the sum over polyakov loop windings @xmath144 will subsume part of the @xmath5-dependent potential @xmath148 in any sector with @xmath149 . the sum over @xmath144 s will still discretize the eigenvalues , and the remaining potential will depend on @xmath140 through the term @xmath150 . it would be fascinating to analyze such monopole states , but the discretization of eigenvalues takes this problem outside the scope of the current work . from now on we restrict ourselves to monopoles whose nonzero gno charges are all different . for simplicity , let us consider @xmath151 and @xmath152 for @xmath153 ; the generalization to multiple nonzero charges will be trivial . the eigenvalue conjugate to @xmath154 now has special status , while the other eigenvalues @xmath155 can be treated just like in the previous subsection . in fact , the density of these @xmath156 eigenvalues is unaffected by the dynamics of @xmath157 at large @xmath4 , and the density @xmath158 is determined by the very same saddle - point calculation used to find @xmath112 in the monopole - free case . the special eigenvalue @xmath157 thus decouples , and the remaining eigenvalues are integrated over to give a multiplicative factor of @xmath159 , defined in eq . . the contribution of these eigenvalues to the partition function with monopoles represents the resummed single - trace fluctuations on top of monopole states , and in the low-@xmath81 limit it does not affect the casimir energy @xmath40 . the remaining integral should be understood as an integral over all excitations of the matter dressing the monopole . in other words , the integral over the remaining eigenvalue represents the sum over all monopoles with the given gno charge . this type of integral over a single eigenvalue and its relation to nonperturbative effects has , in fact , been observed in matrix models describing quantum gravity @xcite . the partition function is thus given by with the above integral can be solved using steepest descent . before we proceed to find saddles of @xmath160 , we note that cs - matter is invariant under the parity transformation @xmath161 , @xmath162 , and the above potential clearly respects this . from now on we take @xmath163 ; any saddle @xmath164 we find at a fixed positive @xmath61 will correspond to a saddle @xmath165 at @xmath166 . for the same reason we also take @xmath167 . the saddle point equation is it is easy to find a complex saddle of the form @xmath168 , where @xmath169 is positive and large at small @xmath170 . to see that this is consistent and to find @xmath169 , we substitute this ansatz and use @xmath171 in the saddle point equation , getting and consequently which is indeed large as @xmath129 . the value of the effective potential at this saddle point is we can now read off the casimir energy @xmath40 from @xmath172 , giving the advertised result it can be checked that the steepest descent through this point in the complex plane is in the direction parallel to the real axis , and the initial contour can be deformed in the needed way without complications . there are no indications of instability around this saddle point . the contribution from the rest of the contour can be analytically shown to behave as @xmath173 for some coefficients @xmath174 and @xmath175 . limit this contribution overwhelms the saddle point value and dominates the integral . this is confirmed by numerical evaluation of the integral . there are no inconsistencies here because we are really taking the double - scaling limit in which @xmath81 is always large , as described in the third point at the beginning of this subsection . i thank guy gur - ari and ethan dyer for numerous discussions and help with understanding the behavior of this integral away from the saddle point . ] this term comes from parts of the contour far away from the saddle point , and we believe it can be understood as the partition function of highly excited monopole states the ones in which the dressing matter is not in its ground state but rather in states so energetic that their free energy eclipses the casimir energy . we now ( very schematically ) show how this contribution from excited monopoles can come about . consider the integral over monopole states with dressing matter excited to an energy @xmath176 above the ground state . their contribution to the partition function is if @xmath81 is infinitesimal ( meaning @xmath132 on a spatial @xmath16 of unit radius ) , the above integral can be approximated just by @xmath177 . however , as stressed in point 3 ) above , we work with @xmath178 and with @xmath128 taken before @xmath129 . thus , states with energies that differ by less than @xmath81 will not be suppressed relative to each other , and the cumulative effect of all these states can be enough to overwhelm the exponential suppression afforded by @xmath129 . consider high - energy excitations with @xmath179 , the density of states appropriate for three - dimensional theories . a saddle point calculation shows that integrating over sufficiently high energy modes in the above integral gives we take this as evidence that highly excited monopoles can , _ in principle _ , sum to a number that cancels out the @xmath177 prefactor and gives the above formula for a small positive @xmath174 . only at @xmath82 would it be possible to get @xmath174 of order one , so this is consistent with the expectation that our total result , @xmath180 , is analytically connected to @xmath181 that we would have gotten at @xmath126 without the double - scaling limit , had we been able to carry out that calculation . a careful study of monopole excitations would give us the correct density of states and allow us to check this conjecture . if the monopole state has several different nonzero gno charges , the same saddle - point equation is solved for each eigenvalue conjugate to these charges . the resulting casimir energy can thus be recorded as the general answer for a monopole with few nonzero gno charges : this result stops being valid when there are so many nonzero gno charges that the collective dynamics of their eigenvalues can no longer be neglected when studying the remaining eigenvalue density @xmath158 . moreover , if we had so many different charges , some of them would have charge comparable to @xmath4 , thus invalidating the assumption that the magnetic flux in each thermal cell is negligible . understanding high - flux monopole states remains an open problem . we now possess a fair amount of information about the spectrum of operators at strong coupling in @xmath58 cs - fermion theories . at low scaling dimensions we only have the tower of higher - spin @xmath58 single - traces and multi - traces with @xmath61-independent dimensions . going to higher dimensions , we encounter the lightest disorder operators of dimension @xmath182 , and from there on we find more and more disorder operators with dimensions given by ; there are also the monopoles with excited matter dressing whose dimensions we can not calculate using the above method . ( disorder operators with several gno charges equal to @xmath183 are not expected to have dimensions lower than the one with just one nonzero gno charge @xmath183 , but they might appear just above it in the spectrum ; these have different topological charge , however , and we do not expect them to have nontrivial overlaps . ) there are no baryon operators in @xmath58 theories , so the disorder operators with @xmath184 are the lowest - dimension nonperturbative operators . what can these operators map to under the large @xmath4 cs - matter bosonization dualities ? the dual to the stronly coupled cs - fermion theory of rank @xmath185 is the weakly coupled cs - wilson - fisher theory of rank @xmath186 . ( recall that dual pairs of theories have ranks related by @xmath187 . ) all single - trace operators match across the duality , so the fermionic disorder operator can not be dual to any combination of bosonic single - traces . moreover , global symmetries must map across the duality , and the single - flux disorder operator is charged with a topological @xmath188 symmetry @xcite . if the dual theory has gauge group @xmath189 , the only candidate in the weakly coupled bosonic theory is the bosonic disorder operator . if the dual theory is @xmath190 , the only reasonable candidate is a baryon operator based on level - rank duality of pure cs theory @xcite , we may expect that @xmath191 maps to @xmath189 while @xmath192 maps to @xmath190 , meaning that disorder operators must map to disorder operators and baryons must map to baryons . however , in this speculative section we remain open to the possibility that @xmath191 maps to @xmath190 . can a bosonic baryon be dual to a fermionic disorder operator ? the powers @xmath193 of derivatives must be inserted so that the levi - civita symbol does not antisymmetrize the baryon to zero . their positioning is similar to populating the fermi sea with fermions in the disorder operator s matter dressing this is a pleasant feature but should not be taken to mean much , as the fermi sea picture is not meaningful at strong coupling . more interestingly , though , this operator is charged under a baryon current @xmath194 , and in an @xmath190 theory this current is the natural ( and only ) candidate dual to the topological current @xmath188 of disorder operators in @xmath191 cs - fermion theories . a mapping between monopoles and baryons has been observed in some supersymmetric theories in three dimensions @xcite . even if some monopoles map to baryons under bosonization , the corresponding dimensions do not match up completely . the lowest bosonic baryon has dimension proportional to @xmath195 at weak coupling @xcite . the lowest fermionic disorder operator has dimension proportional to @xmath196 . their dimensions differ by a factor of @xmath197 . to make dimensions match without ruining the mapping of @xmath2 charges , we must postulate that the fermionic disorder operator is bosonized to a bosonic baryon _ and _ a sea of singlets needed to raise the dimension to the right number . in particular , a way that this can be done is to state that the lowest - dimension fermionic disorder operator maps to @xmath198 baryons and @xmath199 antibaryons in a bosonic theory , with @xmath200 at strong fermionic coupling . the @xmath199 baryon - antibaryon pairs can then be written as a combination of multi - traces . a similar multi - trace - dressing trick can be used to match up the dimensions of monopoles in the case of bosonic @xmath189 duals to the @xmath191 cs - fermion theory . however , this leaves open the question of what a single bosonic baryon is dual to . the mismatch of baryon and monopole dimensions that comes from studying the lowest - dimension disorder operators may serve as evidence that bosonization really does map a @xmath201 fermionic theory to a @xmath202 bosonic one , just as one would expect from pure cs level - rank duality . it is particularly interesting to study these operator maps in light of the fact that the two phase transitions in fig . [ fig phases ] are dual to each other @xcite . based on general expectations of analyticity and the evidence from @xcite , at any @xmath61 the gww transition should be induced by baryon - antibaryon pairs that affect the multi - trace density of states . as all single - trace operators are dual to each other , the relations between them that cause the phase transitions should also be dual . the dk transition , being the dual of the gww transition , should thus be induced by relations between single - traces , and in particular it is natural to assume that it arises due to either monopole - antimonopole or baryon - anitbaryon pairs affecting the density of states of single - traces . monopole - antimonopole pairs can drive a phase transition via different ways to enact the ope of disorder operators @xmath7 and @xmath203 of opposite gno charges ; @xmath204 has zero topological charge so we can expect to find higher - spin currents in this ope . in other words , a two - monopole state with zero topological charge may have nontrivial overlaps with multiple multi - trace states , and this may lead to relations between multi - traces . and @xmath58 . the theories with these groups have the same phase structure , but only @xmath58 has topologically charged monopoles , so once again we are led to think of topologically neutral configurations as the ones that drive the phase transition . in the same way , the baryonic phase transition exists in @xmath58 theories in which baryons are not gauge - invariant operators . ] it is thus very natural to conjecture that we can express a baryon - antibaryon pair @xmath205 in a fermionic theory as a pair of nonperturbative bosonic operators . this is consistent with the picture of individual disorder operators mapping to individual baryons or monopoles , and it is now an interesting question of combinatorics ( relegated to future work ) to show whether the pairwise mapping of operators can be factored into mapping individual operators . doing this combinatorics while respecting constraints set by the duality of phase transitions and the known dimensions of lowest - lying nonperturbative operators seems likely to reveal fascinating patterns in operator maps at finite @xmath4 . if this is possible , we should also be able to conclusively decide whether the bosonization duality maps @xmath191 to @xmath189 , to @xmath190 , or to something else entirely . another question of interest is the bosonization of disorder operators with zero topological charge . based on our calculations , at least some of these are also stable saddles of the path integral ( e.g. the dimension of the operator with @xmath206 is @xmath207 ) . thus they should not map to single- or multi - trace operators . a possible option is that they map to other disorder operators of zero topological charge . finally , a good check for the speculations given in this section would come from studying all the currently unknown quantum numbers of these nonperturbative operators . in particular , understanding bosonic disorder operators would be of great interest . the methods in this paper are not applicable to monopole states in cs - 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nanotechnology , as well as biology , biophysics and chemistry are using or studying setups and objects which are smaller and smaller . in these systems , one is usually interested in mean values , but thermal fluctuations play an important role because their amplitude are often comparable to the mean values . this is for example the case for quantities such as the energy injected in the system or the energy dissipated by the system . these fluctuations can lead to unexpected and undesired effects : for instance , the instantenous energy transfer can be reversed by a large fluctuation , leading energy to flow from a cold source to a hot one . these events , although rare , are quantitatively studied by the recent fluctuations theorems ( fts ) . these theorems give fluctuation relations ( frs ) that quantify the probability of these rare events in systems which can be arbitrarily far from equilibrium . fts have been demonstrated in both deterministic systems @xcite and stochastic dynamics @xcite . experiments searching for frs have been performed in dynamical systems @xcite , but interpretations are very difficult because a quantitative comparison with theoretical prediction is impossible . other experiments have been performed in stochastic systems described by a first order langevin equation : a brownian particle in a moving optical trap @xcite and an out - of - equilibrium electrical circuit @xcite in which existing theoretical predictions @xcite were verified . interesting comments on the langevin equation can be found in @xcite . in the present article , we study a thermostated harmonic oscillator described by a second order langevin equation . we experimentally search frs for the work done by an external operator and for the heat dissipated by the system , and present analytical derivations of fts based on experimental observations . this paper is organized as follows . in section [ sec : system : description ] , we present the experimental system , write its energy balance to define the work given to the system together with the heat dissipated . we then introduce the fluctuation relations ( frs ) and the fluctuation theorems ( fts ) . in sections [ sec : tftexp ] , [ sec : ssftrampexp ] and [ sec : sinusexp ] , we present experimental results on the fluctuations of first the work and then the heat . a short discussion on experimental results in given in [ sec : expconc ] . then , in sections [ sec : worktheo ] and [ sec : heattheo ] , we present some analytical derivations of fts based on hypothesis inspired by experimental observations . we compare these analytical predictions to the experimental observations and finally conclude in section [ sec : conc ] . our system is a harmonic oscillator and we measure the non - equilibrium fluctuations of its position degree of freedom . the oscillator is damped due to the viscosity of a surrounding fluid that acts as a thermal bath at temperature @xmath0 . our oscillator , depicted in fig . [ fig : pendulum]a , is a torsion pendulum composed of a brass wire ( length @xmath1 @xmath2 , width @xmath3 @xmath2 , thickness @xmath4 @xmath5 ) and a glass mirror glued in the middle of this wire ( length @xmath6 @xmath2 , width @xmath7 @xmath2 , thickness @xmath8 @xmath2 ) . the elastic torsional stiffness of the wire is @xmath9 @xmath10 . it is enclosed in a cell filled by a water - glycerol mixture at @xmath11 concentration . the system is a harmonic oscillator with resonant frequency @xmath12 @xmath13 and a relaxation time @xmath14 @xmath15 . @xmath16 is the total moment of inertia of the displaced masses ( _ i.e. _ the mirror and the mass of displaced fluid ) @xcite . the damping has two contributions : the viscous damping @xmath17 of the surrounding fluid and the viscoelasticity of the brass wire which can be neglected here . the angular displacement of the pendulum @xmath18 is measured by a differential interferometer @xcite . the measurement noise is two orders of magnitude smaller than thermal fluctuations of the pendulum . @xmath19 is acquired with a resolution of @xmath20 bits at a sampling rate of @xmath21 @xmath13 , which is about 40 times @xmath22 . we drive the system out - of - equilibrium by forcing it with an external torque @xmath23 by means of a small electric current @xmath24 flowing in a coil glued behind the mirror ( fig . [ fig : pendulum]b ) . the coil is inside a static magnetic field . the displacements of the coil and therefore the angular displacements of the mirror are much smaller than the spatial scale of inhomogeneity of the magnetic field . so the torque is proportional to the injected current : @xmath25 ; the slope @xmath26 depends on the geometry of the system . the angular displacement @xmath18 of this harmonic oscillator is very well described by a second order langevin equation : @xmath27 where @xmath28 is the thermal noise , delta - correlated in time of variance @xmath8 and @xmath29 the boltzmann constant and @xmath0 the temperature of the system which is the one of the surrounding fluid . the fluctuation dissipation theorem ( fdt ) gives a relation between the amplitude of the thermal angular fluctuations of the oscillator at equilibrium and its response function . for a harmonic oscillator , the equilibrium thermal fluctuation power spectral density ( psd ) is : @xmath30 where @xmath31 . using fdt ( eq . [ eq : fdt ] ) , we measure the coefficient @xmath26 and test the calibration accuracy of the apparatus which is better than @xmath32 . more details on the set - up can be found in @xcite . when the system is driven out of equilibrium using a deterministic torque , it receives some work and a fraction of this energy is dissipated into the heat bath . multiplying eq . ( [ eq : langevin_oscillator ] ) by @xmath33 and integrating between @xmath34 and @xmath35 , we obtain a formulation of the first law of thermodynamics between the two states at time @xmath34 and @xmath35 ( eq . ( [ eq : energy_conservation ] ) ) . the change in internal energy @xmath36 of the oscillator over a time @xmath37 , starting at a time @xmath34 , is written as : @xmath38 where @xmath39 is the work done on the system over a time @xmath37 : @xmath40 and @xmath41 is the heat given to the system . equivalently , @xmath42 is the heat dissipated by the system . @xmath36 , @xmath39 and @xmath41 are defined as energy in @xmath43 units . the internal energy is the sum of the potential energy and the kinetic energy : @xmath44^{2 } + \frac{1}{2 } c \theta(t)^2 \right\}. \label{eq : udef}\ ] ] the heat transfer @xmath41 is deduced from equation ( [ eq : energy_conservation ] ) ; it has two contributions : @xmath45^{2}{\mathrm{d}}t ' + \frac{1}{k_b \ t } \int_{t_i}^{t_i+\tau } \eta(t ' ) \frac{{\mathrm{d}}\theta}{{\mathrm{d}}t}(t ' ) { \mathrm{d}}t ' . \label{eq : qdef}\end{aligned}\ ] ] the first term corresponds to the opposite of viscous dissipation and is always negative , whereas the second term can be interpreted as the work of the thermal noise which have a fluctuating sign . the second law of thermodynamics imposes @xmath46 to be positive . we rescale the work @xmath39 ( the heat @xmath41 ) by the average work @xmath47 ( the average heat @xmath48 ) and define : @xmath49 ( @xmath50 ) . the brackets are ensemble averages . in the present article , @xmath51 , respectively @xmath52 , stands for either @xmath53 or @xmath54 , respectively @xmath39 or @xmath41 . there are two classes of fts . the _ stationary state fluctuation theorem _ ( ssft ) considers a non - equilibrium steady state . the _ transient fluctuation theorem _ ( tft ) describes transient non - equilibrium states where @xmath37 measures the time since the system left the equilibrium state . a fluctuation relation ( fr ) examines the symmetry of the probability density function ( pdf ) @xmath55 of a quantity @xmath51 around @xmath56 ; @xmath51 is an average value over a time @xmath37 . it compares the probability to have a positive event ( @xmath57 ) versus the probability to have a negative event ( @xmath58 ) . we quantify the fr using a function @xmath59 ( symmetry function ) : @xmath60 the _ transient fluctuation theorem _ ( tft ) states that the symmetry function is linear with @xmath51 for any values of the time integration @xmath37 and the proportionality coefficient is equal to @xmath8 for any value of @xmath37 . @xmath61 contrary to tft , the _ stationary state fluctuation theorem _ ( ssft ) holds only in the limit of infinite time ( @xmath37 ) . @xmath62 the questions we ask are whether fluctuation relations for finite time satisfy the two theorems and what are the finite time corrections . in a first time , we test the correction to the proportionality between the symmetry function @xmath63 and @xmath51 . in the region where the symmetry function is linear with @xmath51 , we define the slope @xmath64 : @xmath65 . in a second time we measure finite time corrections to the value @xmath66 which is the asymptotic value expected by the two theorems . to the applied torque @xmath67 . ] for the transient fluctuation theorem , we choose the torque @xmath67 depicted in fig . [ fig : lineartorque]a ) . it is a linear function of time : @xmath68 with @xmath69 @xmath70 and @xmath71 @xmath72 . the value of @xmath73 is chosen such that the mean response of the oscillator is of order of the thermal noise , as can be seen in fig . [ fig : lineartorque]b ) where @xmath19 is plotted during the same time interval of fig . [ fig : lineartorque]a ) . the system is at equilibrium at @xmath74 ( @xmath75 pn.m and @xmath76 pn.m @xmath77 ) . in this section the starting time @xmath34 of integration of all quantities defined before ( @xmath39 , @xmath36 and @xmath41 ) is @xmath74 . so the work is : @xmath78 @xmath79 , @xmath80 @xmath81 and @xmath82 @xmath83 plotted as a function of @xmath37 . b ) pdfs of @xmath53 for various @xmath84 : 0.31 @xmath79 , 1.015 @xmath81 , 2.09 @xmath83 and 4.97 @xmath85 . continuous lines are theoretical predictions with no adjustable parameters . c ) corresponding functions @xmath86 . the straight continuous line is a line with slope @xmath8 . d ) pdfs of @xmath80 for two values of @xmath84 : 4.97 @xmath79 and 8,96 @xmath81 . e ) corresponding pdfs of @xmath54 . continuous lines are gaussian fits . f ) corresponding functions @xmath87 . the straight continuous line is a line with slope @xmath8 . ] in fig . [ fig : tft]a ) , we represent the time average ( @xmath88 ) of the power injected into the system , the internal energy difference @xmath89 and the time average ( @xmath88 ) of the power dissipated by the system . @xmath88 and @xmath89 are linear in @xmath37 after some short relaxation time @xmath90 defined in the langevin equation : for @xmath84 smaller than @xmath8 , some oscillations around the linear behavior can be seen . the average value of work @xmath47 is therefore quadratic in @xmath37 and is equal to @xmath91 @xmath43 for @xmath92 . the difference between @xmath47 and @xmath93 corresponds to the mean value of dissipated heat @xmath46 ( eq . ( [ eq : qdef ] ) ) . as can be seen in fig.[fig : tft]a ) , @xmath94 is larger than @xmath93 for all times @xmath37 . the average of the dissipated power ( @xmath95 ) is therefore positive for all times @xmath37 as expected from the second principle . for @xmath37 larger than several @xmath90 , the dissipated power is constant and equal to a few @xmath43 per second because @xmath96 and @xmath89 have the same slope after some @xmath90 . so we have the following behavior : the work done by the external work is used by the system to increase its internal energy but a small amount of energy is lost at a constant rate by viscous dissipation and exchange with thermostat . the probability density functions ( pdfs ) @xmath97 of @xmath53 is plotted in fig . [ fig : tft]b ) for different values of @xmath84 . four typical value of @xmath37 are presented : the first is smaller than the relaxation time and the last equals five relaxation times ; the results are the same for any value of @xmath37 . the pdfs of @xmath53 are gaussian for any @xmath37 . we observe that @xmath53 takes negative values as long as @xmath37 is not too large . the probability of having negative values of @xmath53 decreases when @xmath37 is increased . from the pdfs , we compute the symmetry functions . they are plotted in fig . [ fig : tft]c ) as a function of @xmath53 . they all collapse on the same linear function of @xmath53 for any @xmath37 , which implies they all have the same slope @xmath98 . the straight line in fig [ fig : tft]c ) has slope @xmath8 . within experimental error bars , @xmath98 is equal to @xmath8 for all time @xmath37 . therefore work fluctuations for a harmonic oscillator under a linear forcing satisfy the tft . we checked that this property is true for other values of @xmath73 and @xmath99 . the pdfs of @xmath80 are plotted in fig . [ fig : tft]d ) for two values of @xmath84 : they are not symmetric and have exponential tails . the pdfs of @xmath54 can be seen in fig . [ fig : tft]e ) for the same values of @xmath84 . they are qualitatively different from the ones of the work . we have plotted in the same figure the gaussian fit of the two pdfs of the dissipated heat . it is clear that the pdfs of @xmath54 are not gaussian . extreme events of @xmath54 are distributed on exponential tails . these tails can be interpreted noticing that @xmath100 and @xmath36 have exponential tails . the variance of the pdfs of @xmath54 is also much larger than the variance of the pdfs of @xmath53 . we plot on fig . [ fig : tft]f ) symmetry functions @xmath101 for the same times @xmath84 . only the behavior of large events can be analyzed here because the variance is much larger than the mean @xmath102 . as it can be seen in fig . [ fig : tft]f , @xmath101 is not proportional to @xmath54 , therefore tfr is not satisfied for finite time . within experimental resolution , @xmath101 is constant for extreme events and equal to @xmath6 . this behavior can be interpreted by writing for large @xmath54 , @xmath103 where @xmath104 and @xmath105 are the decrease rate on the exponential tails . each coefficient depends on @xmath37 . there is a simple expression of @xmath101 for large fluctuations : @xmath106 in fig . [ fig : tft]c ) , the pdfs of @xmath54 are symmetric around the mean value for the two values of @xmath37 . it is not the case for small @xmath84 . thus we can conclude that @xmath107 and that the symmetry function is so equal to the constant : @xmath108 . as it can be seen in fig . [ fig : tft]e ) , the pdfs become more and more gaussian when @xmath37 tends to infinity . it is expected that for infinite time , the pdf of @xmath54 is a gaussian . thus , tft appears to be satisfied experimentally in the limit of infinite @xmath37 . our interesting finding is that , for @xmath41 tft if not valid for any times . we call a steady state a state in which both forcing and response to the forcing do not depend on the initial time @xmath34 , but only on @xmath37 . this implies that @xmath109 i ndependant of @xmath34 ; and so is @xmath110 . if the torque drifts along time , the mean of @xmath111 is linear with @xmath35 . thus we have to change the definition of the work done on the system to be in a steady state . this is equivalent to a galilean change of reference frame . the work is now defined as : @xmath112 \frac{\textrm d \theta}{\textrm dt}(t ' ) dt ' . \label{eq : work_ssft_def}\ ] ] with this definition , the forcing is @xmath113 and the response to the forcing @xmath114 . when we impose a forcing linear in time ( @xmath115 ) , the first condition ( @xmath116 independent of @xmath34 ) is satisfied . the second ( @xmath110 independent of @xmath34 ) is also satisfied if @xmath117 , _ i.e. _ after a transient state . thus the system is on a steady state . we remark that , in the transient state , this definition of the work reduces to the usual one , because @xmath118 @xmath70 . the average of @xmath39 is quadratic in @xmath37 for any value of @xmath84 . there are no oscillations in time for small @xmath84 . the pdfs of @xmath53 are gaussian for any value of @xmath84 ( fig . [ fig : ssftramp]a ) . probability of negative values is high and decreases with @xmath37 , like in the transient case . the symmetry functions @xmath119 are again proportional to @xmath53 ( fig . [ fig : ssftramp]b ) but the slope @xmath120 is not equal to @xmath8 for smaller @xmath37 and tends to @xmath8 for @xmath121 only , as can be seen in fig . [ fig : ssftramp]c . thus we obtain a fluctuation relation for the work done on the system in this steady state and this relation satisfy the ssft . the slope at finite time is slightly oscillating at a frequency , close to @xmath122 . for various @xmath84 : 0.019 @xmath79 , 0.31 @xmath81 , 2.09 @xmath83 and 4.97 @xmath85 . b ) corresponding functions @xmath123 . c ) the slope @xmath98 of @xmath123 is plotted versus @xmath37 ( @xmath124 : experimental values ; continuous line : theoretical prediction eq.([eq : epsilon_linear ] ) with no adjustable parameters ) . ] for various @xmath84 : 0.019 @xmath79 , 0.31 @xmath81 , 2.09 @xmath83 and 4.97 @xmath85 . b ) corresponding functions @xmath123 . c ) the slope @xmath98 of @xmath123 is plotted versus @xmath37 ( @xmath124 : experimental values ; continuous line : theoretical prediction eq.([eq : epsilon_linear ] ) with no adjustable parameters ) . ] the heat dissipated during this linear forcing has a behavior very similar to the one observed in the transient case ( section [ sec : heattft ] ) . we can so transpose here all what we said in section [ sec : heattft ] . we now consider a periodic forcing @xmath125 . this is a very common kind of forcing but it has never been studied in this context . using fourier transform , any periodical forcing can be decomposed in a sum of sinusoidal forcing . we explain here the behavior of a single mode . we choose @xmath126 @xmath70 and @xmath127 @xmath13 . this torque is plotted in fig . [ fig : sinustorque]a . the mean of the response to this torque is sinusoidal , with the same frequency , as can be seen in fig . [ fig : sinustorque]b . we studied other frequencies @xmath128 . the system is clearly in a steady state . we choose the integration time @xmath37 to be a multiple of the period of the driving ( @xmath129 with @xmath130 integer ) . the starting phase @xmath131 is averaged over all possible @xmath34 in order to increase statistics ; in the remaining of this section , we drop the brackets @xmath132 . . ] integrated over @xmath130 periods of forcing , with @xmath133 ( @xmath134 ) , @xmath135 ( @xmath124 ) , @xmath136 ( @xmath137 ) and @xmath138 ( @xmath139 ) . b ) the function @xmath140 measured at @xmath141 @xmath13 is plotted as a function of @xmath142 for several @xmath130 : @xmath143 ; @xmath144 @xmath145 ; @xmath146 . for these two plots , continuous lines are theoretical predictions with no adjustable paramaters ( eq . ( [ eq : ssft_sinus_mean_pdf_work ] ) and eq . ( [ eq : ssft_sinus_var_pdf_work ] ) ) . c ) the slopes @xmath147 , plotted as a function of @xmath130 for two different driving frequencies @xmath128 = 64 hz ( @xmath124 ) and 256 hz ( @xmath134 ) ; continuous lines are theoretical predictions from eq . ( [ eq : epsilon_sinus ] ) with no adjustable parameters . ] integrated over @xmath130 periods of forcing , with @xmath133 ( @xmath134 ) , @xmath135 ( @xmath124 ) , @xmath136 ( @xmath137 ) and @xmath138 ( @xmath139 ) . b ) the function @xmath140 measured at @xmath141 @xmath13 is plotted as a function of @xmath142 for several @xmath130 : @xmath143 ; @xmath144 @xmath145 ; @xmath146 . for these two plots , continuous lines are theoretical predictions with no adjustable paramaters ( eq . ( [ eq : ssft_sinus_mean_pdf_work ] ) and eq . ( [ eq : ssft_sinus_var_pdf_work ] ) ) . c ) the slopes @xmath147 , plotted as a function of @xmath130 for two different driving frequencies @xmath128 = 64 hz ( @xmath124 ) and 256 hz ( @xmath134 ) ; continuous lines are theoretical predictions from eq . ( [ eq : epsilon_sinus ] ) with no adjustable parameters . ] the work is written as a function of @xmath130 , the number of periods of the forcing : @xmath148 the pdfs of @xmath142 are plotted in fig . [ fig : sinuswork]a . work fluctuations are gaussian for all values of @xmath130 as in previous cases . thus symmetry functions are again linear in @xmath142 ( fig . [ fig : sinuswork]b ) . the slope @xmath147 is not equal to @xmath8 for all @xmath130 but there is a correction at finite time ( fig . [ fig : sinuswork]c ) . nevertheless , @xmath147 tends to @xmath8 for large @xmath130 , so ssft is satisfied . the convergence is very slow and we have to wait a large number of periods of forcing for the slope to be @xmath8 ( after @xmath149 periods , the slope is still @xmath150 ) . this behavior is independent of the amplitude of the forcing @xmath73 and consequently of the mean value of the work @xmath151 . the system satisfies the ssft for all forcing frequencies @xmath128 but finite time corrections depends on @xmath128 , as can be seen in fig . [ fig : sinuswork]c . ( @xmath134 ) and @xmath152 ( @xmath124 ) . in the next plots , the integration time @xmath37 is a multiple of the period of forcing , @xmath153 , with @xmath133 ( @xmath134 ) , @xmath135 ( @xmath124 ) , @xmath136 ( @xmath137 ) and @xmath138 ( @xmath139 ) . continuous lines are theoretical predictions with no adjustable parameters . b ) pdfs of @xmath36 . c ) pdfs of @xmath54 . d ) symmetry functions @xmath101 . e ) the slope @xmath154 of @xmath101 for @xmath155 , plotted as a function of @xmath130 ( @xmath134 ) . the slope @xmath147 of @xmath119 plotted as a function of @xmath130(@xmath124 ) . continuous line is theoretical prediction . ] ( @xmath134 ) and @xmath152 ( @xmath124 ) . in the next plots , the integration time @xmath37 is a multiple of the period of forcing , @xmath153 , with @xmath133 ( @xmath134 ) , @xmath135 ( @xmath124 ) , @xmath136 ( @xmath137 ) and @xmath138 ( @xmath139 ) . continuous lines are theoretical predictions with no adjustable parameters . b ) pdfs of @xmath36 . c ) pdfs of @xmath54 . d ) symmetry functions @xmath101 . e ) the slope @xmath154 of @xmath101 for @xmath155 , plotted as a function of @xmath130 ( @xmath134 ) . the slope @xmath147 of @xmath119 plotted as a function of @xmath130(@xmath124 ) . continuous line is theoretical prediction . ] we first do some comments on the average values . the average of @xmath36 is obviously vanishing because the time @xmath37 is a multiple of the period of the forcing . @xmath156 and @xmath157 have consequently the same behavior and they are linear in @xmath37 , as can be seen in fig . [ fig : sinusheat]a ) but the pdfs of heat fluctuations @xmath158 have exponential tails ( fig . [ fig : sinusheat]c ) . this can be understood noticing that , from eq . ( [ eq : qdef ] ) , @xmath159 and that @xmath36 has an exponential pdf independent of @xmath130 ( fig . [ fig : sinusheat]b ) . therefore , in a first approximation , the pdf of @xmath54 is a convolution between an exponential distribution ( pdf of @xmath36 ) and a gaussian distribution ( pdf of @xmath53 ) . symmetry functions @xmath160 are plotted in fig . [ fig : sinusheat]d ) for different values of @xmath130 ; three different regions appear : \(i ) for large fluctuations @xmath158 , @xmath160 equals @xmath6 . when @xmath37 tends to infinity , this region spans from @xmath161 to infinity . \(ii ) for small fluctuations @xmath158 , @xmath160 is a linear function of @xmath158 . we then define @xmath154 as the slope of the function @xmath160 , _ i.e. _ @xmath162 . this slope is plotted in fig . [ fig : sinusheat]e ) where we see that it tends to @xmath8 when @xmath37 is increased . so , ssft holds in this region ii which spans from @xmath163 up to @xmath164 for large @xmath37 . \(iii ) a smooth connection between the two behaviors . we observe that @xmath147 matches experimentally @xmath154 , for all values of @xmath130 ( fig . [ fig : sinusheat]e ) . so the finite time corrections to the ft for the heat are the same than the ones of ft for work : @xmath165 . these regions define the fluctuation relation from the heat dissipated by the oscillator . the limit for large @xmath37 of the symmetry function @xmath101 is rather delicate and we will discuss it in section [ sec : heat_sinus_theo ] . in the previous sections , we have presented experimental results on a harmonic oscillator driven out of equilibrium by an external deterministic forcing @xmath23 . we operated with two different time - prescriptions : one in which @xmath23 is a linear function of time , and one in which @xmath23 is a sinusoidal function of time . the energy injected into the system is the work @xmath166 of the torque @xmath23 . the pdfs of the work @xmath166 are gaussian whatever the time prescription of @xmath23 is , and work fluctuations satisfy a tft ( @xmath23 linear in time ) and a ssft ( @xmath23 linear or sinusoidal in time ) . the energy dissipated by the system is represented by the heat @xmath167 , and we measured it using the first principle of thermodynamics ( eq . [ eq : qdef ] ) . heat probability distributions are not gaussian and are very different from the ones of the work . they nevertheless satisfy a ssft in both the case of a sinusoidal forcing and a linear forcing . but they do not satisfy a tft in the case of a linear forcing , because the symmetry functions are not linear for all values of dissipated heat @xmath54 . in the next two sections , we use some experimental evidences to derive analytical expressions of the pdfs of work and the heat exchanged on an arbitrary time interval @xmath37 . we then derive fts together with their finite time corrections . in this section , we derive the analytical expression of the pdf of the work given to the system , and defined as the work of the torque applied to the pendulum , which is either linear or sinusoidal in time . experimentally , we observed that the pdfs are always gaussian , so we restrict our task to deriving expressions for the first two moments of the work distribution . to do so , we use experimental observations on the fluctuations of the angle @xmath18 , as described in section [ sec : exp_observations ] below . we then compute in section [ sec : pdf_w ] the mean and the variance of the work @xmath168 in the different experimental situations , and then write formally the corresponding fluctuations relations , from which we obtain analytical expressions of the finite time corrections to the fluctuation theorems . we discuss here the angular fluctuations . we decompose the angle @xmath18 into a mean value @xmath169 and a fluctuating part @xmath170 , writing @xmath171 . the mean value corresponds to an ensemble average . it is obtained experimentally by averaging over realisations of the forcing , and it is presented in fig . [ fig : lineartorque ] and [ fig : sinustorque ] . a first experimental observation is as follows . the measured mean response @xmath169 is exactly equal to the solution of the deterministic second order equation obtained when removing the noise term ( @xmath172 ) in the langevin equation ( [ eq : langevin_oscillator ] ) . we checked this from our data , and found this way a value of the calibration @xmath26 ( see section [ sec : pendulum ] ) in perfect agreement with the one obtained from the application of the fluctuation dissipation theorem . a second experimental observation concerns the probability distribution of @xmath170 in out - of - equilibrium conditions . we know and observed that at equilibrium , @xmath170 has a gaussian distribution with variance @xmath173 , and the associated momentum @xmath33 has fluctuations @xmath174 which also have a gaussian distribution , with a variance @xmath175 . we observe that the statistical properties of angular fluctuations @xmath170 when a torque @xmath67 linear in time is applied are the same as the statistical properties at equilibrium , when no torque is applied . in figure [ fig : fluct_ang_comparison]a , we plot the pdf of @xmath170 measured at @xmath176 together with the gaussian fit of the pdf at equilibrium ( continuous line ) . the two curves matches perfectly within experimental accuracy . thus we conclude that the external driving does not perturb the equilibrium distribution of angular fluctuations , so we use : @xmath177 when the torque is applied ( @xmath134 ) , compared with a gaussian fit of the pdf at equilibrium ( continuous line ) . b ) the measured spectrum of @xmath170 ( @xmath134 ) is compared with the prediction of fluctuation dissipation theorem in equilibrium ( continuous line ) . ] the third experimental observation concerns time correlations . in figure [ fig : fluct_ang_comparison]b , we plot the power spectral density function of @xmath170 when applying an external forcing ( @xmath134 ) . we compare it to the prediction of fluctuation dissipation theorem at equilibrium ( eq . [ eq : fdt ] ) computed using the oscillator parameters . the two spectra are identical , so we can confidently use for our system a description in terms of a second order langevin dynamic where the noise term is not perturbed by the presence of the driving . from the power spectral density function of @xmath18 ( eq . [ eq : fdt ] ) , we derive the autocorrelation function @xmath178 of @xmath170 during a time interval @xmath37 . it is the same at equilibrium and out of equilibrium , and decreases exponentially : @xmath179 where @xmath180 and @xmath181 is defined by @xmath182 and @xmath183 . thus we observe experimentally that when we drive the system out of equilibrum , the angular fluctuations @xmath170 are identical ( with respect to the expressions above ) to those at equilibrium . we verify the same properties for the sinusoidal time prescription of the torque , and use equilibrium expression for the correlation function in the next sections . in figs . [ fig : tft ] , [ fig : ssftramp ] and [ fig : sinuswork ] , we see that the pdfs of the work are gaussian for any integration time @xmath37 and whatever the forcing is . so these distributions are fully characterized by their mean value @xmath47 and their variance @xmath184 . the external torque @xmath23 is determistic , so the mean value of the work done on the system can be written as : @xmath185 we have defined @xmath186 . the value @xmath187 depends on the time - prescription of the torque we apply to the oscillator . choosing @xmath188 describes the linear ramp and @xmath189 corresponds to the sinusoidal forcing . the variance of the pdfs is : @xmath190 this expression involves the autocorrelation function of the angular speed @xmath174 , ( @xmath191 ) . using the expression of the autocorrelation function of angular fluctuations ( @xmath192 , eq . ( [ eq : autocorrelation ] ) ) , we can calculate exactly the expression of @xmath193 : @xmath194 we have calculated the mean value and the variance of the pdfs in the three situations of interest : stationary and transient cases with a forcing linear in time , and stationary case with a forcing sinusoidal in time . details and results can be found in the appendix . in all of the cases , we compare the theoretical pdfs and the symmetry functions with the experimental results . we have plotted in fig . [ fig : tft ] , [ fig : ssftramp ] and [ fig : sinuswork ] our theoretical pdfs and the corresponding symmetry functions with no adjustable parameters . within experimental error bars , our analytical and experimental results are in excellent agreement . @xmath119 is linear in @xmath53 because the pdfs of the work are gaussian . we now want to calculate analytically the corrections to the slope @xmath98 for finite time @xmath37 . for a gaussian distribution , the symmetry function is : @xmath195 the expression of the slope @xmath98 uses only the mean value and the variance of the gaussian distribution . we define @xmath196 , where the correction @xmath197 is a decreasing function of @xmath37 . we obtain @xmath198 for the transient case , which is in agreement with a tft . for the two steady states , there are corrections to the value @xmath8 ; we find : a. linear forcing @xmath199 . \label{eq : epsilon_linear}\ ] ] b. sinusoidal forcing @xmath200 exact values of the coefficients @xmath201 are given in the appendix . these two expressions are in perfect agreement with experimental results as can be seen in fig . [ fig : ssftramp ] and fig . [ fig : sinuswork ] . these corrections depend on the kind of forcing and it is difficult to predict their form for an arbitrary forcing . nevertheless , the two situations we consider are useful as building blocks of such an arbitrary forcing , and they provide a very nice test of our method . we now determine an analytical expression of the pdf of the dissipated heat . to do so , we make the same hypothesis as in the case of the work ( see section [ sec : exp_observations ] above ) , and we complete them by additional assumptions to simplify our derivations . we are interested by pdfs of the heat for integration time @xmath37 large compared to @xmath90 , so that exponential corrections which are scaling like @xmath202 can be neglected . in the case of sinusoidal forcing , this is correct after @xmath203 or @xmath204 periods of forcing ( @xmath205 ) . within this assumption , @xmath206 and @xmath207 are independent , and so are @xmath208 and @xmath209 . additionally , as the equation of motion of the oscillator is second order in time , @xmath18 and @xmath210 are independent at any given time @xmath211 . we use the technique proposed in @xcite . to obtain the pdf @xmath212 of the heat , we define its fourier transform , the characteristic function , as @xmath213 we then write @xmath214 using eq . ( [ eq : qdef ] ) as : @xmath215 where @xmath216 is the joint distribution of the work @xmath168 , @xmath18 and @xmath217 at the beginning and at the end of the time interval @xmath37 . this distribution is expected to be gaussian because @xmath168 is linear in @xmath33 and additionally @xmath18 , @xmath33 and @xmath168 are gaussian . the details of the calculation are given in the appendix . the fourier transform of the pdf of dissipated heat can be exactly calculated : @xmath218 \right.\right . \nonumber \\ & + \left.\left . \frac{-16\cos(\varphi)^2 + 4 + 4 i s(4\cos(\varphi)^2 + 1)}{1+s^2}\right)\right\}. \nonumber \label{eq : fourierpdfheat_linear}\end{aligned}\ ] ] as far as we know , there is no analytic expression for inverse fourier transform of this function , so for the pdf of dissipated heat . however we can do some comments . this expression is very similar to the one found in the case of a brownian particle @xcite . the factor @xmath219 is the fourier transform of an exponential pdf and this is directly connected to the exponential tails of the pdf . moreover the pdf is not symmetric around its mean , because there is a non vanishing third moment . in this expression , only two terms depend on @xmath37 . for large @xmath37 , this expression reduces to : @xmath220 this expression will turn out to be similar to the one obtain with a sinusoidal forcing , as we will comment in the next section . both expressions depend on the non - dimensional factor @xmath221 defined as : @xmath222 all moments of the distribution of @xmath223 are linear with @xmath224 and @xmath225 . so @xmath224 compares the mean value of the angular speed to the root mean square of the angular speed fluctuations . this coefficient @xmath224 increases when the system is driven further from equilibrium . we consider it as a measure of the distance to equilibrium . in our system @xmath221 is positive , but smaller than @xmath8 , so we are out - of - equilibrium but not very far from it ( @xmath226 ) . just like in the experiments , we choose the integration time @xmath37 to be a multiple of the period of the forcing , so @xmath227 and therefore @xmath228 . within this framework , we find that the pdf of @xmath229 is exponential : @xmath230 it is independant of @xmath37 because @xmath229 depends only on @xmath18 and @xmath217 at times @xmath34 and @xmath35 which are uncorrelated . this expression is in perfect agreement with the experimental pdfs for all times ( see fig . [ fig : sinusheat]b ) . some algebra then yields for the characteristic function of @xmath167 : @xmath231 the characteristic function of heat fluctuations is therefore the product of the characteristic function of an exponential distribution ( @xmath232 ) with the one of a gaussian distribution ( @xmath233 ) . thus the pdf of heat fluctuations is nothing but the convolution of a gaussian and an exponential pdf , just as if @xmath168 and @xmath229 were independent . the inverse fourier transform can be computed exactly : @xmath234 \ , , \label{eq : pdfq}\end{aligned}\ ] ] where @xmath235 stands for the complementary erf function . in fig . [ fig : sinusheat]c , we have plotted the analytical pdf from eq . ( [ eq : pdfq ] ) together with the experimental ones , using values of @xmath236 and @xmath237 from the experiment and no adjustable parameters . the agreement is perfect for all values of @xmath130 , _ i.e. _ for any time @xmath37 . from eq . ( [ eq : pdfq ] ) , we isolate three different regions for @xmath87 : \(i ) if @xmath238 , then @xmath239 . this domain of @xmath240 corresponds to fluctuations larger then three times the average value . the pdf has exponential tails , corresponding to an exponential distribution with a non - vanishing mean . \(ii ) if @xmath241 , then @xmath242 with @xmath243 . in this domain , values of the heat are small and heat fluctuations behave like work fluctuations . the slope @xmath244 is the same as the one found for work fluctuations . the exact correction to the asymptotic value 1 is plotted in fig . [ fig : sinusheat]e and again it describes perfectly the experimental behavior . \(iii ) for @xmath245 , there is an intermediate region connecting domains ( i ) and ( ii ) by a second order polynomial : @xmath246 . these three domains offer a perfect description of the three regions observed experimentally ( fig . [ fig : sinusheat]d ) . now , we examine the limit of infinite @xmath37 in which ssft is supposed to hold . to do so , we distinguish two variables : the heat @xmath223 or the normalized heat @xmath247 . their asymptotic behavior are different because the average heat @xmath237 depends on @xmath37 , more precisely it is linear in @xmath37 . we discuss first @xmath223 . the asymptotic shape of the pdf of @xmath223 ( eq . ( [ eq : pdfq ] ) ) for large @xmath37 is a gaussian whose variance is @xmath236 , the variance of the pdf of @xmath168 . thus , the pdf of @xmath223 coincides with the pdf of @xmath168 for @xmath37 strictly infinite . as we have already shown , work fluctuations satisfy the conventional ssft ; therefore heat fluctuations also satisfy the conventional ssft ( eq . ( [ eq : ssft ] ) ) . we have found three different regions separated by two limit values : the mean and three times the mean . but in the limit of large times @xmath37 , the pdf shrinks and only region ( ii ) is relevant . region ( ii ) corresponds to small fluctuations and it is bounded from above by @xmath248 with the average @xmath237 being linear in @xmath37 . so all the behavior of the fluctuations of @xmath223 for large @xmath37 lays in region ( ii ) where the symetry function is linear and ssft holds . we turn now to the normalized heat @xmath247 . as the average value of @xmath223 is linear in @xmath37 , rescaling by @xmath237 is equivalent to a division by @xmath37 ; the mean of @xmath247 is then @xmath8 . this normalization makes the two limit values constant . the boundary between regions ( ii ) and ( iii ) is @xmath249 and the boundary between ( iii ) and ( i ) is @xmath250 . the function @xmath87 is not linear for large values of @xmath251 but it is linear only in region ( ii ) , for @xmath252 , _ i.e. _ for small fluctuations . so ssft is satisfied only for small fluctuations but not for all values of @xmath247 , and we obtain for @xmath247 a fluctuation relation which prescribes a symetry function that is non - linear in @xmath247 . these two different pictures , in terms of @xmath223 or @xmath247 , results from taking two non - commutative limits differently . the first description using @xmath223 implies that the limit @xmath37 infinite is taken before the limit of large @xmath223 . the second description does the opposite . however , the probability to have large fluctuations decreases with @xmath37 and experimentally , for large @xmath37 , only the region ( ii ) can be seen , and it is the region in which where ssft holds . as we have done in the case of the linear forcing , we introduce a non - dimensional factor @xmath221 such as : @xmath253 the moments of the distribution of @xmath41 are linear with @xmath224 and , like the linear torque , @xmath221 is equal to the amplitude of @xmath33 divided by @xmath254 . we consider it also as a measure of the distance to equilibrium . in our system @xmath221 is positive , but smaller than @xmath8 , so we are out - of - equilibrium but not very far it : here @xmath255 . in this paper , we have studied the fluctuations of energy input and energy dissipation in a harmonic oscillator driven out of equilibrium . this oscillator is very well described by a second order langevin equation . we have performed experiments using a torsion pendulum driven out of equilibrium following a stationary protocol in which either the torque increases linearly in time , or oscillates at a given frequency . we have also studied transient evolutions from the equilibrium state . we have defined the work given to the system as the work of the torque applied during a time @xmath37 . accordingly we have defined the heat dissipated by the pendulum during this time @xmath37 , by writing the first principle of thermodynamics between the two states separated by time @xmath37 . fluctuations relations are obtained experimentally for both the work and the heat , for the stationary and transient evolutions . we have experimentally observed that angle fluctuations of the brownian pendulum have the same statistical and dynamical properties at equilibrium and for any non - equilibrium driving . from this observation , we have derived expressions for the probability density functions of the work and the heat . in our system , fluctuations of the angle are gaussian , and so are fluctuations of the work @xmath53 . so the symmetry functions @xmath119 of the work are linear , and we have calculated exactly the time - correction to the proportionality coefficient between @xmath119 and @xmath53 . these corrections match perfectly the experimental results , both in the case of a forcing linear in time , and sinusoidal in time . we have also computed the analytic expression of the fourier transform of the pdfs of the dissipated heat . for the sinusoidal forcing , we have obtained for the first time an analytic expression of the pdf of the heat . this expression is in excellent agreement with the experimental measurements . for a torque linear in time , the pdf of the heat has no simple expression but its fourier transform gives insight on the behavior of the symmetry function of the heat . it is very similar to the one obtained in the case of a first order langevin dynamics @xcite . we emphasize here that our analytical derivations are strongly connected to experimental observations on the properties of the noise ; and are therefore different from any previous theoretical approach . we have introduced a dimensionless variable @xmath221 which we think is a measure of the distance from equilibrium : the average dissipation rate is proportional to @xmath221 , and it increases when the system is further from equilibrium . @xmath221 is also proportional to the strength of the driving and in the fluctuation relations , it gives a proper unit to measure the amplitude of fluctuations . so @xmath221 plays the same role as the dissipation coefficient ( the viscosity in our case ) in the fluctuation dissipation theorem at equilibrium . we have an expression of @xmath221 for the two different time - prescriptions we have used . these expressions can be generalized : @xmath256 the numerator corresponds to the solution of the langevin equation when removing the thermal noise term ( @xmath257 ) . the denominator corresponds to the variance of thermal fluctuations of angular speed @xmath258 . we thank g. gallavotti for useful discussions . this work has been partially supported by anr-05-blan-0105 - 01 . in this section , we will calculate the mean and the variance of the work given to the system in the cases : 1 . transient state , linear forcing 2 . steady state , linear forcing 3 . steady state , sinusoidal forcing [ tftramp]the torque is : @xmath259 . the mean value of the angular displacement is the solution of eq . [ eq : langevin_oscillator ] : @xmath260 for the work done on the system , the pdfs are gaussian for all integration time @xmath37 . the mean of the pdf of @xmath168 for a given @xmath37 is : @xmath261 \label{eq : meanworkramp}\end{aligned}\ ] ] and its variance is : @xmath262 , \label{eq : varianceworkramp}\end{aligned}\ ] ] @xmath263 [ ssftramp]the torque is : @xmath259 . the mean value of the angular displacement is the solution of eq . [ eq : langevin_oscillator ] after some @xmath90 . thus the exponential term is vanished : @xmath264 for the work done on the system , the pdfs are gaussian for all integration time @xmath37 . the mean of the pdf is : @xmath265 and the variance is : @xmath266 . \label{eq : varianceworkrampstat}\end{aligned}\ ] ] from this , we deduce : @xmath267 where @xmath268 [ ssftsinus]the torque is @xmath269 . the mean value of the angular displacement is : @xmath270 where @xmath271 for the work done on the system , the pdfs are gaussian for all integration time @xmath37 . the mean of the pdf is : @xmath272 and the variance is : @xmath273 where @xmath274 @xmath275\,.\label{eq : ssft_sinus_var_pdf_work}\end{aligned}\ ] ] [ appendixheat ] in this section , we will calculate the fourier transform of the pdf of the dissipated heat in two cases : 1 . linear forcing 2 . sinusoidal forcing we introduce non - dimensional parameters in order to simplify calculations : @xmath276 the mean value and the variance of @xmath277 and @xmath278 can be simply expressed : @xmath279 where @xmath221 is a non - dimensional value . integrating by part , the work @xmath168 can be rewritten : @xmath280-\frac{(d\omega_0)^2}{2}\left[(t_i+\tau)^2-t_i^2\right]+w^{*},\nonumber\\ w^ { * } & = & -(d\omega_0)\int_{t_i}^{t_i+\tau}\tilde{x}(t'){\mathrm{d}}t'.\end{aligned}\ ] ] with these definitions , we obtain : @xmath281 and @xmath282 . like the distribution of @xmath168 , the distribution of @xmath283 is gaussian for all values of @xmath37 and we find : @xmath284 for notational convenience , we introduce a five dimensional vector : @xmath285 . @xmath216 is gaussian and is so fully characterized by the covariance matrix @xmath286 defined as : @xmath287 where @xmath288 denotes the complex conjugate of @xmath289 . so the distribution @xmath216 is written : @xmath290 where @xmath291 denotes the transpose of @xmath289 . we suppose that the integration time is larger than the relaxation time . within this assumption , @xmath206 and @xmath207 are independent , and so are @xmath292 and @xmath293 . as the equation of motion of the oscillator is second order in time , @xmath18 and @xmath33 are independent at any given times @xmath211 . with these hypotheses , we get : @xmath294 the other coefficients of the covariance matrix are : @xmath295 the fourier transform of the pdf of the heat will be calculated here . we define two quantities : @xmath296 one can write @xmath223 as : @xmath297 . the fourier transform can be so written : @xmath298 we use a new variable defined as : @xmath299 with this definition , the argument in the exponential @xmath23 can be rewritten : @xmath300.\end{aligned}\ ] ] changing the integration variable to @xmath301 yields : @xmath302 to make eq . [ eq : gen_heat_lin_exp ] into an explicit expression for @xmath303 , the inverse of matrix @xmath304 is required in the expression for @xmath305 and its determinant . these are obtained as follows . we find : @xmath306 the determinant of the matrix is @xmath307 . for the inverse of this matrix , we get : @xmath308 we now have the material needed to calculate @xmath305 . we find : @xmath309.\end{aligned}\ ] ] the analytic expression of the fourier transform of the pdf of the heat dissipated during a linear forcing is : @xmath310 \right.\right . \nonumber \\ & + \left.\left . \frac{-16\cos(\varphi)^2 + 4 + 4 i \,s(4\cos(\varphi)^2 + 1)}{1+s^2}\right)\right\}.\end{aligned}\ ] ] [ ap : sinus_heat ] we will determine in a first time the gaussian joint distribution @xmath216 of @xmath168 , @xmath207 , @xmath206 , @xmath293 and @xmath292 . for notational convenience , we introduce a five dimensional vector : @xmath311 . the pdf @xmath216 is fully characterized by the covariance matrix @xmath286 . @xmath312 where @xmath313 denotes the complex conjugate of @xmath289 . we suppose that the integration time is larger than the relaxation time . within this assumption , @xmath206 and @xmath207 are independent , and so are @xmath292 and @xmath293 . as the equation of motion of the oscillator is second order in time , @xmath18 and @xmath33 are independent at any given times @xmath211 . with these hypotheses , we get : @xmath314 the covariance matrix is so a diagonal matrix : @xmath315 @xmath36 is a function of the positions and velocities at the beginning ( @xmath34 ) and at the end ( @xmath35 ) . thus , @xmath36 and @xmath39 can be considered as independent . the pdf of @xmath41 is so the convolution between the pdf of @xmath39 which is gaussian and the pdf of @xmath36 : @xmath316 we first calculate exactly the pdf of the variation of internal energy . we have shown that @xmath207 , @xmath206 , @xmath293 and @xmath292 are independent . the fourier transform of the pdf is : @xmath317 where @xmath318 and @xmath319 . the distribution of @xmath18 is gaussian with variance @xmath320 . the distribution of @xmath321 and the distribution of @xmath322 are the same : @xmath323 the fourier transform of this distribution is : @xmath324 . this distribution is the same for @xmath321 and @xmath322 at @xmath34 and @xmath35 . thus the fourier transform of the variation of internal energy is @xmath325 and the probability is : @xmath326 as @xmath36 and @xmath39 are independent , the fourier transform of the dissipated heat can be calculated : @xmath327 this expression can be inversed because it is simply the convolution between a gaussian distribution and an exponential distribution . so we find eq . ( [ eq : pdfq ] ) . 0 d. j. evans and d. j. searles , _ phys . e _ * 50 * , 1645 ( 1994 ) . d. j. evans , d. j. searles , _ advances in physics _ * 51 * ( 7 ) 1529 ( 2002 ) . d. j. evans , d. j. searles and l. rondoni , _ phys . e _ * 71 * ( 5 ) 056120 ( 2005 ) . r. van zon , s. ciliberto , e. g. d. cohen , _ phys . lett . _ * 92 * ( 13 ) 130601 ( 2004 ) . r. van zon and e. g. d. cohen , _ phys . * 91 * ( 11 ) 110601 ( 2003 ) ; _ phys . e _ * 67 * 046102 ( 2003 ) ; _ phys . * 69 * 056121 ( 2004 ) . g. m. wang , e. m. sevick , e. mittag , d. j. searles , d. j. evans , _ phys . rev . lett . _ * 89 * 050601 ( 2002 ) ; g. m. wang , j. c. reid , d. m. carberry , d. r. m. williams , e. m. sevick , d. j. evans , _ phys . e _ * 71 * 046142 ( 2005 ) .

the formal expressions of equilibrium statistical mechanics strictly apply only to ergodic systems that are in thermodynamic equilibrium . thus these expressions only strictly apply to systems which are at the global free energy minimum given the system hamiltonian and the macroscopic thermodynamic state variables ( number of particles , temperature and pressure or density ) . for such systems gibbsian equilibrium statistical mechanics provides an exact prescription for how to calculate the various thermodynamic quantities @xcite . however , these prescriptions are routinely applied to systems that are not in true thermodynamic equilibrium ( for example to metastable liquids @xcite , glasses @xcite , polymorphs @xcite and allotropes ) . it is often observed empirically that within experimental uncertainties many expressions for thermodynamic quantities yield consistent results . in the present paper we provide arguments for why many of the results of equilibrium statistical mechanics can be applied to such time independent nondissipative nonequilibrium systems . we also point out some of the limits inherent in the application of the formulae of equilibrium statistical mechanics to such systems . we choose to study the isothermal isobaric ensemble @xcite ( externally regulated pressure and temperature ) . the methods and reasoning we use here can be directly transferred to other ensembles such as the canonical ( fixed volume and externally regulated temperature ) . the gibbs free energy @xmath0 , which is the thermodynamic potential for the isothermal isobaric ensemble , is related to the partition function @xmath1 by the equation @xmath2 and the partition function is given by the integral@xmath3,\label{del0}\ ] ] where @xmath4 is the phase space vector describing the coordinates @xmath5 and momenta @xmath6 , of all the @xmath7 particles in the system , @xmath8 is the thermodynamic pressure , and @xmath9 where @xmath10 is boltzmann s constant and @xmath11 is the temperature . the integration domain @xmath12 provides limits for both integrals and extends over all the available phase space @xmath13 . this is @xmath14 for every component of the generalized momentum , @xmath15 for the volume @xmath16 , and over the volume for the cartesian coordinates of the particles . since the system hamiltonian @xmath17 is single valued , so too is the partition function and in turn the free energy . if we require the distribution function of a single thermodynamic phase it is necessary that other phases do not contribute significantly to the partition function . the full integration domain @xmath12 may include states that are characteristic of crystalline states or fluids states . in the thermodynamic limit this does not cause problems because , as we shall see , the partition function will be _ completely _ dominated by those microscopic domains that have the lowest free energy . however the application of these formulae to allotropes or metastable systems does present a problem . the standard equilibrium statistical mechanical expressions for variables such as the enthalpy @xmath18 , the average volume @xmath19 and second order quantities such as the specific heat at constant pressure @xmath20 may all be computed from a knowledge of the partition function eq . [ del0 ] or equivalently the thermodynamic potential eq . [ g0 ] . if other phases of lower free energy exist this computation ( from eq . [ del0 ] as written ) will strictly speaking be incorrect . it is well known that the formulae for thermal properties such as entropy , free energy , temperature and specific heat do _ not _ hold for _ dissipative _ nonequilibrium systems outside the linear response regime @xcite . in this paper we examine the question of whether they are correct for any _ nondissipative _ nonequilibrium systems such as allotropes , metastable systems or history dependent glasses . we provide a statistical mechanical theory of time independent , nondissipative , nonequilibrium systems . the theory is based on the fact that these systems are nonergodic and individual sample systems comprise ergodic domains that do not span all of phase space . we show that if these domains are robust with respect to small changes in thermodynamic state variables , a successful statistical mechanical treatment of these nonequilibrium systems can be given . we provide direct evidence , from molecular dynamics simulations on a model glass former , that the resulting statistical mechanical formulae are satisfied within empirical errors . finally we provide an independent test of the two key elements of our theory : boltzmann weights within the phase space domains and the robustness of those domains . it happens that these two elements are the necessary and sufficient conditions for the application of the transient fluctuation relation to finite thermodynamic quenches ( in temperature or pressure ) for such systems @xcite . while the application of thermodynamics to a single time averaged system is usually straightforward the application to a ensemble , whose members may be locked in different phase space domains , can require modification to the standard formulae . in the case of glasses our treatment has some similarities with the energy landscape approach of stillinger and weber @xcite . however there are significant differences ; our treatment makes no reference to the inherent structure and imposes no a priori knowledge of the inter - domain relative population levels . the energy landscape approach has been extended to account for the phenomena of ageing or history dependence by the addition of a fictive parameter @xcite . sciortino has convincingly shown that the addition of a single fictive parameter is inadequate to deal with glasses , which may have different properties at the same temperature and pressure if they are prepared by a different protocol ( different history dependence ) @xcite and poses the challenge to recover a thermodynamic description `` by decomposing the ageing system into a collection of substates '' . the treatment we present here succeeds in doing just that by providing a rigorous development of equilibrium statistical mechanics and thermodynamics for ensembles of systems where the phase space breaks up into ensembles of domains whose inter - domain dynamics is nonergodic and whose inter - domain population levels may not be boltzmann weighted . a dynamical system in equilibrium has the properties that it is nondissipative and that its macroscopic properties are time independent . thus the n - particle phase space distribution function , @xmath21 , must be a time independent solution to the liouville equation @xcite,@xmath22 where @xmath23 is the phase space compression factor @xcite obtained by taking the divergence of the equations of motion ( see eq . [ phase - space - compression ] ) and @xmath24 is the extended phase space vector which consists of @xmath25 and may include additional dynamical variables such as the volume @xmath16 . since the system is assumed to be nondissipative both the ensemble average @xmath26 and the time average @xmath27 of the phase space compression factor ( which is directly proportional to the rate at which heat is exchanged with the fictitious thermostat ) are zero . the time independent solution to eq . [ liouville - eq0 ] depends on the details of the equations of motion . equilibrium solutions to eq . [ liouville - eq0 ] for the equations of motion , suitable for use in molecular dynamics simulations , are compatible with gibbsian equilibrium statistical mechanics @xcite . microscopic expressions for mechanical properties like the pressure , the internal energy , the enthalpy and the volume can be derived without reference to gibbsian statistical mechanics and indeed can be proved to hold for nonequilibrium systems including nonequilibrium dissipative systems . there are two ways in which the formulae derived from gibbsian equilibrium statistical mechanics can break down . the most obvious way is that the relative weights of microstates may be non - boltzmann and the exponential factor @xmath28 $ ] , may be replaced by some other function ( either the exponential function itself may be modified as in tsallis statistics @xcite or the hamiltonian may be modified to some new function @xmath29 ) . in either circumstance the standard expressions for the thermal quantities derived from equilibrium gibbsian statistical mechanics will not be valid . this certainly happens in dissipative nonequilibrium systems where the distribution function is not a time independent solution to eq . [ liouville - eq0 ] . in deterministic nonequilibrium steady states the phase space may break down into ergodically separated domains ( each of which will be fractal and of lower dimension than the ostensible phase space dimension . this is a consequence of dissipation . ) however for these steady states , the domains are always exquisitely sensitive to macroscopic thermodynamic parameters since they are strange fractal attractors @xcite . often a deterministic _ nonequilibrium _ steady state approaches a unique fractal attractor . as time progresses the distribution function collapses ever closer to ( but never reaching ) the steady state attractor . the second way that these expressions may fail is that the system may become nonergodic . in this case three things happen . a ) most obviously time averages no longer equal full ( domain @xmath12 ) ensemble averages . b ) if we take an initial microstate the subsequent phase space trajectory will span some phase space domain @xmath30 where the initial phase is labeled @xmath31 . in this case for nondissipative nonequilibrium systems where the domains are robust ( i.e. small changes in thermodynamic state parameters , to leading order do not change the domain ) the standard equations of equilibrium statistical mechanics may continue to be valid but in a slightly modified form . we will examine this in some detail below . c ) given robust domains the population densities between each domain may well depend on the history of the system . macroscopic _ history can be expected to condition the ensemble s set of initial _ microstates _ @xmath32 from which the macroscopic material is formed . this in turn can be expected to condition the set of nonergodic domains @xmath33 that characterize the ensemble . for a macroscopic sample spanning a single ergodic domain @xmath30 , the free energy @xmath34 then satisfies only a local extrema principle and thus looses much of its thermodynamic meaning . we use the constant pressure nos - hoover equations of motion by combining the nos - hoover feedback mechanism with the so - called sllod or dolls equations of motion , which are equivalent for dilation . it is known that these equations of motion do not produce artifacts in the systems linear response to an external field and that to leading order the effect on the dynamical correlation functions is at most @xmath35 , where @xmath7 is the number of particles @xcite . the equations of motion are,@xmath36 where @xmath37 is the position , @xmath38 is the momentum and @xmath39 is the force on the @xmath40 particle , @xmath41 is the particle mass , @xmath42 is the barostat time constant , @xmath43 is the thermostat time constant , @xmath11 is the input temperature , @xmath8 is the input ( thermodynamic ) pressure and the instantaneous ( mechanical ) pressure is given by @xmath44 . because these equations of motion have additional dynamical variables the extended phase space vector is @xmath45 . in order to obtain the equilibrium distribution function we first define the hamiltonian , in the absence of any external fields , dilation @xmath46 or thermostats @xmath47 , as @xmath48 , where @xmath49 is the total inter - particle potential energy . to proceed further we identify the extended hamiltonian as @xmath50 and then obtain the phase space compression factor@xmath51 where the index @xmath52 sums over the components of the cartesian position and momentum vectors . using the heisenberg streaming representation ( rather than the more usual schrdinger representation eq . [ liouville - eq0 ] ) of the liouville equation@xmath53=-\lambda\left({\mbox{\boldmath $ \gamma$}}^{\prime}(t)\right),\label{liouville - streaming}\ ] ] we can obtain the particular time independent solution for the distribution function@xmath54 where @xmath55 is the instantaneous enthalpy . the second exponential on the rhs of eq . [ dist0 ] with @xmath56 and @xmath57 in the argument , which has no dependence on the input temperature @xmath11 or the input pressure @xmath8 , is statistically independent from the rest of the distribution function , which is the standard equilibrium isothermal isobaric distribution . we can normalize eq . [ dist0 ] by integrating over all space to obtain the thermodynamic equilibrium distribution function , @xmath58 where the standard isothermal isobaric distribution function is@xmath59 it should be emphasized that the derivation of eq . [ dist0 ] says nothing about the existence or otherwise of any domains . these must be considered when normalizing eq . [ dist0 ] and thus eqs . [ dist1 ] & [ dist - standard ] are only valid in thermodynamic equilibrium . if we wish to use eq . [ dist0 ] outside thermodynamic equilibrium we must consider domains . we can also use the so called sllod equations of motion @xcite to apply strain rate controlled couette flow ( planar shear ) to our equations of motion . the necessary modifications to the first two lines of eq . [ eom ] result in@xmath60 where @xmath61 is the strain rate and the last three lines of eq . [ eom ] remain unchanged . as we have stated in the introduction , the full phase space domain includes phase points from many different thermodynamic phases ( gases , liquids and crystals ) . in the thermodynamic limit this does not cause problems . to understand this suppose we can label microstates to be in either of two possible thermodynamic phases 1 or 2 bound by two phase space domains @xmath62 and @xmath63 . by assumption we are not presently considering the possibility of co - existence . the system is assumed to be ergodic : atoms in one thermodynamic phase can in time , transform into the other phase . assume the two thermodynamic phases have different free energies : @xmath64 is the gibbs free energy of the first phase and @xmath65 is that of the second phase . for a sufficiently large @xmath7 the free energy eq . [ g0 ] is an extensive variable . we may thus express the partition function as the sum of contributions from the two phases@xmath66 where the lower case @xmath67 on the second line is used to represent the intensive free energies which do not change with system size @xmath7 in the thermodynamic limit . if @xmath68 is less than @xmath69 then in the thermodynamic limit , @xmath70 , the only significant contribution to the partition function @xmath1 will be due to the `` equilibrium '' phase namely phase 1 . thus although the free energy defined in eq . [ del0 ] , is given by an integral over all of phase space @xmath12 , in the thermodynamic limit this integral can be approximated to arbitrary precision , as an integral over the domain that includes the most stable phase . suppose @xmath62 includes only crystalline phases and @xmath63 includes only amorphous phases and further suppose a particular crystalline phase has a lower free energy that any amorphous phase . according to eq . [ del0 ] , we should calculate the free energy by integrating over all crystalline and all amorphous phases . in practice in the thermodynamic limit we can compute the free energy to arbitrary accuracy by integrating eq . [ del0 ] , only over that part of phase space within which the thermodynamically stable state resides . if we consider a nonergodic system that according to different preparative protocols can be formed in either phase 1 or phase 2 . after preparation , because the system is nonergodic both phases are kinetically stable indefinitely . by restricting the phase space integrals for the free energy to those domains that contain the kinetically stable phase we can compute the free energy of that phase . however , although it may be possible to formally assign free energies to nonergodic systems , these free energies clearly fail to satisfy any global extremum principle . as we will show these partition functions can be used formally to yield first and second order thermodynamic quantities by numerical differentiation . the metastable domain is a subset of the thermodynamic equilibrium domain which contains all possible atom positions including ones belonging to the metastable phase . within a single domain the system is , by construction , ergodic . thus for almost all microstates @xmath31@xmath71@xmath30 , ensemble averages , of some variable @xmath72 , @xmath73 , equal time averages @xmath74 , for phase space trajectories that start at time zero , @xmath75 microscopic expressions for mechanical variables may be used as a test of ergodicity in nondissipative systems which are out of equilibrium . ( note a nondissipative system does not on average exchange heat with any thermal reservoir with which it has been in contact for a long time . ) in the case of metastable fluids or allotropes we may introduce a single restricted domain and by construction the system remains ergodic within this domain . a _ gedanken _ experiment can be used to justify the boltzmann weighting and the applicability of the zeroth law of thermodynamics for such systems . consider a double well potential with an inner and outer potential well . if the barrier between the inner and outer wells is much greater than @xmath76 , so that over the duration of observation ( which is much greater than any relaxation time in the ergodically restricted subsystem ) no particles cross the barrier , then the system considered as a double well system , will be , by construction , nonergodic . for systems composed of particles that are solely found in the inner potential well , our hypotheses are that the distribution of states in the inner well will be given by a boltzmann distribution taken over the inner domain only and that if such a system is in thermal contact with a body in true thermodynamic equilibrium , then the temperature of the ergodically restricted system must equal that of the system in true thermodynamic equilibrium . we can justify these hypotheses by considering a fictitious system that only has the inner potential well and in which the potential function is positive infinity for all separations that are greater than the inner well ( this includes the position of the outer well ) . in accord with gibbsian statistical mechanics the distribution of states is canonical over this ( single well ) potential . furthermore the zeroth law of thermodynamics will apply to this single well system . now if we dynamically generate the outer well , all the particles locked inside the inner well can not `` know '' that the outer well has been formed so their dynamics will be completely unchanged by the time dependent generation of the new outer well . the generation of an inaccessible outer well will not alter the distribution of states in the inner well nor will it cause any flow of heat to the equilibrium heat bath surrounding the system . this provides a compelling physical justification for our domain hypotheses over a single ergodic sub domain of phase space . in order to recover many of the basic relationships of gibbsian statistical mechanics it is also necessary that the system appears to be in dynamical equilibrium , i.e. @xmath77 , for some small @xmath78 , over the longest observation time @xmath79 . we use the definition of the partition function eq . [ del0 ] as before but now the domain @xmath30 in the integral is over a single contiguous hypervolume in the configuration space of the generalized position coordinate @xmath5 * * * * and volume @xmath16 . the domain over the generalized momentum @xmath6 and multipliers @xmath56 and @xmath57 remains unchanged . we then obtain the gibbs free energy by use of eqs . [ g0 ] & [ del0 ] . thus far all we have altered is our definition of the domain . in changing the definition of the domain we have opened a potential problem for gibbsian statistical mechanics . if we change the temperature or the pressure of the system the domain may also change . if the domain changes this may make a contribution to the derivatives of the partition function , eq . [ del0 ] , and the direct connection with the standard outcomes of macroscopic thermodynamics will be lost . thus the domains need to be robust with respect to changes in thermodynamic state variables . there are three means by which a system could have robust domains . the first and most obvious is that the domain does not change when the pressure or the temperature changes , @xmath80 , where @xmath81 is a thermodynamic state variable and @xmath82 is the other thermodynamic state variables . when we lower the temperature only the inverse temperature @xmath83 in eq . [ dist0 ] changes and when we change the pressure only the parameter @xmath8 changes . if the domain s boundary is determined by a surface on which @xmath84 always has a very high value it will remain unchanged under infinitesimal changes in @xmath8 or @xmath83 . we will refer to a surface domain that does nt change with the state variables as completely robust . the second way is that the distribution function is always identically zero on the domain boundary , @xmath85 where @xmath86 is the surface of the domain @xmath30 . because the domain is contiguous ( required for it to be ergodic ) it must have a single connected surface . such a domain will be robust . the third way the domain can be robust is less restrictive and allows for the possibility that the domain does change when the thermodynamic variables are changed substantially . if @xmath87 is an infinitesimal change in a thermodynamic state variable then , @xmath88 where @xmath82 denotes the other thermodynamic state variables , we require that @xmath89 for first order thermodynamic property formulae to be correct @xmath90 for second order property formulae to be correct etc . obviously this third way will be satisfied in the first two cases as well . later in the paper we will introduce an independent test of domain robustness . however , if a system was not robust then we would expect that small changes in the state variables would change the macroscopic properties of the sample permanently - it would be as though the preparation history of the sample was continuing even for small changes in the state variable . quite obviously if we produce huge changes in the state variables we will of course permanently change the properties of the system because we permanently deform the ergodic domain . experience shows however that very many nondissipative nonequilibrium systems are quite robust with respect to small changes in state variables . all that is required for fluctuation formulae for first second and third order thermodynamic quantities to be valid , is that the domains be unchanged , to first second or third order , by infinitesimal changes in the state variables . obviously a robust domain is an ideal construct . however on the typical time scale of interest , which is usually orders of magnitude less than the time scale on which the system will change to a new phase of lower free energy , this can be a very good approximation . we are now able to recover most of the standard results of gibbsian equilibrium statistical mechanics . for example we may calculate the enthalpy @xmath91 from the partition function @xmath1 , eq . [ del0 ] , as@xmath92 here we are considering an ensemble of systems which occupy a single ergodic domain @xmath30 . since this domain is self ergodic the ensemble average is equal to the corresponding time average . the term on the rhs of the second line is obviously the average value of the instantaneous enthalpy , @xmath93 , obtained by using the equilibrium distribution function , eq . [ dist - standard ] or equivalently eq . [ dist1 ] with the integration limits restricted to the domain @xmath30 , where @xmath8 is the externally set thermodynamic pressure . we can also obtain expressions for the average volume @xmath19 and the constant pressure specific heat @xmath20 by taking the appropriate derivatives of the partition function eq . in other ensembles we can use the same procedure to find other variables e.g. the internal energy , the average pressure and the constant volume specific heat in the canonical ( _ n_,_v_,_t _ ) ensemble . an important outcome is that this description remains compatible with macroscopic thermodynamics . here the gibbs free energy is defined as,@xmath94 where @xmath95 is the internal energy and @xmath96 is the entropy . if we take the derivative of eq . [ gibbs thermo ] with respect to one of the isobaric isothermal ensembles conjugate variables @xmath97 while keeping the others fixed we obtain@xmath98 we now write down the microscopic equilibrium equation for the gibbs entropy@xmath99 it is an easy matter to show that eqs . [ g0 ] , [ del0 ] & [ gibbs entropy ] are consistent with the two derivatives given in eq . [ gibbs conj deriv ] . given our condition of a robust boundary we thus have a form of gibbsian statistical mechanics for metastable states which remains in agreement with macroscopic thermodynamics . we now wish to consider an ensemble of systems which is prepared from an initial ergodic ( usually high temperature ) equilibrium ensemble . there is some protocol @xmath100 , which involves a temperature quench or a sharp pressure increase etc , which breaks the ensemble into a set of sub - ensembles characterized by different macroscopic properties . after the protocol @xmath100 , has been executed we allow all the ensemble members to relax to states which are macroscopically time independent - to within experimental tolerances . we assume that the ensemble can be classified into a set of sub - ensembles @xmath101 whose macroscopic properties take on @xmath102 distinct sets of values . for the longest observation times available a macroscopic system classified as an @xmath103 system is not observed to transform into a @xmath83 system , and vice versa . the full ensemble of systems is thus non - ergodic . however , in each individual sub - ensemble , say sub - ensemble @xmath103 , the constituent members are ergodic ( by construction ) . thus we can partition the full phase space into a set of domains , @xmath33 . from the arguments given above ( in section b ) , after the relaxation of initial transients , we expect to observe a boltzmann distribution of states within an individual domain which is therefore independent of the quench protocol . however the distribution between domains can not be expected to be boltzmann distributed and will instead be dependent on the quench protocol . within a given ensemble the proportion of ensemble members ultimately found in domain @xmath30 is given by a weight @xmath104 which is subject to the constraint @xmath105 we can calculate the full _ ensemble _ average of some macroscopic property @xmath72 as,@xmath106 since the full ensemble of states is nonergodic the phase space breaks up into disjoint domains which in themselves are ergodic . thus each domain may be identified by any point in phase space , @xmath13 , that is a member of it so the subscript @xmath103 is a function of the phase vector @xmath107 allowing the following expression for the distribution function@xmath108 where @xmath109 if @xmath110 and @xmath111 otherwise , and @xmath112 the entropy is given by @xmath113 and using eq . [ multi - d - dist ] we have the following expressions for the multidomain entropy,@xmath114=\sum_{\alpha=1}^{n_{d}}w_{\alpha}s_{\alpha}-k_{b}\sum_{\alpha=1}^{n_{d}}w_{\alpha}\ln(w_{\alpha}).\label{s - multi - d}\ ] ] the term @xmath115 is the inter - domain entropy , which is maximized by an even distribution of ensemble members over all domains , while @xmath86 is the intra - entropy of domain @xmath103 considered as a single _ n_-particle system . if we substitute eq . [ multi - d - dist - raw ] into eqs . [ s - multi - d ] we find that , @xmath116 where @xmath117 . combining eq . [ gibbs thermo ] with eq . [ s - multi - d - g ] we obtain the following expression for the gibbs free energy @xmath118=\sum_{\alpha=1}^{n_{d}}w_{\alpha}g_{\alpha}-s_{d}t.\label{g - multi - d}\ ] ] it is easy to verify that if we hold the local domain weights fixed and then vary the temperature or the pressure that eqs . [ s - multi - d ] & [ g - multi - d ] are compatible with eqs . [ gibbs conj deriv ] . this means that if we have a fixed number of robust domains , whose population levels or weights are non - boltzmann distributed , eqs . [ g - multi - d ] & [ s - multi - d ] provide a direct microscopic link to standard macroscopic thermodynamics . on the extremely long time scale the weighting functions @xmath119 may vary and the system will tend towards the direction where the free energy eq . [ g - multi - d ] is reduced . without the inter - domain entropy term , @xmath120 , eq . [ g - multi - d ] would be minimized when the domain with the lowest free energy has all the ensemble members in it . it turns out that eq . [ g - multi - d ] is minimized when all the domain weights are boltzmann distributed , ie when@xmath121 here ( i.e. upon obeying eq . [ weights boltzmann ] ) the entropy and free energy given by eqs . [ s - multi - d ] & [ g - multi - d ] coincide with the standard equilibrium expressions so the free energy must be a minimum . to prove this we use eq . [ g - multi - d ] and we remove the first weight @xmath122 , so that the constraint eq . [ w - constraint ] is respected while the remaining weights are independent . this means that the free energy can be written as @xmath123 . the constrained partial derivatives are then @xmath124 using the fact that for boltzmann weights , eq . [ weights boltzmann],@xmath125\label{weights - boltz-2}\ ] ] where @xmath126 is the equilibrium free energy , we find that at equilibrium , @xmath127 it remains to prove that this is indeed a minimum . using the same approach for treating the constraint , we continue making the first weight a function of all the others , and obtain@xmath128 using the boltzmann weights it is easy to show that@xmath129\nonumber \\ \frac{\partial^{2}g}{\partial w_{1}\partial w_{\alpha } } & = & \frac{\partial^{2}g}{\partial w_{\gamma}\partial w_{1}}=0\nonumber \\ \frac{\partial^{2}g}{\partial w_{\gamma}\partial w_{\alpha } } & = & \delta_{\gamma\alpha}\frac{k_{b}t}{w_{\alpha}}=\delta_{\gamma\alpha}k_{b}texp[\beta(g_{\alpha}-g_{eq})].\label{hessian-2}\end{aligned}\ ] ] from these results it is easy to see that the hessian matrix @xmath130 is positive definite , thus we have proved the free energy to be a minimum for the case of equilibrium . we can use a knowledge of the multiple domain thermodynamic potential , eq . [ g - multi - d ] , to compute averages . as an example we consider the average enthalpy again,@xmath131 eq . [ enthalpy - multi - d ] can easily be derived from eq . [ gibbs thermo ] and is in essence the same as the first line of eq . [ enthalpy ] . it is straightforward to see that upon using eq . [ g - multi - d ] to calculate the derivative in eq . [ enthalpy - multi - d ] one obtains the average enthalpy as given by eq . [ multi - d - average ] . one can do the same for other quantities such as the specific heat etc . we are now in a position to test various outcomes , using computer simulation , which may be derived by drawing on the previous material . if we start with an ensemble of systems which are initially in equilibrium at temperature @xmath132 and then at time @xmath133 we quench them by setting @xmath134 we can solve the liouville equation eq . [ liouville - streaming ] to obtain@xmath135\label{quench dist}\end{aligned}\ ] ] where @xmath136 . this nonequilibrium distribution function , valid for @xmath137 , explicitly requires the solution of the equations of motion and is of limited utility . however it allows the identification of a formal condition to identify the amount of time , which must elapse after the quench , before eqs . [ dist0 ] , [ dist - standard ] , [ multi - d - average ] or [ multi - d - dist ] can be applied to the ensemble . that is the quantity @xmath55 must be statistically independent of @xmath138 . thus we are interested in the correlation function @xmath139 where @xmath140 this function will equal 1 for a perfectly correlated system , -1 for a perfectly anticorrelated system and 0 for an uncorrelated system . when we consider an ensemble of systems , occupying the various domains to different degrees , we see that eq . [ correlation1 ] may not decay to zero given the trajectories are unable to leave their domains . if the transients , due to the quench , fully decay eq . [ correlation1 ] will decay to zero and eqs . [ dist0 ] & [ dist - standard ] will become valid for the ensemble . if the correlation function , eq . [ correlation1 ] decays to a plateau then we may have a situation where eqs . [ multi - d - average ] and [ multi - d - dist ] are valid . it may be that the material can still slowly age , due to processes , that occur on a time scale which is longer than the one we are monitoring . for a glass we expect that correlation function eq . [ correlation1 ] will not fully decay on a reasonable time scale . if we give the system time to age , such that it appears to be time translationaly invariant , and then compute the following correlation function,@xmath141 where@xmath142 we may observe a full decay . if this occurs eqs . [ multi - d - average ] and [ multi - d - dist ] will be valid . to compute this correlation function @xmath143 trajectories are produced and for each of these the averages , @xmath144 , where @xmath72 is an arbitrary dynamical variable , appearing in eq . [ correlation2 ] are approximately obtained by time averaging . when the system is ergodic and time translationaly invariant these two correlation functions , eqs . [ correlation1 ] & [ correlation2 ] , will give the same result . however for a nonergodic system @xmath145 will decay to zero on a reasonable time scale while @xmath146 will not . rather @xmath146 may decay to some plateau on a reasonable time scale and then decay on a much slower time scale . the preceding sections then rest on this clear separation of time scales in the correlation function eq . [ correlation1 ] . for metastable fluids and allotropes this separation is so extreme that we probably can not observe , even the early stages of , the later slow decay on any reasonable experimental time scale . further for these systems there will be only a single domain and thus they appear ergodic . for glasses some signs of the later decay can often be observed , however it is still very much slower than the initial decay . in the field of glassy dynamics the initial decay is often called the @xmath83 decay and the slower long time decay is often called the @xmath103 decay . as the glass is further aged this second stage decay is observed to slow down dramatically while the early decay does not change very much @xcite . to allow the hypothesis of local phase space domains to be tested we will now introduce several relations whose derivation draws upon the equilibrium distribution function eq . [ dist - standard ] . we will also discuss the effect of the phase space domains on these relations . first we introduce the configurational temperature@xmath147 where @xmath148 is a @xmath149 dimensional vector representing the inter - particle forces on each atom @xmath150 . this relation is easily derived from eq . [ dist - standard ] ( set @xmath72 to @xmath151 , integrate by parts and drop the boundary terms ) and will remain valid for nonergodic systems where eq . [ weights boltzmann ] is not obeyed . the relation involves the spatial derivative of the force , which is not zero at the cutoff radius for the potential we use in our simulations ( see below ) . this along with finite size effects can result in a small disagreement between eq . [ config temp ] and the kinetic temperature for our system when in equilibrium . equilibrium fluctuation formulae may be easily derived from eq . [ multi - d - average ] , see ref . @xcite for some examples of how this is done . we consider how the average enthalpy changes with the temperature at constant pressure,@xmath152\label{cp - regular}\ ] ] where @xmath153 is the constant pressure specific heat . the calculation of the rhs of eq . [ cp - regular ] using the ensemble average given by eq . [ multi - d - average ] is subtle . if we simply calculate @xmath154 and @xmath155 with the use of eq . [ multi - d - average ] and then plug the results into eq . [ cp - regular ] we obtain , what we will refer to as , a _ single domain average _ which does not give us the correct change in the average enthalpy for a history dependent equilibrium ensemble . this is because the average @xmath155 superimposes across the different domains while the quantity @xmath156 contains spurious cross terms . if we derive the heat capacity by taking the second derivative of the gibbs free energy for the multiple domain distribution function , eqs . [ g - multi - d ] & [ enthalpy - multi - d ] , with respect to the temperature , @xmath157 we obtain the following @xmath158,\label{cp - concat}\ ] ] where the quantity @xmath159 is obtained for each domain separately ( here @xmath160 represents an average taken where all ensemble members are in the @xmath161 domain ) . we will refer to this as a _ multidomain average _ which is consistent with the nonergodic statistical mechanics and thermodynamics that we have introduced here . it is obvious that both a single and multiple domain average will give the same result in the case of thermodynamic equilibrium and metastable equilibrium ( single domain ) . the transition from the single domain average producing the correct result to an anomalous result is symptomatic of an ergodic to a history dependent nonergodic transition . if we consider a large macroscopic system ( the super system ) to be made of @xmath162 independent subsystems the multidomain average remains self - consistent . to see this we apply eq . [ cp - concat ] to fluctuations in the super system and then we inquire how this relates to fluctuations in the subsystem . the enthalpy of one instance of the super system will be given by @xmath163 . in principle an ensemble of super systems can be prepared by applying the same history dependent macroscopic protocol to all members of this ensemble . due to the statistical independence of the subsystems , upon taking an ensemble average of super systems , we have @xmath164 for all @xmath165 . here the average @xmath166 is taken over the ensemble of super systems and the @xmath161 subsystem in each super system is identified by its location . it is then easy to show that the specific heat obtained from the ensemble average eq . [ cp - concat ] of the super system is equivalent to that obtained from the subsystem due to the two quantities @xmath167 and @xmath168 possessing identical cross terms which cancel each other out ( as a result of the independence of the subsystems ) upon applying eq [ cp - concat ] . if we ignore finite size effects , due to assuming the equivalence of ensembles , the constant volume specific heat is related to the constant pressure specific heat by the equation@xmath169 we may also obtain an expression for the constant volume specific heat @xmath170 by deriving equilibrium fluctuation formula for each of the derivatives appearing in eq . [ cv ] in lieu of directly measuring them . we may then obtain a single domain expression for @xmath170 which does not work for the history dependent glass and also a correctly weighted ensemble average ( multidomain average ) which does . this is completely analogous to what has been shown in detail for @xmath20 . as the equations are unwieldy , and their derivation ( given an understanding of the @xmath20 case ) is straightforward , we will not reproduce them here . the application of the evans - searles transient fluctuation theorem @xcite to the systems treated in this paper provides a sharp test of the assumptions used to develop the theory given in this paper . the theorem describes a time reversal symmetry satisfied by a generalized entropy production , namely the so - called dissipation function . the precise mathematical definition of this function requires a knowledge of the dynamics and also of the initial distribution function . the three necessary and sufficient conditions for the fluctuation theorem to be valid are that the initial distribution is known ( here we assume the distribution is boltzmann weighted over some initial domain of phase space ) , that the dynamics is time reversible ( all the equations of motion used here are time reversible ) and lastly that the system satisfies the condition known as ergodic consistency . when applied to the systems studied here this requires that the phase space domains should be robust with respect to the sudden changes imposed on the system and that the number of inter - domain transitions remain negligible on the time over which the theory is applied . if any one of these three conditions fails then the theorem can not be applied to the system and the corresponding fluctuation relation will not be satisfied@xcite . we can use the fluctuation theorem to obtain relations for how the system responds upon suddenly changing the input temperature or pressure for a system , which is initially in equilibrium as specified by eq . [ dist0 ] or [ dist1 ] . firstly we consider a change in the pressure , while holding the temperature fixed , by changing the input variable @xmath8 in eq . [ eom ] ( thermodynamic pressure ) to @xmath171 at time @xmath133 for a system initially in equilibrium with @xmath172 . the probability density @xmath173 of observing a change in volume of @xmath174 relative to a change of equal magnitude but opposite sign is then given by@xmath175 to derive this expression we have had to assume that the intra - domain populations are boltzmann distributed according to eq . [ dist0 ] . ergodic consistency requires that for any initial phase space point @xmath176 that can be initially observed with nonzero probability , there is a nonzero probability of _ initially _ observing the time reversal map @xmath177of the end point @xmath178 , ( ie @xmath179 such that @xmath180@xmath181@xmath182 ) . this condition obviously requires that the phase space domains remain robust and the number of inter - domain transitions remain negligible for at least a time @xmath183 , after the pressure ( or temperature ) quench . if we sample all or our initial @xmath133 , @xmath172 configurations from the one trajectory which remains locked in a single domain even after the quench , we expect eq . [ tft - p ] to be valid . if we prepare an ensemble of initial configurations using the same protocol we still expect eq . [ tft - p ] to remain valid even with different domain weightings @xmath184 , as defined in eqs . [ multi - d - average ] & [ multi - d - dist ] , provided the domains are robust over the time @xmath183 appearing in eq . [ tft - p ] . note that suddenly reducing the pressure by a very large amount could result in a breakdown of the robustness condition . [ tft - p ] may be partially summed to obtain what is referred to as the integrated fluctuation theorem @xmath185 for the case where we change the input temperature @xmath11 in eq . [ eom ] while holding the input pressure @xmath8 fixed we obtain a relation for fluctuations in the extended instantaneous enthalpy @xmath186 . we start with a system initially in equilibrium at temperature @xmath187 and we then subject it to a temperature quench by changing the input temperature in eq . [ eom ] to @xmath188 , at time @xmath133 , while holding the input pressure fixed . the probability density @xmath189 of observing a change in instantaneous enthalpy @xmath190 relative to a change of equal magnitude but opposite sign is then given by@xmath191 note that if we suddenly increase the temperature by a very large amount we could expect to violate the robustness or the negligible inter - domain transition condition . in common with eq . [ tft - p ] we expect that this expression will be valid when all initial configurations are sampled from a single common domain and also when sampled from multiple arbitrarily populated domains under the assumption that the domains are robust and the number of transitions are negligible over time @xmath183 . this equation may also be partially summed to obtain@xmath192 for our simulations we use a variation on the kob andersen glass former @xcite featuring a purely repulsive potential @xcite . the pairwise additive potential is@xmath193 & \;\;\;\forall\;\;\ ; r_{ij}<\sqrt[6]{2}\,\sigma_{\alpha\beta}\nonumber \\ u_{ij}(r_{ij})=0\;\;\;\;\;\ , & & \;\;\;\forall\;\;\ ; r_{ij}>\sqrt[6]{2}\,\sigma_{\alpha\beta},\label{potential}\end{aligned}\ ] ] where the species identities of particles @xmath194 and @xmath195 , either @xmath196 or @xmath72 , are denoted by the subscripts @xmath103 and @xmath83 . the energy parameters are set @xmath197 , @xmath198 and the particle interaction distances @xmath199 , @xmath200 . the energy unit is @xmath201 , the length unit is @xmath202 and the time unit is @xmath203 with both species having the same mass @xmath41 . the composition is set at @xmath204 , the number of particles are @xmath205 , the pressure is set to @xmath206 and the temperature unit is @xmath207 . the time constants are set at @xmath208 and @xmath209 . note that the energy parameters are slightly different to the potential we used in ref . the equations of motion were integrated using a fourth order runge - kutta method @xcite . the time step used was @xmath210 and sometimes @xmath211 for very low temperatures . from previous work on binary mixtures we know the basic reason why this system is vary reluctant to crystallize @xcite . the chosen nonadditivity of the species _ interaction makes the mixture extremely miscible ; consider the present value of @xmath200 relative to the additive value of @xmath212 . this effect dominates over the choice of the energy parameters . due to this extreme miscibility the relatively large composition fluctuations necessary , about the average composition of @xmath213 , to form the crystal phases ( either the pure species _ a , _ @xmath214 , fcc crystal or the binary , @xmath215 , cscl crystal ) are strongly suppressed and crystallization is strongly frustrated . with @xmath216.[fig-1 ] ] we identify the nominal glass transition by calculating the diffusion coefficient as a function of temperature . this is shown for both species in fig . [ fig-1 ] on a logarithmic plot demonstrating how the diffusion coefficient approaches zero critically , @xmath217 where @xmath218 is the critical exponent , with a nominal glass transition temperature of @xmath216 . it would be , perhaps , more customary to obtain a nominal glass transition temperature by analyzing the critical divergence of the viscosity . given that the stokes einstein relation is strongly violated upon approaching the glass transition one might be concerned that this would give a very different result . however the violation of the stokes einstein relation can largely be attributed to the exponent @xmath218 being different between the viscosity and the diffusion coefficient rather than the nominal glass transition temperature @xmath219 @xcite . the correlation functions given in eqs . [ correlation1 ] & [ correlation2 ] were calculated from ensembles of @xmath220 independent simulations at the two temperatures given in fig . [ fig-2 ] ( @xmath221 and @xmath222 ) . in all cases the systems were subject to an instantaneous quench , from an initial equilibrium at @xmath223 , by changing the value of the input temperature in eq . the system was then run for a significant time , in the case of the glass ensemble @xmath224 , in an attempt to age it . of course the longest time that can be accessed in a molecular dynamics simulation is rather short , and so the system is not very well aged , but we are still able to meaningfully treat it as a time invariant state . each of the @xmath220 independent simulations was interpreted as being stuck in its own domain @xmath30 and the correlation functions were calculated for each of these domains using time averaging . the time averaging was approximately @xmath220 times longer than the longest time @xmath225 that the correlation functions were calculated out to . obviously in the limit of an infinite number of independent simulations and the case where the domains are robust we will obtain the exact multidomain average given by eq . [ multi - d - average ] . we assume our limited ensemble of simulations is representative of this . the data from each domain ( independent simulation ) was then used to obtain the correlation functions eqs.[correlation1 ] & [ correlation2 ] as seen in fig . [ fig-2 ] . & [ correlation2 ] as a function of logarithmic time for the temperature @xmath221 . the calculation of the function was started after the fluid had been given time to equilibrate . the strong agreement between the two correlation functions is indicative of ergodicity . + b ) the instantaneous enthalpy correlation functions as defined by eqs . [ correlation1 ] & [ correlation2 ] as a function of logarithmic time for the temperature @xmath222 . the calculation of the correlation function was started at various times after the quench , all approximately at @xmath226 , in an attempt to age the system . the difference between the correlation functions is indicative of nonergodicity . notice that @xmath145 reaches a near full decay between @xmath227 , while @xmath146 reaches a non - decaying plateau . [ fig-2 ] ] at the higher temperature @xmath221 it can be seen that the two functions are equivalent demonstrating how the system is ergodic . it can also be seen that the correlation function has decayed on a time scale of @xmath228 which is therefore ( by eq . [ quench dist ] ) the time scale on which the ensemble becomes accurately represented by eq . [ dist - standard ] with only one domain @xmath12 which does not necessarily extend over all phase space . the oscillations , which can be seen in the correlation function , are due to both ringing in the feedback mechanisms of eq . [ eom ] and the frequency dependent storage component of the bulk viscosity . the statistical uncertainty in the correlation function becomes larger than these oscillations somewhere between a time of @xmath227 . if we constructed an experiment where the pressure was regulated by a piston and a spring , the correlation functions , eqs.[correlation1 ] & [ correlation2 ] , would depend on the details of the piston and spring parameters in a similar way to the simulations dependence on the details of the feed back mechanism . when the system is quenched to the lower temperatures ( @xmath222 ) ergodicity is lost and we obtain a glass . the complete decay of the first correlation function , @xmath146 , eq . [ correlation1 ] may no longer occur because the individual trajectories remain stuck in local domains which have different average values for @xmath229 . this is similar to the much studied density correlation function @xcite ( the intermediate scattering function ) which decays to a finite plateau for a glass or more generally a solid material . on the other hand there is nothing to stop the second correlation function , @xmath145 , eq . [ correlation2 ] , from fully decaying when the system is nonergodic . if the second correlation function fully decays whilst the first is only able to decay to a plateau then we have a situation where eqs . [ multi - d - dist ] & [ multi - d - average ] are valid as can be seen from eq . [ quench dist ] . in fig . [ fig-2 ] it can be seen that @xmath146 does indeed fail to decay while @xmath145 comes very close to fully decaying at the time where @xmath146 reaches the plateau . the reason @xmath145 does nt fully decay here is due to the fact that the inter - domain transition rates , while small , are not exactly zero . if the system had been aged more extensively this problem would be significantly reduced . this effect is exacerbated by the time averaging , used to form the averages for each trajectory , being two orders of magnitude longer than the longest time the correlation function was calculated to . the effect of the state slowly evolving due to finite inter - domain transition rates is too small to seriously compromise the modeling of the system as obeying eqs . [ multi - d - average ] & [ multi - d - dist ] and thus we have obtained direct evidence for the validity of these equations . the height of the plateau for @xmath146 will depend on the history of the system , i.e. the protocol used to prepare the ensemble . we move on to a comparison between the kinetic temperature and the configurational temperature , the results of which may be seen in fig . [ fig-3]a ) . ( controlled directly by the nos hoover thermostat ) and the configurational temperature given by eq . [ config temp ] as a function of the kinetic temperature . the results for the system undergoing couette flow ( shear ) are for a constant strain rate of @xmath230 . + b ) the heat capacity calculated using the equilibrium fluctuation formula by the single domain averaging method eq . [ cp - regular ] and the multidomain method eq . [ cp - concat ] . also shown is data obtained by numerically differentiating the enthalpy by central difference . at the temperatures above the peak the three types of averages give very similar results . also shown is equivalent equilibrium fluctuation formula data for the constant volume specific heat eq . [ fig-3 ] ] the input temperature ranges from @xmath231 to @xmath232 . also shown are results for the system , undergoing constant planar shear , eq . [ sllod ] , with a strain rate of @xmath230 . at the higher temperatures we see a very small relative discrepancy between the two types of temperatures , which we attribute to the discontinuity in the first spatial derivative of the inter - particle force at the cutoff radius and to finite size effects . these effects appear to diminish a little at lower temperatures . for temperatures above t = 1.5 the chosen strain rate has no significant effect on the configurational temperature indicating that our system is in the linear response domain @xcite . at the lowest temperatures , well below the glass transition temperature , we observe good agreement between the configurational and input temperatures for the system without shear . this provides further evidence of our assertion that the system obeys boltzmann statistics in the glass eqs . [ multi - d - average ] & [ multi - d - dist ] . however , at low temperatures , the system that is undergoing shear shows an increasing relative discrepancy between the two temperatures . at low temperatures the system leaves the linear response domain @xcite demonstrating the fundamental difference between the nonequilibrium distribution of the history dependent glassy state and that of a strongly driven steady state . if we drive the system hard enough , at any given temperature , we can always make a disagreement between the two types of temperature due to the steady state no longer being accurately represented by a boltzmann distribution i.e. due to a break down in local thermodynamic equilibrium . when the system is not driven by an external field we have been unable to observe any difference in the two temperatures by deeply supercooling a glass forming mixture apart from the initial transient decay immediately following the quench , which falls off surprisingly rapidly . in fig [ fig-3]b ) results are presented for the heat capacities ( the specific heat multiplied by the volume ) at both constant pressure @xmath233 and constant temperature @xmath234 . details of the protocol used to obtain this data is given in the end note @xcite . the estimates from the multidomain average are compared with those from the single domain average . the results from the multidomain averages exhibit the well - known peak , which is a signature of the onset of the glass transition , and has been observed directly by calorimetry in many experiments on real glass forming materials @xcite . the temperature , where the peak is observed , depends on the history of the system . no peak is observed for the single domain averages which continue to increase as the temperature is lowered . while the two methods for forming averages give the same results at temperatures above the peak , they diverge at temperatures below the peak . it is the multidomain average that gives results consistent with the actual calorimetric behavior of the system . this may be seen in the figure by comparing the data which has been computed by numerically differentiating the enthalpy using central difference . at the peak neither the central difference ( due to rapid rate of change ) or the multidomain average data ( due to a lack of domain robustness ) are reliable and they show significant differences . however below the peak they once again show quantitative agreement providing strong evidence that the domains are robust in this region . and the solid line is@xmath235 and b ) a sudden temperature change , where the symbols are @xmath236 and the solid line is@xmath237 . results from an ensemble of independent initial systems and in addition from a single initial trajectory ( with a time of @xmath238 being computed between each transient trajectory ) are shown for a total of @xmath239 pressure or temperature changes . the serial results have been shifted up , for clarity , by adding 0.2 to the data [ fig-4 ] ] if we substantially increase the duration the time averages ( for each domain ) are constructed on , the peak will be shifted to lower temperatures as previously shown @xcite . this requires the time average to be constructed over some two orders of magnitude more time than the decay time for the correlation function in eq . [ correlation1 ] . this is necessary in order to obtain enough independent samples for a meaningful estimate of the variance , of the instantaneous enthalpy appearing in eq . [ cp - regular ] . for a large macroscopic system we would expect that the specific heat measured over the entire ensemble would differ very little to that measured from any one of its members . we are now in a position to make an unambiguous interpretation of the peak in the specific heat . the peak is observed at the temperature where the system leaves metastable equilibrium and enters a history dependent state that requires averages to be computed by eq . [ multi - d - average ] rather than by direct use of eq . [ dist - standard ] . the calculation of both @xmath229 and @xmath240 will be different for each domain . if we use time averaging to calculate these variables on a time scale that falls within the plateau region for eq . [ correlation1 ] , see fig . [ fig-2]b , then the amount of time chosen to form the average is not critical . the peak occurs because the various ensemble members have become locked in local domains on the time scale that we are able to access . near the peak itself these domains are not expected to be robust . the multidomain average , eq . [ cp - concat ] , gives the heat capacity for a glass with robust domains . at the lowest temperatures the heat capacity reaches the beginning of a plateau , fig . [ fig-3]b . for the constant volume heat capacity @xmath234 this plateau ( within uncertainties due to finite size effects ) has a value that is consistent with the dulong - petit law @xcite , as would be expected for an amorphous solid where the potential energy surface can be modeled as harmonic upon transformation to the orthogonal independent basis set . this is exactly what we would expect from our local domain model at low temperatures . testing the integrated transient fluctuation theorem ( itft ) for a sudden pressure change eq . [ ift - p ] and a sudden temperature change eq . [ ift - t ] provides further evidence that the boltzmann distribution may be used to accurately describe intra - domain statistics in the glassy state and also that in a properly aged glass , the domains are robust with respect to the pressure and temperature changes studied here , fig . [ fig-4 ] . these equations remain valid whether we subject an ensemble of simulations ( multidomain ) to a quench or we sample from a single trajectory ( single domain ) , which remains stuck in a single domain . the accuracy with which these relations are satisfied is powerful independent evidence for the applicability of our assumptions to the systems studied here . the fact that over the times shown in fig . [ fig-4 ] , the itft does indeed yield correct results directly implies that , within experimental tolerance of the data , the phase space domains must be robust and the number of inter - domain transitions must be negligible . unlike the specific heat fluctuation formula this requires that the domains are robust to finite changes of the state rather than infinitesimal changes . thus , given that we have aged the glass sufficiently that domains are robust the number of transitions are negligible over the longest time the fluctuation formulae are computed , we expect eqs . [ ift - p ] & [ ift - t ] to be correct . if we wished to apply the steady state fluctuation theorem matters become more difficult @xcite . we have presented a rigorous development of statistical mechanics and thermodynamics for nonergodic systems where the macroscopic properties are sensibly time independent and the phase space for the ensemble is partitioned into robust domains . using computer simulation we have carried out various tests on a glassy system and shown that apart from the immediate vicinity of the glass transition , the computed results are consistent with our theory . while the intra - domain populations are individually boltzmann distributed , the inter - domain populations are not . a correlation function whose decay to zero requires global boltzmann weighting , has been derived and it has been shown that it decays on a reasonable time scale for ergodic systems but not for nonergodic systems . a second correlation function which decays to zero if the intra - domain populations are boltzmann distributed but globally the inter - domain populations are not , has also been defined . we have developed expressions for obtaining averages in a multiple domain ensemble and shown how single domain averages , which always give correct results in metastable equilibrium , can give spurious results in a history dependent nonergodic ensemble . the statistical mechanics and thermodynamics developed here allow the derivation of expressions for multidomain ensemble averages which give the correct results for time nondissipative nonequilibrium ensembles . the fundamental origin of the peak in the specific heat near the glass transition has been unambiguously shown to be a signature of a transition from metastable equilibrium to a nonergodic multi - domain ensemble . we have shown that the transient fluctuation relations for temperature and pressure quenches provide independent tests of the fundamental hypotheses used in our theory : that intra - domain populations are individually boltzmann distributed , that except in the immediate vicinity of the glass transition the domains are robust with respect to small but finite variations in thermodynamic state variables , and that the inter - domain transition rates are negligible .

gas cooling plays a key role in galaxy formation . according to the hierarchical structure formation model , galaxies form in gravitationally collapsed dark matter halos . the dissipative baryonic matter cools by emitting radiation and condenses into halos to form galaxies @xcite . subsequently stars form through the radiative cooling of interstellar medium ( ism ; e.g. , * ? ? ? the cooling rate of gas varies with its chemical abundance , because different atoms and molecules have different cooling rates . the chemical abundance of cosmic gas evolves as a function of time and environment . in the early universe , the primordial gas consists of mainly hydrogen ( mass fraction of @xmath3 ) and helium ( mass fraction of @xmath4 ) with a negligible amount of light metals ( e.g. * ? ? ? * ) . throughout this paper , all the elements heavier than helium are collectively called ` metals ' . most metals are produced inside stars and spread out to the space by galactic winds driven by supernova ( sn ) explosions . in the metal - enriched gas , the total cooling rate is significantly enhanced relative to that of the primordial gas by the atomic emission lines from recombination of ionised metals . metals may influence galaxy formation in two different ways . firstly , the sf efficiency is increased owing to the shorter gas cooling time . star formation generally takes place in dusty metal - enriched ism , therefore , the enhanced cooling rate by metals would boost up the sf efficiency . however , the energy / momentum feedback by sne may suppress the subsequent star formation after the initial starburst by heating up the ambient gas . this complicates the situation , and it is not clear whether the net effect of metal cooling and sn feedback would be negative or positive . the feedback process is highly nonlinear , therefore , a direct numerical simulation would be a useful tool to explore the effects of feedback and metal cooling . secondly , the metals dispersed into the igm by sn feedback also enhance the cooling of igm . this can lead to the increase of igm accretion onto galaxies , because colder gas can sink into galaxies more easily . both of the above two effects may affect the galaxy growth and cosmic sf history significantly . therefore it is essential to include the effects of metal cooling and chemical enrichment by feedback in the studies of galaxy formation and evolution , and capture the complex two - way interactions between galaxies and igm . in addition , metal enrichment also affects the equation of state by altering the mean molecular weight of gas , which influences the hydrodynamic calculation of gas thermal state . cosmological hydrodynamic simulations are widely used in the studies of galaxy formation and cosmic sf history ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? most of these works included the treatments of radiative cooling by h & he , star formation , and sn feedback . however , some of the simulations did not include the effects of metal cooling and/or chemical enrichment by galactic wind . furthermore , the effects of metal cooling on galaxy growth and cosmic sf history has not been explored in detail using cosmological hydrodynamic simulations and presented in the literature , as it is costly to run large cosmological simulations with and without the effect of metal cooling . in this paper , we investigate the effects of metal cooling and metal enrichment on galaxy growth and cosmic sf history using a series of cosmological hydrodynamic simulations with and without metal cooling . the aim of this paper is to single out the effects of metal cooling among our simulations . this work can be regarded as our initial step towards the long - term goal of developing more complete cosmological hydrodynamic code with physically motivated models of star formation and feedback . the paper is organised as follows . in [ sec : method ] , we describe our simulation method focusing on how we implement the metal cooling . in [ sec : global ] , we study the metal cooling effects on the global properties , such as cosmic sfr , phase space ( @xmath5 vs. @xmath6 ) distribution of gas , and the evolution of four phases ( hot , warm - hot , diffuse , and condensed ) of baryons . we then study the galaxy mass functions ( [ sec : mf ] ) and gas mass fractions ( [ sec : gasfrac ] ) . finally , we discuss and summarise our findings in [ sec : summary ] . we use the updated version of the tree - particle - mesh ( treepm ) smoothed particle hydrodynamics ( sph ) code gadget-2 @xcite for our cosmological simulations . the gravitational dynamics is computed by a treepm algorithm , which uses a particle - mesh method @xcite for the long - range gravitational force and a tree method for the short - range gravitational force @xcite . this hybrid algorithm makes the gravitational force calculation faster than a tree method and allows better force resolution than a pm method in dense regions . the gas dynamics is computed by an sph method . the sph is particularly useful if the simulation needs to resolve large dynamical range , which is an inevitable requirement for the study of galaxy formation in a cosmological context . therefore a treepm - sph simulation can provide a fast and high resolution calculation for both gravitational dynamics and hydrodynamics . the gadget-2 code adopts the entropy - conservative formulation @xcite , which alleviates the overcooling problem that previous sph codes suffered from . our basic simulations include radiative cooling and heating processes for hydrogen and helium using a method similar to @xcite . an external uv background radiation is treated as a spatially uniform photoionising radiation @xcite , and modified to match the ly@xmath7 forest observations @xcite . implementing star formation and feedback from first principles is not feasible in current cosmological simulations , because the spatial & mass - scales of molecular clouds are not resolved . star formation and sn feedback are represented by the subgrid multiphase ism model developed by @xcite . in the multiphase scheme , a single gas particle represents both hot and cold gas . the stars are formed in the cold portion when the density exceeds a given threshold , @xmath8 , which is derived self - consistently within the multiphase ism model . the related parameters , such as the normalisation of gas consumption time and evaporation efficiency of cold gas , are set to satisfy the empirical kennicutt - schmidt law @xcite . the sn feedback returns some fraction of the cold gas to the hot phase , and increases the thermal energy of the hot gas . as an extension to the multiphase ism model , the simulation includes a phenomenological model for sn - driven galactic wind @xcite . the galactic wind is particularly important for distributing the metals produced by sne into the igm . we use the strong kinematic wind with a velocity of 484kms@xmath9 . it has been shown that this model produces favourable results for the luminosity function of lyman - break galaxies at the bright - end @xcite and the hi column density distribution function @xcite at @xmath1 , when compared to the runs without the wind . however , @xcite pointed out the problems of this galactic wind model by comparing with the observations of civ absorption lines in quasar spectra , and suggested that the momentum - driven wind model @xcite is a more viable model , which can carry more metals with lower wind velocities . in this paper we choose not to modify our galactic wind model in order to single out the effects of metal cooling , and to allow direct comparisons to the previous works @xcite . here we focus on the effects of metal cooling on galaxy growth , while @xcite focused on the civ statistics of the igm . the metallicity of gas particles are also tracked by the code , assuming a closed box model for each gas particle . the yield ( @xmath10 ) of 0.02 is assumed . in principle , there could be a time delay between sn explosions and chemical enrichment of the ambient gas . unfortunately , current cosmological simulations do not have sufficient resolution to track the detailed mixing process of metals . in our simulations , we ignore this time delay and assume an instantaneous mixing within each gas particle . this assumption is reasonable , because the overall time step of the simulations is longer than the mixing time - scale on small scales . ideally we would like to track individual metals and compute the cooling rate of each element at each time - step of the simulation . this approach has been used to study the evolution of individual galaxies @xcite . for cosmological simulations , it requires a large memory to track individual metals and the computation becomes exceedingly expensive ( but see * ? ? ? * ; * ? ? ? * for such an attempts ) . our simulations collectively track the total mass fraction of metals ( @xmath11 ) as the measure of metallicity . we obtain the chemical abundance and cooling rates for a given @xmath12 and temperature by interpolating the table given in ( * ? ? ? * hereafter sd93 ) for the standard collisional ionisation equilibrium model , and create a lookup table . metal cooling is implemented in the full range of metallicity in the table from sd93 , from [ fe / h]=-3 to [ fe / h]=1 . in addition , the metal cooling outside of this range is computed by extrapolating the table . we require that the extrapolation does not go lower than the primordial cooling rate . the primordial abundance pattern is used where [ fe / h ] @xmath13 @xcite , and the solar abundance pattern is used where [ fe / h ] @xmath14 @xcite . for @xmath15 [ fe / h ] @xmath16 , the abundance pattern is computed by interpolating between the primordial and solar abundance patterns . note that , [ fe / h ] = @xmath17 - @xmath18 , where @xmath19 is the number density . figure [ fig : xyz_table ] shows the relationships between the metal parameters ( [ fe / h ] , @xmath20 , @xmath21 , and @xmath22 ) and the total metallicity @xmath12 . because the abundance pattern is fixed for a given @xmath12 , [ fe / h ] is a monotonically increasing function of @xmath12 . the main difference between the solar and the primordial abundance pattern is the oxygen or @xmath7-particle enhancement . in the early phase of metal enrichment , type ii sne are the main source of metals and they tend to generate more @xmath7-particles than sn ia . therefore , it is expected that a low metallicity gas to have more @xmath7-particles than the high metallicity one . as mentioned above , we use two different chemical abundance pattern ( at low- and high - end of the @xmath12 values ) to model this evolution in chemical composition . we compute the oxygen abundance based on the table 4 of sd93 . in our code , we incorporate the metal cooling effect by adding the additional cooling contribution from metals on top of the primordial cooling rate , which is computed in a similar fashion as described in @xcite . metal enrichment changes the mean molecular weight @xmath23 , which is needed to compute the temperature and internal energy of gas . in order to compute @xmath23 , we need to know the electron number density @xmath24 , which varies with metallicity . we obtain the values of @xmath24 from sd93 in the same way as we obtained the metal cooling rates . our electron number density estimate ignores the electrons from photoionised metals by the uv background radiation , but we expect that the number of electrons from ionised metals will be small . varying the mean molecular weight @xmath23 also influences the sf threshold density @xmath8 in the multiphase ism model of @xcite , as well as the cold and hot gas fractions of multiphase gas particles . in this paper , we consider two different treatments of @xmath8 . in the ` constant @xmath8 ' model , we take the value of @xmath8 as in the original formulation by @xcite and do not modulate @xmath8 by the varying @xmath23 . in the ` varying @xmath8 ' model , we take into account of the modulation of @xmath8 by the change in @xmath23 . we use three series of simulations with varying box size and resolution . the effects of metal enrichment and metal cooling are tested by comparing the runs listed in table [ tbl : simulation ] . the effects of galactic wind are also tested by turning on and off the wind . the adopted cosmological parameters of all simulations are @xmath25 , which mostly concur with the results of the wilkinson microwave anisotropy probe ( wmap1 ) @xcite . [ cols="<,^,^,^,^,^,^,<,^",options="header " , ] our naming conventions for the simulations are as follows . the first part of the run name denotes the particle number and box size used for the simulation series . the run with the extension of ` mc ' ( for ` metal cooling ' ) adopts the ` constant @xmath8 ' model for metal cooling , the run with the extension ` mv ' ( for ` metal , varying ' ) adopts the ` varying @xmath8 ' model , the run with the extension ` nw ' has no galactic wind , and the run without any extension uses the cooling rates for the primordial chemical composition ( i.e. , only h and he ) . the ` n216l10 ' series is the fiducial simulation set in our study . we implement one simulation without galactic wind ( ` n216l10nw ' run ) . the ` n216l10 ' run has the same physical models as the ` q4 ' run used in @xcite and @xcite with no metal cooling . to examine the resolution effect , we implement lower resolution simulations , the ` n144l10 ' series . owing to the comparably small box size and missing long wavelength perturbations , we evolve the n144l10 and n216l10 series only down to @xmath26 . to evolve the simulation toward lower redshifts , we need a simulation with a larger box size . therefore we implement the ` n288l34 ' series , which has a larger box size than other series , but a lower resolution . this series is evolved down to @xmath0 . the runs within the same series use the identical initial condition , which makes the comparison more robust and enables halo - by - halo comparison if needed . in cosmological simulations , dark matter , gas , and stars are represented by particles , and these particles are monte carlo representations of matter distribution . in order to study galaxy formation using a cosmological simulation , we need a working definition for simulated galaxies . in this paper the simulated galaxies are defined as isolated groups of star and gas particles , identified by a simplified variant of the algorithm developed by @xcite . in more detail , the code first computes a smoothed baryonic density field to identify candidate galaxies with high density peaks . the full extent of these groups are found by adding gas and star particles to the groups in the order of declining density . if all @xmath27 is the minimum number of gas and star particles that constitute one isolated group . in this paper we set @xmath28 . ] nearest neighbour particles have lower densities , this group of particles is considered as a new group . if there is a denser neighbour , the particle is attached to the group to which its nearest denser neighbour already belongs to . if two nearest neighbours belong to different groups and one of them has less than @xmath27 particles , these the two groups are merged . if two nearest neighbours belong to different groups and both of them has more than @xmath27 particles , the particles are attached to the larger group , leaving the other group intact . in addition , the gas particles in groups should be denser than @xmath29 . in this paper we are not concerned with groups without star particles . the fraction of such groups are very small , therefore they do not affect our conclusions . % by metal cooling throughout the entire redshift range . ] , but for the n288l34 series , which goes down to @xmath0 . the n216l10mc run is also shown for a resolution comparison . the green dot - dashed line ( hs03 ) is for the analytic model of ( * ? ? ? * eq.(2 ) ) . ] figure [ fig : sfr_n216l10 ] shows the effect of metal cooling and galactic wind on the cosmic sf history for the n216l10 series . the comparison of n216l10mc and n216l10 runs clearly shows that metal cooling enhances the cosmic sfr throughout all redshifts by @xmath30% . previously , ( * ? ? ? 13 ) argued that the sfr density is enhanced by metal cooling mostly at lower redshifts and hardly at high-@xmath31 , however , our simulations suggest otherwise . ( see [ sec : summary ] for more discussion on this point . ) metallicity needs to be higher than @xmath32 to noticeably increase the gas cooling rate , because the metal cooling rate is much smaller than that by h and he at @xmath33 . the metallicity of diffuse igm is generally lower than @xmath34 at @xmath35 , and the metal cooling effect on igm is small . however the local sfr could be enhanced significantly even at high-@xmath31 when the ism is enriched to @xmath36 , which can be easily achieved after a few events of sn ii . our results suggest that star formation at high-@xmath31 can also be enhanced by metal cooling , and not just at low-@xmath31 . we also find that the peak of the cosmic sf history hardly shifts by the introduction of metal cooling , consistently with the estimate by @xcite . the n216l10mv run has a lower sfr at @xmath37 and a higher sfr at @xmath38 than the n216l10mc run . this is because at high-@xmath31 , the value of @xmath8 is higher in the ` mv ' run than in the ` mc ' run due to lower metallicity and mean molecular weight , leading to a larger fraction of gas not being able to satisfy the sf criteria in the n216l10mv run than in the n216l10mc . the opposite is true at @xmath39 . figure [ fig : sfr_n216l10 ] also shows that the sfr in the n216l10nw run is higher than other runs by an order of magnitude . this is because the gas can cool very efficiently without being ejected by the galactic wind . although the current galactic wind models in cosmological simulations are not well established yet and need to be improved , both theoretical and observational studies evidently suggest the important role of galactic wind feedback @xcite . as expected , the n144l10mc run has lower sfr densities at @xmath40 compared to the n216l10 series . this is because a lower resolution run is unable to resolve low - mass halos that would otherwise forms stars in them . when the star formation at high-@xmath31 is underestimated , some of the gas remain unconverted into stars until lower redshift , resulting in a higher sfr at @xmath41 in the n144l10mc run than in the n216l10mc run . figure [ fig : sfr_n288l34 ] shows the cosmic sf history down to @xmath0 for the n288l34 series . again , the sfr density in the n288l34mc run is systematically higher than that in the n288l34 run at all redshifts . moreover , the difference in sfr density between the two runs increases at @xmath39 . by @xmath42 the igm metallicity becomes high enough to noticeably increase the cooling rate , and the accretion of igm onto galaxies becomes more efficient , which later fuels the star formation . our results suggest that the cosmic sfr density is enhanced by metal cooling at @xmath43 by both of the following two effects : 1 ) more efficient igm accretion onto galaxies due to higher metallicity of igm , and 2 ) increased sf efficiency due to higher cooling rate with the contribution from metal cooling . in figure [ fig : sfr_n288l34 ] the analytic model of ( * ? ? ? * eq.(2 ) , hereafter h&s model ) is also shown in the green dot - dashed line . this model is a result of many cosmological sph simulations with increasing resolution , and the authors gave a theoretical interpretation to the empirical fitting formula for the cosmic sfr density . since our runs adopt a lower value of @xmath44 than the original value used to derive the h&s model ( @xmath45 ) , we modify the h&s model by rescaling the @xmath46 parameter in their eq.(2 ) and eq.(47 ) . this reduces the sfr density at @xmath47 by @xmath48% , but not much at @xmath49 . the results of the n288l34 and n216l10mc runs agree well with the h&s model at @xmath41 and @xmath50 , respectively . this is expected , because the n288l34 run adopts the same physical models as in the original simulations that were used to derive the h&s model . at high-@xmath31 , both runs fall short compared to the h&s model owing to insufficient resolution . the metal enrichment and metal cooling also change the phase space distribution ( @xmath51 diagram ) of cosmic gas . figure [ fig : phase.n216l10 ] shows the state of cosmic gas at @xmath1 for the n216l10 series . as shown in panel ( @xmath52 ) , the baryons in the universe can be broadly categorised into four different phases according to their overdensity and temperature : ` hot ' , ` warm - hot ' , ` diffuse ' , and ` condensed ' @xcite . the first two phases are mostly the shock - heated gas in clusters and groups of galaxies . the ` diffuse ' phase is mostly the photoionised igm with lower temperature , which can be observed as the ly@xmath7 forest . the tight power - law relationship between @xmath5 and @xmath6 at @xmath53 and @xmath54k is governed by the ionisation equilibrium @xcite , where @xmath55 is the mean density of baryons at @xmath56 . the ` condensed ' phase is the high density , cold gas in galaxies . because the primordial cooling curve has a sharp cutoff at @xmath57k , the temperature of condensed phase decreases only down to this temperature in figure [ fig : phase.n216l10]a , b . the characteristic finger - like feature at @xmath58 is due to the pressurisation of the star - forming gas by sn feedback in the multiphase ism model of @xcite . a fitting formula for this effective equation of state was derived by @xcite , which is shown by the blue dashed lines . above the sf threshold density @xmath8 , the gas particles in the simulation go into the multiphase mode and are allowed to form stars . the most noticeable and interesting change in the @xmath51 plot by the metal cooling is in the distribution of ` condensed ' gas . star formation and sn explosions occur in the multiphase star - forming gas , therefore the immediate effect of metal cooling appears in the condensed phase . the presence of gas at @xmath59k with metal cooling results from the effective temperature calculation for the multiphase medium , and not from the direct gas cooling down to @xmath59k . according to the multiphase scheme in @xcite , the star forming gas consists of two phases : hot gas ( @xmath60k @xmath61 @xmath62k ) and cold gas ( @xmath63k ) . the effective temperature for the multiphase gas is computed from the internal energy of hot and cold gases , which are derived using the cold gas fraction . the cold gas fraction is computed from the cooling rate of hot gas and the gas density . the original gadget imposes the effective temperature of @xmath64k at the sf threshold density . in our metal cooling scheme , however , the cooling rate of hot gas increases , and the cold gas fraction increases . therefore , the multiphase gas above the sf threshold density with higher metallicity could have lower effective temperature of @xmath59k in our implementation . in the n216l10mc run , the value of @xmath8 was kept fixed as the original value , which causes a sharp edge at @xmath65 for the multiphase gas . in the n216l10mv run , the value of @xmath8 was varied according to the change in mean molecular weight , therefore the condensed gas has a spread in both temperature and density near @xmath65 . galactic wind returns the metal - enriched , star - forming gas to the hot and low density igm , as evidenced in the @xmath66 diagrams . this metal - enriched , low - density ( @xmath67 ) gas is absent in the n216l10nw run , which clearly shows the necessity of galactic wind to explain the metallicity observed in ly@xmath7 forest . the phase distribution of igm in the n216l10 , n216l10mc , and n216l10mv runs are very similar . this is presumably because the effects of metal cooling is not so significant yet in the igm at @xmath68 . as we will discuss in the next section , this is not be the case at @xmath2 . the connection between the four phases of cosmic gas is important in galaxy formation , and it is beneficial for us to study their evolution for a better understanding of metal cooling effects . a conventional view is that the diffuse gas is shock - heated to hot or warm - hot phase , which later cools down to the condensed phase @xcite . @xcite also suggested that some diffuse gas can migrate to the condensed phase without being shock - heated , which they called the ` cold mode ' accretion . figure [ fig : evolv4 ] shows the evolution of mass fractions of different phases . all of our simulations show that the diffuse phase continuously decreases from high-@xmath31 to low-@xmath31 , while the ` warm - hot + hot ' and ` condensed ' phases continuously increase , consistently with the general expectations . the igm ( i.e. , hot + warm - hot + diffuse ) is accreted onto galaxies , to become the condensed phase . the diffuse component ( panel @xmath52 ) decreases from @xmath69% at @xmath70 to @xmath71% at @xmath0 . the ` warm - hot + hot ' component ( panel @xmath72 ) increases from almost zero at @xmath70 to @xmath73% at @xmath0 . the mass fraction of the total igm ( panel @xmath74 ) decreases from @xmath75% at @xmath70 to @xmath76% at @xmath1 . this reflects the increase of the condensed phase ( panel @xmath77 ) from a few percent at @xmath70 to @xmath78% at @xmath1 . these mass fractions are in general agreement with the results of @xcite . the comparison between n288l34 and n288l34mc runs shows interesting differences . at @xmath68 , the mass fractions of all four phases are similar in the two runs . however , at @xmath2 , the igm mass fraction continues to decrease in the n288l10mc run , while it becomes almost constant in the n288l10 run . a similar behaviour is seen for the condensed phase , and the n288l10mc continues to increase its condensed mass . this suggests that the metal cooling enhances the igm accretion onto galaxies mostly at @xmath39 , which supplies the fuel for star formation . the cooling of warm+hot phase into diffuse phase is also enhanced , leading to a larger mass fraction of diffuse component in the n288l34mc run than in the n288l34 run . ( the same is true for n216l10mc and n216l10 runs . ) the n216l10nw run shows stark differences from all other runs . it has a significantly lower igm mass fraction , and a very large mass fraction in the condensed phase . this shows that the igm is mostly enriched by galactic wind , and that most of the masses are returned into the ` warm - hot + hot ' phase , but not to the diffuse phase . galaxy stellar mass function ( gsmf ) tells us how the stellar mass is distributed in different galaxies , and its redshift evolution reflects the growth of structure in the hierarchical universe . figure [ fig : mf_star ] compares the gsmf in different simulations . we only show the range of @xmath80 , which corresponds to the limiting mass of 32 star particles in n216l10 runs . we chose 32 star particles as our resolution limits because all mass function start to turn over around this mass range . panel ( @xmath52 ) shows that the run with no galactic wind ( n216l10nw ) completely overestimates the gsmf , and the run with lower resolution ( n144l10 ) underestimates the number of lower mass galaxies at @xmath81 relative to the n216l10 run . panel ( @xmath72 ) shows that the metal cooling increases the masses of galaxies by @xmath82% at @xmath1 . as a result , both the n216l10mc and n216l10mv runs show a slight increase in the number of galaxies at the massive end , with a slightly stronger enhancement in the n216l10mv run . figures [ fig : mf_star]c , d demonstrate that the metal cooling also changes the redshift evolution of gsmf . in the n216l10mc run , the peak of mass function is shifting more toward the massive side than in the n216l10 run at @xmath1 . from @xmath83 to @xmath1 , the low - mass galaxies merge to form more massive galaxies , increasing the number of galaxies with @xmath84 furthermore . it also seems that the formation of low - mass galaxies ( @xmath81 ) is enhanced by metal cooling at early times ( @xmath85 ) . however we find that the behaviour of mf at the low - mass end is somewhat dependent on the threshold density for the grouping . as mentioned in [ sec : groups ] , our grouping code imposes a threshold gas density of @xmath29 for a gas particle to be part of a galaxy . when we lower this threshold density , we find that the two mfs for the n216l10 and n216l10mc converged , as more particles are incorporated into galaxies at the outskirts of galaxies . metal cooling increases the gas density in galaxies , enabling more low - mass galaxies to satisfy the gas density threshold and to be identified as simulated galaxies . therefore the behaviour of mf at @xmath86 is somewhat dependent on the choice of the outer threshold density of galaxies . figure [ fig : mf_comp ] compares the gas and baryonic ( star + gas ) mfs for the n216l10 series at @xmath1 ( panels @xmath52 & @xmath72 ) , and the redshift evolution of the baryonic mf from @xmath87 to @xmath1 in the n216l10 ( panel @xmath74 ) and n216l10mc ( panel @xmath77 ) runs . here we show only the range of @xmath88 , which corresponds to the limiting mass of 32 gas particles . panel ( @xmath52 ) shows that more gas is converted into stars in the runs with metal cooling ( n216l10mc and n216l10mv ) compared to the n216l10 run , while panel ( @xmath72 ) shows that the baryonic mf is similar in all the runs . as we discussed in [ sec : evolv4 ] , the igm accretion onto galaxies is not so much enhanced yet before @xmath89 due to relatively low igm metallicity , which explains the similarity in the total baryonic mf . in the runs with metal cooling , the peak of gsmf is shifted toward higher mass , whereas the peak of gas mf is shifted toward lower mass . this suggests that the enhanced gsmf at @xmath1 is due to the increased sf efficiency by metal cooling within the galaxies . again , the apparent enhancement in the number of low - mass galaxies in the n216l10mc run compared to the n216l10 run is somewhat dependent on the threshold density of grouping , therefore it should be interpreted with caution . as the accretion of igm onto galaxies become more efficient at @xmath2 due to the increased igm cooling by the metals , we expect that the total baryonic mass of galaxies would be more enhanced at @xmath0 than at @xmath1 . figure [ fig : mf_comp_n288l34 ] shows the mfs at @xmath1 and @xmath0 in the n288l34 series , and it clearly demonstrates that this expectation is true . here we show only the range of @xmath91 , which corresponds to the limiting mass of 32 gas particles . the general trend in the three ( star , gas , and total baryon ) mfs is similar to that we saw in figure [ fig : mf_comp ] , although the difference at @xmath1 between the n288l34 and n288l34mc runs is slightly smaller than in the n216l10 series due to poorer resolution . figure [ fig : mf_comp_n288l34]f shows most prominently the enhancement of the total baryonic mf at @xmath0 in the n288l34mc run owing to the metal cooling . in panels ( @xmath92 ) & ( @xmath93 ) , the n288l34 run actually has a longer tail at the most massive - end than the n288l34mc run . the reason for this feature is not fully clear , but it may be related to the balance between igm accretion and feedback . owing to metal cooling , igm accretion rate increases in the mc run , and the sfr is also enhanced , leading to a stronger feedback . the amount of mass loss is greater in low mass galaxies , but the net heating of igm is more significant for massive galaxies . the mc run has stronger feedback , therefore its feedback heating may become more significant than igm accretion for very massive galaxies . the significant igm heating suppresses the growth of massive - end of mass function from @xmath1 to @xmath0 and results in shorter tail for the mc run at @xmath0 as shown in figure [ fig : mf_comp_n288l34]e , f . for a fixed galaxy baryonic mass , the enhancement of star formation by metal cooling would decrease the gas mass fraction , @xmath94 . figure [ fig : gfrac_evol_n216l10 ] shows @xmath95 as a function of galaxy stellar mass for the n216l10 series . panel ( @xmath52 ) shows the data only from n216l10mc run at @xmath1 , and each data point corresponds to a simulated galaxy . to characterise the distribution , we compute the following two quantities in each logarithmic stellar mass bin : ` average ' and ` median ' . the ` average ' is the ratio of total gas mass to total baryonic mass for all the galaxies in each mass bin , i.e. , @xmath96 . the ` median ' case is simply the median of @xmath95 values in each mass bin . both quantities show a similar trend , however , there is a sharper drop - off at @xmath97 for the ` median ' case in figure [ fig : gfrac_evol_n216l10]a . this mass - scale corresponds to 32 star particles in the n216l10 series . we find that there are many galaxies with @xmath98 above this mass - scale , which causes the sharp drop - off in the ` median ' line . below this limiting mass , galaxies are not resolved well , which results in an underestimate of star formation and an overestimate of @xmath95 . if we had a higher resolution simulation with finer particle masses , this limiting mass - scale would shift to a lower mass . therefore the location of this sharp drop - off is currently determined by the resolution of our simulation . however , dark matter halos would stop forming stars at some lower limiting halo mass , if we had an infinitely high - resolution simulation , this lower limit to the galaxy mass is presumably determined by the photoevaporation of gas by the uv background radiation @xcite . recent works suggest that star formation could be suppressed by the uv background in halos with @xmath99 at @xmath89 . galaxies with @xmath100 would reside in halos with @xmath101 . the n216l10 series would resolve such a halo with @xmath102 dark matter particles , and its mass resolution is actually close to the astrophysical limit for dwarf galaxy formation at @xmath89 . therefore the sharp drop - off in @xmath95 at @xmath100 may not be so far from the true answer . we find that @xmath103 increases with decreasing @xmath104 at all redshifts , regardless of metal cooling and wind effects . this trend is qualitatively consistent with current observations . @xcite estimate the gas fraction as a function of stellar mass using the rest - frame uv - selected star - forming galaxies at @xmath105 , and show that the gas fraction decreases with increasing stellar mass . and using the mean gas and stellar mass , they find the average @xmath106 , which agrees with the predicted gas fraction for massive galaxies in our simulation . in addition , @xcite reported that the average neutral gas fraction is @xmath107 for the local dwarf galaxies selected from the sloan digital sky survey . in our n216l10 series with metal cooling , the ` average ' @xmath95 reaches 0.6 for galaxies with @xmath108 at @xmath1 & 4 . figures[fig : gfrac_evol_n216l10]b , c , d show the redshift evolution of @xmath95 for the n216l10 series . in the runs with metal cooling ( n216l10mc and n216l10mv ) , @xmath95 is lower than in the n216l10 run by @xmath30% at all redshifts for galaxies with @xmath109 . this result suggests that the metal cooling reduces @xmath95 owing to more efficient star formation . the values of @xmath95 seem to be more convergent at the massive - end ( @xmath110 ) . in addition , we find that galaxies in the n216l10nw run are the most gas - rich , because almost no galactic gas is returned to the igm . figures [ fig : gfrac_evol_n216l10]b , c , d also show that @xmath95 is higher at higher redshifts . for example , the well - resolved galaxies with @xmath111 have @xmath112 at @xmath87 , but @xmath113 at @xmath1 . as we saw in figure [ fig : mf_star]d , there is a considerable growth in gsmf from @xmath87 to @xmath1 . together with the decrease in the gas fraction , these results suggest that most of the gas accreted during @xmath114 has been converted into stars . figure [ fig : mgfrac_resolution]a highlights the redshift evolution of @xmath95 from @xmath87 to @xmath0 in the n288l34mc run . the gas mass fraction clearly decreases with decreasing redshift as the gas is converted into stars . the rate of decrease is greater at @xmath115 than at @xmath116 , with @xmath117 & 0.2 for @xmath118 , & 1 , respectively , for galaxies with @xmath119 . figure [ fig : mgfrac_resolution]b shows the effect of resolution on @xmath95 . the location of the drop - off in @xmath95 shifts to lower masses as the resolution is increased from n144l10mc to n216l10mc run . the location of the drop - off is at around our resolution limit the vertical lines in _ panel ( b)_. therefore the values of @xmath95 are overestimated in unresolved galaxies . using cosmological hydrodynamic simulations with metal enrichment and metal cooling , we studied their effects on galaxy growth and cosmic sfr . owing to metal cooling , the sfr density increases about 20% at @xmath1 and about 50% at @xmath0 . our results suggest that metal cooling enhances the star formation through two different processes : 1 ) more efficient conversion of local ism into stars ( i.e. , increase of local sf efficiency ) , and 2 ) the increase of igm accretion onto galaxies . the former process is in effect essentially at all times at @xmath120 , because the local ism can be instantaneously enriched by sn explosions . this process enhances the sfr , but does not noticeably change the total baryonic mass of galaxies . the latter process , on the other hand , can increase the total baryonic mass of galaxies , as well as enhancing the overall sfr density by supplying more gas for star formation . this process becomes effective only at lower redshifts ( @xmath39 ) , because it takes some time to enrich the igm up to @xmath121 . in this paper , we used two different prescriptions for star formation in the runs with metal cooling : 1 ) the ` constant @xmath8 ' scheme and 2 ) the ` varying @xmath8 ' scheme . the value of @xmath8 was fixed in the former scheme , while it was modulated according to the metallicity in the latter scheme . both schemes show similar increase of cosmic sfr density ( see figure [ fig : sfr_n216l10 ] ) , which implies that the overall sf enhancement by metal cooling is not so sensitive to the choice of @xmath8 ( but with some differences as we discussed in [ sec : cos_sfr ] ) . the multiphase ism model for star formation by @xcite contains two free parameters : @xmath8 and the normalisation of gas consumption time - scale , @xmath122 . the values of these two parameters were originally chosen to match the empirical kennicutt - schmidt law @xcite using simulations of isolated disk galaxies . since we did not change the value of @xmath122 in the runs with metal cooling ( as well as @xmath8 in the ` mc ' runs ) , it is possible that our simulations may now violate the kennicutt law . however , as we show in figure [ fig : kenni ] , the plots of @xmath123 vs. @xmath124 for the n216l10 and n216l10mc runs are not so different from each other . this can be understood as follows . as figure [ fig : evolv4]c showed , metal cooling increases the density and lowers the temperature of star - forming gas . the fundamental scaling relationship between sfr and cold gas density does not need to change when the metal cooling is introduced ; the star - forming gas particles would simply slide upward along the kennicutt law , @xmath125 . in the case of n216l10mv run , we varied @xmath8 according to the gas metallicity , primarily scattering it to lower values because higher metallicity increases the cooling rate and lowers @xmath8 . therefore more gas particles become eligible to form stars and @xmath124 is scattered upward above the kennicutt law for a given value of @xmath123 , resulting in a broader deviation from the power - law relationship of the kennicutt law near @xmath126 . at higher values of @xmath123 , @xmath124 is enhanced , but still within the range of the kennicutt law . this difference can be alleviated by adjusting the sf timescale ( see * ? ? ? * ) , however , we try to emphasise the effect of metal cooing on sfr by keeping this parameter the same in all the runs . further investigations of the hi aspect of our simulations is beyond the scope of the present paper , and we will present the results elsewhere . upon evaluating the effect of metal cooling , @xcite considered two different gas phases where the cooling become important . one is the diffuse gas in galactic halos , which must radiate its thermal energy in order to collapse onto the high - density ism , and the other is the multiphase star - forming gas . they argued in their 5.2 that the cosmic sfr density at @xmath127 would not be affected by the metal cooling very much , because at high-@xmath31 cooling is so efficient that the gas in diffuse atmospheres of halos cools nearly instantly , even without any metal cooling . they expected their model results to be largely independent of metal enrichment , because in this regime the evolution of sfr density is driven by the fast gravitational growth of the halo mass function . for the star - forming multiphase ism , they argued that the parameters of their model ( e.g. cold gas evaporation efficiency and gas consumption time - scale ) can be adjusted such that the normalisation of the kennicutt law can be maintained , yielding to first order unaltered model predictions . however , our simulations , in particular the comparison between the n216l10 and n216l10mc run , suggest that the metal cooling can enhance the sfr density even at @xmath127 by @xmath128% . this is because the gas density becomes higher and temperature becomes lower ( see figure [ fig : evolv4]c ) by responding to the enhanced cooling rate by metals . in summary , we consider that our study serves to demonstrate the importance of metal enrichment and metal cooling in galaxy formation , beyond our original motivation to perform a simple numerical study . our study highlights the role of metals at low-@xmath31 and high-@xmath31 to enhance the star formation in the universe , and demonstrates that the metals increase the sf efficiency in the star - forming ism , as well as enabling more igm to accrete onto galaxies and fuel star formation . we expect that metal cooling would also alter some of the galaxy properties such as the mass - metallicity relationship ( e.g. , * ? ? ? * ) and specific sfr ( e.g. , * ? ? ? we plan to investigate these issues in the future , as well as alternative models for star formation and feedback . we thank v. springel for allowing us to use the updated version of gadget-2 code for our study , and for useful comments on the manuscript . kn is grateful for the hospitality of institute for the physics and mathematics of the universe , university of tokyo , where part of this work was done . we are also grateful to the anonymous referee for constructive comments which improved this paper . this research was supported in part by the national aeronautics and space administration under grant / cooperative agreement no . nnx08ae57a issued by the nevada nasa epscor program , by the national science foundation through teragrid resources provided by the san diego supercomputer center ( sdsc ) , and by the president s infrastructure award at unlv . the simulations were performed at the unlv cosmology computing cluster and the datastar at sdsc . d. n. , verde l. , peiris h. v. , komatsu e. , nolta m. r. , bennett c. l. , halpern m. , hinshaw g. , jarosik n. , kogut a. , limon m. , meyer s. s. , page l. , tucker g. s. , weiland j. l. , wollack e. , wright e. l. , 2003 , apjs , 148 , 175

early - type galaxies in clusters exhibit a linear color - magnitude ( cm ) relation indicating that bright galaxies are systematically redder than their faint cluster companions ( visvanathan & sandage 1977 ) . this remarkable relation shows very small scatter ( @xmath3 magnitude ) in high precision photometry of local clusters such as coma and virgo ( bower , lucey & ellis 1992a , 1992b , hereafter ble92 ) and can be extended to clusters at medium - to - high redshift ( @xmath4 ) ( ellis et al 1997 , stanford , eisenhardt & dickinson 1998 ) . a first attempt at explaining the universality of the cm relation involves using the age of each galaxy as the main determinant of its color . ageing stellar populations redden progressively as stars with decreasing initial mass evolve off the main sequence . therefore , if the colors of cluster galaxies are purely controlled by age , the small scatter about the cm relation implies a nearly synchronous star formation process for all galaxies of a given mass , while the slope of the cm relation implies systematically older ages for more massive galaxies . as shown most recently by kodama & arimoto ( 1997 ) , such a picture is highly unlikely because it does not preserve the slope nor the magnitude range of the cm relation in time . another important factor that affects the colors of stellar populations is metallicity . at fixed age , a more metal - rich stellar population will appear redder and fainter than a more metal - poor one ( e.g. , worthey 1994 ) . hence , increasing metallicity at fixed age has a similar effect on colors as increasing age at fixed metallicity . this is usually referred to as the _ age - metallicity degeneracy _ ( worthey 1994 ) . several studies have shown that cm relation of cluster elliptical galaxies could be primarily driven by metallicity effects ( larson 1974 ; matteuci & tornamb 1987 ; arimoto & yoshii 1987 ; bressan , chiosi & tantalo 1996 ; kodama & arimoto 1997 ) . the physical mechanism usually involved is that of a galactic wind : supernovae - driven winds are expected to be more efficient in ejecting enriched gas , and hence in preventing more metal - rich stars from forming , in low - mass galaxies than in massive galaxies with deeper potential wells . although age is generally assumed to be the same for all galaxies in these studies , this has not been proven to be an essential requirement . in fact , scenarios in which e / s0 galaxies progressively form by the merging of disk galaxies ( schweizer & seitzer 1992 ) in a universe where structure is built via hierarchical clustering also predict that the cm relation is driven primarily by metallicity effects ( kauffmann & charlot 1998 ) . moreover , age effects could be important if , for example , there is sufficiently strong feedback from early galaxy formation to bias the luminous mass distribution of subsequent generations of galaxies by the heating of intergalactic gas . in this paper we present a new , more model - independent approach for evaluating the full range of ages and metallicities allowed by the spectro - photometric properties of early - type galaxies in clusters . the method is based on the construction of age - metallicity diagrams constrained by the colors of early - type galaxies in the nearby coma cluster and in 17 clusters observed with the _ hubble space telescope _ ( _ hst _ ) at redshifts up to @xmath5 ( stanford et al . such an analysis has hitherto been hindered because of the lack of both accurate stellar libraries for different metallicities and reliable morphological information on cluster galaxies at medium - to - high redshifts . our results can subsequently be reframed into specific theories of galaxy formation , since they will be indispensable for any model that seeks to produce galaxies resembling those actually observed . in 2 we present the spectral evolution models used in this paper . the cluster sample is described in 3 . in 4 we construct the age - metallicity diagrams allowed by the observations , and in 5 we compute the corresponding ranges in mass - to - light ratio and in several commonly used spectral indices . we discuss our main conclusions in 6 . we compute the spectral evolution of early - type galaxies using the latest version of the bruzual & charlot ( 1998 ) models of stellar population synthesis . these span the range of metallicities @xmath6 and include all phases of stellar evolution , from the zero - age main sequence to supernova explosions for progenitors more massive than @xmath7 , or to the end of the white dwarf cooling sequence for less massive progenitors . in addition , the models predict the strengths of 21 stellar absorption features computed using the worthey et al . ( 1994 ) analytic fitting functions for index strength as a function of stellar temperature , gravity and metallicity . this constitutes the standard `` lick / ids '' system that is often used as a basis for spectral diagnostics in early - type galaxies . the resulting model spectra computed for stellar populations of various ages and metallicities have been checked against observed spectra of star clusters and galaxies ( bruzual et al . 1997 ; bruzual & charlot 1998 ) . the uncertainties in the models are discussed in charlot , worthey & bressan ( 1996 ) . these can reach up to 0.05 mag in rest - frame @xmath8 , 0.25 mag in rest - frame @xmath9 and a 25% dispersion in the @xmath10-band mass - to - light ratio . with these uncertainties in mind , we will concentrate more on understanding the trends seen in the observations than on inferring absolute age and metallicity values . it is worth noting that the most massive elliptical galaxies exhibit [ mg / fe ] ratios in excess of that found in the most metal - rich stars in the solar neighborhood ( by @xmath11 dex ; see worthey , faber , & gonzalez 1992 ) . while this may limit the accuracy of the predicted mg@xmath12 indices of bright elliptical galaxies , the recent models of bressan et al . ( 1998 ) convincingly show that an enhancement in light elements at fixed total metallicity has virtually no effect on the other spectrophotometric properties of model stellar populations . we approximate model early - type galaxies by instantaneous - burst stellar populations . the reason for this is that we aim at constraining the age and metallicity ranges of stars dominating the light of early - type galaxies , whose photometric properties are well represented by instantaneous - burst populations . in fact , this is true even if the galaxies underwent subsequent small amounts of star formation or if the epoch of major star formation was extended over several billion years ( e.g. , fig . 1 of charlot & silk 1994 ) . the predicted colors of our models at fixed age and metallicity agree well with the results of more refined calculations including the effects of infall and galactic winds for corresponding values of the mean age and metallicity ( kodama & arimoto 1997 , bressan , chiosi & tantalo 1996 ) . for example , adopting metallicities matching the luminosity weighted metallicities @xmath13 in table 2 of kodama & arimoto yields @xmath14 and @xmath9 colors that agree to better than 0.05 mag with the results from these authors at an age of 15 gyr . such a discrepancy is well within the errors of current population synthesis models ( charlot et al . 1996 ) . in the remainder of the present paper , the initial mass function ( imf ) is taken from scalo ( 1986 ) and is truncated at 0.1 and 100@xmath15 . we use the above models to compute the locations in the age metallicity diagram of stellar populations satisfying specified spectro - photometric properties . figure 1 shows four such age metallicity diagrams corresponding to imposed values of the @xmath14 and @xmath9 colors and mg@xmath16 and h@xmath17 spectral indices . these quantities are chosen here because they can be constrained by many observations of early - type galaxies ( 3 and 5 ) . in each panel , the models satisfying the same value of the spectro - photometric property of interest are related by a continuous line , different lines corresponding to different imposed values . with this definition , the slope of a line in the age metallicity diagrams indicates the relative sensitivity of the color or index under consideration to age and metallicity . vertical lines would correspond to a sensitivity purely to age , and horizontal lines to a sensitivity purely to metallicity . figure 1 then shows immediately that the @xmath14 and @xmath9 colors and mg@xmath16 index depend more strongly on metallicity than on age , while the h@xmath17 index depends more strongly on age than on metallicity . we will return to this point in 4 and 5 . the relative dependence of the spectro - photometric properties of instantaneous - burst populations on age and metallicity has been previously investigated by worthey ( 1994 ) . he used the parameter @xmath18 at fixed color or index to represent the ratio of the change @xmath19 in age needed to counterbalance a change @xmath20 in metallicity in order to keep that color or index unchanged . the difference between worthey s and our approach is that he computed a single effective value of @xmath18 for each spectro - photometric property , while the different lines in figure 1 show the behavior of the @xmath18 slope for different values of the color or index under consideration . for comparison , the arrow in each panel of figure 1 indicates the @xmath21 vector obtained by worthey ( 1994 ) . in each case , the general agreement with the mean slope of the lines is good . table 1 gives a more quantitative comparison between worthey s ( 1994 ) and our results . we computed linear fits to all lines in figure 1 and then a linear fit between the derived slopes and their corresponding color or index value . the slopes and zero points of these relations for each spectro - photometric property are listed in columns ( 2 ) and ( 3 ) of table 1 . we then evaluated @xmath18 for four values of the @xmath14 and @xmath9 colors ( and corresponding model predictions for the mg@xmath16 and h@xmath17 indices ) matching the properties of early - type galaxies at four magnitudes @xmath22 , @xmath23 , @xmath24 and @xmath25 along the local cm relation ( ble92 ; see 3 ) . the agreement with worthey s ( 1994 ) predictions is seen to be of the order of @xmath26 % . it is worth noting that our generalized fits deviate significantly from worthey s `` 3/2 rule '' which takes the @xmath27 ratio to be 1.5 for any color . c|cc|cccc|c property & slope & zero point & @xmath28= 17.5 & @xmath28= 19.0 & @xmath28= 20.5 & @xmath28= 22.0 & worthey + [ @xmath14 ] & 0.444 & 0.653 & 1.19 & 1.24 & 1.30 & 1.35 & 1.5 [ @xmath9 ] & 0.886 & @xmath311.004 & 1.59 & 1.69 & 1.79 & 1.89 & 1.9 [ mg@xmath16 ] & 0.118 & 0.976 & 1.33 & 1.36 & 1.40 & 1.45 & 1.7 [ h@xmath17 ] & @xmath310.092 & 0.645 & 0.47 & 0.48 & 0.49 & 0.50 & 0.6 observational constraints on the photometric properties of early - type galaxies are taken from the recent sample of stanford , eisenhardt & dickinson ( 1998 ) . the sample consists of 19 clusters in the redshift range @xmath34 . these were extracted on the basis of available _ hst _ imaging from a larger , heterogeneous sample of 46 clusters drawn from a variety of optical , x - ray and radio - selected samples . the 19 clusters studied by stanford et al . ( 1998 ) were imaged in the near infrared @xmath35 , @xmath36 and @xmath37 passbands . exposure times in all passbands were chosen to achieve 5-@xmath38 detection of objects with the spectral energy distribution of an unevolved present - day elliptical galaxy down to 2 mag fainter than @xmath39 ( corresponding to an apparent limiting magnitude @xmath40 and 20.0 at @xmath41 and 0.895 , respectively ) . for 17 out of these 19 clusters , photometry is also available in two optical passbands , referred to as _ blue _ and _ red _ , that were tuned as a function of redshift to span the 4000 break in the galaxy rest frame spectra . this subsample of 17 clusters is of considerable interest to us because it sets the most useful constraints on the spectral properties of galaxies . stanford et al . morphologically classified the galaxies in these clusters on the basis of _ hst _ wide field and planetary camera 2 ( wfpc2 ) images . they did not attempt to distinguish between e and s0 galaxies , and for the purpose of the present analysis we also consider these galaxy types together as a single early - type class . table 2 lists the name , redshift , _ blue _ and _ red _ passbands and number of e / s0 galaxies for each of the 17 clusters in the sample . on the average , cluster membership is expected to be secure for over 85% of the e / s0 galaxies in the sample , as estimated either from morphologically - dependent number counts or from statistical field corrections ( see stanford et al . 1998 for a thorough analysis ) . furthermore , all clusters exhibit a tight cm relation with a slope showing no significant change out to @xmath5 . this has been taken as evidence that all galaxies shared a common history of star formation ( stanford et al . 1998 ; kodama et al . 1998 ; but see kauffmann & charlot 1998 ) . we note that there is an open question about the membership of some morphologically - selected early - type galaxies with faint magnitudes and colors far bluer than the cm relation in the clusters ( e.g. , kodama et al . this can be appreciated most readily from deep _ hst _ wfpc2 imaging by ellis et al . ( 1997 ) of three of the clusters in table 2 , f1557.19tc , cl 0016 + 16 and j1888.16cl . although these faint ( @xmath42 ) galaxies may be field contaminants , for completeness we should not abandon the possibility that they are blue cluster early - type galaxies that violate the cm relation . thus , they are also included in our analysis . it is worth pointing out that stanford et al . ( 1998 ) find no significant difference between the photometric properties of early - type galaxies in clusters of different richness or x - ray luminosity at a similar redshift . while this finding does not necessarily imply that all early - type galaxies were assembled early ( kauffmann & charlot 1998 ) , it strongly supports analyses like the one presented in 4 in which constraints on photometric evolution are drawn from comparisons of the properties of galaxies in clusters at different redshifts . also , to avoid any bias near the bright end of the cm relation , we have checked that the brightest cluster galaxies in the sample of table 2 match the @xmath37-band absolute luminosity versus redshift relation published by aragn - salamanca et al . ( 1998 ) . finally , to tighten the models at @xmath43 , we use the cm relation derived by ble92 for e / s0 galaxies in the coma cluster . this takes the forms @xmath44 and @xmath45 , with standard deviations of 0.055 mag and 0.065 mag in @xmath14 and @xmath9 colors , respectively . to define the absolute magnitude scale of the models , the recession velocity of the coma cluster is taken to be @xmath46 km s@xmath47 ( han & mould 1992 ) . we adopt @xmath48 km sec@xmath47mpc@xmath47 and @xmath49 , except when otherwise indicated . the distance modulus to the coma cluster is therefore 35.41 mag . the small scatter of the cm relation at fixed luminosity has been used by ble92 to constrain the age range of early - type galaxies in clusters ( see also ellis et al . they computed the maximum spread in star - formation epoch allowed by the observed scatter in @xmath14 color according to the rate of color change @xmath50 predicted by population synthesis models . the basis of this argument can be understood from figure 2 . the top panel shows the @xmath14 color evolution of a stellar population in the time interval @xmath51 $ ] as a function of age @xmath52 for several metallicities , while the bottom panel shows the inferred @xmath50 evolution as a function of @xmath52 . since @xmath50 decreases with model age @xmath52 , ble92 found that the older the galaxies , the larger the allowed spread in star - formation epoch . on the basis of this analysis , ble92 favored a major star - formation epoch at @xmath53 for early - type galaxies in clusters , spread over a period of roughly 1 gyr . it is crucial to realize that ble92 s argument utilizing @xmath50 is based on the _ a priori _ requirement that at fixed luminosity , all early - type galaxies in the cm relation have the same metallicity . in fact , the analysis was conducted using population synthesis models for uniquely solar metallicity . as the present paper demonstrates , however , at fixed luminosity the photometric constraints on cluster galaxies allow a wide range of ages and metallicities ( 4 ) . hence , the @xmath54 argument already includes restrictive hypotheses on the ages and metallicities of early - type galaxies with respect to the full allowed ranges . it is not surprising , therefore , that ble92 s conclusions on the epoch of major star formation for early - type galaxies in clusters are unnecessarily constraining when compared to the results of our more complete analysis below . lccccc cluster & @xmath55 & _ blue _ @xmath31 _ red _ & n(e / s0 ) & & & ac 118 & 0.308 & @xmath56 & 38 ac 103 & 0.311 & @xmath56 & 32 ms 2137.3 - 234 & 0.313 & @xmath56 & 21 cl 2244 - 02 & 0.330 & @xmath56 & 24 cl 0024 + 16 & 0.391 & @xmath56 & 39 gho 0303 + 1706 & 0.418 & @xmath56 & 38 3c 295 & 0.461 & @xmath57 & 25 f1557.19tc & 0.510 & @xmath57 & 29 gho 1601 + 4253 & 0.539 & @xmath57 & 42 ms 0451.6 - 0306 & 0.539 & @xmath57 & 51 cl 0016 + 16 & 0.545 & @xmath57 & 65 j1888.16cl & 0.560 & @xmath57 & 38 3c 220.1 & 0.620 & @xmath57 & 22 3c 34 & 0.689 & @xmath57 & 19 gho 1322 + 3027 & 0.751 & @xmath58 & 23 ms 1054.5 - 032 & 0.828 & @xmath58 & 71 gho 1603 + 4313 & 0.895 & @xmath59 & 23 we now use the models described in 2 and the observational constraints outlined in 3 to compute the regions allowed in age metallicity space for early - type galaxies in clusters . we proceed as follows . we consider each cluster in table 2 , with redshift @xmath55 , and search for all models which can match the photometric properties of galaxies in that cluster at @xmath55 and those of coma galaxies at @xmath43 . in practice , we start by considering models that span a full range of metallicities and with ages at @xmath60 that range between the age of the universe and the lookback time to redshift @xmath55 . out of these models , we select all those matching @xmath14 and @xmath9 colors of galaxies in coma within the scatter of the observed cm relation ( 3 ) . for a set of @xmath14 and @xmath9 colors this defines a range of possible @xmath10 magnitudes . we then compute the predicted apparent @xmath37 magnitudes and _ blue_@xmath61 , _ red_@xmath61 , @xmath62 and @xmath63 colors of the selected models at the cluster redshift @xmath55 . to decide whether a model is acceptable , we compare the predicted photometric properties at @xmath55 with the observations . for this purpose , the observed cm relation of each cluster in table 2 was divided into four apparent @xmath37 magnitude bins , roughly 1 mag wide , allowing good sampling from the brightest to the faintest galaxies ( see 3 ) . for each bin , the standard deviation around the relation was computed in _ blue_@xmath61 , _ red_@xmath61 , @xmath62 and @xmath63 colors . this procedure allows us to account for the increased scatter of the observed cm relations towards fainter magnitudes . a model is retained if it simultaneously falls within @xmath64 of the cm relation in all four colors . the main reason for adopting a @xmath64 criterion is that we must allow for known uncertainties in the spectral evolution models . uncertainties in the predicted optical / infrared color evolution over lookback times of a few billion years can reach a few tenths of a magnitude in current population synthesis models ( charlot et al . this is several times larger than the typical scatter around the cm relation for the clusters of table 2 ( see fig . 5_a _ of stanford et al . 1998 and table 1 of kodama et al . 1998 ) . figures 3_a _ , 3_b _ and 3_c _ show the results of our analysis for the clusters in the redshift ranges @xmath65 , @xmath66 and @xmath67 , respectively . metallicity space is parameterized in terms of the formation redshift @xmath68 and the iron abundance computed as [ fe / h]@xmath69 , with @xmath70 and @xmath71 . we separate galaxies into four luminosity bins at @xmath43 by defining apparent magnitude bins in coma centered on @xmath72 , 14.25 , 13.5 and 12.75 . these correspond roughly to absolute @xmath10 luminosities @xmath73 , @xmath74 , @xmath75 and @xmath76 , respectively . the four panels in figures 3_a_3_c _ separately show the results for each of the four luminosity bins . at fixed luminosity , figure 3 indicates that the observations of all 17 clusters roughly constrain similar regions in @xmath77})$ ] space . this is a consequence of the similarity of the cm relation among all clusters in the sample . an important discriminant , however , is that the condition @xmath78 allows more recent formation epochs and higher metallicities for galaxies in low - redshift clusters than for those in high - redshift clusters . as expected , different luminosities imply different allowed values of @xmath68 and [ fe / h ] . since faint galaxies on the cm relation are bluer than bright galaxies , they can be modelled on average by more metal - poor , and to some extent younger , stellar populations . we also note that the metallicity range of model galaxies in figure 3 , @xmath79 } \simlt + 0.2 $ ] , compares well with the range observed in nearby e / s0 galaxies ( e.g. , worthey , faber & gonzalez 1992 ) . the upper panel of figure 4 shows the areas corresponding to all four luminosity bins superimposed on a similar diagram for the cluster cl0016 + 16 . this cluster , at @xmath80 , has one of the best - defined cm relations in the sample ( 65 e / s0 galaxies ; see table 2 ) . figure 4 readily shows that a pure age sequence can not account for the photometric properties of early - type galaxies because a single horizontal line can not cross all four areas simultaneously in @xmath81})$ ] space . alternatively , different combinations of age and metallicities can accommodate the data . the simplest assumption of a pure metallicity sequence ( vertical line ) requires that the bulk of stars in galaxies formed at redshifts @xmath53 , otherwise one can not account for the photometric properties of the brightest galaxies . this confirms earlier conclusions by ellis et al . ( 1997 ) based on similar data for cl0016 + 16 . as figure 4 demonstrates , however , there is no requirement that the bulk of stars in all early - type galaxies in this cluster formed at the same epoch . if the most metal - rich galaxies form most recently , as might be expected from simple chemical enrichment arguments , then figure 4 implies that the dominant stellar populations of all early - type galaxies in cl 0016 + 16 were already in place at redshifts @xmath82 . on the other hand , a wide range of scenarios are also allowed in which faint metal - poor galaxies form as recently as @xmath83 , i.e. , more recently than bright metal - rich galaxies . the lower panel of figure 4 shows that the alternative cosmology @xmath84 km sec@xmath47mpc@xmath47 and @xmath85 would lead on average to slightly larger @xmath68 values for model galaxies because of the lower age of the universe implied at redshift 1 . we emphasize that the constraints on the formation redshift and metal abundance of the stellar populations of early - type galaxies derived from figure 4 apply to cl 0016 + 16 only . as figure 3_a _ shows , constraints on clusters at lower redshifts are consistent with formation redshifts as small as 0.6 for even the brightest , reddest galaxies . hence , the apparent lack of evolution of the slope and scatter of the cm relation in clusters does not imply by itself a common epoch of star formation for all early - type galaxies . instead , what is strictly implied is that new galaxies joining the cm relation at low redshifts must be more metal - rich than their older cluster companions of similar luminosity . the required age metallicity evolution inferred from figure 3 corresponds roughly to a 25% increase in [ fe / h ] from @xmath86 to @xmath87 , i.e. , over a period of 2.5 gyr for the assumed @xmath88 km sec@xmath89mpc@xmath47 and @xmath49 . we note that adopting longer timescales of star formation instead of an instantaneous burst for early - type galaxies in clusters would increase the formation redshift of the first stars in model galaxies with respect to the results of figure 3 . whatever the adopted history of star formation , however , the values of @xmath68 in figure 3 are robust limits on the redshift of the last major event of star formation in early - type galaxies ( see 2 ) . finally , we have explored the possibility that the faint morphologically - selected early - type galaxies with very blue colors in cl 0016 + 16 were actual cluster members ( see 3 ) . the inferred @xmath68 and [ fe / h ] ranges for these objects is indicated by the heavy shaded region in figure 4 . as expected , the blue colors imply that the galaxies must be both young and metal - poor . we may speculate that they are young objects which `` violate '' the main sequence cm relation , but which could eventually evolve towards it as they age and undergo chemical enrichment . however , spectroscopic confirmation as well as a deeper search for faint blue outliers in clusters at higher redshifts are needed before we can draw any definitive conclusions as to their true nature . the above analysis has enabled us to determine the areas allowed in @xmath90})$ ] space for the stellar populations of e / s0 galaxies in clusters . while we emphasize that real galaxies may occupy only part of these allowed areas , it is interesting to compute the implied evolution with redshift of other observable properties of early - type galaxies such as mass - to - light ratios and spectral indices . since we do not know which subareas real galaxies might occupy in figure 3 , in the following we investigate properties averaged over the entire allowed areas . we first investigate the mass - to - light ratios of the model galaxies selected in figure 3 . for each cluster , we compute the logarithmic mean of @xmath91 in each luminosity bin at the cluster redshift by averaging over the @xmath81})$ ] area constrained by the observations . since absolute values of @xmath91 depend sensitively on the assumed low - mass end of the imf , we adopt for all clusters the arbitrary normalization @xmath92 at @xmath93 and focus our investigation on the dependence of the mass - to - light ratio on luminosity . the mean values of @xmath94 computed in this way are shown as triangles in figure 5 . since the mass - to-@xmath10 light ratio of a stellar population increases at increasing age and metallicity , brighter galaxies in figure 5 generally tend to have larger @xmath94 than fainter galaxies . this is not always true , however , as can be understood from the dispersion in the results of figure 3 . also shown in figure 5 are the results from two recent observational studies of the fundamental plane for cluster e / s0 galaxies . jrgensen , franx , & kjrgaard ( 1996 ) have parameterized the mass - to - gunn @xmath95 luminosity ratio , @xmath96 , in terms of a combination of half light radius and central velocity dispersion from observations of a large sample of 226 e / s0 galaxies in 10 nearby clusters . they conclude that the half light radius has a negligible effect on determinations of @xmath96 and show that the data follow the simple relation @xmath97 with a scatter of only 25% . the central velocity dispersion @xmath38 is tightly correlated with luminosity via the faber - jackson relation ( faber & jackson 1976 ) . we use ble92 s calibration of this relation based on observations of early - type galaxies in the coma cluster in order to relate @xmath98 to @xmath99 in figure 5 . we then adopt the mean @xmath100 colors of galaxies along the cm relation to convert jrgensen et al.s ( 1996 ) result into an expression involving the mass - to-@xmath10 luminosity ratio . this yields @xmath101 , corresponding to an increase of less than 3% in logarithmic slope with respect to the relation derived in gunn @xmath95 . the horizontal shading in figure 5 then indicates the range of slopes allowed after accounting for the scatter of jrgensen et al.s ( 1996 ) data around the mean relation . an alternative constraint on the observed range of mass - to-@xmath10 luminosity ratios can be obtained from the relation recently derived by graham & colless ( 1997 ) between half light radius , central velocity dispersion and mean surface brightness of early - type galaxies . their analysis is based on accurate fitting of the @xmath10-light profiles of 26 e / s0 galaxies in the virgo cluster using both homologous ( @xmath102 ) and non - homologous ( @xmath103 , with @xmath104 a free parameter ) radial dependence laws . the results indicate a slight but systematic breaking of the generally assumed homology in the sense that @xmath104 increases with increasing half - light radius . we can use the virial theorem and faber - jackson relation to reexpress the results of graham & colless ( 1997 ) in terms of a relation involving @xmath91 and @xmath98 . the outcome is shown in the upper and lower panels of figure 5 for non - homologous and homologous light profiles , respectively . in each case , the heavy solid line indicates the mean relation and the slanted shading represents the allowed range of slopes when the uncertainties quoted by graham & colless are included . figure 5 shows that the mass - to - light ratios of model galaxies are all within the range constrained by current observations . the models appear to be mostly compatible with the graham & colless ( 1997 ) results for non - homologous light profiles . however , we point out that the model values correspond to purely stellar mass - to - light ratios , whereas there is some observational evidence for the presence of dark halos around early - type galaxies . this is inferred via the studies of hi kinematics ( e.g. franx et al . 1994 ) , x - ray emission ( e.g. forman et al . 1994 ) , radial velocities of planetary nebulae and globular clusters ( e.g. mould et al . 1990 , hui et al . 1995 ) , gravitational lensing ( e.g. maoz & rix 1993 ) and measurements of the shape of the stellar line - of sight velocity distribution ( e.g. carollo et al . 1995 ) . more recently , rix et al . ( 1997 ) have analyzed the velocity profile of the elliptical galaxy ngc 2434 and show that roughly half the mass within an effective radius is dark . hence , one must remain cautious in interpreting trends in the mass - to - light ratios of early - type galaxies on the basis of pure population synthesis models . for completeness , we also compute the strengths of several commonly used spectral indices implied by the above photometric study for early - type galaxies in clusters . most observations in this domain have concentrated on measurements of h - balmer , magnesium and iron - dominated indices of the lick system ( 2 ) such as h@xmath17 , mg@xmath12 , mg@xmath16 and mgfe ( worthey et al . 1992 ; bender , burstein & faber 1993 ; gonzalez 1993 ; davies , sadler & peletier 1993 ; ziegler & bender 1997 ) . one of the main motivations for studies of this type is to identify a pair of indices including one mostly sensitive to age and one mostly sensitive to metallicity in order to break the age metallicity degeneracy from photometric colors . sensitive h@xmath17 index , however , can be significantly contaminated by nuclear emission in e / s0 galaxies , for which the corrections are uncertain ( e.g. , gonzalez 1993 ) . it is therefore deemed preferable to study higher - order balmer indices such as h@xmath105 ( e.g. , worthey & ottaviani 1997 ) . also , the apparent systematic increase of [ mg / fe ] from faint to bright e / s0 galaxies challenges analyses based on models with solar - scaled abundance ratios ( e.g. , worthey et al . 1992 ; tantalo , bressan & chiosi 1997 and references therein ) . worthey ( 1995 ) shows that the metal - sensitive index c4668 of the lick system appears to be significantly less enhanced with respect to iron than mg in bright e / s0 galaxies . thus , c4668 represents a good alternative to mg - dominated indices for reliable model investigations of early - type galaxy spectra ( e.g. , kuntschner & davies 1998 ) . by analogy with our approach in 5.1 , we compute mean spectral indices for each of the four luminosity bins in each cluster of our sample . the results are shown in figure 6 for the mg@xmath12 , h@xmath17 , mg@xmath16 , mgfe , c4668 and h@xmath106 indices of the lick system . as expected , there is a general correlation between index strength and luminosity , which is more pronounced for metal - sensitive ( mg@xmath12 , mg@xmath16 , mgfe and c4668 ) than for age - sensitive ( h@xmath17 , h@xmath106 ) indices . the reason for this is that luminous galaxies in figure 3 are on average found to be significantly more metal - rich and slightly older than faint galaxies . also , figure 1 shows that the sensitivity to age of h - balmer indices such as h@xmath17 is significantly weakened after @xmath107 gyr , when a and b stars have evolved off the main sequence . for comparison , the solid and dashed lines in figure 6 show the locations of passively evolving model galaxies with @xmath108 and the metallicities @xmath109 and @xmath110 , respectively . the mean index strengths of cluster galaxies appear to evolve roughly along these lines , supporting the consistency of the stellar populations with apparently passive evolution . in fact , ziegler & bender ( 1997 ) find that the evolution from @xmath43 to @xmath111 of the correlation existing between the mg@xmath16 index strength and central velocity dispersion of e / s0 galaxies is consistent with passive evolution . it should be noted that the model indices computed in figure 6 are global indices averaged over the emission from all stars in a galaxy . observations of nearby e / s0 galaxies , however , reveal systematic variations of the strengths of spectral indices between the central and outer regions ( e.g. , worthey et al . we have shown that the tight photometric constraints on early - type galaxies in clusters allow relatively wide ranges of ages and metallicities for the dominant stellar populations . in particular , the small scatter of the cm relation out to redshifts @xmath2 does not necessarily imply a common epoch of star formation for all early - type galaxies . it requires , however , that galaxies assembling more recently be on average more metal - rich than older galaxies of similar luminosity . in this context it is interesting to mention that , based on the spectral indices of nearby e / s0 galaxies , worthey , trager & faber ( 1996 ) favor younger ages for more metal - rich galaxies than for metal - poor ones at fixed velocity dispersion . the results of our unbiased analysis therefore define the boundaries in age and metallicity that must be satisfied by theoretical studies aimed at explaining the formation and evolution of early - type galaxies in clusters . the constraints obtained here on the age and metallicity ranges of e / s0 galaxies are consistent with conventional models in which the galaxies all form monolithically in a single giant burst of star formation at high redshift ( e.g. , kodama et al . 1998 , and references therein ) . in fact , this implies that regardless of the true ages and metallicities of early - type galaxies within the allowed range , their photometric properties will always be consistent with _ apparently _ passive evolution of the stellar populations . as figure 6 shows , this consistency even extends to spectral index strengths . our results are also consistent with scenarios in which e / s0 galaxies are formed by the merging of disk galaxies ( schweizer & seitzer 1992 ) in a universe where structure is built through hierarchical clustering ( kauffmann 1996 ; baugh , cole & frenk 1996 ; kauffmann & charlot 1998 ) . for such scenarios , figure 3 constrains the metallicity and epoch of the last major event of star formation in e / s0 galaxies and their progenitors ( see 2 and 4 ) . the ages and metallicities of cluster e / s0 galaxies predicted by hierarchical models are found to be consistent with these constraints ( kauffmann & charlot 1998 ) . to better assess the origin of e / s0 galaxies in clusters one therefore needs to appeal to observational constraints other than their spectro - photometric properties . for example , conventional models of e / s0 galaxy formation are being challenged by the paucity of red galaxies found at high redshifts in deep surveys ( kauffmann , charlot , & white 1997 ; zepf 1997 ) . also , morphological distinction between e and s0 galaxies and the evolution of the morphology - density relation out to moderate redshifts appear to point to different formation epochs for e and s0 galaxies ( dressler et al . 1997 ) . the tightness of the cm relation is proof of a stable process in the assembly of cluster early - type galaxies . however , as we move towards greater redshifts , a drastic change is expected at lookback times that approach the formation of the first e / s0 galaxies . this change can arise as a systematic blueing , an increased scatter or a slope flattening in the cm relation ( e.g. , aragn - salamanca et al . 1993 ; charlot & silk 1994 ; kauffmann & charlot 1998 ) . an interesting question is raised by the presence of morphologically - selected early - type galaxies with very blue colors in clusters at moderate redshifts ( 3 and 4 ) . if these objects are true cluster members , our analysis shows that they could be young metal - poor galaxies that will later join the cm relation . hence , we need to probe deeper down the galaxy luminosity function in distant clusters in order to assess whether these objects can have any fundamental bearing on the origin of early - type galaxies . we thank adam stanford for sending us a machine readable list of the cluster photometry used in this paper . thanks the cnrs and the institut dastrophysique de paris for hospitality and financial support , and also acknowledges a ph.d . scholarship from the `` gobierno de cantabria . 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the structure , properties and reactions of nuclei and nuclear matter depend on the dynamics of the nucleons @xcite . this is the reason why the derivation of the nuclear forces is probably the most important problem of nuclear physics . after the discovery of quantum chromodynamics ( qcd ) the fundamental theory of strong interactions a solid theoretical understanding of the nuclear force should be grounded on qcd , either directly or indirectly . lattice qcd represents the direct , computational derivation : the interaction of quarks and gluons is not analytically solvable at the distances that are characteristic for nuclear physics but it is numerically solvable at the expense of huge computational resources . recent progress in this front is exciting @xcite . the indirect derivation requires to explain the nuclear interaction without explicitly solving qcd . yet qcd must enter indirectly in the picture . otherwise we will end up with a phenomenological description instead of a theoretical explanation . physics as a science depends to a great extent on the existence of scale separation in nature . one can describe the properties of atoms without explicit knowledge of the composite nature and internal structure of the nuclei within . the nucleus is indeed much smaller than the atom containing it , i.e. there is an excellent separation of scales . analogously , one can describe the dynamics of nucleons and pions without knowing the details of the strong interaction of the quarks and gluons inside them . however the average distance of nucleons in a nucleus about @xmath0 is not that different from the size of the nucleon or the wavelength of the quarks and gluons inside , maybe @xmath1 . without a clear separation of scales the development of satisfactory theoretical explanations to physical phenomena becomes more difficult . as a consequence the description of nuclei is less clear and more involved than that of atoms . effective field theories ( efts ) are the standard theoretical tool to exploit the separation of scales of a physical system with the intention of building the most general description of it at low energies @xcite . if we call the low energy scale @xmath2 and the high energy scale @xmath3 , an eft provides a power expansion in terms of @xmath4 of all the physical quantities of a system . for that one considers first all the possible interaction terms in the lagrangian that are compatible with the low energy symmetries of the system . then one orders the infinite feynman diagrams obtained in the previous step according to their expected size . the method by which we estimate the size of the diagrams is called power counting . while writing the diagrams is trivial , their power counting is not . the connection of the eft to the fundamental theory at the scale @xmath3 is provided by renormalization , the core idea of eft . in its standard formulation renormalization deals with ultraviolet ( uv ) divergences in the feynman diagrams of the eft . to remove the divergences one includes an uv cut - off and allows the couplings in the lagrangian to depend on the cut - off . if the calculation of the observable quantities of the eft is independent of the cut - off then the eft is renormalizable . power counting is decided according to how we have to arrange calculations to remove the divergences at each order in the expansion . wilsonian renormalization @xcite provides an alternative but equivalent formulation . here the starting point is the independence of observables with respect to the cut - off . in this case it is the calculation of the couplings under the assumption of cut - off independence that leads to the size of these couplings at low energies and to their power counting @xcite . this is referred to as renormalization group : the focus is on the evolution of the couplings as the cut - off changes , not on the divergences . in wilsonian renormalization the cut - off runs from the high to the low energy scale , from @xmath3 to @xmath2 . this is counterintuitive from the standard point of view , where the cut - off runs from @xmath2 to @xmath3 with the purpose of finding out whether there are uv divergences . yet they are equivalent . the cut - off can either run to the ultraviolet or the infrared ( ir ) . as far as the observables are independent of the cut - off we end up with identical power countings . the starting point in wilsonian renormalization can be either an eft or the fundamental theory . the advantage in the first case is that power counting can be determined without a complete order - by - order calculation of observables . in the second case there is the possibility of evolving a fundamental theory from @xmath3 to @xmath2 , which amounts to uncovering the eft by means of a concrete calculation . of course this is only possible in the few cases where the fundamental theory is known or easily solvable ( a nice example can be found in ref . ) . this manuscript is dedicated to wilsonian renormalization in nuclear eft @xcite : even though it is less well - known than the standard idea of removing divergences , it can provide a clearer interpretation of power counting and the role of the cut - off in eft . in nuclear physics the eft usually contains nucleon and pion fields that are constrained by chiral symmetry , a low energy symmetry of qcd that is exact in the limit of massless @xmath5 , @xmath6 and sometimes @xmath7 quarks . the problem is that historically renormalization has been only well understood in the case of systems that are perturbative @xcite . this is the case in hadron physics for processes involving at most one baryon , where chiral perturbation theory ( chpt ) @xcite , the standard eft for low energy hadronic processes , is used . but in nuclear physics the existence of the deuteron and the virtual state ( the @xmath8 singlet ) , not to mention the few thousand known nuclei , indicates that the nuclear force is non - perturbative . besides there is the additional problem that eft entails nuclear forces that are strongly divergent at short distances . thus it is not a surprise that progress in nuclear eft has been full of unexpected turns and controversies . recently we have begun to have a solid grasp of the non - perturbative renormalization of the eft potentials @xcite and how to organize the power counting in this situation @xcite , but even these advances have been the subject of debate @xcite . here we will review power counting from the perspective of the renormalization group . historically events have unfolded in a zig - zag pattern . weinberg made the first proposal for a nuclear eft @xcite , which includes the iteration of the eft potential ( at least at lowest order ) . this serves to capture the non - perturbative character of nuclear interactions but in exchange requires non - perturbative renormalization . as previously said , this has been the source of a few surprises . kaplan , savage and wise ( ksw ) discovered a subtle but nonetheless serious inconsistency with the weinberg proposal @xcite . these authors also developed a new formulation of nuclear eft , the ksw counting @xcite , which is free from that inconsistency . however the convergence of the ksw counting in the triplet partial waves happened to be unsatisfying to say the least @xcite . the community turned back to the weinberg proposal in search for phenomenological success @xcite . but later nogga , timmermans and van kolck @xcite discovered that the weinberg proposal contains a new , more conspicuous inconsistency at the lowest order : it is not renormalizable in some p- and d - waves . new developments about renormalizability followed @xcite that made finally possible a consistent nuclear eft with good convergence properties @xcite . despite these advancements , there is an ongoing debate about whether these problems are relevant and whether it would be simply more sensible to reinterpret renormalizability for non - perturbative problems in a different way @xcite . we will not discuss these new developments , except for a brief comment . here we are mostly concerned about the derivation of eft power counting from a specific set of renormalization tools , which happen to be more than enough to make nuclear eft work at the theoretical level . from this perspective the previous ideas , though interesting , do not appear to be totally necessary . this manuscript is organized as follows : in sect . ii we introduce wilsonian renormalization for the particular case of non - relativistic scattering of two particles . part of it is general and part of it is specific to nuclear physics . we also discuss the relationship of power counting with the anomalous dimension of couplings and the relationship between wilsonian renormalization and the more standard approach of removing ultraviolet divergences . in sect . iii we extend the results beyond the two - nucleon system , in particular to the deuteron electroweak reactions and to the three - body problem . finally we summarize our conclusions . we also include an appendix discussing the derivation of a particular equation in this manuscript . here we illustrate how wilsonian renormalization works for non - relativistic s - wave scattering @xcite . the starting point is a `` fundamental theory '' . for a non - relativistic two - body system the equivalent of a fundamental theory is the non - relativistic potential @xmath9 . to obtain the scattering amplitudes we solve the schrdinger equation at finite momentum @xmath10 @xmath11 where @xmath12 is the reduced wave function , @xmath13 the reduced mass , @xmath10 the center of mass momentum and @xmath14 the underlying potential , which we assume to be known at all distances . as we are considering s - wave scattering there is no centrifugal term . we solve this equation with the regular boundary condition at the origin @xmath15 finally the phase shift can be extracted from the asymptotic wave function @xmath16 wilsonian renormalization works as follows . in a first step we include a cut - off @xmath17 as a separation scale @xmath18 we will consider that the physics at distances shorter than the cut - off @xmath19 is unknown . of course if we cut the potential for @xmath19 the physical observables will change . we want to prevent this from happening . in a second step we include a new piece in the potential that counteracts the loss of information from having a cut - off and keeps the observables unchanged . this extra piece is the contact - range potential , which can take many parametrizations . for simplicity we choose the following form for the contacts @xmath20 that is , an energy - dependent delta shell potential . now we solve the schrdinger equation with the `` renormalized '' potential @xmath21 for distances below @xmath17 we have a free schrdinger equation @xmath22 with the regular solution @xmath23 for distances above @xmath17 we have the original schrdinger equation , i.e. eq . ( [ eq : schro - fundamental ] ) . finally at @xmath24 the delta - shell potential @xmath25 generates a discontinuity in the first derivative of the wave function that takes the form @xmath26 where @xmath27 refers to @xmath28 , with @xmath29 . a derivation can be found in [ app : delta ] . this is the renormalization group equation ( rge ) for the contact - range coupling @xmath30 . the rge we have written above is exact : the starting point is the full potential @xmath9 and we want to check what type of contact interaction we have to include to account for the existence of a cut - off radius @xmath17 . for @xmath31 we have the boundary condition @xmath32 : we know the potential at all distances and there is no need for the contact - range couplings . as we increase the cut - off radius , we will need non - vanishing @xmath33 couplings to account for the missing physics . the reason we are interested in wilsonian renormalization is because we want to know how physics looks like at large distances in general . we want to describe phenomena at low energies regardless of which is the fundamental theory at high energies . with wilsonian renormalization we can build a theory for distances larger than the cut - off ( @xmath34 ) that is equivalent to the fundamental theory for momenta @xmath35 . in this context it is useful to define the soft and hard scales @xmath2 and @xmath3 . the soft scale @xmath2 is the characteristic momentum of the low energy physics we want to describe , while the hard scale @xmath3 is the natural momentum scale of the fundamental theory . if we solve the rge for @xmath36 we will be able to derive the kind of generic low energy theory we are interested in . to solve the rge and obtain the contact range couplings we simply have to make an ansatz for the wave function @xmath12 . the simplest case is provided by a theory in which the underlying potential has a finite range set by the hard scale @xmath3 @xmath37 with this is mind we see that the wave functions for @xmath38 are given by @xmath39 where @xmath40 is the phase shift of the fundamental potential @xmath9 . therefore the solution of the rge for @xmath41 is @xmath42 for finding the running of the individual @xmath33 couplings we expand the rge in powers of @xmath43 . we first take into account that @xmath9 is a finite - range potential and the effective range expansion applies ( for @xmath44 ) @xmath45 where @xmath46 is the scattering length , @xmath47 the effective range and @xmath48 the shape coefficients . now we expand and get the set of equations @xmath49 plus analogous equations for the higher order couplings . in the equations above @xmath50 and @xmath51 are polynomials of the cut - off @xmath17 and the effective range coefficients . they are easy to calculate but they are not included here because their exact form is inconsequential for the analysis that follows . the previous equations are generic solutions for an arbitrary finite range potential @xmath9 . however the counting of the couplings @xmath33 as @xmath52 depends on which is the size of the effective range coefficients and in particular the scattering length . in general the size of the effective range coefficients is known to scale according to the range of the potential ( therefore the name ) : @xmath53 the exception is the scattering length @xmath46 that can take any value , more so if there is non - perturbative physics . thus we distinguish two possibilities : @xmath54 the first one is a scattering length of natural size and the second an unnaturally large scattering length , which is what happens for instance if there is a bound state near the threshold . if the scattering length is of order @xmath55 we are entitled to expand in powers of @xmath56 because @xmath41 . we obtain @xmath57 \ , , \end{aligned}\ ] ] that is , @xmath58 scales as @xmath59 . if we analyze now the subleading couplings @xmath33 , we find for @xmath60 @xmath61 + c_2^r(r_c ; a_0 ) \ , , \end{aligned}\ ] ] that is , @xmath62 scales as @xmath63 . in the equation above @xmath64 is a `` redundant '' piece of the coupling @xmath60 that does not contain information about a the effective range @xmath47 . the function of @xmath64 is to absorb the cut - off dependence that the @xmath65 coupling generates at finite energy . the @xmath64 piece of @xmath60 is inessential for power counting . for @xmath66 we have @xmath67 + c_4^r(r_c ; a_0 , r_0 ) \ , , \end{aligned}\ ] ] which scales as @xmath68 and where @xmath69 is analogous to @xmath64 , only that it absorbs the residual cut - off dependence of @xmath65 and @xmath60 . for the higher order couplings we have @xmath70 . the other possibility is that the scattering length is large : @xmath71 . now the cut - off and the scattering length can have the same size and we are not allowed to expand in powers of @xmath56 . if we solve the rge for @xmath65 we obtain @xmath72 which means @xmath73 , an enhancement of one power of @xmath74 . for the @xmath62 coupling we get @xmath75 which entails @xmath76 , an enhancement of two powers of @xmath74 . for the @xmath77 coupling we find @xmath78 and in general for the @xmath30 we have @xmath79 which implies a @xmath80 enhancement over the natural case . the first implementation of this type of rg analysis of the couplings in nuclear eft is due to birse , mcgovern and richardson @xcite , who formulated the rges in momentum space . instead of imposing the invariance of the phase shifts with respect to the regulator , their analysis requires the invariance of the full off - shell t - matrix . for contact - range interactions both conditions are equivalent : on - shell renormalization implies off - shell renormalization . probably this is the case too for finite - range potentials ( it has been proved for potentials that have power - law divergences near the origin @xcite ) . the analysis of the rges in momentum space is pretty convoluted though . the analysis of the coordinate space rges of ref . is simpler as it only depends on the schrdinger equation and its solutions . it connects the rges with the cut - off dependence of the observables after an arbitrary number of contact - range operators are included . but at the same time it neglects how the rges relate to the power counting of the couplings . the purpose of this section has been to close this gap and to translate the rg analysis of ref . from momentum to coordinate space , attempting to make its interpretation clearer in the way . the question we wanted to answer is : what kind of low energy theory does one derive from the rges ? the answer involves two ingredients . the first is a non - relativistic potential for the low energy theory , the effective potential : @xmath81 that is , the contact - range potential that compensates the cut - off dependence . the fundamental potential @xmath9 does not enter into the effective potential for the simple reason that it vanishes at large distances ( @xmath38 ) . the second ingredient is the size of the couplings , that we have already calculated from the rges . as a consequence of the scaling properties of the couplings we can write the effective potential as a power series in @xmath4 . let us consider the example of a theory with a natural scattering length , for which we have @xmath82 if we take into account the typical factors of @xmath83 and the reduced mass that are common in non - relativistic scattering , the previous scaling allows to rewrite the @xmath30 couplings as @xmath84 where @xmath85 is a number of @xmath86 . now we plug this expression into the effective potential . we arrive to @xmath87 that is , a power series in @xmath4 . for large scattering length we have instead @xmath88 with @xmath89 and @xmath90 for @xmath91 . in this case the expansion of the potential reads @xmath92 \ , .\end{aligned}\ ] ] independently of the power counting of the couplings we end up with a series that converges for @xmath93 . this idea of arranging the effective potential as a power series extends to every physical quantity we can think of . it is a fundamental concept in eft , the reason why calculations are systematic . we can predict observable quantities up to a given degree of accuracy , that is , up to a given power of the expansion parameter @xmath94 . the calculations are organized as to only include the operators that contribute within the accuracy goals we have set up in the first place . we can illustrate this concept with the phase shift . in perturbation theory the phase shift is expanded as @xmath95 \ , , \end{aligned}\ ] ] i.e. the born approximation followed by second and higher order perturbation theory . that is , we have written a coordinate space version of the lippmann - schwinger equation . in the expression above @xmath96 is a green function , which we can take to be @xmath97 \ , .\end{aligned}\ ] ] if the scattering length is natural , the size of the born and second order term are @xmath98 \ , , \\ \frac{2 \mu}{k}\,\langle v g_0 v \rangle & = & { \left(\frac{q}{m } \right)}^2\,f_2(k r_c)\ , { \left [ \sum_{n=0}^{\infty } c_{2n}(r_c)\ , { \left ( \frac{q}{m } \right)}^{2n } \right]}^2 \ , , \end{aligned}\ ] ] where @xmath99 , @xmath100 is simply a compact notation for the first and second order of the perturbative series and @xmath101 , @xmath102 are functions that encode the cut - off dependence . if we continue we will find that for higher order perturbations we have @xmath103}^r \ , , \end{aligned}\ ] ] where @xmath104 refers to the number of insertions of the potential @xmath9 . the eft expansion for the phase shift starts at @xmath4 leading order ( lo ) from now on and second order perturbation theory carries an additional factor of @xmath4 over the born approximation . analogously each additional iteration of the potential involves an extra power of @xmath4 . putting all the pieces together , the @xmath105 calculation only contains @xmath65 at tree level , the next - to - leading order ( @xmath106 ) calculation requires to include two iterations of @xmath65 , the next - to - next - to - leading order ( @xmath107 ) calculation contains @xmath60 at tree level and three iterations of @xmath65 . higher orders calculations are set up in a similar fashion . for the large scattering length case the evaluation of the perturbative series for the tangent of the phase shift leads to @xmath108 \ , , \\ \frac{2 \mu}{k}\,\langle v g_0 v \rangle & = & f_2'(k r_c)\ , { \left [ c_0(r_c ) + \sum_{n=1}^{\infty } c_{2n}(r_c)\ , { \left ( \frac{q}{m } \right)}^{2n } \right]}^2 \ , , \\ \frac{2 \mu}{k}\,\underbrace{\langle v g_0 \dots g_0 v \rangle}_{\mbox{$r$ iterations of $ v$ } } & = & f_r'(k r_c)\ , { \left [ c_0(r_c ) + \sum_{n=1}^{\infty } c_{2n}(r_c)\ , { \left ( \frac{q}{m } \right)}^{2n } \right]}^r \ , .\end{aligned}\ ] ] now the eft expansion of the phase shift begins at order @xmath109 , which is the @xmath105 for this power counting ( i.e. the @xmath105 is defined differently for each scaling of the couplings ) . it is also apparent that the @xmath105 calculation contains all the iterations of the @xmath65 coupling . that is , @xmath65 is non - perturbative at @xmath105 . however all the other couplings are perturbative : @xmath60 enters at tree level at @xmath106 , @xmath77 at @xmath110 and @xmath30 at @xmath111 . a systematic exposition of the diagrams involved in the calculation of the amplitudes ( for natural and unnatural scattering length ) can be found in ref . . the previous analysis can be extended to the possibility that the potential can be separated into a short and long range piece @xmath112 the range of @xmath113 scales as @xmath3 , while the range of @xmath114 as @xmath2 . to determine the effect of a long range potential on the power counting we follow the previous steps : introduce a cut - off and include a contact - range potential to keep physics unchanged . the couplings @xmath30 of the contact - range potential can be calculated from the rge , i.e. eq . ( [ eq : rge ] ) . for @xmath41 the short - range potential vanishes and the wave functions that enter into the rge are solutions of @xmath115 with the boundary conditions @xmath116 with @xmath40 the phase shift of the full potential @xmath117 . we are interested in the wave functions in the distance range @xmath118 . the condition @xmath38 is necessary if we use a wave function that is a solution of the long range potential @xmath114 . a soft cut - off let s say @xmath119 is perfectly acceptable . but a excessively soft cut - off of the order of @xmath120 is not : for this choice of the cut - off the long range potential @xmath114 vanishes and we end up with the power counting of a pure short range potential . everything that is left is to calculate the wave functions for the long range potential . in general the form of the solution of the wave function will take the form @xmath121 where @xmath122 and @xmath123 are two linearly independent solutions of @xmath114 and @xmath124 a coefficient that selects the particular linear combination . if we expand the solutions in powers of @xmath43 @xmath125 and the coefficient @xmath124 as @xmath126 we end up with the following expressions for the running of the @xmath30 couplings @xmath127 plus analogous expressions for the higher order couplings , where the wave functions and their derivatives are understood to be evaluated at @xmath24 . in the expression above , @xmath128 is a polynomial of @xmath129 and @xmath130 , @xmath131 and its derivatives that encodes the residual cut - off dependence . a few general comments might be of help at this point . first : the coefficient @xmath124 can be thought of as the analogous of the ere in the presence of a long range potential . what this means is that the set of coefficients @xmath132 , @xmath133 , etc . , will scale according to inverse powers of @xmath3 . the exception is @xmath129 , which could take any value if @xmath113 is non - perturbative . second : if the long range potential is perturbative , the wave functions will coincide with the free wave functions at tree level in perturbation theory . the couplings will also accept a perturbative expansion , but at tree level will coincide with the couplings of the short - range case . thus the power counting does not change if the long range potential is perturbative . in nuclear physics the longest range piece of the interaction is the one pion exchange ( ope ) potential . this potential can be written as @xmath134 \ , \vec{\tau}_1 \cdot \vec{\tau}_2 \ , , \end{aligned}\ ] ] where @xmath135 and @xmath136 are the spin and isospin operators acting on the nucleon 1(2 ) , @xmath137 is the pion mass , @xmath138 the nucleon mass and @xmath139 is a mass scale that characterizes the strength of the ope potential ( its value is of the order of @xmath140 ) . the tensor operator is defined as @xmath141 , while @xmath142 and @xmath143 refer to the spin - spin and tensor components of the potential @xmath144 we will ignore the complications coming from the tensor operator and will concentrate on the fundamentals : ( i ) @xmath145 vanishes in the singlet and ( ii ) @xmath142 and @xmath143 behave as a @xmath146 and a @xmath147 potential respectively . before analyzing the power counting with ope it will be helpful to comment on the role of @xmath139 . notice that we have written the ope potential as @xmath148 where @xmath149 is a polynomial that contains a @xmath150 , @xmath151 and @xmath147 term , where all the terms have three powers of @xmath152 . the analogy with the contact - range potential is clear , more so if we write @xmath25 as @xmath153 where we can appreciate that the @xmath154 factor in @xmath155 plays the same role as the @xmath65 coupling in @xmath156 . that is , if we count @xmath139 as @xmath3 the ope potential will be perturbative @xcite . on the contrary if we count @xmath139 as @xmath2 the ope potential will be non - perturbative @xcite . we are interested in the later case : as we have already pointed out , if ope is perturbative the counting is the same as that of a pure short - range potential . the bottom line is that non - perturbative ope goes along with the assumption that @xmath139 is a light scale . the idea that non - perturbative ope requires the existence of an additional light scale ( besides the obvious choices such as the external momenta , the pion mass or the inverse of the scattering length ) was probably _ explicitly _ realized in ref . for the first time . in ref . one can see how to include this scale in the rg equations and what kind of consequence it has for the power counting . the @xmath146 potential , which corresponds to the ope potential in the @xmath8 singlet , is the easiest to analyze . here we are not going to enter into the specific details of how to do the detailed analysis . we merely comment that the power counting is unchanged with respect to the case where there is no long range potential @xcite . that is , there are two possible arrangements of the power counting : a natural one , in which the couplings scale as @xmath70 and an unnatural one , in which the couplings scale as @xmath157 and @xmath158 . the reason for that is that the @xmath146 potential is not strong enough as to modify the behaviour of the wave functions in the distance window @xmath159 . even if the strength of the potential is such as to generate a low lying bound state , the wave functions are only substantially modified for @xmath160 . however this cut - off range is not of interest for power counting because we are already making the assumption that @xmath161 . for the @xmath162 triplet the potential behaves as an attractive @xmath147 for @xmath163 , which induces important changes in the scaling of the couplings triplet is a coupled channel and the ope is a matrix : the tensor operator contains an attractive and repulsive eigenvalue , and the attractive one happens to have a bigger impact on power counting . ] . the first thing to notice is that there is not anymore a natural and unnatural power counting . the solutions of the wave function are all equally fine - tuned : there is not a more natural or preferred solution ( see ref . for a different conclusion though ) . the reason is that the attractive @xmath147 potential has no unique solution in quantum mechanics : the choice of the solution inherently depends on the existence of short range physics , which is the only responsible for fixing the wave function . every linear combination of independent wave functions is equally acceptable . for @xmath163 the wave function can be written as @xcite @xmath164 \ , , \label{eq : uk_lo}\end{aligned}\ ] ] where @xmath165 is a dimensionless number and @xmath166 is a phase the semiclassical phase that characterizes the particular solution we are dealing with . the value of @xmath166 depends on the short - range physics . the changes in the counting are the following : @xmath167 this result requires a careful examination of the scaling properties of @xmath166 , which are not trivial ( the derivation is not contained here but will be included in a future publication ) . as a matter of fact the counting with an attractive tensor force is more similar to nda than to that of a short range potential with large scattering length . for a repulsive singular potential the analysis is analogous with the exception of a few details . the wave function is @xmath168 \nonumber \\ & \times & \left [ 1 + \mathcal{o}(\sqrt{\lambda_{\rm nn } r } , \,k^2 r^2\ , , m r ) \right ] \ , , \end{aligned}\ ] ] with @xmath169 a coefficient that depends on the short - range physics . it is expected to be small and as happened with @xmath170 its scaling properties are important in the detailed analysis , yet they are not trivial . the scaling of the coupling now is @xmath171 that is , the only difference with the attractive case is the scaling of the @xmath65 coupling . however the previous scaling properties are difficult to interpret . it is sensible to expect that the importance of short - range physics within eft depends on the long - range dynamics . the attractive @xmath147 potential complies with this expectation : as a consequence of the strong attraction the wave function is enhanced at short distances , which in turn enhances the short - range couplings . for the repulsive @xmath147 potential we expect the contrary to happen , that the size of the @xmath30 couplings diminishes . what happens is precisely the contrary , which is puzzling to say the least . with this we have finished the discussion of power counting for the moment . the types of power counting and the physical situations to which they correspond are summarized in table [ tab : counting ] . of course they are not the only types of power counting that can be built , but they are for sure the most relevant ones for nuclear eft . now i will try to show how to rederive these counting rules with other methods . in particular i will consider the calculation of anomalous dimensions , ultraviolet renormalizability and residual cut - off dependence . summary of the power counting for s - wave two - body scattering . the table indicates when the coupling enters as a power of @xmath2 ( and the relative order in parenthesis ) . in the text we have considered the case of a pionless and a pionful eft . for pionless the scaling of the couplings depends on the size of the scattering length . for pionful the scaling is identical to the pionless case if either one of these conditions is met : ( i ) pion exchanges are perturbative ( ii ) pion exchanges are non - perturbative but there is only the central piece . if the tensor piece is non - perturbative the scaling of the couplings will be modified with respect to the previous cases . we show this in the table by indicating whether the long - range potential is zero ( @xmath172 ) , perturbative ( @xmath173 ) or non - perturbative ( @xmath174 ) and then the type of long - range potential ( @xmath146 or @xmath147 ) . finally in the last row we indicate the size of a perturbation of @xmath65 , which determines whether the power counting is infrared stable or unstable ( see discussion around eq . [ eq : dc0 ] ) [ tab : counting ] there is a very interesting simplification in the above calculations : it is enough to take into account the cutoff dependence of @xmath30 to guess its scaling @xcite . more specifically we refer to the cut - off dependence for @xmath175 ( @xmath176 ) , a condition that will remarkably simplify the discussion below . if we consider a two - body system with natural scattering length the cut - off dependence of the @xmath30 couplings is trivial @xmath177 \ , , \\ c_{2n}(r_c ) & = & \frac{2\pi}{\mu}\,a_0 ^ 2\,v_n \,\times\ , \left[1 + \mathcal{o}(\frac{r_c}{a_0 } ) \right ] \ , , \quad \mbox{for $ n \geq 1$,}\end{aligned}\ ] ] while for a system with a large scattering length we have @xmath178 \ , , \\ c_{2n}(r_c ) & = & \frac{2\pi}{\mu}\,r_c^2\,v_n \ , \times \ , \left [ 1 + \mathcal{o}(\frac{r_c}{a_0 } ) \right ] \quad \mbox{for $ n \geq 1$,}\end{aligned}\ ] ] where to simplify the notation we have taken @xmath179 . that is , the power - law dependence on the cut - off matches the enhancement of the coupling . the rule is simple : if @xmath180 for @xmath181 the size of @xmath30 for @xmath52 is @xmath182 , a @xmath183 enhancement . equivalently , in momentum space , if @xmath184 for @xmath185 the size of @xmath30 for @xmath186 is @xmath182 . why is that so ? actually the idea can be better explained with a momentum space cut - off . if the couplings scale as @xmath187 with respect to the radial cut - off @xmath17 , in momentum space they will scale as or @xmath188 . ] @xmath189 which simply amounts to take into account that @xmath190 . this scaling property implies that the couplings follow a rge of the type @xmath191 = 0 \ , , \end{aligned}\ ] ] where the dots refer to corrections involving smaller powers of @xmath188 . for the moment we will assume that this rge is valid in the cut - off window @xmath192 . if we ignore the dots the solution is straightforward @xmath193 with @xmath194 and @xmath195 two arbitrary cut - offs . therefore with a boundary condition we can get the running of the couplings for arbitrary @xmath188 this boundary condition is the value of the couplings at @xmath196 . at this scale we expect the couplings to scale with @xmath3 ( we do not expect @xmath2 to play a role at high energies , which means that @xmath3 is the only relevant scale ) , which implies @xmath197 as a consequence @xmath198 which is the expected enhancement for @xmath30 . in short , the scaling of the coupling decides the power counting . this idea is not new and has appeared in different contexts . in the ksw counting @xcite the hard scale can be deduced from the running of the @xmath199 coupling : the scaling of @xmath199 changes when @xmath188 approaches @xmath139 , which happens to be the hard scale in ksw instead of @xmath188 . it is also worth noticing that ksw does not use a cut - off regularization , but a variant of dimensional regularization . ] . recently it has been applied in nuclear eft for the analysis of reactions on the deuteron @xcite . there is the issue of where the rge of the @xmath30 couplings comes from , which is related to the calculation of the power @xmath200 that appears in it . the starting point in wilsonian renormalization is to include a cut - off and then require observable quantities to be independent of the cut - off @xmath201 where @xmath202 is the wave function and @xmath203 an operator corresponding to an observable . notice that here we are demanding the matrix element to be independent of the cut - off . actually this condition is too strong observables are the square modulus of matrix elements but in most situations it will work . if we are in the cut - off window @xmath192 we can substitute the wave function and the operator by the corresponding ones in the eft @xmath204 moreover the operator @xmath205 can be divided into a contact- and finite - range piece @xmath206 now we can rewrite @xmath207 which tell us that the contact - range piece has two functions : to absorb the cut - off dependence of the finite - range piece and to directly contribute to the matrix element . had we used the full wave function @xmath202 and the full operator @xmath203 instead of the eft ones , the contact would have only been there to absorb the cut - off dependence ( the contacts vanish for @xmath208 ) . but within the eft description the contacts must have a non - trivial contribution to the observables regardless of the cut - off . the reason is that the finite - range piece of the eft potential / wave function / operator does not correspond to the fundamental potential / wave function / operator . the bottom line is that for the contact - range operators we can distinguish between a piece that directly contributes to observables and a piece that absorbs cut - off dependence @xmath209 where the superscript @xmath210 and @xmath211 stand for `` direct '' and `` residual '' . this distinction is analogous to the one that we made previously for the running of the @xmath30 in short - range theories . each of these pieces follows a different rge @xmath212 corresponding to their different roles within eft . notice that we are assuming that the distinction between @xmath213 and @xmath214 exists . this is not clear ( if we want the definitions to be unambiguous ) , but we are only using this distinction to simplify the arguments here . we can write a contact - range operator @xmath215 as a coupling times a polynomial involving the light scales in momentum space @xmath216 where @xmath217 is the coupling and @xmath218 is the polynomial , which can be regularized ( hence the subscript @xmath188 ) . if we include this general form in the rge for the `` direct '' piece we arrive to @xmath219 & = & 0 \ , .\end{aligned}\ ] ] what is left is to determine the cut - off dependence of the matrix element of the polynomial , which in general will take the form @xmath220 \ , , \end{aligned}\ ] ] where the form of the corrections follow from the assumption that the rge are valid in the region @xmath221 , that is , from the analyticity of the rge which in turn implies that we can write a power series on @xmath222 and @xmath223 . the previous discussion is rather general and concerns any observable that receives a direct , linear contribution from a contact - range operator . in the case of two - body scattering the matrix element we are interested in is the t - matrix @xmath224 which is not receiving a linear contribution from the contact - range physics , at least at first sight . the t - matrix is the solution of the lippmann - schwinger equation @xmath225 which is a convenient way of rewriting the schrdinger equation for a scattering problem . in eft we can expand the t - matrix and the potential as power series @xmath226 or more concisely as @xmath227 that is , a lo contribution plus a subleading correction . the interesting thing here is that the subleading correction to the t - matrix is perturbative and is given by @xmath228 \\ & = & \langle \psi_{\rm lo } | \delta\,v_{\rm eft } | \psi_{\rm lo } \rangle + \mathcal{o}\left [ { \left(\delta\,v_{\rm eft}\right)}^2 \right ] \ , , \end{aligned}\ ] ] where in the second line @xmath229 is the lo wave function ( @xmath230 ) . if we ignore the iteration of @xmath231 , make the separations @xmath232 and follow the steps previously described , we end up with the rge @xmath233 the solution of this rge depends on the evaluation of the matrix element of the contact - range potential . the details depend on the particular representation for the contacts . for a delta - shell representation in coordinate space @xmath234 the evaluation is trivial @xmath235 with @xmath12 the @xmath236 reduced wave function . we can concentrate on the evaluation of a particular coupling @xmath30 , in which case we obtain @xmath237 where @xmath238 is the zero - energy @xmath236 reduced wave function . the rge for the coupling @xmath30 reads @xmath239 = 0 \ , , \end{aligned}\ ] ] which means that the running is determined by the power - law dependence of the wave function for @xmath240 . in a purely short - range theory the running of the @xmath30 couplings is easy to compute . the zero - energy wave function is @xmath241 where @xmath242 is a normalization factor that is arbitrary ( it does not affect the running of @xmath30 ) . we remind the reader that we are interested in the region @xmath243 . if the scattering length is natural ( @xmath244 ) , we can take @xmath245 and rewrite the wave function as @xmath246 \ , .\end{aligned}\ ] ] therefore the rge for the couplings is @xmath247 = 0 \ , , \end{aligned}\ ] ] as a consequence @xmath248 , in agreement with the previous determination . if the scattering length is large ( @xmath71 ) , we set the normalization to @xmath249 to express the wave function as @xmath250 which leads to the rge @xmath251 = 0 \ , .\end{aligned}\ ] ] that is , the couplings scale as @xmath252 . there is a point of explain here : the counting of @xmath65 can not always be determined with this method . the reason is that we are calculating the scaling of the perturbative piece of the potential . if @xmath65 is perturbative in the first place we will obtain the correct scaling this is the case in a short - range theory with natural scattering length , where we get @xmath253 . on the contrary if @xmath65 is non - perturbative the ideas presented here do not apply . we know that @xmath65 is enhanced by @xmath254 if the scattering length is large . yet the application of this method to @xmath65 is not useless : it gives us information about the scaling of a small , perturbative change of @xmath65 @xmath255 where @xmath256 is enhanced as @xmath80 . that is , the perturbation @xmath256 is of lower order than the original unperturbed coupling @xmath65 . how is that so ? the meaning of this enhancement for @xmath257 is that systems with large scattering lengths are fine - tuned @xcite . a minor change in @xmath65 generates a large change in the scattering length . in particular @xmath258 which entails @xmath259 . from the rg flow perspective the natural solution represents a stable fixed point of the rg equations and the large scattering length solution an unstable fixed point @xcite . that is , the running of the @xmath65 coupling eventually behaves as a constant as @xmath260 . but if the scattering length is large @xmath65 will scale as @xmath261 only as far as @xmath262 . the scaling of @xmath256 for the different cases that we are considering can be consulted in table [ tab : counting ] . the extension to the pionful eft is trivial but requires a case - by - case discussion . if pion exchanges are perturbative ( i.e. subleading ) , the power counting is exactly the same as in the short - range case . the reason is that the @xmath105 wave functions are identical to the short - range case . if pion exchanges are non - perturbative ( i.e. leading ) , the power counting depends on whether the potential is @xmath146 ( central ) or @xmath147 ( tensor ) . for central ope the power counting is again as for a short - range potential because the wave functions behave either as @xmath263 or as @xmath104 for @xmath240 . however for attractive tensor ope the power counting changes . using the wave function written in eq . [ eq : uk_lo ] we find @xmath264 } + \dots \right ] = 0 \ , , \end{aligned}\ ] ] where the dots account for power - law corrections , which are at least of order @xmath265 ( @xmath266 ) . it is worth noting that the corrections to the wave function were computed in ref . for the @xmath267 triplet . it can be checked that they do not affect the counting . the conclusion is that the @xmath30 are bigger than expected by a factor of @xmath268 . the title of this section makes mention of the _ anomalous dimension_. what is that ? the concept is easy to understand . let us assume that we have a physical quantity ( operator , coupling , observable ) @xmath269 where @xmath270 depends on the light scale @xmath2 , on the cut - off @xmath188 and the hard scale @xmath3 . we can define several types of dimensions for @xmath270 . the most obvious one is the canonical dimension @xmath6 , which can be related to the rescaling @xmath271 that is , the canonical dimension refers to how @xmath270 changes with a change of physical units . another type of dimension we can define is the power counting dimension , which refers to a rescaling of @xmath2 ( and @xmath272 only @xmath273 it is important to notice that while the canonical dimension of a physical quantity is unique , the power counting dimension is not . rather a physical quantity is a superposition of contributions with different power counting dimensions @xmath274 the inclusion of @xmath188 among the things we rescale for the power counting dimension seems counter - intuitive at first , but it is natural once we consider the argument about the rg evolution of cut - off dependent quantities from @xmath196 to @xmath275 . finally the anomalous dimension can be defined as @xmath276 which is exactly the kind of power - law dependence on the cut - off that we have been studying along this section . thus we can restate that the anomalous dimension of a coupling is what determines its power counting . wilsonian renormalization is not the most popular or widely understood method of analyzing power counting in efts . this honor corresponds to ultraviolet ( uv ) renormalizability , in which contact - range couplings are included to absorb divergences in feynman diagrams . quantum electrodynamics ( qed ) provides a good illustration of this idea for a quantum field theory ( qft ) that only contains marginal or relevant operators for @xmath277 , while for a marginal operator the coupling runs either as a constant , as @xmath278 or more generally as something that is not power - law . ] , i.e. what is traditionally known as a _ renormalizable _ qft . at this point it is important to mention that nowadays after the discovery of efts renormalizability is understood in a broader sense . yet for efts the application of this principle is simple : we begin by considering the matrix element of an eft operator between eft wave functions @xmath279 the operator contains a finite- and a contact - range piece . for the moment we will ignore the contact - range piece because we want to use these operators to remove divergences in the finite - range piece . thus we consider @xmath280 now we expand this matrix element in powers of @xmath94 as before . to simplify the analysis we only take into account the @xmath105 wave functions @xmath281-th order contribution , include a cut - off @xmath188 and check whether the matrix element @xmath282 is finite for @xmath208 . if not , we include contact - range contributions until the matrix element @xmath283 , the contact is counted as being of this order . we can illustrate the idea in non - relativistic scattering , where the relevant matrix element is @xmath284 where @xmath285 represents the @xmath105 reduced wave function and @xmath17 ( @xmath286 ) is the radial cut - off . in the formula above a contribution to the finite - range potential is said to be of order @xmath287 when it contains @xmath287 powers of the light scales in the momentum space representation @xmath288 where @xmath2 includes @xmath289 , @xmath290 , the pion mass @xmath137 in pionful nuclear eft and/or other scales depending on the particular eft we are dealing with . the expression @xmath291 refers to an arbitrary ratio of light scales ( for instance , @xmath292 and @xmath293 in nuclear eft ) and @xmath294 is a non - polynomial function that we must compute from the eft lagrangian , but which exact form is not important at this point . if we fourier - transform this expression into coordinate space ( and assume for simplicity that the potential is local ) , we find @xmath295 where @xmath291 refers to @xmath296 in nuclear physics . the point is that we know the uv behaviour of the eft potential . provided we have the @xmath105 wave functions we can analyze the matrix element @xmath297 for divergences and decide which contacts to include . the complete analysis can be found in refs . . here we merely comment on the results . if the @xmath105 wave function comes from the @xmath146 potential , perturbative and wilsonian renormalization lead to identical power countings . this is also true if the @xmath105 wave function comes from a purely contact - range potential . on the contrary if the @xmath105 potential is of the @xmath147 type and attractive there is a small , yet significant difference between perturbative and wilsonian renormalization . removing the divergences only requires the @xmath30 to enter at order @xmath298 , in contrast with @xmath299 from rge . the apparent scaling of the couplings is thus @xmath300 , i.e. an extra suppression of @xmath301 with respect to the wilsonian renormalization value @xmath302 . if the @xmath105 @xmath147 potential is repulsive the matrix elements for scattering are always finite and no contact interaction is required . however we will not discuss this problem here . back to the attractive @xmath147 potential the reason for the mismatch probably has to do with the @xmath43 expansion of the @xmath105 wave function , which induces a contamination of @xmath303 into the @xmath30 coupling . in fact the @xmath43 expansion of the @xmath105 wave function reads @xmath304\end{aligned}\ ] ] with @xmath165 , @xmath129 , @xmath132 , @xmath133 , etc . numerical coefficients and where @xmath170 now is independent of energy wave functions for the rg equations we included an energy - dependent semiclassical phase ( see eq . [ eq : uk_lo ] ) , instead of an energy - independent one like here . the reason is that here we are writing the @xmath105 wave functions , in which only the energy - independent @xmath65 operator contributes , while there we were writing the generic solution of the @xmath147 potential with arbitrary short - range physics . ] . we can see that each two powers of @xmath10 imply half a power of @xmath139 as a consequence the couplings that make the matrix element of the potential finite are not the standard @xmath30 s but implicitly contain @xmath305 half integer powers of @xmath139 . that is , they have a different operator structure : @xmath306 instead of @xmath307 . in the same way that it is useful to make the distinction @xmath308 we can also write @xmath309 to make the different structure of these couplings explicit . in this notation the @xmath310 s happen to be enhanced by @xmath311 , just as the @xmath30 s . however the drawback of this explanation is that unlike the @xmath312 coupling , the proposed @xmath310 couplings do not have a clear interpretation at the lagrangian level . the analysis of the residual cut - off dependence of the matrix elements is another method for determining the power counting @xcite . first we will review the theoretical basis for this idea : for that we consider a matrix element for which all uv divergences have been removed at the arbitrary order @xmath13 . the matrix element still contains a residual cut - off dependence that vanishes for @xmath313 : @xmath314 where @xmath315 refers to the contact - range potential that renormalizes the order @xmath13 calculation , while @xmath316 and @xmath317 are coefficients . the point is that the residual cut - off dependence indicates that the next new higher - order coupling of the contact - range potential enters at order @xmath318 . what is the reason for that ? let us assume that the next divergence indeed enters at order @xmath318 . the softest divergence that we are expected to find in the matrix elements of the potential is logarithmic , thus @xmath319 where the previous matrix element is the one for the finite - range potential at order @xmath318 plus the number of contact - range couplings that is expected at order @xmath13 . notice that the value of these couplings change order - by - order , but their number only changes at the order at which a new @xmath30 coupling is included : we have written @xmath320 with a prime to indicate this fact . as the divergence of the finite - range potential is @xmath321 , it is not difficult to infer that going one order down translates into a residual cut - off dependence of @xmath17 while going one order up gives rise to a @xmath322 divergence . equivalently , if we move @xmath323 orders down the expansion the residual cut - off dependence will be @xmath324 , from which the previous conclusion about power counting follows . after this an example might be the best way to illustrate the method . the easiest one is that of a contact - range theory with a large scattering length . if we solve @xmath325 at @xmath105 for the delta - shell short - range potential that we have been using , we obtain the result @xmath326\ , k^2 + \mathcal{o}(k^4 ) \ , , \end{aligned}\ ] ] where the residual cut - off dependence if of order @xmath17 . the conclusion is that the next counterterm is one order below @xmath65 . that is , @xmath60 enter at @xmath327 . if we now proceed to compute @xmath325 at @xmath106 we find calculation includes @xmath60 at first order perturbation theory . otherwise the residual cut - off dependence will be different . ] @xmath328\ , k^4 + \mathcal{o}(k^6 ) \ , .\end{aligned}\ ] ] the residual cut - off dependence is now of order @xmath261 : @xmath77 enters two orders below @xmath60 , that is @xmath110 . strictly speaking residual cut - off dependence is a constructive process and we can use it to determine the location of only the next coupling that enters in the theory , but not more . if we want to find the order of @xmath329 we must first compute the @xmath110 amplitudes that contain @xmath77 and from this extract the residual cut - off dependence . alternatively , we can always rely on the natural expectation that @xmath329 should enter two orders below @xmath77 . finally it is interesting to check the predictions of this idea for the tensor force . on general grounds we expect the cut - off dependence of a @xmath236 calculation of the phase shift with attractive tensor ope to be @xcite @xmath330 which after integration leads to a residual dependence of @xmath331 . this indicates that @xmath60 enters at @xmath332 in agreement with the previous determinations . the central point of this section has been to review how we can derive power counting in wilsonian renormalization . the application of renormalization group analysis ( rga ) to nuclear eft , though sometimes considered a bit arcane , can lead to interesting insights . to illustrate the idea we have taken non - relativistic s - wave scattering as an example and shown in detail how to derive well - known facts about power counting in the two - body sector that we review in table [ tab : counting ] . we have used two equivalent rg formulations . the first is the standard one in which the starting point is a `` fundamental theory '' : we include a cut - off in the theory and then evolve it from the ultraviolet to the infrared . as a result we find a physical theory the eft that is equivalent to the fundamental theory at low energies . the eft incorporates the familiar counting rules that we already know , for instance the enhancement of the couplings when the scattering length is large . the second is a more streamlined formulation in which we do not directly evolve the eft from the fundamental theory and instead use a convenient shortcut to determine the size of the effective couplings . this shortcut is the calculation of the anomalous dimension of the couplings , which turns out to be relatively easy , at least in the two - body case . as we will see , this is also the case for reactions of external probes on two - body states and for the three - body problem in pionless . other important point is the relationship between rga and more standard techniques of determining the power counting . by more standard techniques we refer to ultraviolet renormalization and residual cut - off dependence . in principle we expect all derivations to be equivalent . in practice this equivalence has to be shown by means of concrete calculations . the results indicate the direct equivalence with rge in the absence of singular pion exchanges , i.e. in the absence of the tensor force . if the tensor force is present there is an apparent contradiction though : the @xmath30 couplings seem to be more demoted in ultraviolet renormalization than in rga or in residual cut - off dependence . this disagreement can be explained as a contamination of the @xmath30 couplings with the @xmath139 scale . in other words , what we call @xmath30 in ultraviolet renormalization is not really the @xmath30 coupling , but rather a coupling with the structure @xmath333 instead of the expected @xmath334 . this distinction is in fact analogous to the one that is usually made between @xmath65 and @xmath335 . there are a few open problems that we have not addressed though . the most obvious example is the power counting of the triplet channels where the ope potential is a repulsive @xmath147 . the rg evolution of the couplings indicates that , with the exception of @xmath65 , the scaling of the couplings is identical to that of the attractive @xmath147 potential . this conclusion agrees with a previous rga of the ope potential @xcite , but it is counterintuitive to say the least . if the long - range physics is repulsive we expect that the short - range physics will play a lesser role at low energies because the repulsive long - range physics acts as a potential barrier . that is why some authors prefer to use naive dimensional analysis in this case @xcite . other problem that is related to the previous one is what happens with coupled channels such as the @xmath267 deuteron channel . in this latter case three different countings have been proposed @xcite . even in the attractive triplet channels there are two proposals about the scaling of the @xmath65 coupling : does it enter at @xmath236 ( @xmath336 ) or at @xmath337 ( @xmath338 ) ? the rga of birse @xcite assumes that @xmath65 is @xmath337 in the attractive triplet channels . but here we have taken the view that @xmath65 is @xmath236 : it has to be there because the @xmath236 wave function of a non - perturbative attractive triplet is not well defined without the inclusion of short - range physics . we find it worth noticing that the perturbation @xmath256 of this coupling is @xmath337(@xmath338 ) , which is where @xmath65 is predicted to be by birse s rga @xcite . that is , the cause of the disagreement seems to be that ref . overlooks the presence of short - range physics in the @xmath236 wave functions : the @xmath65 coupling is implicit in the choice of a semiclassical phase , i.e. the choice of @xmath170 in eq . ( [ eq : uk_lo ] ) . one last problem is the scaling of the @xmath30 couplings in the @xmath8 singlet . the standard counting @xcite is that the piece of @xmath30 that carries physical information enters at @xmath339 : @xmath60 at @xmath327 , @xmath77 at @xmath340 , @xmath329 at @xmath341 , etc . long and yang @xcite get to a different conclusion instead : the @xmath30 s enter at @xmath342 : @xmath60 at @xmath327 , @xmath77 at @xmath343 , @xmath329 at @xmath340 , etc . the conclusion is a bit puzzling : according to ref . , dimensional regularization with minimal subtraction leads to a stronger enhancement of the @xmath30 couplings than cut - off regularization other aspect to discuss is the interpretation of the cut - off in eft . the rg equations use a cut - off in the region @xmath192 . this raises the question of whether the cut - off should stay below the breakdown scale , as happens in the rga . the answer is _ not necessarily_. the rg equations are formulated with the limits @xmath344 and @xmath345 in mind to uncover the scaling of the couplings this is a formal requirement to make the analysis easier , not a practical requirement in eft calculations . efts are rg - invariant : the cut - off does not appear in the observable quantities that we compute , only in the intermediate calculations leading to the eft predictions . that is , the cut - off is kept low in rga with the intention of making the scaling of the contact - range couplings as obvious as possible . once the rga is done the only constraints about the size of the cut - off are practical ones . one of these constraints is the existence of residual cut - off independence . in most eft calculations we do not include all the couplings that are required to achieve exact rg independence . we only include the couplings that carry physical information at the order we are considering . there are two reasons for doing this : first , exact rg independence is not well - defined if we are making calculations in an eft at a given order . the systematic eft error is always present and rg independence must be understood within this error . the second reason is that exact rg independence requires the inclusion of what we have called here the redundant couplings , i.e. the @xmath346 piece of the couplings in the rg equations . these redundant operators can be calculated and included explicitly in a few specific cases : the ksw counting @xcite and pionless eft with pds @xcite . but on more general cases this is unpractical and not really necessary . it is easier to keep the residual cut - off dependence under control by a judicious choice of the cut - off . for this condition to be true it is usually enough for @xmath188 to be of the order of the hard scale , though the exact details will depend on the regulator . other important thing is to stress that a power counting is merely an ideal organization of the size of the interactions of a theory . they are derived under the assumption that the scale separation is large and that we can clearly classify all scales either as soft ( @xmath2 ) or hard ( @xmath3 ) . however the real physical world is not necessarily like that . what do we do if we have a two - body system with a scattering length that is neither small nor large ? the point is that what we obtain with rga is just an approximation to a more complex reality . in particular other power countings are possible beyond the ones we have discussed here . for instance in ref . van kolck developed a counting for two - body systems in which the scattering length is tiny . other possibility is when both the scattering length and the effective range are large , a case which can be useful for the description of low lying s - wave resonances or even for the @xmath8 singlet to improve the convergence @xcite . that means that we are entitled to curb the counting rules in view of practical physical information of the system . the limit is theoretical consistency : the eft must be equivalent to the fundamental theory at low energies , which means that renormalizability must be respected . the principles of renormalization work in the same way for operators different than the two - body potential . the advantage of calculating the anomalous dimension is that we can extend the idea seamlessly to any other problem . the point is to have a coupling and a polynomial contact - range operator , to evaluate their matrix element and to demand rg invariance at the end , @xmath347 actually the whole process amounts to nothing more than following a recipe . the only thing we have to do is to choose the wave functions and the operators that are appropriate for the particular physical process we are studying . let us consider the case of a reaction involving a external probe and the deuteron ( or more generally the two - nucleon system ) . in this case the contact - range operators of the theory involve two nucleons and one ( or more ) external fields . the external fields we are interested in are pions , photons and neutrinos . in principle the initial and final wave functions are the product of a two - nucleon wave function and the wave function of zero , one or more external probes @xmath348 where @xmath349 refers to the probes , with the index @xmath350 . the contact - range operator involves a coupling and a polynomial of the momenta of the nucleons and the external probes . in the plane wave basis it reads @xmath351 where @xmath352 ( @xmath353 ) is the center - of - mass momentum of the initial ( final ) two - nucleon system and @xmath354(@xmath355 ) the momenta of all the incoming ( outgoing ) probes involved in the operator . in general the polynomial @xmath356 will involve spin and isospin degrees of freedom , but for the moment we will ignore them . how does one evaluate the matrix elements ? if we consider that the wave functions of the external probes are plane waves , the evaluation of the contact - range operator yields latexmath:[\ ] ] with @xmath478 a small positive number . the evaluation of the following pieces is direct @xmath479 while the two remaining pieces vanish in the @xmath29 limit @xmath480 the reason being that the integrand is bounded in the region around @xmath17 . we can see that while @xmath12 is continuous at @xmath24 , @xmath481 develops a discontinuity . putting the pieces together for @xmath29 we arrive at @xmath482 finally , expanding @xmath483 in powers of @xmath43 we obtain eq . ( [ eq : rge ] ) . nplqcd collaboration ( s. r. beane , e. chang , w. detmold , h. w. lin , t. c. luu , k. orginos , a. parreno , m. j. savage , a. torok and a. walker - loud ) , _ phys . * d85 * ( 2012 ) 054511 , http://arxiv.org/abs/1109.2889[arxiv:1109.2889 [ hep - lat ] ] . nplqcd collaboration ( s. r. beane , e. chang , s. d. cohen , w. detmold , h. w. lin , t. c. luu , k. orginos , a. parreno , m. j. savage and a. walker - loud ) , _ phys . _ * d87 * ( 2013 ) 034506 , http://arxiv.org/abs/1206.5219[arxiv:1206.5219 [ hep - lat ] ] . hal qcd collaboration ( s. aoki , t. doi , t. hatsuda , y. ikeda , t. inoue , n. ishii , k. murano , h. nemura and k. sasaki ) , _ ptep _ * 2012 * ( 2012 ) 01a105 , http://arxiv.org/abs/1206.5088[arxiv:1206.5088 [ hep - lat ] ] . j. polchinski , effective field theory and the fermi surface , in _ theoretical advanced study institute ( tasi 92 ) : from black holes and strings to particles boulder , colorado , june 3 - 28 , 1992 _ , ( 1992 ) . [ arxiv : hep - th/9210046 [ hep - th ] ] . t. s. park , l. e. marcucci , r. schiavilla , m. viviani , a. kievsky , s. rosati , k. kubodera , d. p. min and m. rho , _ phys . * c67 * ( 2003 ) 055206 , http://arxiv.org/abs/nucl-th/0208055 [ arxiv : nucl - th/0208055 [ nucl - th ] ] .

scattering experiments are amongst the most powerful tools to obtain information on the microscopic structure of matter . in solid state physics , conventional elastic scattering of photons , electrons , and neutrons are the standard methods to study the precise spacing and the location of atoms in solids . one obtains this information by measuring the angular distribution of the diffracted particles which depends on the microscopic lattice structure of the studied solid . in conventional x - ray and electron diffraction experiments , the incident particles interact with all the electrons in the sample and , since the majority of the electrons is located close to the nucleus , these experiments yield the averaged position of the atoms . the situation is very similar for neutron diffraction , for which scattering occurs from the nuclei directly . in addition to the sensitivity to the lattice structure , neutron and non - resonant x - rays also interact with the magnetic moments and therefore information on the magnetic structures in solids can be derived . however , in many of the currently most intensively studied systems not only the behaviour of lattice and spins needs to be studied . also the charge and orbital degrees of freedom often play an essential role . in some materials a collective electronic ordering of spins , charges , and orbitals occurs which typically only affects a small fraction of the valence electrons . these phenomena can lead to novel exciting ground states . for instance , complex electronic phenomena involving the cooperative ordering of various electronic degrees of freedom are discussed intensively in relation with such outstanding phenomena like high - temperature superconductivity in cuprates@xcite or the colossal magneto - resistance in manganites@xcite . as we will describe later on , charge and orbital orders are very difficult to observe with the aforementioned traditional scattering techniques . a new experimental probe , which enables to observe complex electronic order was therefore urgently needed . resonant ( elastic ) soft x - ray scattering ( rsxs ) provides exactly this : a highly sensitive probe for spacial modulations of spins , charges , and orbitals in complex materials . this unique sensitivity is achieved by merging diffraction and x - ray absorption spectroscopy ( xas ) into a single experiment , where the scattering part provides the information of spatial modulations and the xas part provides the sensitivity to the electronic structure . more precisely , rsxs close to an absorption edge involves virtual transitions from core levels into unoccupied states close to the fermi level and these virtual transitions depend strongly on the spin , charge and orbital configuration of the resonant scattering centers . the ordering of spins , charges , and orbitals typically results in electronic superlattices with periodicities of several nanometers , which matches very well with the wavelengths of soft x - rays lying between between @xmath0 6 to 0.6 nm corresponding to photon energies between @xmath0 200 ev to 2000 ev . moreover , since the virtual excitations in rsxs are related to specific core level excitations , the excitation energy of which changes from element to element , the method is element specific like xas . thus rsxs , different from magnetic neutron scattering can probe magnetic structures related to specific elements . the price paid for the unique sensitivity offered by rsxs is a very limited ewald sphere and a rather short photon penetration depth limiting studies to only the topmost 100 atomic layers or , depending on the resonance used , even less . but these limitations are very often compensated by the gain in sensitivity , which is extremely important when turning to magnetic and electronic properties of samples characterized by a very small amount of contributing material , i.e. , for studying thin films , nanostructures as well as surfaces and interfaces . as a consequence of the broken translational symmetry at the surface , nanosystems can show ordering phenomena which differ locally or macroscopically from those of the respective bulk systems , and even completely new phenomena can arise . for such systems , rsxs has been established as a very powerful and unique tool to study complex electronic ordering phenomena on a microscopic scale . the need for such studies has strongly increased within the last decade since the fabrication of high quality samples with macroscopic properties tunable by composition , strain , size and dimensionality has become possible , raising the hope for future multifunctional heterostructures characterized by so far unexpected and unexplored novel material properties and functionality . although rsxs offers many important ingredients to study condensed matter , it has only recently developed its full power . the reason for this is that intense soft x - ray sources with tunable energy only became available with the advent of 2nd and 3rd generation synchrotron radiation facilities . furthermore those soft x - rays are absorbed in air and therefore the diffraction stations have to operate in vacuum , which poses another complication and made the development of rsxs challenging . finally , as mentioned above , the penetration depth into the solids is of the order of nanometers which means that surface effects may become important . thus in many surface sensitive systems the surfaces should be prepared under ultra - high vacuum ( uhv ) conditions or at least should be kept under uhv during the measurements . this review covers the recent progress in rsxs and gives a survey of the application of the technique . after the introduction , the principles of rsxs are presented , followed by a description of the experimental development . the main part is then devoted to the application of rsxs in the study of magnetic structures , charge order , and orbital order in thin layers , artificial structures , interfaces , and correlated systems . the review concludes with a summary and an outlook . elastic resonant x - ray scattering combines x - ray spectroscopy and x - ray diffraction in one single experiment . roughly speaking , x - rax diffraction provides the information about the spatial order , while the spectroscopic part provides the sensitivity to the electronic states involved in the ordering . a first qualitative understanding of this strongly enhanced sensitivity to electronic order can be gleaned from the schematic illustration of the resonant scattering process shown in figure[fig : rsxs_scheme ] : the incoming photon virtually excites a core electron into the unoccupied states close to the fermi level , thereby creating the so - called intermediate state @xmath1 of the resonant scattering process . this virtual transition depends very strongly on the properties of the valence shell and , therefore , results in the tremendously enhanced sensitivity of resonant x - ray scattering to electronic ordering . the state @xmath1 then decays back into the ground state @xmath2 and a photon with the same energy as the incoming one is re - emitted . this combination of spectroscopy and diffraction will be described in the following . in this section , aspects of the diffraction of x - rays by a crystal will be summarized . for more detailed and extensive descriptions of this topic the reader is referred to the literature@xcite . + + in the following we will consider a crystal that is formed by a perfectly periodic arrangement of lattice sites , which act as scattering centers for the incident x - ray field . the scattering from site @xmath3 in the crystal is described in terms of a scattering length @xmath4 , which is also called the form factor , and can be represented as@xcite @xmath5 the first two terms @xmath6 and @xmath7 represent the non - resonant charge and magnetic scattering , respectively , where the scattering described by @xmath6 , which is proportional to the total number of electrons of the scatterer , is called thomson scattering . @xmath8 denotes the so - called dispersion correction , which is not only a function of the photon energy @xmath9 , but also of the polarisation of the incoming ( @xmath10 ) and scattered beam ( @xmath11 ) . this correction becomes very important close to absorption edges and describes the resonant scattering processes illustrated in figure[fig : rsxs_scheme ] . and @xmath12 , respectively . the whole crystal can be described by discrete translations of the unit cell with its basis . the relative phase of x - rays scattered by the two sites of the basis is @xmath13 ( right panel).,width=283 ] physically the scattering length @xmath4 describes the change in amplitude and phase suffered by the incident wave during the scattering process : a scatterer with scattering length @xmath4 exposed to an incident plane wave @xmath14 causes a scattered radial wave @xmath15 , which , far away from the scattering center , can be approximated by a plane wave . in order to calculate the intensity of the total scattered wave field with wave vector @xmath16 , all the radial waves have to be summed with the correct relative phases . as illustrated in figure[fig : scatter_scheme ] , the geometric relative phase of two scatterers is given by @xmath17 , where @xmath18 is the scattering vector and @xmath19 is the difference in position . the total intensity detected in a distant detector is therefore @xmath20 where @xmath21 is the vector pointing to the origin of unit cell @xmath22 and @xmath23 is the position of the scatterer @xmath4 measured from that origin . @xmath24 is called the unit cell structure factor and describes the interference of the waves scattered from the different sites within a unit cell . the lattice sum @xmath25 is due to the interference of the scattering from the different unit cells at @xmath21 . its @xmath26-dependence therefore provides information about the number of sites scattering coherently , i.e. , it is related to the correlation length of the studied order . for an infinite number of coherently scattering unit cells , @xmath27 , where @xmath28 represents the reciprocal lattice . this corresponds to the well - known laue condition , which states that a reflection can only be observed if @xmath29 is a reciprocal lattice vector . the treatment described above is known as the kinematic approximation . this approximation is valid as long as the intensity of the scattered wave is much weaker than that of the incidence wave , which means that the interaction between the incoming and outgoing waves or multiple scattering events can be neglected . since in resonant x - ray diffraction experiments one usually deals with weak superlattice reflections , this approximation is valid in most cases . if , however , the scattered wave becomes too strong , then one has to resort to the so - called dynamical theory of diffraction . this description will not be presented here and the interested reader is referred to the literature@xcite . and @xmath16 , is vertical in this example . incoming and outgoing polarisations are denoted as @xmath30 and @xmath31 , respectively , where @xmath32 indicates a polarisation perpendicular to the scattering plane and @xmath33 refers to a polarisation parallel to the scattering plane . the polarisation of the scattered radiation can be determined using the polarisation analyzer . @xmath34 define the reference frame in which the polarisations , wave vectors and spin - directions will be expressed.,width=283 ] + + a typical geometry of a resonant scattering experiment is shown in figure[fig : exp_scheme ] . an incoming photon beam with defined wave vector @xmath35 impinges on the sample and is scattered elastically into the direction defined by @xmath16 , corresponding to a scattering vector @xmath18 . in the so - called specular geometry the incoming and the outgoing beam are at an angle @xmath36 with the sample surface . in this case , the scattering angle between @xmath35 and @xmath35 is @xmath37 . the wave vectors of the incident and scattered beam define the scattering plane , which is vertical in the present example . polarisation directions parallel or perpendicular to this scattering plane are referred to as @xmath33- and @xmath32-polarisation , respectively . correspondingly , the incoming and outgoing polarisations in figure[fig : exp_scheme ] are denoted as @xmath30 and @xmath31 . an important aspect of resonant x - ray scattering is given by the polarisation dependence of @xmath38 at resonance . this can cause the polarisation of the outgoing beam to be different from that of the incoming beam . a typical example , which is frequently encountered in resonant scattering experiments , is a change of the incoming polarisation from @xmath39 to an outgoing polarisation @xmath40 ; so - called @xmath41-scattering . such changes in the polarisation contain important information about the symmetry of the studied order , as will be discussed in more detail later on . controlling both the incoming and the outgoing polarisation can therefore be a big advantage . the incoming polarisation can be routinely controlled by modern insertion devices like elliptical undulators . the determination of the outgoing polarisation in rsxs is currently less common , but can be achieved using a polarisation analyzer as sketched in figure[fig : exp_scheme ] . by switching between the two configurations shown in the figure , for example , it is possible to observe either the @xmath40- or the @xmath42-component of the scattered beam . in the following discussion of the various case studies , a number of different scan - types will be described , which we will introduce briefly here : * radial scan or @xmath43 scan . this is a scan along the direction defined by the scattering vector @xmath44 , which therefore provides information about the correlations along @xmath44 . in practice this scan is done by rotating the sample and the detector in steps of @xmath45 and @xmath46 about @xmath47 , respectively , which is equivalent to changing @xmath36 of the incoming and outgoing beam by the same amount . * transverse scan . this is a scan perpendicular to the direction defined by @xmath44 . it hence contains information about correlation in the plane perpendicular to @xmath44 , but is also often affected by the sample mosaic . a transverse scan can be done by rotating the sample about @xmath48 or @xmath47 , while keeping the detector position fixed . * azimuth scan . this is another way of investigating the polarisation dependence of the resonant scattering . in this type of scan , the resonant scattered intensity is recorded while rotating the sample by an angle @xmath49 about the scattering vector @xmath44 ( cf . figure [ fig : exp_scheme ] ) . azimuthal scans will be described in more detail in sec.[sec : pol_dep ] . * energy scan . here the photon energy dependence of a given reflection at @xmath44 is measured . in many cases this measurement is performed by scanning the photon energy , while recording the scattered intensity at @xmath18 . since both @xmath35 and @xmath16 change with the photon energy , this means that for each photon energy the scattering geometry has to be adjusted such as to keep @xmath44 constant . in the following sections[sec : lightmatter]-[sec : rsxsandxas ] the resonant scattering length and its relation to x - ray absorption will be described in a rather formal way . readers who are mainly interested in the application of the method , can skip this theoretical part and continue with sec.[sec : hannon ] . readers who would like to learn more about the theoretical description of the interaction between light and matter can find a very nice treatment with more details in ref.@xcite . + + in general , the scattering of photons caused by matter is due to the interactions between the electromagnetic wave and the particles in a solid . in the present case only the interactions between the electromagnetic wave and the electrons in a solid are important . the corresponding coupling term @xmath50 is obtained by replacing the momentum operator @xmath51 in the free electron hamiltonian @xmath52 by @xmath53 , which gives @xmath54 here , @xmath51 and @xmath55 correspond to the _ l_th electron of the atom . furthermore , @xmath56 for a transversal radiation field @xmath57 has been used . note also , that the spin dependent part of @xmath50 , which is proportional to and , has been neglected , because these relativistic terms scale as @xmath58 and hence are very small compared to the terms in eq.[hint ] . the quantized electromagnetic field that couples to the electrons can be expressed as @xmath59 where @xmath60}$ ] and @xmath61}$ ] . the operators @xmath62 and @xmath63 , respectively , create and annihilate a photon with polarisation @xmath64 and wave vector @xmath65 . since the first term in eq.[hint ] is proportional to @xmath66 it is linear in @xmath62 and @xmath63 and therefore describes processes involving the emission and absorption of one photon . in other words , the @xmath66 term changes the number of photons by @xmath67 . the second term in eq.[hint ] is proportional to @xmath68 and is therefore quadratic in the @xmath62 and @xmath63 . as a result , this term changes the number of photons by 0 or @xmath69 . + + as introduced above , the scattering of a photon from a specific site @xmath3 in the crystal is described by the scattering length @xmath4 , which is determined by the interactions given by eq.[hint ] . the most important terms entering @xmath4 can be calculated by means of time dependent perturbation theory up to second order in @xmath50 . in the following we will briefly sketch the main steps of this calculation and will present the final result . a detailed derivation of the expression for @xmath4 can be found , for example , in ref.@xcite . since in the following we will consider the scattering from a single site , we will drop the site index @xmath3 . the following scattering event will be considered : a single photon with wave vector @xmath65 and polarisation @xmath10 impinges on a lattice site in the initial state @xmath2 and is scattered into a state with wave vector @xmath70 and polarisation @xmath11 . since we are dealing with elastic scattering , the lattice site will remain in state @xmath2 ; i.e. latexmath:[\[\label{scatter } only one photon exists and , therefore , the only non - vanishing contributions of @xmath50 ( eq.[hint ] ) to the scattering process must contain one creation and one annihilation operator corresponding to @xmath72 and @xmath73 , respectively . the terms first order in @xmath50 are given by the matrix element @xmath74 . since the interaction term @xmath66 is linear in the @xmath63 and @xmath75 , it changes the net number of photons and therefore does not contribute to @xmath76 . only the term proportional to @xmath77 contains products of @xmath78 and @xmath63 , which do not change the number of photons and , hence , only this term gives non - vanishing contributions to @xmath76 . while the @xmath66 term does not contribute to the first order matrix element @xmath76 , there are contributions of this term in second order perturbation theory , which are of the same order of magnitude as @xmath76 . these second order contributions have to be considered as well . the second order matrix element @xmath79 involves two successive @xmath80 interactions . the dominant contributions to @xmath79 are due to terms , where the incoming photon is annihilated first and then the scattered photon created : @xmath81 . here , @xmath1 denotes an intermediate state of the system without a photon . taken together , the terms given by @xmath76 and @xmath79 yield the following expression for the differential cross - section @xmath82 , i.e. , the probability that a photon is scattered into the solid angle @xmath83 : @xmath84 in this equation , @xmath85 is the classical electron radius , @xmath86 corresponds to the fourier amplitude at @xmath87 of the charge density , @xmath88 ( @xmath89 ) is the ground state ( intermediate state ) of the scatterer ( cf . figure[fig : rsxs_scheme ] ) , @xmath90 ( @xmath91 ) is the energy of @xmath88 ( @xmath89 ) , @xmath92 is the life time of @xmath1 and @xmath93 is the current operator that describes the virtual transitions between @xmath88 and @xmath89 . equation[kheqn ] is the famous kramers - heisenberg formula applied to elastic scattering of photons . by definition the differential cross - section is related to the scattering length by @xmath94 , i.e. , the above equation also provides the expression for @xmath4 . the first term proportional to @xmath95 describes the non - resonant thomson scattering of photons from the total charge density @xmath96 . the scattering due to the thomson term is given by the first order matrix element @xmath76 , which describes the direct scattering of a photon caused by the @xmath77 interaction . this non - resonant scattering scales with the total number of electrons and , hence , usually does not provide high sensitivity to electronic ordering phenomena , which typically affect only a very small fraction of @xmath96 . the second order matrix element @xmath79 results in the second term in eq.[kheqn ] . this term describes the resonant scattering processes of the kind illustrated in figure[fig : rsxs_scheme ] . the matrix elements in the nominators describe the virtual @xmath2@xmath97 @xmath1@xmath97 @xmath2 processes , which correspond to the two successive @xmath80 interactions . close to an absorption edge , the photon energy @xmath98 is close to some of the @xmath99 , in which case the corresponding contribution to the sum becomes large and can completely dominate the scattering . according to the above discussion , the scattering length @xmath100 can be written as @xmath101 where @xmath102 represents the non - resonant thomson scattering and @xmath103 is the anomalous dispersion correction . equation[eq : scattkh ] corresponds exactly to eq.[eq : f_general ] given above with the magnetic scattering neglected . + + in many cases , the current operator @xmath104 is treated in the dipole approximation , which means that for @xmath105 one can use @xmath106 and therefore @xmath107 . in the dipole approximation the dependence of @xmath103 on @xmath65 and @xmath70 is therefore neglected . furthermore , since @xmath108=-2i\hbar { \bf p}$ ] , one can replace the momentum operators @xmath51 appearing in @xmath104 by the commutator @xmath109 $ ] . this together with the resonant term in eq.[kheqn ] then yields the following expression for the resonant scattering length : @xmath110 here , @xmath111 is the dipole operator . the polarisation dependence in the above equation can also be expressed using the tensor @xmath112 , which is defined by @xmath113 . this tensor formalism is often found in the literature and will also be used in the following . + + as can be already realized in figure[fig : rsxs_scheme ] , the excitation from the ground state into the intermediate state corresponds to an x - ray absorption process . therefore , there should be a relation between the resonant scattering length @xmath103 and the x - ray absorption cross section . that such a direct relation indeed exists can be seen in the following way : firstly , @xmath103 can also be written more compactly as @xmath114^+ \ , \mathcal{g}(\hbar \omega)\ , [ { \bf e}.{\bf j}(-{\bf k})]\,|g\rangle\ ] ] using the greens function @xmath115 note that @xmath116 is an operator , which is related to the hamiltonian @xmath117 describing the intermediate state , i.e. , the state with the core - hole . secondly , the transition probability per unit time @xmath118 for the x - ray absorption is given by fermi s golden rule : @xmath119 as explained in sec.[sec : lightmatter ] , only the @xmath66 is linear in the photon annihilators and , hence , it is the only term active in the x - ray absorption process . the effective vector field @xmath120 , which couples @xmath2 to @xmath121 can be expressed as@xcite . @xmath122 the action of the operator @xmath63 on the multiphoton state yields the factor @xmath123 , where @xmath124 is the number of photons with wave vector @xmath35 and polarisation @xmath125 . furthermore , @xmath118 together with the photon flux @xmath126 gives the the absorption cross - section @xmath127 . this relation together with eqs.[eqn : fermi ] and [ aabsorp ] gives the following result : @xmath128^+ \ , \mathcal{g}(\hbar \omega)\ , [ { \bf e}.{\bf j}(-{\bf k})]\,|g\rangle\,\right]\end{aligned}\ ] ] since the greens function entering @xmath129 is also given by eq.[eqn : green ] , the comparison of eqs.[eqn : dfgreen ] and [ eqn : abs ] shows that @xmath130\ ] ] for @xmath131 and @xmath132 . this important relation is known as the optical theorem . it enables one to determine the imaginary part of the resonant scattering length from x - ray absorption . once @xmath133 $ ] is known , the corresponding real part can be calculated by means of the kramers - kronig relations , which read @xmath134&=&\frac{2}{\pi } \mathcal p \int_0^{\infty } \frac{\epsilon'{\rm im}[\delta f(\epsilon')]}{\epsilon'^2-\epsilon^2 } d\epsilon'\quad { \rm and}\nonumber\\ { \rm im}[\delta f(\epsilon)]&= & -\frac{2\epsilon}{\pi } \mathcal p \int_0^{\infty } \frac{{\rm re}[\delta f(\epsilon')]}{\epsilon'^2-\epsilon^2 } d\epsilon'\ , , \end{aligned}\ ] ] with @xmath135 the principal part @xmath136 of the integral and @xmath137 the photon energy . edge for yba@xmath138cu@xmath139o@xmath140 . the xas spectra exhibit a characteristic polarisation dependence , which is caused by the local symmetries of the different cu sites in the lattice . the right panel shows the resonant structure factors as a function of energy and polarisation , which were deduced from a kramers - kronig analysis . it can be seen that the energy dependent lineshape of @xmath103 depends strongly on the polarisation . ( reprinted from @xcite . copyright 2011 , american physical society.),width=321 ] using these relations the energy dependence of @xmath141 can be calculated , where @xmath125 is the polarisation of the absorbed photon . this is demonstrated in figure[fig : kkanalysis ] , were such an analysis is shown for the high - temperature cuprate superconductor yba@xmath138cu@xmath139o@xmath140 . it is important to realize however , that the above analysis allows one to obtain only matrix elements of the form @xmath142 . the matrix elements @xmath143 with @xmath144 can therefore not directly be determined by x - ray absorption . in principle it is possible , however , to measure the x - ray absorption with 3 different polarisations with respect to the crystal axes and 6 other linear combinations of those . this enables to retrieve all 9 matrix elements of @xmath112 , by solving a set of linear equations . the scattering length can also be related to the index of refraction @xmath3 . we will only briefly discuss the origin of this relation . readers interested in the detailed derivation are referred to the literature@xcite . the relation between @xmath3 and @xmath100 is essentially given by the fact that the transmission of a beam through a plate of some material can be described in two equivalent ways : ( i ) behind the plate there will be an additional phase shift that was suffered by the wave during the passage through the material of the plate . at normal incidence , this phase shift is simply given by @xmath145 , where @xmath146 and @xmath147 are the the wave vector of the light and the thickness of the plate , respectively . ( ii ) the plate can also be considered as a set of scatterers , which cause an additional scattered wave behind the plate . the total wave field behind the plate is therefore a superposition of the incident wave and the scattered wave , where the latter is related to the scattering length @xmath100 of the scatterers within the plate . only identical scattering centers are assumed . since both descriptions must give the same result for the wave field behind the plate one can deduce that @xmath148 where @xmath149 is the number density of the material . since @xmath103 is a complex number , the above eq.[eq : nfrelation ] relates the real and imaginary part of @xmath100 to the dispersion terms @xmath150 and @xmath151 of @xmath152 . + + the general expressions derived in the previous sections are useful to understand the general process of resonant diffraction . however , for the analysis of experimental data it is necessary to evaluate @xmath103 for a specific situation . the simplest case for which explicit expressions can be derived , is the one of a free atom with a magnetic moment . in this situation only the magnetic moment breaks the spherical symmetry of the free atom . for this special case the resonant scattering length in the dipole approximation can be expressed as@xcite @xmath153 where @xmath154 is the unit vector defining the local spin moment and the @xmath155 ( @xmath156 ) are photon energy dependent resonance factors . these factors depend on the products @xmath157 appearing in eq.[eqn : fdipole ] and define the strength of the resonant scattering . there are 3 different terms in eq.[eqn : hannon ] : the first term does not depend on the spin direction and corresponds to non - magnetic resonant scattering . this term simply adds to the thomson scattering in eqs.[eq : f_general ] and [ kheqn ] , which has the same polarisation dependence . the second term is proportional to @xmath158 and is linear in @xmath154 and therefore describes magnetic resonant scattering . by virtue of the optical theorem this part of @xmath103 can be related to the x - ray magnetic circular dichroism in absorption@xcite . the third term is proportional to @xmath159 , i.e. , this term is quadratic in @xmath154 and hence also corresponds to resonant magnetic x - ray scattering . this scattering is related to the linear dichroism in x - ray absorption@xcite . equation[eqn : hannon ] is very often used for the analysis of resonant scattering from magnetically ordered materials@xcite . however , it is important to remember that this expression has been derived for a free atom , where only the magnetic moment breaks the otherwise spherical symmetry . + + whenever an atom is embedded in a crystal , the chemical environment will break the spherical symmetry that was used to derive eq.[eqn : hannon ] . this equation has therefore to be considered as an approximation for describing magnetic scattering in a crystal lattice . for instance , the x - ray absorption of a non - magnetic and non - cubic material will change with the polarisation @xmath10 pointing along the inequivalent lattice directions . the optical theorem therefore immediately implies that @xmath160 also varies with @xmath125 . this polarisation dependence is not captured by eq.[eqn : hannon ] , because in the non - magnetic case , this formula implies @xmath161 , which is isotropic and the same for all @xmath162 . in the following we will provide a simplified description of the effects described by carra and thole@xcite , and by haverkort@xcite . in particular , we will restrict ourselves to the dipole approximation and to magnetic scattering terms that are linear in the spin direction @xmath163 . this is sufficient to illustrate the effects of the local environment , but in general the higher order terms can be relevant as well . for the complete expressions including higher order terms in @xmath164 the reader is referred to ref.@xcite . in the most general case of a magnetic site in a low symmetry environment , the tensor @xmath165 ( cf . [ sec : dipoleapprox ] ) , describing the polarisation and energy dependence of the resonant scattering has the form : @xmath166 with complex energy dependent matrix elements @xmath167 . however , depending on the local point group symmetry of the scatterer , this tensor can be simplified and especially in high symmetries the simplifications are significant : in the extreme case of a free non - magnetic atom with spherical symmetry one has @xmath168 with the identity matrix @xmath169 . if the spherical symmetry is broken only by the spin moment of the scatterer , the first two terms of eq.[eqn : hannon ] translate to @xmath170 it is this expression that is very often used for the analysis of resonant x - ray scattering data . it is important to know in which situations this approximate expression can be used and in which situation it yields wrong results . we therefore discuss two other cases to clarify this point . first , we consider a local environment of @xmath171-symmetry . in this situation @xmath112 can be expressed as @xmath172 comparing eqs.[eqn : fsph ] and [ eqn : foh ] it can be seen , that the spherical approximation may still be used , as long as the magnetic term linear in @xmath164 dominates the resonant scattering signal . we mention that already for @xmath171-symmetry the terms quadratic in @xmath164 differ from eq.[eqn : hannon ] . as a second example we consider a local @xmath173 symmetry . depending on the deviation from @xmath171 , the resonant scattering length can now deviate significantly from the spherical approximation : @xmath174 as can be seen in the above equation , the energy dependence of the magnetic resonant scattering now depends on the spin direction , i.e. , depending on the direction of @xmath164 , @xmath175 and @xmath176 contribute with different weights . this effect is therefore given by the difference between @xmath175 and @xmath176 . it is also important to note that even the non - magnetic scattering in eq.[eqn : fd4h ] is no longer isotropic . this means that for local symmetries lower than cubic the non - magnetic resonant scattering also becomes polarisation dependent , in agreement with the qualitative arguments given in the beginning of this section . we already mentioned that the polarisation dependence of resonant x - ray scattering provides important information about the symmetry of the studied ordering . in this section we will provide a simple example for this , namely the resonant magnetic scattering of a bcc antiferromagnet . in this case the polarisation dependence of the resonant scattering allows one to determine the directions of the ordered moment . + + before we discuss our example , it is useful to introduce an efficient way to describe polarisation dependent resonant scattering . the resonant scattering length as a function of the incoming and outgoing polarisation can be expressed , using a @xmath177 scattering matrix @xmath178@xcite : @xmath179 the matrix elements of @xmath180 therefore correspond to @xmath181- , @xmath41- , @xmath182- and @xmath183-scattering of lattice site @xmath3 . using these @xmath180 , the corresponding structure factor matrix @xmath184 can be calculated according to eq.[eq : kinematic ] : @xmath185 . for the special case described by eq.[eqn : hannon ] , the scattering matrix as a function of the local spin - direction reads@xcite : @xmath186 where the spin - direction @xmath187 is expressed with respect to the @xmath34 and @xmath36 denotes the scattering angle ( cf . figure[fig : exp_scheme ] ) . - and @xmath183-scattering , where the spin directions are @xmath188 and @xmath49 is the rotation angle about the ( 00l)-scattering vector . the lower two panels display the azimuthal dependences of the @xmath41- ( left ) and @xmath183-scattering ( right ) for the spins pointing along @xmath189 . in this example , the polarisation dependence of the resonant scattering enables the spin - directions to be determined . @xmath190 in all cases.,width=321 ] . + + in order to illustrate the polarisation dependence of resonant magnetic x - ray scattering , we consider the example of a bcc antiferromagnet . the unit cell contains two inequivalent lattice sites : one at @xmath191 with @xmath192 and one at @xmath193 with @xmath194 . for the sake of simplicity we assume that the two lattice sites differ only in the local spin moment and are identical otherwise . note that the above coordinates refer to the so - called crystal frame , which is defined by the crystal axes . we will consider the ( 001)-reflection for which @xmath195 . since the system is cubic and only the terms linear in @xmath196 contribute to resonant @xmath197 , we can use the expression given in eq.[eqn : foh ] which is proportional to @xmath198 , i.e. , the @xmath199 are simply given by the term proportional to @xmath200 in eq.[eqn : soh ] . this then yields : @xmath201 in this equation @xmath49 is the azimuthal angle and describes a rotation of the sample about @xmath202 as indicated in figure[fig : exp_scheme ] . when the sample is rotated about @xmath202 , the crystal frame and with it the spin directions rotate within the @xmath203-plane . in the current setting the spin directions are along @xmath47 for @xmath204 . the square modulus of the matrix elements of @xmath184 is proportional to the intensities in the different scattering channels . for instance , the intensities in the @xmath41- and the @xmath183-channels are @xmath205 and @xmath206 , respectively . the calculated azimuth - dependences for @xmath207 and @xmath208 are shown in the top panel figure[fig : azi ] . the azimuthal dependence of @xmath207 is due to the fact that @xmath209 , i.e. , this intensity is maximal if the spin directions are parallel to the scattering plane . similarly , it is easy to see that @xmath210 , which means that @xmath208 maximal for spins aligned perpendicular to the scattering plane . figure[fig : azi ] also shows the azimuthal dependences of @xmath207 and @xmath208 for spins along @xmath211 . it can be seen that the azimuthal dependence changes drastically with the inclination angle @xmath212 between @xmath213 and @xmath154 . in this simple example of a cubic antiferromagnetic system , the polarisation dependence of the resonant scattering provides a powerful means to identify magnetic scattering and to determine spin - directions : ( i ) in a cubic system , the observation of @xmath41-scattering directly implies magnetic scattering . ( ii ) analyzing the azimuthal dependences enables to determine the direction of the spins . however , great care must be taken when dealing with non - cubic systems . in this case , @xmath41-scattering no longer unambiguously identifies magnetic scattering . a very prominent example is the orbitally ordered and magnetically disordered phase of lamno@xmath139 , where the @xmath41-scattering is caused by orbital ordering and not by spin order@xcite . the @xmath41-scattering in this case is related to the polarisation dependence of the first term in eq.[eqn : fd4h ] . in materials with low local symmetries , the analysis of the azimuthal dependences is also more involved and the cubic approximation might lead to wrong results@xcite . one of the most outstanding properties of resonant x - ray diffraction is its greatly enhanced sensitivity to electronic ordering phenomena . this sensitivity is a spectroscopic effect , which is related to the resonance factors @xmath214 of the resonant scattering length . it should be noted , however , that resonant scattering involves transitions into unoccupied states , i.e. , these experiments probe modulations of the unoccupied states close to the fermi level . besides the high sensitivity , the inherent element selectivity is a second advantage of resonant x - ray scattering : by tuning the photon energy to a specific absorption edge , the scattering is usually dominated solely by the scatterers at resonance . in the following we very briefly summarize some of the main spectroscopic effects at the o @xmath215 , the tm @xmath216 and the re @xmath217 edges , which will be relevant for the results presented in section[sec : casestudies ] . a much more complete account of spectroscopic effects in core - level spectroscopies , including the core - levels described below , can be found in @xcite . + + in the tm oxides the o @xmath218 states play an important role in the chemical bonding . it is therefore of great interest to investigate their role for electronic ordering phenomena . by performing a scattering experiment at the o @xmath215 edge , the scattering process involves @xmath219 transitions , i.e. , virtual excitations directly into the @xmath218 valence shell . in this way , the resonant scattering at the o @xmath215 edge becomes very sensitive to spatial modulations of the o @xmath218 states . there is no spin - orbit interaction of the @xmath220 core - hole and the spin - orbit interaction is usually weak in the @xmath218 final state also . resonant scattering at the o @xmath215 edge is therefore typically rather insensitive towards the ordering of spin moments . however , the hybridisation of the o @xmath218 levels with the tm @xmath221 states can induce magnetic sensitivity also here as discussed in section[sec : multiferroics ] . in addition to this , and in contrast to the tm @xmath216 and re @xmath217 edges described below , the @xmath220 core hole does not produce a strong multiplet splitting in the intermediate state . up to date a number of experiments on different materials , including the cuprates and the manganites , have demonstrated a strong resonant enhancement of superlattice reflections at the o @xmath215 edge , revealing the active role of the oxygen @xmath218 states in the electronic order ( cf . section[sec : chargeorder ] ) . nearly all the resonances observed so far are peaked at the low - energy side of the oxygen @xmath215 edge , showing that the probed electronic modulations mainly involve the electronic states close to the fermi level . + + at the transition metal @xmath216 edges the resonant scattering process involves @xmath223 transitions . this makes it possible to study ordering phenomena related to the @xmath221 valence states of the tm . the strong spin - orbit interaction of the @xmath218 core - hole splits the x - ray absorption spectra into the so - called @xmath224 edge at lower energies ( @xmath225 core hole ) and the @xmath226 edge at higher energies ( @xmath227 core hole ) . this is shown for the cu @xmath216-edge in figure[fig : kkanalysis ] . furthermore , the coupling of the @xmath218 core - hole with a partially filled @xmath221-shell in the intermediate state , results in a very pronounced multiplet structure of the @xmath216 edges not only in absorption , but also in resonant scattering , as can be inferred from the optical theorem . the spin - orbit interaction causes a greatly enhanced sensitivity towards spin - order@xcite , which , by means of the optical theorem , is directly related to the magnetic circular and linear dichroism in absorption@xcite . rsxs at the tm @xmath216 edges therefore allows the study of magnetic order of the tm - sites . in addition other electronic degrees of freedom of the @xmath221 electrons , like orbital and charge can be probed at these edges in a very direct way , since the virtual transitions depend directly on the configuration of the @xmath221 shell . + + as will be described in detail in section[sec : magneticstructureofthinfilms ] , rsxs at the re @xmath217 edges plays an important role in determining the magnetic structure of thin films containing res . at the re @xmath217 edge , the resonant scattering process involves @xmath228 transitions . one therefore probes the @xmath229 states of the re elements , which play the most important role for their magnetism . in the @xmath230 final state of xas there is a strong spin - orbit coupling that splits the absorption edge into the @xmath231 edge at lower energies ( @xmath232 core hole ) and the @xmath233 edge at higher energies ( @xmath234 core hole ) . furthermore , the coupling between the core - hole and the valence shell , and the spin - orbit interaction acting on the 4f - electrons , causes a multiplet structure . both the @xmath217 splitting and the multiplet structure are also reflected in rsxs as dictated by the optical theorem . similar to the case of the tm @xmath216 edge , the spin - orbit splitting in the intermediate state of the rsxs process again results a greatly enhanced magnetic sensitivity , which enables the study of the magnetic order in extremely small sample volumes . a particular feature of the re spectra is the small crystal field splitting . hence , the spectral shape at the @xmath217 resonances is largely determined by the atomic multiplet structure independent of the crystal symmetry . effects of the local symmetry , as discussed in section[sec : haverkort ] are therefore not so pronounced as in case of @xmath221 tm , which permits an interpretation of rsxs data without recourse to detailed multiplet calculations@xcite . rsxs experiments require photon energies in the range between 200 ev and 2000 ev and substantially benefit from a control of the incident light polarisation , in particular linear in the scattering plane as well as in the perpendicular direction ( @xmath32 and @xmath33 in figure[fig : exp_scheme ] ) . but also circular polarisation is important , as will be discussed in connection with cycloidal magnetic structures in section[sec : multiferroics ] . these conditions are met by a number of beamlines at third - generation synchrotron radiation sources worldwide . they typically have elliptical undulators as radiation sources , often of apple - ii type . beamlines are designed using grating monochromators and focusing mirrors operated in grazing incidence . an account of the beamline layout at the berlin synchrotron bessy ii is given in@xcite . as pointed out already , soft x - rays are subject to strong absorption by ambient atmosphere . unlike diffractometers for hard x - ray resonant scattering , apparatus for resonant scattering in this energy range must therefore be designed as vacuum instruments , while permitting scattering geometries as shown in figure[fig : exp_scheme ] . for many applications , even ultra - high vacuum ( uhv ) conditions are preferred , because the x - ray absorption length in the materials studied can be as short as 200 in the vicinity of soft x - ray resonances@xcite , rendering the method quite surface sensitive at small scattering angles . therefore , cleanliness of the sample surface can become an issue , and particularly thin - film reflectivities at low temperatures will be modified by growing overlayers due to gas absorption . while resonant diffraction using hard x - rays could make use of conventional diffractometers , instrumentation for soft x - ray diffraction therefore has only recently become available . one of the first soft x - ray diffractometers was developed for studies of thin polymer films and worked in a vacuum of 10@xmath235 torr@xcite . motivated by the high magnetic sensitivity of the soft x - ray resonances , and following pioneering experiments@xcite , the development of vacuum instrumentation for diffraction experiments was initiated . reflectometers developed at synchrotron radiation sources for the characterization of optical elements were modified for the study of magnetic thin - film materials@xcite . as soft x - ray resonances were found to be very useful for the study of complex ordering in correlated electron systems like charge or orbital order@xcite dedicated diffractometers were developed that are now operational at various synchrotron radiation sources@xcite . [ scale=0.8 , trim= 0 0 0 0 , angle= 0 ] hawthorn2011 - 1 for the design of these instruments , two approaches are employed . in order to maintain the full flexibility in terms of scattering geometries and sample rotation , diffractometers as known from hard x - ray applications were built from uhv - compatible materials and mounted in a uhv chamber . this requires vacuum - compatible motors and sophisticated mechanical design but maintains the full functionality of a conventional diffractometer . the first instrument of this type was commissioned at the national synchrotron light source@xcite . recently , a new diffractometer with a horizontal scattering geometry was launched at beamline 10id-2 of the canadian light source@xcite . the instrument is shown in figure[hawthorn2011 - 1 ] and represents a 4-circle diffractometer , with sample ( @xmath236 ) and detector ( @xmath237 ) rotation as well as sample tilt ( @xmath238 ) and azimuthal rotation ( @xmath239 ) . sample manipulation is complemented by @xmath240 , @xmath241 , and @xmath242 stages . situated inside a vacuum chamber , it operates at a base pressure of @xmath243 torr . samples can be cooled with a closed - cycle he cryostat connected to the sample holder by flexible cu braids . avoiding a rigid mechanical connection to the cryostat has the advantage that temperature - dependent studies can be carried out without substantial movement of the sample due to the thermal contraction of the sample holder . sample position was reported to be stable within 100@xmath244m@xcite . the base temperature achieved in this instrument is 18 k. the @xmath237 rotation carries various photon detectors , such as a axuv 100 silicon photodiode , which works very reliably over a large dynamic range of many orders of magnitude , which is particularly required for reflectivity measurements . for the detection of weak signals a channeltron typically coated with a low workfunction material such as kbr can be used . the instrument also has a multi channel plate ( mcp ) with resistive anode readout for the two - dimensional detection of scattering signals . [ scale=1 , trim= 0 0 0 0 , angle= 0 ] beale2010a-1 other designs at least partially avoid in - vacuum motors and complex mechanics and have the drives at atmosphere . figure[beale2010a-1 ] shows the experimental station rasor operated with a vertical scattering geometry at beamline i10 of the diamond light source . here sample and detector rotation are driven by motors and gears outside the vacuum using a double differentially pumped rotary feed - through . also sample tilt as well as the translational alignment @xmath245 are actuated from outside the vacuum , while sample azimuth involves an in - vacuum drive@xcite . a new in - vacuum spectrometer has been installed on beamline x1a2 at the nsls , brookhaven national laboratory . this spectrometer employs a 6-circle geometry with both horizontal and vertical detector rotations , enabling the detector to be placed anywhere within the hemisphere of the vacuum vessel . this detector configuration dispenses with the need for a @xmath238 rotation of the sample and greatly simplifies the sample rotation hardware enabling a much shorter cryostat to aid in stability and the future use of more complicated sample environments . unique to this instrument is the capability of performing both surface soft x - ray scattering measurements and , through the use of zone plate optics , soft x - ray nano - diffraction by raster scanning the beam over the sample to measure a real space image of the scattering@xcite . [ scale=0.5 , trim= 0 0 0 0 , angle= 0 ] bessydiffractometer-1 other designs of soft x - ray diffractometers do not use a conventional four - circle geometry and in this way benefit from avoiding in - vacuum motors and mechanics . such an instrument is operated at beamline ue46-pgm1 of the berlin synchrotron bessy ii ( see figure[bessydiffractometer-1 ] ) . the basis of the diffractometer is a large differentially pumped two - circle rotary feedthrough driven outside the vacuum by two modified huber rotation stages . working in horizontal scattering geometry , the instrument combines very high mechanical precision with the capability of carrying high loads , particularly on the detector circle . the latter consists of a large disc that comprises several uhv flanges up to a bore size of 100 mm diameter . detectors of various degrees of complexity can be assembled on dn 100 cf flanges and can then be easily transferred and removed from the diffractometer ( see figure[bessydiffractometer-2 ] ) . presently , the instrument is equipped with a conventional axuv 100 si photodiode , a slit carousel allows in - situ adaption of the resolution . the whole detector is mounted on a linear feed - through , which resembles an @xmath246 rotation of the detector and readily compensates for a missing @xmath150 tilt of the sample . the instrument can operate under uhv conditions . pressures in the low @xmath247 mbar range are reached even without baking the chamber by operating the sample cryostat together with a liquid nitrogen cryogenic trap for a few days . various sample holders can be introduced via a cf flange with a bore of 90 mm diameter ( see figure[bessydiffractometer-2 ] ) . for standard experiments , samples are mounted on a janis he flow cryostat . here , base temperatures of @xmath248k are easily achieved and with proper shielding 3 k is reliably obtained . it is to be noted , however , that in this temperature region , heating by the x - ray beam is non - negligible and must be taken into account , e.g. , by adequately reducing the photon flux . [ scale=0.5 , trim= 0 0 0 0 , angle= 0 ] bessydiffractometer-2 in rsxs experiments , the requirements for sample environment do not only include vacuum conditions and low temperatures . an important parameter for the study of magnetic materials is an external magnetic field . while fields of the order of 100 gauss can be obtained by small coils in the vicinity of the sample , larger fields require substantial effort . several apparatus were developed in recent years that are equipped with normal - conducting coil systems , achieving magnetic fields of @xmath249 tesla . the alice instrument operated at bessy ii has water - cooled coils outside the vacuum , and the magnetic field is guided into the chamber by an iron yoke@xcite . fields at the sample position can reach 0.8 tesla . the resoxs instrument now operated at soleil has a four - pole magnet arrangement that can be rotated and allows high flexibility in the direction of the field with respect to the sample@xcite . here , @xmath250 tesla are achieved . a recent reflectometer for magnetic measurements developed at the max - planck - institut stuttgart provides a magnetic field of 0.65 tesla@xcite . dedicated soft x - ray diffractometers equipped with superconducting coils that provide magnetic fields considerably larger than one tesla are presently not available . however , a superconducting magnet that is rotatable inside a vacuum chamber and allows for selected scattering geometries with magnetic fields up to 7tesla is operated at beamline ue46-pgm1 of bessy ii . while sample environment is one challenge of soft x - ray diffraction experiments , x - ray detection is now also a focus of instrumental development . two - dimensional detectors can enhance the efficiency of measurements substantially and some of the instruments are already equipped with vacuum - compatible x - ray ccd cameras that are mounted on the @xmath251-arm of the diffractometer and can be rotated inside the vacuum . instruments with that option are operated at the swiss light source@xcite and the national synchrotron light source@xcite . in order to obtain the full information that can be obtained from a soft x - ray diffraction experiment , polarisation analysis of the scattered x - ray beam is also desirable . the method is routinely used in hard x - ray non - resonant and resonant diffraction@xcite and uses single crystals with a suitable lattice spacing as analyzers ( cf . figure[fig : exp_scheme ] ) . with a bragg diffraction angle corresponding to the brewster angle ( @xmath252 ) at the given photon energy , x - rays with @xmath33 linear polarisation are ( ideally ) completely suppressed . rotating the analyzer crystal around the x - ray beam scattered from the sample allows to characterize the polarisation of the scattered x - rays . [ scale=1 , trim= 0 0 0 0 , angle= 0 ] staub2008 - 1 using the same method with soft x - rays usually requires artificial multilayer structures with suitable lattice spacings . the technique was initially applied in polarimeters to characterize the polarisation state of synchrotron radiation@xcite . recently , it was also integrated in soft x - ray diffractometers . the first instrument to be equipped with a polarisation analysis was reported by staub et al.@xcite . polarisation analysis is performed by an artificial w / c multilayer with graded layer spacings . by linear translation the layer spacing can be chosen to meet the bragg condition at the brewster angle for a given resonance energy . the efficiency of such a multilayer at the first - order diffraction peak is typically of the order of a few percent , as can be inferred from the reflectivity curve shown in figure[staub2008 - 1 ] . despite this substantial loss in intensity , polarisation analysis in soft x - ray diffraction is feasible and can be employed to study complex phenomena such as orbital currents@xcite . recently , an advanced polarisation analysis has become available with the previously mentioned rasor instrument@xcite . here , even a full rotation of the analyzer multilayer around the scattered x - ray beam is possible ( see figure[beale2010a-2 ] ) . [ scale=1 , trim= 0 0 0 0 , angle= 0 ] beale2010a-2 following the prediction of very strong magnetic scattering cross sections at the lanthanide @xmath217 and 3@xmath253-transition metal @xmath216 absorption thresholds @xcite , historically , rsxs was first employed to study magnetic ordering in thin metallic films and multilayers . the first demonstration of the strength of magnetic rsxs has been reported by kao et al . , who observed changes in the specular reflectivity of a ferromagnetic 35 - thin single - crystalline fe film upon magnetization reversal@xcite . exploiting linear polarised x - rays , they found asymmetry ratios @xmath254 as large as 13% at the fe @xmath216 resonance and even stronger effects have been reported for scattering of circular polarised x - rays from a 37thin co film with asymmetry ratios up to 80% close to the co @xmath216 resonance as shown in figure[p_kao94 ] @xcite . here @xmath255 and @xmath256 are the measured specular reflectivity for the two opposite in - plane magnetization directions of this experiment . the asymmetry ratios strongly depend on the scattering angle , which represents additional structural information on the magnetic order . reflectivity of a 37 thin co film as a function of grazing angle measured at the maximum of the co @xmath257 edge . the solid and dashed lines denote the two opposite magnetization directions , the dots represent the resulting asymmetry ratio , which shows a strong angular dependence and amounts up to 80% . ( reprinted with permission from @xcite . copyright 1994 , american physical society.),width=264 ] these two pioneering experiments revealed the huge potential of rsxs combining the spectroscopic power of x - ray magnetic circular dichroism ( xmcd ) with structural information . even without exploiting the latter aspect , a large number of studies used the strong dichroic effects of the scattered intensities to extract information usually obtained by x - ray absorption techniques @xcite , like monitoring element - specific magnetic hysteresis loops by measuring the field dependence of the circular dichroism of the reflected intensity for fixed photon energies and angular positions . such studies avoid the experimental difficulties of xmcd performed in total electron yield ( tey ) or fluorescence yield ( fy ) mode , i.e. , a limited probing depth and the influence of external fields and charging in the tey mode or weak signals connected with the fy mode . while these first experiments focused on thin single films , tonnerre et al . presented the first demonstration of magnetic rsxs from a multilayer system consisting of a repetition of ag(11)/ni(17.5 ) double layers with antiferromagnetic ( afm ) or ferromagnetic ( fm ) coupling of the ni layers depending on external magnetic field . from this sample either pure half - order magnetic bragg reflections ( afm coupling ) or huge magnetic circular dichroism on the corresponding chemical superstructure reflections ( fm coupling ) could be observed . by analyzing the magnetic scattering intensities , a magnetic scattering strength at the ni @xmath216 edge of the order of ten times the electron radius @xmath258 was deduced ( see figure[p_tonnerre95 ] ) , i.e. , a value very close to the non - resonant charge scattering strength @xcite . imaginary part of magnetic rsxs at the ni @xmath216 edge ( open circles ) obtained from a ni / ag multilayer compared to ni magnetic circular dichroism data ( solid line ) in electron units . ( reprinted with permission from @xcite . copyright 1995 , american physical society . ) ] the strong enhancement of the scattering cross section at resonance is intrinsically accompanied by a huge change of the optical constants @xmath152 ( see section[sec : rsxsandxas ] ) , with @xmath151 and @xmath150 being of the order of @xmath259 . as a consequence , dispersion and absorption effects can no longer be neglected but often have to be included in a quantitative analysis of rsxs data . consequently , a large number of studies have been dedicated to a quantitative characterization of soft x - ray resonances , where the optical parameters are typically not well known and the tabulated values @xcite can not be applied . experimentally , the easiest access to the optical constants is by measuring the absorption coefficient by transmission @xcite , tey , or fy methods and subsequent calculation of the real part of the index of refraction by applying the kramers - kronig relation ( cf . section[sec : rsxsandxas ] ) . however , the accuracy of the individual methods of measuring the absorption coefficient are often limited by various experimental uncertainties such that it is advantageous to acquire additional information independently as has been done by measuring the faraday rotation of linear polarised light passing through an fe / cr multilayer @xcite or the energy - dependent bragg - peak displacement of fe / v superlattice reflection @xcite . in particular , it has been shown that the analysis of polarisation- and energy - dependent position , width and intensity of a superstructure reflection can be used to determine the full set of optical parameters at resonance @xcite . while these first studies focused on the @xmath221 tm @xmath216 resonances , detailed characterization of the lanthanide @xmath217 resonances has been performed later revealing even stronger resonance effects @xcite . figure [ p_peters2004 ] shows the resonant scattering length through the gd @xmath217 resonance calculated on the basis of polarisation dependent absorption obtained in transmission through a thin gd@xmath260fe@xmath261 film . here , @xmath262 represents the resonant charge scattering contribution of the order of 500@xmath263 , which is almost 10 times stronger than off - resonant charge scattering ( cf . eq.[eqn : hannon ] in which the photon energy dependent resonant factors are called @xmath264 , i.e. , @xmath265=@xmath264 ) . even more impressive , the circular dichroic scattering contribution @xmath266 ( @xmath200 in eq.[eqn : hannon ] ) is as large as 200@xmath263 , i.e. , a magnetic scattering strength 3 times stronger than off - resonant charge scattering . resonant amplitudes at the gd @xmath217 edges . shown are the complex charge @xmath262 , circular magnetic @xmath267 , and linear magnetic @xmath268 , atomic scattering factors as function of energy in units of @xmath263 . top : imaginary parts , bottom : real parts . right axis : approximate atomic cross sections in @xmath269 using a fixed wavelength for e=1200 ev . dash - dotted line : high energy limit of the atomic scattering amplitude z=64 . ( reprinted with permission from @xcite . copyright 2004 , american physical society . ) ] with this extraordinarily high magnetic sensitivity , the study of magnetic structures of thin films down to only a few magnetic layers became feasible . + + the power of magnetic rsxs to study afm bragg reflections from thin single films of only a few magnetic layers has been demonstrated for thin holmium metal films @xcite . metallic holmium possesses the largest magnetic moment of all elements coupled by oscillatory long - range rkky interactions . below the bulk ordering temperature of 132k , ho develops ferromagnetic order within the closed - packed planes of the hexagonal closed packed ( hcp ) structure but with a certain angle between the moments of neighbouring planes resulting in a long - period afm structure along the [ 001 ] direction with an incommensurate temperature - dependent period of about 10 monolayers at 40k . in a magnetic diffraction experiment , this magnetic structure causes magnetic superstructure reflections well separated from the crystallographic reflections ( 0 0 2l @xmath270 ) with @xmath271 at 40k as observed from ho single crystals by neutron scattering@xcite . the potential of magnetic off - resonant and magnetic hard x - ray resonant scattering has been demonstrated in a series of pioneering magnetic structure studies on ho by gibbs et al.@xcite . in rsxs experiments at the ho @xmath272 soft x - ray resonance , due to the long period of the magnetic structure , the @xmath273 magnetic reflection of ho is well within the ewald sphere . exploiting the huge magnetic scattering strength of rsxs , this magnetic structure could be readily studied in thin films down to the thickness of a single magnetic period @xcite as can been seen from the intense and well developed magnetic superstructure reflections obtained from a 11 monolayer thin ho metal film shown in figure[p_weschke2004 ] . top : ( a ) magnetic reflection of an 11 ml - thin ho film at various temperatures , recorded at the ho @xmath272 resonance . ( b ) square root of the integrated intensity ( solid squares ) and half widths of the rocking curves of the magnetic reflection ( open squares ) . the solid lines represent the result of a simultaneous power - law fit analysis to determine @xmath274 . bottom : @xmath275 as a function of ho film thickness @xmath253 , including data from mbe films ( solid circles ) and films grown in situ on w(110 ) ( open circles).the inset shows a comparison to the behaviour of ferromagnet gd . ( reprinted from @xcite . copyright 2004 , american physical society . ) ] while for rsxs the relation between magnetic ordering parameters and integrated diffraction peak intensity can be more involved , the integrated superstucture peak intensity of ho has been shown to be in good approximation a measure of the helical afm order parameter@xcite and thus allows to determine the ordering temperatures for various film thicknesses @xmath253 . for small @xmath253 , a finite - size scaling of @xmath275 would have been anticipated and was indeed observed . however , this study found a much faster decrease of @xmath275 with @xmath253 ( see figure[p_weschke2004 ] ) than known from simple antiferromagnets and ferromagnets @xcite . the observed modified finite - size effect in the ordering temperature is characterized by an offset thickness @xmath276 representing a minimum sample thickness below which no helical order can be established . this finding has been related to the long period of the afm structure@xcite , in line with results obtained for the long - period spin - density wave of cr in cr / fe superlattices @xcite . as shown in figure[p_weschke2004 ] , even above @xmath275 , finite magnetic intensity could be observed characterized by a diverging peak width with increasing temperature . these remaining broad intensity distributions are caused by short - range magnetic correlations persisting above @xmath275 , with the peak width being an inverse measure of the magnetic correlation length . this observation shows the huge potential of rsxs for studying magnetic short - range correlations in films of only a few monolayer thickness , even with the perspective to employ coherent scattering as discussed later . the helical magnetic structure of bulk ho was already well established by neutron scattering . for some materials , however , no large single crystals can be synthesized , which limits the capability of neutron scattering for magnetic structure determination . in such cases , rsxs can provide detailed information on the magnetic moment directions exploiting the polarisation dependence of the scattered intensity as discussed in section[sec : pol_dep ] . here , the example of thin epitaxial ndnio@xmath277 ( nno ) films @xcite is presented . in this work the element specificity of rsxs to study the magnetic structure of ni and nd moments in nno separately is exploited . nno shows a metal - to - insulator transition at about 210k . at the same temperature afm ordering occurs . from the observed ( 0.5 0 0.5 ) magnetic reflection , initially , an unusual collinear up - up - down - down magnetic structure connected with orbital ordering of the ni@xmath278 @xmath279 electrons has been proposed on the basis of neutron powder diffraction data @xcite . these results have been challenged by the later observation of charge disproportionation without any indication for orbital order using hard x - ray diffraction @xcite . however , off - resonant and resonant scattering at the ni @xmath215 edge can suffer from weak signals and only indirect sensitivity to orbital order . therefore the ( 0.5 0 0.5 ) reflection has been studied by rsxs in the vicinity of the ni @xmath216 and nd @xmath280 resonance . figure [ p_scagnoli2008 ] shows the observed azimuthal dependence of the scattered intensity compared to model calculations based on the formalism developed by hill and mcmorrow@xcite . analogous to the case of the simple bcc antiferromagnet ( cf . figure[fig : azi ] ) , either a collinear up - up - down - down or a non - collinear magnetic structure was assumed . azimuthal angle dependence of the ( 0.5 0 0.5 ) magnetic reflection of ndnio@xmath139 measured at the ni @xmath224 and nd @xmath280 edge . the solid and dotted lines correspond to model calculations assuming a non - collinear and a collinear up - up - down - down magnetic structure , respectively . ( reprinted with permission from @xcite . copyright 2008 , american physical society . ) ] the results clearly reveal a non - collinear magnetic structure of the ni and nd moments at low temperatures . in addition , the observed polarisation dependence of scattered intensity , including polarisation analysis of the scattered light at the ni @xmath216 resonance , has been found to be inconsistent with orbital order and the energy - dependent line shape of the reflection could only be reproduced assuming charge order . thus these studies fully support a picture of combined charge and non - collinear magnetic order of the ni sites but are inconsistent with orbital order suggesting that the metal - insulator transition is driven by charge disproportionation in nno . at this point , it is appropriate to stress again a caveat : the model calculations for nno used eq.[eqn : hannon ] , which strictly applies to the case , in which only the spin breaks the otherwise spherical local symmetry . this approach is commonly used and yields a reasonable data interpretation in many cases . nonetheless , it has to be kept in mind that this is an approximation which might not always be valid in a real material ( cf . section[sec : haverkort ] ) . in particular , for systems with symmetry lower than cubic , the standard treatment can yield completely wrong results . this is demonstrated in a study on nacu@xmath138o@xmath138 by leininger et al . nacu@xmath138o@xmath138 belongs to the class of edge - sharing copper - oxide chain compounds which have attracted considerable interest in the past years due to their diversity of ground states . the related compound @xmath281 , e.g. has been found to exhibit ferroelectricity . at present it is not clear whether this ferroelectricity is caused by the complex magnetic structure or the strong tendency of li and cu intersubstitution . a non - collinear spiral antiferromagnetic structure was derived by neutron diffraction on polycrystalline samples and nmr measurements on nacu@xmath138o@xmath138@xcite in line with results from susceptibility measurements @xcite . serving as an isostructural and isoelectronic reference compound without complicated intersubstitution of alkaline and @xmath221 tm ions@xcite , nacu@xmath138o@xmath138 is only available in form of tiny single crystals . therefore , rsxs is the method of choice to study details of the magnetic structure . however , according to the simple treatment of magnetic rsxs data , the azimuthal angle dependence of the magnetic reflection suggested a magnetic structure made of only one magnetic moment direction in stark contrast to the results of the susceptibility measurements and neutron powder diffraction data . this inconsistency could be explained on the basis of the strong anisotropy observed in the polarisation dependent absorption probability . the anisotropy is caused by a selection rule precluding excitations into the partially occupied planar @xmath282 orbitals of the @xmath283 ions which modifies the polarisation dependent magnetic scattering strength according to the kramers - heisenberg formula and hence masks the azimuthal angle dependence of the magnetic signal . these findings are in excellent agreement with the theoretical predictions by haverkort et al.@xcite , which shows that in general the local orbital symmetry has to be taken into account for magnetic structure determination by magnetic rsxs . the first application of rsxs to study long - period afm structures in bulk materials has been reported by wilkins et al.@xcite showing the high potential of rsxs to study magnetic structures and , even more importantly , other electronic ordering phenomena in bulk oxides . for magnetic structure determination , however , neutron scattering will stay an unrivaled method . on the other hand , despite the limited ewald sphere , the potential of magnetic rsxs for exploring magnetic structure has been clearly shown and its importance will increase with growing interest in thin transition metal oxide films and superstructures that can be obtained with very high quality . films of @xmath284mno@xmath139 perovskites represent a prominent example , where long period complex magnetic order is strongly coupled to ferroelectricity . such a thin improper multiferroic has been indeed studied for the first time by rsxs very recently by wadati et al.@xcite . diffraction scans with momentum transfer perpendicular to the magnetic stripes in fepd films for left ( dotted ) and right ( solid ) circularly polarised x - rays and resulting difference signal below . insets : right : asymmetry ratios for the first and second order magnetic reflections . left : magnetic force microscope image of stripe domains and schematic magnetization profile.(reprinted with permission from @xcite . copyright 1999 , american association for the advance of science . ) ] + + while rsxs is particularly useful in case of long - period magnetic structures , rsxs can also provide detailed information on the microscopic magnetic properties of ferromagnets and simple antiferromagnets , in particular when applied to artificial structures that produce an additional periodicity of the order of the wavelength of soft x - rays . such structures are also naturally obtained by the formation of domains in chemically homogeneous materials , as has been observed for thin films of fepd . ferromagnetic fepd films are characterized by perpendicular magnetic anisotropy leading to the formation of well ordered stripe domains as shown in figure[p_durr99 ] . in order to reduce the stray field outside the sample , closure domains can be formed at the surface , which link the magnetizations of the neighbouring stripes , resulting in an overall domain structure of chiral nature . x - ray scattering experiments usually are not sensitive to the phase of the scattered wave and , hence , can not easily distinguish left and right handed magnetic structures . however , for circular polarised incident light , the resonant magnetic structure factor differs for structures of different sense of rotation @xcite giving rise to circular dichroism in the scattered intensity not connected with absorption effects but depending on the sense of rotation of a magnetic structure . this property has been exploited by drr et al.@xcite to study closure domains in thin fepd films by rsxs . here , the regular domain pattern at the surface does not cause superstructure reflections in terms of a single peak but rod - like off - specular magnetic scattering intensity with a maximum intensity at a transverse momentum transfer determined by the inverse of the mean distance of the stripe domains as shown in figure [ p_durr99 ] . the intensity of these off - specular magnetic rods showed pronounced circular dichroism , in this way proving the existence of the above mentioned closure domains . by modeling the q - dependent magnetic rod intensity , shown in figure[p_dudzik00 ] , the depth of the closure domains could be estimated to be about 8.5 nm in a 42 nm thick film @xcite . rod scan of the magnetic satellite peak from stripe domains in fepd thin films together with model calculations showing the influence of closure domain depth : solid line : no closure domains , dashed line : closure depth 110 , dotted line : closure depth 90 .(reprinted with permission from @xcite . copyright 2000 , american physical society . ) ] as shown in the latter example , rsxs is capable of studying ordering phenomena with spatial resolution . the information on spatially varying ordering phenomena is contained in the dependence of scattered intensity on the chosen incident energy and momentum transfer which can be very sensitive even to tiny modifications of order . hence , rsxs has been applied very successfully to obtain magnetization depth profiles of thin films and multilayers with very high depth resolution . in the past years , three different ways have been introduced to extract depth dependent information on magnetic ordering , exploiting 1 . the change of the sample volume probed through the resonance due to strong incident energy - dependence of the photon penetration depth @xcite , 2 . the sensitivity of the shape of a magnetic reflection from very thin samples on the spatial varying magnetic ordering parameter @xcite , and 3 . the incident energy- and momentum - transfer dependent reflectivity @xcite . top : ( a ) and ( b ) magnetic diffraction data characterizing the fm / helical afm phase transition of 180ml dy / w(110 ) upon cooling down ( open symbols ) and warming up ( filled symbols ) for two different photon energies ; ( c ) widths of magnetic peaks in the directions parallel and perpendicular to the film plane recorded at 7780 ev ( squares ) and at 1305 ev ( circles ) . bottom : magnetic depth profile of a 180ml thick dy film during the first order phase transition from the fm to the helical afm phase . ( reprinted from @xcite . copyright 2006 , american institute of physics . ) ] while the first two approaches allow to draw qualitative conclusions from the raw data without involved data analysis they , however , require a magnetic reflection inside the ewald sphere at resonance . approach ( i ) has been applied to study depth - dependent afm ordering in thin dy metal films grown on w(110 ) . metallic dy develops a long - period helical afm superstructure below @xmath275=179 k , which undergoes a first order transition into a ferromagnetic structure at about 80 k. this transition is characterized by a pronounced thickness dependence of its hysteretic behaviour , i.e. , the hysteresis smears out with decreasing film thickness . the long period afm structure gives rise to magnetic superstructure reflections well separated from the structural bragg reflections similar to those observed from ho metal described above , which can be easily measured in the vicinity of the dy @xmath217 resonance . through this resonance not only the magnetic scattering strength varies but also the probing depth of the photons due to the energy dependence of the imaginary part of the index of refraction . figure [ p_ott2006 ] shows the temperature dependent intensity of this afm reflection through the afm - fm transition measured for two different photon energies , i.e. , with different probing depths of the photons . distinctly different hysteretic behaviour was observed , which readily indicates a pronounced depth dependence of the growth of the helical afm order with temperature . from modeling these energy - dependent hysteresis loops , taking the energy - dependent absorption into account , a temperature - dependent depth profile through this first order phase transition could be extracted shown in figure[p_ott2006 ] @xcite . obviously , upon heating the helical order starts to grow in the near surface region of the film and develops deeper towards the w interface with temperature . this behaviour has been attributed to pinning of the fm phase at the interface by strain effects . interestingly , also at the top surface , a phase of modified afm order characterized by a tendency towards ferromagnetism survives over a large range of temperature . this way of achieving depth - dependent information requires a film thickness much larger than the minimum photon penetration depth at resonance and can hence not be applied to ultra - thin samples . in this latter case , however , according to approach ( ii ) , the shape of a superstructure reflection measured at a fixed energy already contains the information on the spatial modulation of magnetic order , as has been shown for magnetic scattering from the magnetic semiconducter eute @xcite . below @xmath275=9.8k , eute develops a simple afm structure with ferromagnetically ordered ( 111 ) planes but alternating magnetization along the [ 111 ] direction . by virtue of the strong magnetic eu @xmath217 resonance and the very high quality of the epitaxially grown [ 111 ] eute films , a very well resolved magnetic half order reflection could be observed from a film of only 20 monolayers , with a peak intensity in the order of almost 1% of the incident intensity . this magnetic reflection is already comparable to the off - resonant charge scattering from the superlattice peak of figure[fig : multilayer ] . at resonance , this magnetic reflections appears very close to the brewster angle where the charge scattering background for @xmath33 polarised incident light is almost completely suppressed . this ideal combination of magnetic period length and resonance energy in eute therefore leads to a magnetic signal - to - background ratio of several orders of magnitude , allowing measurements of the magnetic superstructure reflection with unprecedented quality . as can be seen in figure[p_schierle2008 ] , the intensity of this reflection is not contained in a single peak but distributed over several side maxima caused by the finite size of the system ( so - called laue oscillations ) . the period of these oscillations displays a measure of the inverse number of contributing magnetic layers and the overall envelope functions depends sensitively on the magnetization depth profile . by virtue of the high magnetic contrast in this study , the side maxima could be resolved over a large range of momentum transfer revealing distinct changes with temperature : with increasing temperature the period of the laue oscillations increases and the intensity of the side maxima compared to the central peak decreases . this behaviour signals a non - homogeneous decrease of the magnetic order through the film , such that the effective magnetic thickness decreases with increasing temperatures , i.e. , the order decays faster at the film boundaries . ( colour online ) magnetic ( 0.5 0.5 0.5 ) bragg reflections with magnetic laue oscillations of thin eute films with 44 monolayers ( a ) and 20 monolayers ( b ) thickness . ( c ) data for a 20 monolayer thick film , recorded for various temperatures below ( solid ) and above ( dashed ) @xmath275 ; the photon energy was 1127.5 ev , the eu @xmath231 resonance maximum . ( d ) two selected experimental diffraction patterns ( dots ) and results of calculations , taking magnetization depth profiles into account ( lines ) . ( reprinted from @xcite . copyright 2008 , american physical society.),width=7 ] from a kinematical modeling of the observed intensity distributions , temperature - dependent atomic - layer - resolved magnetization profiles across the entire film could be extracted . even at low temperatures , these profiles displayed reduced order at the film interfaces , which is due to some chemical intermixing . in addition , the profiles reveal a faster decrease of the order close to the film boundaries with temperature , i.e. , a layer - dependent temperature dependence of the magnetic order is observed . due to the half filled 4f shell of eu@xmath285 ions , eute exhibits almost pure spin magnetism caused by strongly localised moments and can therefore be regarded as a heisenberg model system , for which this type of surface - modified order was theoretically predicted@xcite . top : charge intensities ( upper panels ) and magnetic asymmetries ( lower panels ) of the first- and second - order bragg peaks recorded at the co @xmath216 absorption edges from a co@xmath286mnge / au multilayer . the dots represent measured data ; the lines are model calculations as described in @xcite . bottom : structural and magnetic depth profiles of co and mn as determined from the model calculations . ( reprinted with permission from @xcite . copyright 2005 , american physical society . ) ] ( colour online ) simulated xmcd spectra for two lcmo layers on strontium - titanate with thicknesses of ( a ) 6 and ( b ) 50 nm . the presence of a 1 nm nonmagnetic surface layer strongly affects the spectral shape irrespective of the film thickness , making the strong sensitivity of xmcd in reflection geometry to nonmagnetic layers evident . ar - xmcd here denotes the asymmetry value @xmath287 , where @xmath255 and @xmath256 refer to the measured intensities for opposite directions of the external magnetic field . ( reprinted with permission from @xcite . copyright 2007 , american institute of physics . ) ] modeling of q - dependent intensity distributions is not restricted to magnetic superstructure reflections but can also be applied in a more general way to the specular reflectivity from systems without any magnetic or chemical reflection inside the ewald sphere at a soft x - ray resonance , e.g. ferromagnetic films , in this way opening the method to a broader field of applications . this has been successfully demonstrated for hard x - ray resonant scattering@xcite . in the soft x - ray range , in particular at resonance , specular reflectivity typically does not fall off so rapidly and hence can be measured over a large range of momentum transfer due to the strong optical contrast at the interfaces and surfaces . information on ferromagnetic depth profiles usually requires to compare reflectivities measured with different polarisations of the incident x - ray beam or for opposite magnetization directions . interpretation of such reflected intensities is often less intuitive and requires detailed theoretical modeling as well as precise knowledge of the energy - dependent optical parameters . based on an magneto - optical matrix algorithm developed by zak et al.@xcite , several computer codes have been developed within the last decade for this purpose . in the pioneering early experiments , modeling of energy - dependent reflectivity data has been exploited to extract averaged magnetic properties , like the fe magnetic moment in fe@xmath261mn@xmath260 thin films , which undergo a magneto - structural transition at x=0.75 @xcite . later studies improved data analysis to incorporate the effects of magnetic and chemical roughness as well as magnetization depth profiles @xcite . examples of interest are ferromagnetic half - metals , like la@xmath260sr@xmath261mno@xmath277 ( lsmo ) close to @xmath240=0.3 and a number of heusler compounds . half - metals are characterized by a band gap for only one spin direction and therefore are candidates to show 100% spin polarisation at the fermi level , thus being of high interest for spintronic devices . functionality of such materials crucially depends on the magnetic properties at interfaces , which can be strongly altered by extrinsic effects like electronic reconstruction , disorder , roughness , or strain , but also by intrinsic changes caused by the broken translational symmetry . therefore , obtaining depth - dependent information on the ferromagnetic properties of such materials at various interfaces is greatly needed but challenging . the interface magnetization depth profiles for co and mn moments in co@xmath138mnge / au multilayers were obtained from the analysis of element - specific rsxs@xcite . by fitting the xmcd spectra measured at the angular positions of the first three superlattice reflections , the individual magnetization profiles shown in figure[p_grabis2005 ] for the co and mn moments through the magnetic layers could be reconstructed . they are characterized by non - magnetic interface regions of different extension for the upper and lower boundaries and also differ for the two magnetic species leading to a rather complicated behaviour of the magnetization through a magnetic layer . this observation has been attributed to structural disorder caused by strain at the interfaces which affects co and mn spin in a different way since co on a regular mn position keeps its ferromagnetic spin orientation and full moment but mn on a co position has an antiparallel spin orientation and a reduced moment @xcite . with a similar scope , rsxs was applied to thin manganite films . manganites show one of the richest set of phase diagrams among the transition metal oxides and therefore offer a high potential for application in future functional heterostructures . freeland et al.@xcite showed the existence of a thin non - ferromagnetic and insulating surface layer with a thickness of only one mn - o bilayer on a single crystal of layered lsmo by analyzing its energy - dependent reflectivity at the mn @xmath216 resonance combined with xmcd and tunneling probe techniques@xcite . this finding renders the material a natural magnetic tunnel junction . the sensitivity of rsxs to weak magnetic modifications like a thin non - magnetic layer in manganites is nicely demonstrated in a study of la@xmath260ca@xmath261mno@xmath277 ( lcmo ) films for different thicknesses and substrates by valencia et al.@xcite . by analyzing the xmcd spectra , measured on the specular reflectivity for fixed scattering angles , detailed knowledge of the chemical and magnetic surface properties could be obtained . in particular , all samples studied developed a thin surface region ranging from 0.5 to 2 nm with depressed magnetic properties . figure [ p_valencia2007 ] shows simulations of the xmcd spectra depending on the presence of a magnetic dead layer , revealing distinctly different energy profiles irrespective of the film thickness . in the same way , verna et al . very recently reconstructed the magnetization depth profile with high spatial resolution from thin lsmo samples , again , observing a non - ferromagnetic surface region of about 1.5 nm thickness which expands with increasing temperature @xcite . the observation of such non - ferromagnetic dead layers at manganite surfaces for lcmo as well as lsmo single crystals and films of different thickness , independent of the particular epitaxial strain strongly suggests that the observed modified surface magnetization is an intrinsic property of manganite surfaces @xcite . this conclusion is further supported by an xas study exploiting linear dichroism at mn @xmath216 resonance from thin lsmo films which observed an orbital reconstruction , i.e. , a rearrangement of the orbital occupation at the surface consistent with a suppression of the double - exchange mechanism responsible for the ferromagnetic properties @xcite . the latter examples of modified , mostly suppressed magnetic properties at surfaces and interfaces , are important results in view of possible application of materials in heterostructures . however , the proximity of two materials in heterostructures can cause much stronger and more complex changes of magnetic properties at those buried interfaces , which has been studied by rsxs intensively in the last decade . interfaces are presently of high interest since here direct control of material properties is possible by several different mechanisms . in particular , the proximity of two different materials at an interface can generate phenomena not present in the individual materials . while this holds in general for a rich variety of electronic phenomena , it is already true for two adjacent magnetic systems . + + transverse scans normalized to the maximum of the specular reflectivity from a 7 nm co film capped with al recorded at the co @xmath224 resonance . diamonds : average signal from two measurements with opposite magnetization , representing the correlation of the chemical structure . circles : difference signal of the two measurements with opposite magnetization , representing the magnetic contribution . the faster decay with @xmath288 in the latter case yields smaller magnetic roughness . ( reprinted with permission from @xcite . copyright 1996 , american physical society . ) ] ( colour online ) . profile of the optical parameters @xmath150 and @xmath151 across the interface of the mnpd / fe exchange bias system , separated into structural non - magnetic ( red , black ) and various magnetic contributions . rotatable mn moments lead to the dashed green ( @xmath150 ) and dashed blue ( @xmath151 ) curve . dashed - dotted magenta and light - blue curves : same for the pinned mn moments . ( reprinted with permission from @xcite . copyright 2008 , american physical society . ) ] famous examples of interface - generated magnetic phenomena with high technological relevance are giant magneto resistance and exchange bias . the experimental challenge of characterizing such phenomena is due to the extremely small amount of contributing material , the presence of more than one type of magnetic moment , and the buried character of the region of interest . also here , rsxs has been shown to be a very sensitive tool to study such interfaces since its huge charge and magnetic scattering strength is accompanied by a moderate penetration depth , element specificity and access to structural information . one crucial parameter that can influence interface - generated functionality is chemical and/or magnetic roughness , which for a buried interface can be characterized best by scattering techniques measuring the related off - specular scattered intensity distributions . such diffuse scattering contributions , however , are typically several orders of intensity weaker than the specular reflectivity and require an extraordinary magnetic scattering strength to be detectable . the first demonstration of how diffuse rsxs intensities can be exploited to track magnetic roughness was reported by mckay et al . for co films and co / cu / co trilayers @xcite . in this study the diffuse intensity around the specular reflectivity was measured at the co @xmath224 resonance containing contributions from magnetic and chemical roughness . in order to separate the two quantities , the intensity was measured with circular polarised x - rays for two opposite magnetizations of the sample . in this case the sum of the two measurements represents the chemical roughness while the difference was interpreted as a measure of the magnetic roughness . from the intensity and width of such broad intensity distributions , properties like the root mean square roughness ( intensity ) and correlation lengths ( width ) can be extracted independently for the magnetic and the chemical interface . interestingly , the magnetic signal drops much faster with increasing momentum transfer @xmath288 than the chemical intensity as can be seen in figure[p_mckay96 ] . this observation yields a surprising result , namely a much smoother magnetic interface if compared with the chemical roughness . it was explained by a weaker coupling of the co moments in the intermixing region compared to that of the homogeneous part of the film . this means that the moments connected with chemical roughness do not contribute to the signal from the film magnetization and the magnetic interface therefore appears smoother than the chemical one . similar results have been obtained by freeland at al . who studied systematically the relation between magnetic and chemical interfacial roughness by monitoring the diffuse scattering from a series of cofe alloy films with varying chemical roughness@xcite . here , it was also shown that the diffuse intensity can be used to measure magnetic hysteresis loops of bulk and interfacial magnetization independently @xcite . it has been shown later , that the difference signal contains charge - magnetic interference contributions , such that it does not represent the pure magnetic roughness @xcite . the general result , however , that the magnetic and chemical roughness can be very different , is not affected . it was rather supported later by a study of hase et al . , who separated chemical and magnetic roughness in a different way by comparing the diffuse scattering in the vicinities of either a chemical or half - order pure magnetic reflection from an antiferromagnetically coupled [ co / cu]@xmath289 multilayer @xcite . the observation of different roughnesses of the chemical and magnetic interfaces shows that the understanding of macroscopic magnetic phenomena and their relation to roughness can not be simply obtained from the knowledge of the chemical roughness alone . nevertheless , grabis et al . observed almost identical chemical and magnetic roughness parameters in a co@xmath286mnge / au multilayer , showing that different roughness are not a general feature of magnetic interfaces @xcite . theory describing diffuse resonant scattering from chemically and magnetically rough interfaces beyond the kinematical approximation has been published by lee et al.@xcite . the proximity of two different magnetic systems can cause more complex changes of the magnetic properties than just introducing disorder , as discussed in the previous paragraphs . a prominent example , connected with functionality already applied in technological applications is the exchange bias effect . exchange bias is the observation of a shift of the hysteresis along the field axis of a fm material in contact with an antiferromagnet ( af ) after field - cooling through the nel temperature . the general mechanism was identified to be the exchange interaction between the spins of the fm and the af at the interface , which generates an additional unidirectional anisotropy for the fm spins @xcite . however , the original models assuming perfect interfaces failed to describe exchange bias and related phenomena in a quantitative way . rather , the exchange bias strongly depends on details of the magnetic structure at the interface . exploiting xmcd techniques , rich knowledge on the magnetic behaviour of the interface was obtained . in particular the presence of a fm component within the af including a pinned fraction that is related to the exchange bias was observed @xcite . due to the fact that interfaces are buried , techniques not limited to the top sample surface region are of advantageous . therefore , a large number of experiments exploited the xmcd on the specular reflectivity to characterize the element specific behaviour of spins in exchange bias systems @xcite . such studies revealed the complex magnetic situation at the interface connected with exchange bias . in particular , several different types of spins are involved , namely the fm spins , the compensated afm moments as well as uncompensated afm moments that can be divided into a pinned and a rotatable fraction . by exploiting the structural information contained in reflectivity data , detailed characterization of the depth dependence of the different magnetic contributions at the interface could be achieved . by varying the absorption threshold , magnetization direction , photon polarisation and scattering geometry , rsxs can be tuned to be sensitive to almost all the magnetic ingredients of the magnetic interface . in this way roy et al . determined the depth profile of fm and unpinned uncompensated afm spins of the exchange bias system co / fef@xmath286 by analyzing the specular reflectivities measured with constant helicity of the incident photon beam for opposite magnetization directions @xcite . they found the majority of the uncompensated unpinned afm spins in a region of a few nanometers below the interface , characterized by afm coupling to the spins of the ferromagnet . in addition to the magnetization direction , elliptical undulators allow to switch the light helicity , providing a detailed picture of the magnetic structure at the interface . in this way , brck et al . studied the interface of the mnpd / fe exchange bias system@xcite . figure[p_bruck ] shows the scenario resulting from an analysis of respective reflectivity difference curves for either opposite external field directions or light polarisations , separating the pinned and unpinned uncompensated spins within the antiferromagnet . according to this study , the major fraction of the uncompensated afm spins located very close to the ferromagnet is rotatable and antiferromagnetically coupled to the ferromagnet , while the uncompensated afm spins deeper in the afm are predominantly pinned to the af which finally generates the exchange bias effect . similar results were obtained by tonnerre et al . who extended the method to study the interface of a system with perpendicular exchange bias @xcite . ( colour online ) absorption signal and circular dichroic asymmetry @xmath290 observed in specular reflectivity from fe / batio@xmath277 samples through the o @xmath215 ( left ) and ti @xmath216 resonance as well as corresponding element - selected ferromagnetic hysteresis loops ( right ) . they show a magnetic polarisation of the otherwise non - magnetic o and ti ions by the fe at the interface . ( reprinted with permission from @xcite . copyright 2011 , nature publishing group . ) , width=566 ] interesting interface - generated functionality in magnetic heterostructures can emerge already from a magnet in contact with a non - magnetic material . in a very recent study , valencia et al . explored the influence of a ferromagnet in contact with a ferroelectric@xcite . in this experiment , the structural information that can be extracted from scattering experiments was disregarded , still , it can be considered a text - book example showing the sensitivity of xmcd - like experiments exploiting specular reflectivity rather than tey or fy methods to study tiny magnetic effects at buried interfaces . in addition , it represents an important example of interface - generated material properties@xcite . in present solid state research , magnetic control of ferroelectric properties and , in particular , electric control of magnetic order are phenomena with very promising perspectives for applications . unfortunately , most of the known ferroelectrics are either not magnetically ordered or magnetism and ferroelectricity are weakly coupled . exceptions are the so called improper ferroelectrics , where ferroelectricity is induced by complex mostly antiferromagnetic structures , which is discussed for bulk materials in more detail in section[sec : multiferroics ] . these materials , however , are often connected with very low ordering temperatures . in the quest for room - temperature multiferroicity , one way to overcome this limitation is to combine a room - temperature ferromagnet and a room - temperature ferroelectric in a heterostructure . here , the chosen heterostructures consist of fe or co on batio@xmath277 ( bto)@xcite . bto is a robust diamagnetic room - temperature ferroelectric . the study could show that the tunnel - magnetic resistance of fe or co / bto / lsmo trilayers strongly depends on the direction of the ferroelectric polarisation in the insulating diamagnetic bto layer . thus , these heterostructures possess coupled magnetic and ferroelectric , i.e. , multiferroic , properties . element - selective xmcd in reflection geometry could identify the key mechanism : the ferromagnet generates ferromagnetism in the topmost layer of bto by inducing a magnetic moment in the formally non - magnetic o and ti@xmath291 ions as revealed by a seizable asymmetry @xmath290 displayed in figure [ p_valencia2011 ] . the corresponding magnetic interface properties are very weak and hardly detectable by conventional xmcd techniques . exploiting the dichroism in specular reflectivity through the ti @xmath216 and o @xmath215 resonance , valencia et al . have been able to unambiguously prove the existence of ferromagnetic moments at the ti and o ions coupled to the magnetization of the ferromagentic layer . this observation shows the huge advantage of performing xmcd - like experiments exploiting scattered photons to study tiny effects at buried interfaces and , besides these methodological aspects , opens the door for future application of interface - generated room temperature multiferroicity . modified macroscopic and/or microscopic properties at heterostructure interfaces are by far not limited to magnetic behaviour but can be found for other electronic phenomena as well , which offers even more fascinating new perspectives for future nanoscale devices . + + ( colour online ) reflectivity of a 23.2-nm la@xmath286cuo@xmath292 film ( a ) just below and on the mobile carrier peak ( mcp ) at 535 ev , showing the resonant enhancement ; and ( b ) over the full angular range for three different energies . the faster decrease of the red curve compared to the blue one indicates a smoothing of the charge - carrier density at the interface . ( reprinted with permission from @xcite . copyright 2002 , american association for the advance of science . ) ] ( colour online ) energy dependence of ( 0 0 l ) superlattice reflections of smo / lmo multilayers at t=90 k : at l=3 ( line ) , at l=1 ( circles ) , and l=2 ( triangles ) , compared to xas data ( squares , aligned to zero below the edge ) . the strong resonant enhancement in the pre - edge region of the o @xmath215 resonance is indicative of an electronic effect . ( reprinted with permission from @xcite . copyright 2007 , american physical society . ) ] in ultra - thin samples or at heterostructure interfaces completely new phenomena may arise that can not be found in the corresponding bulk materials . an important and fast developing field of research was triggered by the discovery that interfaces of transition metal oxides can be grown with very high structural quality down to a flatness on the atomic level . with these structurally perfect interfaces , fundamental studies of new interface properties have become feasible . examples are the discovery of a highly mobile electron gas @xcite and even superconductivity @xcite at the interface of the two insulating perovskite oxides lao and srtio@xmath277 ( sto ) . a recent example is the observation of insulating and antiferromagnetic properties of lanio@xmath277 ( lno ) in lno / laalo@xmath277 ( lao ) superlattices , as soon as the lno thickness falls below 3 unit cells , while thicker lno samples are metallic and paramagnetic @xcite . finally a major reconstruction of the orbital occupation and orbital symmetry in the interfacial cuo@xmath286 layers at the ( y , ca)ba@xmath286cu@xmath277o@xmath293 / lcmo interface has been observed @xcite . it is a matter of debate to which extent such properties are caused by either electronic reconstruction , strain , oxygen vacancies , or disorder . hence , microscopic understanding of macroscopic interface properties may require detailed knowledge on microscopic electronic properties with high spatial resolution . with its strong sensitivity not only to magnetic but also to charge and orbital order , rsxs is well suited to extract quantitative information on even subtle changes of these electronic degrees of freedom at interfaces . the general strategy here is very similar to the case of magnetic depth profiling : the information on spatially modulated orbital occupation or charge distribution is contained in the reflectivities , measured as a function of momentum transfer and incident photon energy . this has been demonstrated first by abbamonte et al . while studying the carrier distribution in thin films of la@xmath286cuo@xmath292 grown on srtio@xmath277 @xcite . the thin film reflectivities of these samples are characterized by well developed thickness oscillations when measured at the cu @xmath216 resonance , proving high sample homogeneity and surface / interface quality ( see figure[p_abbamonte2002 ] ) . in contrast , with a photon energy corresponding to the mobile carrier peak ( mcd ) , the oscillations vanish at higher angles of incidence . this mobile carrier peak in the o @xmath215 pre - edge region represents a characteristic energy for scattering from the doped carriers with an enhancement of the scattering strength of a single hole by about two orders of magnitude . hence , the reflectivity measured at this photon energy is almost entirely determined by the distribution of the doped carriers . the strong damping of the thickness oscillations at this specific energy is readily explained by assuming a smoothing of the carrier density at the interface towards the substrate . in a very similar way smadici et al . studied the distribution of doped holes in superlattices consisting of double layers of insulating la@xmath286cuo@xmath294 ( lco ) and overdoped la@xmath295sr@xmath296cuo@xmath294 @xcite . while none of these materials is superconducting , the heterostructure shows superconductivity below @xmath297=35k suggesting charge redistribution at the interface . in rsxs with the photon energy tuned to the la @xmath231 resonance , the superstructure gives rise to a series of reflections detectable up to the 5th order . these reflections mainly contain information about the distribution of the sr ions that induce the hole doping . in contrast , at the mobile carrier peak energy only the first superstructure reflection could be observed . this result directly shows that , although being modulated with the superlattice period , the doped hole distribution does not follow that of the sr ions . from a quantitative analysis , an average hole density in the insulating lco layers of 0.18 holes per cu site has been deduced , suggesting that superconductivity occurs in the formally insulating lco layers . in order to clearly separate interface behaviour , often it is of advantage to design a multilayer such that a specific superstructure reflection is interface sensitive , i.e. , its structure factor essentially is given by the difference of the optical properties of an interfacial layer and a bulk - like layer , in that way directly representing the changes at the interface . this approach has been demonstrated first by smadici et al.@xcite who studied the interfacial electronic properties of srmno@xmath277 ( smo ) / lamno@xmath277 ( lmo ) . while the two single materials are a mott insulator ( lmo ) and a band insulator ( smo ) , repectively , ferromagnetic and metallic interface properties had been predicted @xcite and macroscopically observed @xcite . in this experiment , ( 0 0 l ) superstructure reflections were studied in a heterostructure consisting of @xmath298lmo/@xmath299smo double layers . @xmath300 and @xmath3 were chosen such that the l=3 reflection perpendicular to the surface only occurs if scattering from the interfacial and inner mno@xmath286 planes of lmo or smo differ . a pronounced l=3 reflection from this structure could be observed only in the vicinity of the mn @xmath216 resonance and in a very narrow energy region at the onset of the o @xmath215 resonance ( see figure[p_smadici2007 ] ) , i.e. , involving electronic states in the vicinity of the fermi level . the observation of no broken symmetry in the atomic lattice ( i.e. , no l=3 reflection off - resonance ) , but an interfacial reflection induced by the unoccupied density - of - states near @xmath301 gives strong evidence that the interface is characterized by electronic reconstruction . the intensity of the l=3 reflection at the mn @xmath216 resonance was identified to be of magnetic origin from a rough azimuthal dependence of the scattered intensity . comparison of the temperature dependence of the macroscopic properties ( conductivity and magnetization ) with the observed l=3 intensities at the o @xmath215 and mn @xmath222 resonances suggests that metallic and ferromagnetic behaviour is indeed interface generated driven by electronic reconstruction . a very similar approach has been used to characterize also the sto / lao interface in detail very recently by wadati et al.@xcite . ( colour online ) energy scans of the reflectivity from a lno / lao superlattice with constant momentum transfer @xmath302 close to the ( 002 ) superlattice peak using linearly polarised light . the top panel shows the polarisation - dependent experimental data revealing linear dichroism at the ni@xmath224 resonance . the bottom panel shows the corresponding simulated curves for lno layers with ( 1 ) homogeneous orbital occupation within the lno layer stack and ( 2 ) modulated orbital occupation . ( reprinted with permission from @xcite . copyright 2011 , nature publishing group . ) ] while the latter studies revealed interface - driven modifications in a more qualitative way , it has been shown very recently that quantitative information from such superstructure reflections can be derived for each individual atomic plane . this , however , requires detailed modeling of energy- and polarisation - dependent reflected intensities as in the case of magnetic depth profiling discussed above . the studied system is a multilayer made of repeated bilayers of four unit cells of lao and four unit cells of lno . ni in cubic symmetry is characterized by a doubly degenerate @xmath303 level filled with one electron . according to model calculations , epitaxial strain can be utilized to favour the occupation of the in - plane @xmath304 orbital in a superlattice geometry , in this way matching the electronic structure of cuprate high-@xmath305 superconductors , which may be a route to generate high-@xmath305 superconductivity in artificial nanostructures @xcite . while magnetic depth profiling used the contrast obtained by circularly polarised light , light with linear polarisations @xmath33 and @xmath32 ( cf . figure[fig : exp_scheme ] ) was used to measure the linear dichroism of the orbital scattering according to the last term in eq.[eqn : hannon ] . by changing the light polarisation from in - plane to normal to the superlatice plane , the sensitivity to the @xmath306 orbital could be varied . the observed linear dichroism in absorption yields the average difference in occupation of the two @xmath303 orbitals of 5.5% . in addition , the linear dichroism at the second superlattice reflection , sensitive to the difference of the scattering strength from the inner lno unit cell and the interfacial lno unit cell , was monitored . as shown in figure[p_benckiser2011 ] , the observed linear dichroism , here , is much stronger than expected for a homogeneous system with the same orbital occupancy for the inner and interfacial lno layers . hence , an inhomogeneity of the orbital occupation through this 4 unit cells thin lno layer can be readily deduced . modeling the observed energy dependence for the two different linear polarisations , taking the average dichroism as an fixed input , the data could be explained by a distinct higher occupation of the @xmath304 orbitals at the interface ( about 7 % ) compared to the inner layers ( about 4 % ) , which is explained by the reduced gain of kinetic energy for the electrons in the interface layer by hopping across the interface due to the closed shell of the neighbouring al@xmath278 ions . the previous sections demonstrated the capabilities of rsxs for studying ordering phenomena in nanostructures , with particular focus on multilayer samples where interesting physics typically emerge at the interfaces . besides such depth - structured samples , systems can also be lateral structured . this has been discussed for stripe - domains in fepd films above , but can be expanded to artificially designed structures on a nanometer length scale , like regular or irregular line and dot arrays . such regular patterns of magnetic materials are of interest in connection with future data storage technologies . the typical length scales ( nm ) of nanostructures perfectly match the wavelengths of soft x - rays . interestingly , rsxs has not been widely applied to laterally structured samples so far . the first application was presented by chesnel et al . monitoring the magnetization reversal of an regular arrangement of magnetic lines @xcite . these samples have been made of si lines of about 200 nm width and 300 nm height with a line spacing of 75 nm covered by co / pt multilayers . in contrast to the fepd stripe domains discussed above , this structure gives rise to off - specular intensity rods even without magnetism ( see figure[p_chesnel2002 ] ) . analyzing the in - plane momentum transfer @xmath307 as well as the width and the relative intensities of the superstructure - rods of different order yields a detailed characterization of the chemical structure . at the co @xmath216 resonance , additional magnetic information can be extracted from the scattered intensities . here , ferromagnetically aligned neighbouring stripes contribute magnetic intensity to the chemical superstructure rods , while afm aligned lines give rise to pure magnetic satellites at half order positions . measuring hysteresis loops by recording the scattered intensity at various @xmath307 as a function of an external field as shown in figure[p_chesnel2002 ] , yields information about the magnetic coupling of such nanolines and the magnetization reversal process could be modeled in detail . while the hysteresis for @xmath308=0 displays the average magnetic behaviour , as also accessible by macroscopic techniques , the off - specular magnetic rod intensity reflects the magnetic behaviour of the well - ordered part of the sample . the occurrence of significant magnetic intensity at half - order positions for the external field close to the coercive field shows that the demagnetized state of this sample is in fact characterized by an afm alignment of neighbouring lines with a correlation length of about 4 lines . similar studies has been performed on a regular dot array of ferromagnetic permalloy dots @xcite as well as of permalloy rings , where the rsxs intensity distribution has been shown to be sensitive to the possible magnetization states of the single rings , i.e. , vortex or so - called onion states @xcite . resonant scattering at the co @xmath224 resonance from an artificial co / pt multilayer structure on si nanolines separated by 200 nm . upper left corner : chemical superstructure and half - order magnetic superstructure peaks . other panels : evolution of the structural peaks ( a , b , c ) under a perpendicular magnetic field . the bottom panel displays the hysteresis measured on the half - order magnetic peak describing the entire hysteresis loop . ( reprinted with permission from @xcite . copyright 2002 , american physical society . ) ] ordering of mobile charge carriers is a phenomenon observed in a number of transition metal oxides , such as cuprates , nickelates , cobaltates , and manganites . many of the interesting properties seem to be closely related to this kind of order . the most intriguing case is that of the cuprates , as inhomogeneous charge distributions have been discussed for many years in connection with high-@xmath309 superconductivity ( htsc ) in these materials . htsc has fascinated nearly the whole solid state community , and the interest remained even 25 years after the discovery of htsc , since there is no consensus on the mechanism for the superconducting pairing of the charge carriers in these compounds . besides the mechanism of htsc , it is the physics of electron correlations in general , which makes the cuprates so interesting , and charge order is one manifestation . since in many cases , charge ordering involves only a small fraction of the total charge , the high sensitivity of rsxs to scattering from particular electronic states renders the method well - suited to study this phenomenon . + + htsc in cuprates occurs after doping the afm mott - hubbard insulator@xcite with holes or electrons . since the onsite coulomb interaction @xmath310 on the cu sites is considerably larger than the charge transfer energy @xmath311 between cu @xmath221 states and o @xmath218 states , in the undoped system the gap is essentially determined by @xmath311 and the holes are not formed on the cu sites but on the o sites@xcite . the doping - induced mobile holes are shared by four oxygen sites surrounding a divalent cu site with one hole in the @xmath221 shell , in this way forming the antiferromagnetically coupled zhang - rice singlet@xcite state . this state may be considered as an effective lower hubbard band state . the effective upper hubbard band is formed predominantly by cu @xmath312 states hybridised with some o @xmath313 states . the doping , which leads to a controlled metal - insulator transition , occurs by block layers between which the cuo@xmath138 layer are embedded . in la@xmath314sr@xmath315cuo@xmath138 hole doping of the cuo@xmath138 layers occurs via replacing the trivalent la ions in the lao block layers by divalent sr ions . long - range afm order disappears at @xmath240 = 0.05 and the highest superconducting transition temperature @xmath305 is reached at @xmath240= 0.18 . in the overdoped case ( @xmath316 0.18 ) the normal state can be well described by a fermi liquid . for @xmath317 0.15 the normal state properties are far from being understood . there is no agreement what should be the minimal model which contains the basic physics of underdoped cuo@xmath138 layers . several phases are found in this doping regime : an afm insulating phase , the high-@xmath305 superconducting phase , and a charge and spin ordered phase@xcite . regarding the latter two different phases are discussed in the literature : the checkerboard phase and the stripe - like phase . in the stripe - like phase , afm antiphase domains are separated by periodically spaced domain walls along the cu - o directions in which the holes are situated ( see figure[tranquada ] ) . partially motivated by the detection of incommensurate low - energy spin excitations , the existence of a stripe - like state was proposed by theoretical work on the basis of a hartree - fock analysis of the one - band or three - band hubbard model@xcite . the proposed ground state resembles a soliton state in doped conjugated polymers@xcite . the theoretical predictions of a stripe phase were based on a mean - field approximation and the importance of the long range coulomb interaction , not taken into account in the hubbard model , was pointed out@xcite . the phase separation , which seems that the afm background expels holes , was supposed to have a strong influence on the physical properties of doped cuprates : ( i ) a central ingredient to the pairing mechanism in htsc , ( ii ) the non - fermi - liquid behaviour around optimal doping , and ( iii ) the existence of the pseudogap in the underdoped region . the stripe - like structure consists of three concomitant modulations of the spin density , of the charge density , and of the lattice which is coupled to the charge modulation . the spin density modulation has twice the wavelength of the charge and lattice modulation , and in a scattering experiment , these two modulations give rise to superstructure reflections around the afm bragg peaks at ( 0.5@xmath318,0.5,l ) and around the structural reflection at ( @xmath319,0,l ) , respectively , and symmetry related positions ( see figure[tranquada ] ) . assuming half - doped stripes , independent of the average doping concentration , the propagation vector @xmath320 is determined by @xmath321 where @xmath3 is the average stripe distance given in @xmath253-spacings of the lattice . using the same assumption about the constant doping concentration in the stripes , @xmath320 should be equal to 0.5@xmath322 . actually this is observed for @xmath323@xcite , indicating that in real space incommensurate stripes exist , where the stripe distance varies on a local scale and where only the mean distance between the stripes is determined by @xmath322 . in several doped cuprates such as la@xmath314ba@xmath315cuo@xmath138 ( lbco ) , ( la , nd)@xmath314sr@xmath315cuo@xmath138 , and ( la , eu)@xmath314sr@xmath315cuo@xmath138 , the stripe - like order is stabilized near @xmath324 , concomitant with a suppression of superconductivity . it occurs in the so - called low - temperature tetragonal ( ltt ) phase , which is characterized by a corrugated pattern caused by rotation of the cuo@xmath325 octahedra . evidence for static stripes was first detected in an elastic neutron scattering study on the system ( la , nd)@xmath314sr@xmath315cuo@xmath138 , where the superstructure reflection for the spin and the lattice modulation have been detected@xcite . neutron scattering@xcite and non - resonant hard x - ray scattering@xcite monitor the ordering of the charges indirectly by the associated lattice distortion . the reason for this is that these techniques are mainly sensitive to the nuclear scattering and the core electron scattering , respectively . detection of charge order by these methods is hence not fully conclusive , since lattice distortions may also occur with very small or even no charge order of the valence electrons : they may be caused as a result of a bond - length mismatch between different units of a solid ( e.g. planes or chains ) or by a spin density wave alone like , e.g. in chromium metal . on the other hand , rsxs is a method which can directly probe the existence of the charge modulation of the conduction electrons . furthermore , this method enables one to study the wave length of the modulation , the coherence length , the temperature dependence of the order parameter , and in principle also the momentum dependence of the form factor which would give the detailed spatial dependence of the charge modulation . this has been demonstrated for the doped cuprates by rsxs at the o @xmath215 and the cu @xmath216 edges by abbamonte et al.@xcite . in figure[lbco_abba ] we show data from this work on lbco@xcite , comparing x - ray absorption data , measured with the fluorescence method , with the resonance profile , i.e. , the photon energy dependence of the intensity of the charge - order superstructure reflection at energies near a core excitation . edge . green line : xas spectrum for photons polarised parallel to the cuo bond direction . red circles : intensity of the ( @xmath326,0,l ) superstructure reflection at l=0.72 . the enhancement at the mobile charge carrier peak ( mcp ) and the upper hubbard band ( uhb ) demonstrates a significant modulation of the doped hole density . blue line : form factor for scattering from doped holes calculated from xas data . ( b ) data near the cu @xmath224 edge for l=1.47 . ( reprinted with permission from @xcite . copyright 2005 , nature publishing group.),width=283 ] absorption edges at the o @xmath215 and cu @xmath327 edges in cuprates have been extensively studied by electron energy loss spectroscopy ( eels)@xcite and xas@xcite . the o @xmath215 edge spectra of hole - doped cuprates show two pre - peaks : the lower one at 528.6 ev is assigned to transitions into empty o @xmath218 states related to the zhang - rice singlet states . the intensity of this pre - peak , also called the mobile carrier peak ( mcp ) is directly related to the number of the mobile charge carriers . the second pre - peak ( uhb ) at 530.4 ev is related to the upper hubbard band . with increasing doping concentration there is a spectral weight transfer from the upper uhb peak to the mcp peak which is a classical signature of a doped correlated system@xcite , not observed in doped semiconductors . the resonance profile for the ( @xmath328,0,l ) superstructure reflection shown in figure[lbco_abba](a ) for the o @xmath215 edge exhibits a prominent resonance at the mcp , in this way providing a direct link to the doped holes and hence identifying the charge order . interestingly , the reflection also resonates at the uhb energy , which means that also the mottness is spatially modulated . mottness in this context means that the system acts like a mott insulator . the resonance profile for the ( @xmath328,0,l ) superstructure reflection shown in figure[lbco_abba](b ) shows also a strong enhancement near the cu @xmath224 absorption peak . this resonance probably arises from the modulation of the cu lattice@xcite . the charge - order reflections are sharp along h but rod - like along l , indicating quasi - long - range order in the cuo@xmath138 plane but weak coupling between planes . in the work by abbamonte et al.@xcite not only the existence of a charge modulation in lbco has been directly demonstrated but also a quantitative analysis has been presented . analogous to the procedure outlined in section [ sec : rsxsandxas ] , the imaginary part of the form factor can be derived from x - ray absorption data and finally the real part can be derived via a kramers - kronig transformation . the form factor for that part which is related to doped holes is shown in figure[lbco_abba](a ) by the blue curve . one realizes a strong enhancement of the calculated form factor near the two pre - peaks although the line shape does not perfectly agree with the measured resonance profile . according to this analysis , the form factor for the scattering of a doped hole at resonance has a value of 82 , which means that a single hole in this compound scatters like a pb atom off resonance . since the scattered intensity is determined by the form factor squared , this also means that at resonance the scattered intensity of a hole is amplified by a factor of about 7000 . this fact demonstrates the enormous sensitivity of rsxs to the charge modulation of valence electrons which renders this method an ideal tool to study the charge modulation of doped holes in transition metal oxides . the amplitude of the charge modulation was estimated to be 0.063 holes . in a one - band model , which only contains cu sites , the charge modulation for @xmath329 should be close to 0.5 holes . as outlined above , this value is expected for this doping concentration in a simple model in which three cu - o lines are not doped and the fourth line is 50% doped . the relatively small experimental value of 0.063 hole modulation can be explained by the fact that the holes are distributed among many different ( oxygen ) sites . in figure[lbco_abba](b ) xas data at the cu @xmath224 edge together with a resonance profile is presented . here the enhancement of the scattering length was estimated to be about 300 . finally , the temperature dependence of the charge ordering in lbco has been studied yielding a transition temperature @xmath330 = 60 k which is the same as the transition temperature @xmath331 for a lattice transition from the low - temperature orthorhombic ( lto ) to the low - temperature tetragonal ( ltt ) phase . in the latter structure , the cuo@xmath325 octahedra are tilted along the cu - o bond direction stabilizing the stripe formation . a similar rsxs study has been performed on the system la@xmath332eu@xmath333sr@xmath315cuo@xmath334 ( lesco)@xcite . generally , with decreasing ionic radius of the substitutes of la ( here eu ) , the tilt angle @xmath212 in the ltt phase increases leading to a higher lattice transition temperature t@xmath335 . this stabilizes the stripe order and leads to a stronger suppression of high-@xmath305 superconductivity@xcite . charge - order diffraction peaks for lesco could be observed over a range of doping levels @xmath240 . besides information about the phase diagram discussed below , data for different @xmath240 allow to extract more detailed information about the amplitude of the hole doping modulation from the photon energy dependences according to the following considerations . in figure[lesco1_fink ] ( a ) and ( b ) lesco data for @xmath240=0.15 are reproduced , showing xas results , calculated intensities , and a rsxs scattering profile . the calculated intensities were derived from the calculated form factors presented in figure[lesco1_fink ] ( d ) and ( f ) which in turn were evaluated from doping dependent xas data as described above . the resonant scattering intensity of lesco in red as a function of photon energy through the o @xmath336 [ cu @xmath327 ] absorption edge for the stripe superstructure peak . data were taken near the o @xmath215 [ cu @xmath224 ] edges for l=0.75 [ l=1.6 ] . also shown is the x - ray absorption spectrum in black and the calculated scattering intensity in dark blue . ( c ) the real ( solid line ) and the imaginary ( dotted line ) parts of the atomic form factor @xmath100 at the o @xmath215 edge of lsco for @xmath240=0.07 depicted in red and @xmath240=0.15 depicted in blue . ( d ) the real part in red and the imaginary part in blue of the atomic form factor @xmath100 at the cu @xmath327 edge of lsco for @xmath240=0.125 . ( reprinted from @xcite . copyright 2005 , american physical society.),width=7 ] assuming a linear variation of the atomic form factor @xmath339 as a function of the doping concentration @xmath240 one can expand @xmath340= @xmath341 . then the intensity ratio between the lower hubbard band ( lhb ) and the upper hubbard band ( uhb ) is @xmath342_{lhb}/[\delta f(e , x)/\delta x]_{uhb}|^2 $ ] . a remarkable result , typical of a correlated electron system , is that due to a spectral weight transfer , in the xas data the intensity of the uhb decreases proportional to @xmath343 while the intensity of the mcp peak increases proportional to @xmath344@xcite . since the absorption is related to the form factor , in first approximation , the ratio for the form factor of the uhb to the lhb should be near 2 , i.e. , the intensity ratio in the resonant diffraction profile should be four , in agreement with the calculated ratio [ see figure[lesco1_fink ] ( a ) ] , but in clear disagreement with the measured resonance profile presented in the same figure . this discrepancy was explained in terms of a non - linear change of the absorption at the uhb and the mcp observed for doping concentrations @xmath345@xcite . from this analysis it was concluded that the hole doping modulation per cu site was larger than 20 @xmath346 in agreement with the modulation of about 50 @xmath346 derived for lbco mentioned above@xcite . eu@xmath333sr@xmath347cuo@xmath334 near the o @xmath215 edge measured in the fluorescence yield mode in blue . also shown in black is the ( 001 ) superstructure reflection intensity as a function of the photon energy . the inset shows the ( 001 ) reflection measured with a photon energy of 533.2 ev at @xmath348 = 6 k. ( reprinted from @xcite . copyright 2011 , american physical society.),width=283 ] eu@xmath333sr@xmath347cuo@xmath334 normalized to the intensity at @xmath348= 6 k. squares : ( 001 ) reflection measured with photon energies 533.2 ev near the o @xmath215 edge . circles : ( 0.254 0 0.75 ) reflection measured with photon energies 529.2 ev near the o @xmath336 edge . diamonds : ( 0.254 0 1.6 ) reflection measured with photon energies 929.8 ev near the cu @xmath327 edge . ( reprinted from @xcite . copyright 2011 , american physical society.),width=283 ] in figure[lesco2_fink ] we illustrate that other resonances near the o @xmath215 edge can be used to obtain information on the structure of cuprates . there , besides the first two pre - peaks in the xas spectrum , discussed already before , near 533.2 ev a further peak appears due to a hybridisation of o @xmath218 states with rare earth ( @xmath284 = la and eu ) @xmath349 and/or @xmath229 states@xcite . in figure[lesco2_fink ] we also show the photon energy dependence of the ( 001 ) reflection measured in the ltt phase at @xmath348=6 k. in this phase , neighbouring cuo@xmath138 planes are rotated by 90@xmath350 , yielding o sites with different ( rotated ) local environments . as a result , the ( 001 ) reflection becomes allowed at resonance@xcite . the strong resonance at 533.2 ev is due to octahedral tilts , which cause different local environments and affect the hybridisation between the apical o and the @xmath284 orbitals . in the lto phase , neighbouring cuo@xmath138 planes are just shifted , not rotated , with respect to each other . in this case the ( 001 ) reflection remains forbidden even at resonance . using the resonance at 533.2 ev it is possible the detect the lto - ltt phase transition with soft x - rays . this transition is not detectable by the orthorhombic strain ( a - b splitting ) , because the ( 100)/(010 ) reflections can not be reached due to the limited range in momentum space in rsxs . with the help of this resonance feature , it is possible to study the temperature dependence of the structural ltt order and the charge order in one experiment , as shown in figure[lesco3_fink ] . here the temperature dependent rsxs data of lesco @xmath240=0.15 of the stripe superstructure reflection and the ( 001 ) reflection are depicted . these data indicate a first order lto - ltt transition at @xmath331=135 k and a charge order transition at @xmath330=65 k , clearly showing that the two phase transition are well separated . a systematic rsxs study of the doping dependence of charge order yielded the phase diagram for lesco shown in figure[lesco4_fink ] . eu@xmath333sr@xmath315cuo@xmath334 showing transition temperatures for the ltt phase @xmath331 , the antiferromagnetic structure @xmath351 , the magnetic stripe order @xmath352 , the stripe like charge order @xmath330 , and the superconducting transition temperature @xmath305 . closed circles from rsxs experiments@xcite . open circles from ref.@xcite . closed diamond from neutron diffraction data presented in ref.@xcite . ( reprinted from @xcite . copyright 2011 , american physical society.),width=7 ] different from lbco , the lesco results represent the first example in which the lattice transition temperature @xmath331 is so high that the charge order can no longer be stabilized at this temperature and therefore a gap of 55 k exists between @xmath331 and @xmath330 . this is a remarkable result since it demonstrates that the charge order is at least partially electronic in origin . the width of the superstructure reflections , being about five times larger than the experimental resolution , indicate disorder effects and/or glassy behaviour . from the widths the coherence lengths of the charge order have been determined . these lengths are of the order of about 100 lattice constants and increase , at least for smaller @xmath240 , linearly with increasing sr concentration . as a function of doping no maximum of the coherence lengths has been detected for integer numbers of stripe separation in units of the in - plane lattice spacing corresponding to @xmath353 with n being integer . this is different from doped nickelates where a clear maxinum was detected for @xmath354@xcite . the results on the cuprates indicate that in these systems the coherence length is not only related to the commensurability of the stripe lattice with the underlying cuo lattice . furthermore the results point to the fact that the coherence lengths in these systems are not determined by the impurity potential of the doping atoms but is probably related to an increasing tilt angle with increasing sr content . finally the wave vector of the superstructure reflection increases with increasing sr content , at least up to @xmath240=0.125 . since the nesting vector between parallel segments in the fermi surface decreases with increasing doping@xcite , this indicates that stripes are not conventional charge density waves caused by nesting . thus these results point to more strong coupling scenarios for the stripe formation . in a more recent work the stripe modulations in lbco and la@xmath355nd@xmath356sr@xmath357cuo@xmath334 ( lnsco ) were compared using rsxs and hard x - ray scattering@xcite . making use of a two - dimensional detector for rsxs an isotropic coherence length of the charge modulation of the hole density in the cuo@xmath138 planes has been detected . also the strain modulation of the lattice shows an isotropic coherence length . these results are surprising given that the stripes in a single cuo@xmath138 layer are highly anisotropic . in the direction perpendicular to the cuo@xmath138 planes , the stripes are weakly correlated giving rise to the uniform streak of intensity along l. both the charge and the strain modulation is better correlated in lbco than in lnsco . the in - plane hole correlation lengths for lbco and lnsco are @xmath358 and @xmath359 , respectively . similar to lesco , in lnsco the hole density modulation sets in well below the transition into the ltt phase while for lbco @xmath330 is equal to @xmath331 and below @xmath330 the amplitude of the modulation is independent of the temperature . this suggests that the ltt transition occurred at a higher temperature in lbco then it is likely that both the electronic and structural modulations would also have persisted to higher temperatures . a further result from this investigation is that the electronic charge stripe modulation in lbco is 10 times larger in amplitude than in lnsco . this result is surprising since from various other experiments , for lbco , lnsco , and lesco a similar amplitude for the charge modulation is expected . further work is required to solve this puzzle . until now we have reviewed rsxs studies on static stripe - like cdw order in lbco , lnsco and lesco . in all these compounds the cdw is stabilized by an ltt lattice distortion . an important issue in this context is whether static stripes also exist in other two - dimensional cuprates . checkerboard - like static charge order has been reported in ca@xmath314na@xmath315cuo@xmath138cl@xmath138 ( nccoc ) using surface sensitive scanning tunneling experiments@xcite . cdw order could be related to the electron pockets detected in underdoped ybco in quantum oscillation measurements@xcite . the observation of a complete wipe - out of the cu nuclear quadrupole resonance signal at low temperatures in ni substituted ndba@xmath138cu@xmath139o@xmath140@xcite , similar to that of stripe ordered underdoped cuprates , suggests the existence of static stripes in this compound . thus one may think that static stripes are generic to the cuprates and it is very important to look at stripe - like cdw order in other systems which are not stabilized by an ltt lattice distortion . a rsxs study on a tetragonal compound , already discussed in section[sec : newmat ] , has been performed on an excess o doped la@xmath138cuo@xmath292 layer ( @xmath360 ) epitaxially grown on a srtio@xmath139 crystal@xcite . in this work the resonance effects at the pre - edge of the o @xmath215 shell excitation and near the cu @xmath222 edge have been used for the first time to exploit charge modulation in the cuo@xmath138 layers . it was there where it was demonstrated that at the pre - edge of the o @xmath215 shell excitations , the scattering amplitude for mobile holes is enhanced by a factor of 82 which leads to an amplification of diffraction peaks related to these holes by more than 10@xmath361 . extensive non - successful search for superstructure reflections due to a stripe - like cdw lead to the suggestion of an absence of static stripes in this high-@xmath305 superconductor . a further rsxs study on a possible charge modulation in a tetragonal cuprate has been performed on nccoc@xcite . from experiments at the mcp peak at the o @xmath215 edge or at the cu @xmath257 edge no evidence for a checkerboard - like charge modulation has been detected . from this null experiment the authors have concluded that the checkerboard order observed in the sts experiments is either glassy or nucleated by the surface . a further rsxs experiment on ni substituted ndba@xmath138cu@xmath139o@xmath140 was stimulated by the above mentioned cu nuclear quadrupole resonance experiment@xcite . no superstructure reflection due to stripe - like charge order could be detected at low temperatures in this compound@xcite . more rsxs experiments on charge order have been performed on the most studied high-@xmath305 superconductor ybco which contains a cuo@xmath138 double layer and in addition cuo@xmath139 chains along the @xmath362 axis . in a first paper on the ortho - ii yba@xmath138cu@xmath139o@xmath363 phase , in which the cuo@xmath139 chain layers are ordered into alternating full and empty chains , charge order in the planes and in the chains were reported@xcite . this observation was later questioned , partially by the same authors , in a more refined rsxs study on ortho - ii and ortho - viii oxygen ordered ybco@xcite . however , charge order in underdoped ybco was postulated on the basis of nmr studies in high magnetic fields @xcite and was eventually detected in very recent experiments by rsxs @xcite and also with hard x - ray diffraction @xcite . the former study showed by the resonance profile near the cu @xmath224 resonance , that the charge order in fact takes place in the cuo@xmath138 planes rather than the chains . the temperature dependence of the charge order found in both studies clearly revealed competition with superconductivity . this result has important implications for the understanding of the material and the cuprates in general . it readily explains the anomalously low superconducting temperature in underdoped ybco and shows that charge order seems to be a generic feature of cuo@xmath138 planes in layered cuprates . + + among the various mechanisms discussed for high-@xmath305 superconductivity in cuprates , the electronic model based on ladder - like structures in which cu and o atoms are ordered in two chains , coupled by rungs , has been heavily discussed@xcite . in this model , depending on the exchange coupling along the chains and that along the rungs , singlets can exist on the rungs of a doped ladder compound and exchange - driven superconductivity can be formed . depending on the size of parameters or doping concentration , an insulating hole crystal in which the carriers crystallize into a static wigner crystal may also form the ground state . note , this would be an electronic charge density wave ( cdw ) , driven by the coulomb interaction and not by a coupling to the lattice ( peierls transition ) . the competition of the two phases is similar to that believed to occur between ordered stripes and high-@xmath305 superconductivity in two - dimensional cuprates . indeed the only known doped ladder compound sr@xmath364ca@xmath315cu@xmath365o@xmath366 ( scco ) exhibits both phases : superconductivity has been detected for @xmath240= 13.6 below @xmath305=12 k at hydrostatic pressures larger than 3.5 gpa@xcite , while for @xmath240=0 scco exhibits a cdw . therefore and because the interplay between charge and spin degrees of freedom can be easier studied theoretically in quasi one - dimensional systems , experimental studies of charge density modulations in scco by rsxs are extremely important@xcite . scco consists of two different alternating types of copper oxide sheets perpendicular to the @xmath362 axis . one with chains , formed out of edge - sharing cuo@xmath334 plaquettes and one with weakly coupled two - leg ladders , i.e. , two parallel adjacent chains formed out of corner - sharing cuo@xmath334 plaquettes . both the chains and the ladders are aligned along the @xmath367 axis . the sheets are separated by sr / ca ions . since the cuo@xmath334 plaquettes in the chains and those in the ladders are rotated by 45@xmath350 with respect to each other , the ratio of the lattice constant for the ladder @xmath368 to that of the chain @xmath369 is about @xmath370 . this means that one unit cell is composed out of ten cuo@xmath334 plaquettes forming the chains and two times seven equal to 14 cuo@xmath334 plaquettes forming the two - leg ladders . this leads to a `` misfit compound '' which is structurally incommensurate and has a large unit cell along the c axis with the length @xmath367=27.3@xmath371 . according to the chemical formula and assuming that cu is divalent , there are 6 holes per formula unit . the distribution of the holes among chains and ladders is still under debate . earlier polarisation dependent xas measurements for @xmath240=0@xcite suggested that 0.8 holes are on the ladder and the rest is on the chains . replacing the sr ions by the smaller isovalent ca ions , i.e. , by chemical pressure , there is a hole transfer from the chains to the ladders which according to the earlier xas results@xcite , for @xmath240=12 increases the number of holes on the ladders to 1.1 , i.e. , 0.08 holes per cu site . optical spectroscopy@xcite , arpes experiments@xcite and recent xas experiments@xcite showed for high ca replacements values of 0.2 , 0.15 - 0.2 , and 0.31 holes per cu site , respectively , with no sign of convergence with time . in figure[scco_rusyd1 ] we reproduce scco data showing xas spectra for @xmath240=0 and a photon polarisation parallel to @xmath367 together with resonance profiles for various ca concentrations@xcite . in agreement with previous xas data near the o @xmath215 edge@xcite the first pre - peak at 528.4 ev and the shoulder at 528.7 ev were assigned to hole states at the chains and the ladders , respectively . the next peak at 530 ev was interpreted in terms of the uhb . like in other doped cuprates , the first peak at 931.5 ev in the cu @xmath224 xas spectrum is assigned to a transition into empty cu @xmath221 states while the shoulder at 933 ev is related to ligand ( o ) hole states . for @xmath240=0 , a resonance profile measured at a photon energy of 528.7 ev and a wave vector @xmath372 shows a clear resonance for @xmath373=1/5 ( see figures[scco_rusyd1 ] and [ scco_rusyd2 ] ) . this observation together with the fact that away from the ladder hole shoulder no scattering intensity has been detected , signals the existence of a commensurate hole crystal in the ladders with a wave length @xmath374 . at higher ca concentrations ( 10@xmath375 ) a resonance is observed for @xmath373=1/3 ( see figures[scco_rusyd1 ] and [ scco_rusyd2 ] ) . on the other hand , no resonance has been detected for @xmath373=1/4 . this could indicate that the hole crystal is stable for odd , though not even , multiples of the ladder period . the resonance profiles at the cu @xmath224 edge shows , in particular for higher @xmath240 values two peaks , at 930 ev and at 933 ev . the latter peak coincidences with the ligand - hole shoulder and thus indicates the presence of a hole modulation . the former indicates a lattice modulation@xcite which is more pronounced for larger hole ( ca ) concentrations . the hole crystal intensities show both for @xmath373 = 1/5 and 1/3 a strong temperature dependence which points to melting of the hole crystal at higher temperatures and also signals that the modulation is not caused by the misfit between the chains and the ladders . the non - existence of hole crystals with non - integer l values signals that incommensurate hole crystals melt even at very low temperatures . hole crystallization in ladder compounds was predicted by a @xmath376 model@xcite and density matrix renormalization group calculations@xcite . in particular it was found that holes like to pair up on the rungs of the ladder . thus doped rungs are separated by the wave length @xmath377 and the doping concentration @xmath378 per cu site should be then @xmath379 or the number of holes on the ladder per unit cell should be @xmath380 . from these considerations we expect for zero ca concentration @xmath381 @xmath382 and @xmath383 and for @xmath384 @xmath385 and @xmath386 . these values are in good agreement with the values derived in the recent xas evaluation@xcite and thus offer strong support for a pairing of holes along the rungs and thus also support models on the mechanism of high-@xmath305 superconductivity in cuprates based on ladder systems . on the other hand , these results are in striking disagreement with other results on the number of holes an the ladders mentioned above . the absence of a @xmath373=1/4 periodicity was explained in terms of resonant valence bond calculations indicating that the wigner hole crystal is stable for odd , but not for even multiples of the ladder period . further theoretical work showed that the widely used ladder @xmath376 model is not sufficient and has to be supplemented by coulomb repulsion between the holes on neighbouring ladders to explain the existence of a hole crystallization on the ladders . a mean field calculation of the extended model could explain a charge density wave with an odd period@xcite . at the end of the paragraph reviewing charge order in the system scco , we report on rsxs results on the chains of these compounds@xcite . for the undoped compound in which divalent sr was replaced by trivalent la , a superstructure reflection for @xmath387 has been detected for photons with an energy near the cu @xmath224 and @xmath226 edges . since in this compound , no holes can form a charge density wave and since the superstructure reflection was independent of the temperature , it has been interpreted in terms of a pure strain wave formed by the misfit between the chains and the ladders . for doped scco an incommensurate superstructure reflection at @xmath388 has been detected . it shows a remarkable temperature dependence and appears mainly near the ligand - hole shoulder . from this it was concluded that in scco a strain - stabilized charge density wave is formed . + + the first transition metal oxide system in which static stripe - like order was detected was la@xmath314sr@xmath315nio@xmath389 ( lsno ) . in contrast to lsco , upon doping , i.e. , replacing the trivalent la ions by divalent sr ions or by adding oxygen , the nickelates remain insulating except for very high doping concentrations , and superconductivity has not been detected so far . both in the electron diffraction @xcite and neutron diffraction@xcite data , superstructure reflections were observed , which were interpreted in terms of coupled charge and spin - density modulations in the nio@xmath138 planes , similar to the case depicted for the cuprates in figure[tranquada ] . different to cuprates , however , the stripes are rotated by 45@xmath350 . in an orthorhombic unit cell with axes rotated by 45@xmath350 with respect to the ni - o bonds , the magnetic peaks are split about the antiferromagnetic position ( 1,0,0 ) along the [ 100 ] and [ 010 ] directions by @xmath320 , while the charge order are split about the fundamental bragg peaks by 2@xmath320 , as in the case of cuprates . also similar to the cuprates , for hole concentrations @xmath390 , in the nickelates @xmath391 . stripe - like charge and spin order in lsno is observed for 0.15 @xmath392 0.5 . the stripe order is most stable at a doping level @xmath354 where it shows the highest charge and spin ordering temperatures and the longest correlation length . different from the cuprates in la@xmath138nio@xmath334 there are two intrinsic holes on each ni site in the @xmath312 and the @xmath393 state@xcite . edges of la@xmath394sr@xmath333nio@xmath334 for the spin and charge modulation superstructure reflections recorded with @xmath33 and @xmath32 polarised photons . also shown are xas data . the experimental results ( points ) are compared to simulations ( solid line ) . ( reprinted from@xcite . copyright 2005 , american physical society.),width=7 ] rsxs experiments were performed on a la@xmath314sr@xmath315nio@xmath334 single crystal@xcite . the energy profiles at the ni @xmath216 edges , shown in figure[lsno_schuessler ] , exhibit resonances at the charge - order wave vector ( 2@xmath320,0,1 ) and at the spin - order wave vector ( 1-@xmath320,0,0 ) with @xmath320=0.196 in the above notation . a resonance enhancement is also observed at the la @xmath233 edge at 849.2 ev , however , this enhancement is much weaker than at the ni @xmath216 edges indicating that the modulation occurs mainly in the nio@xmath138 planes . the resonances show a strong polarisation dependence , e.g. the intensity of the magnetic superstructure peak for @xmath32 polarisation ) is only 10 @xmath346 of that for @xmath33 polarisation . since only a magnetic moment perpendicular to the polarisation of the incoming light is probed and because in the chosen experimental geometry the @xmath33 polarisation is perpendicular to the stripes , one can conclude that the ni spins are essentially collinear with the stripes . further conclusions can be drawn from a comparison with calculations based on a configuration - interaction model ( see figure[lsno_schuessler ] ) . the difference of about 1 ev for the charge modulation resonances for @xmath33 and @xmath32 polarisation indicates a large energy splitting between the @xmath312 and the @xmath393 levels . the comparison also signals that the holes are going mainly to the o sites on a ligand molecular orbital with @xmath395 symmetry . due to the strong on - site coulomb repulsion of two holes on the ni sites , the ni 3@xmath253 count is not strongly modulated in the stripe structure , ruling out a ni@xmath285/ni@xmath278 charge order scenario . on the contrary , the the situation is very close to that of stripe structures in cuprates , except that in the nickelates there is an additional intrinsic hole in the ni 3@xmath396 states . in summary , the comparison with the rsxs data with the cluster calculations yields interesting results on the stripe structure and the electronic structure of nickelates : ( i ) both the charge order and the spin order resides in the nio@xmath138 layers , ( ii ) the doped holes are mainly located on o sites and the spin of these holes are coupled anti - parallel to the spins of the intrinsic hole on the ni 3@xmath397 states in close analogy to the zhang - rice singlets in the cuprates . + + magnetite , fe@xmath139o@xmath334 , is another prototype correlated transition metal oxide , in which a subtle interplay of lattice , charge , spin , and orbital degrees of freedom determines the physical properties@xcite . it was the first magnetic material known to mankind . verwey discovered in the late 1930 that upon lowering the temperature below the verwey temperature @xmath398=123 k , fe@xmath139o@xmath334 undergoes a first - order transition connected with a conductivity decrease by two orders of magnitude . magnetite has been considered as a mixed valence system . in this compound at high temperatures , the tetrahedral @xmath399 sites are occupied by fe@xmath278 ions while the octahedral @xmath400 sites are occupied by an equal number of randomly distributed fe@xmath285 and fe@xmath278 cations . according to verwey the transition into the low conducting state is caused by an ordering of the fe valency on the @xmath400 sites accompanied by a structural transition . although numerous experimental@xcite and theoretical studies@xcite have been performed on magnetite , no full consensus over all aspects of the order transition exists . with its particular sensitivity to the fe @xmath221 electronic structure , recent rsxs studies at the fe @xmath216 edges contributed significantly to the understanding of the verwey transition@xcite . xas data of fe@xmath139o@xmath334 at the fe @xmath216 edge together with resonace profiles of the ( 0,0,@xmath401 and the ( 0,0,1 ) superstructure reflections are displayed in figure[feo_schlappa ] . xas spectra of fe@xmath139o@xmath334 compared to resonance profiles of the ( 0,0,@xmath402 ) and the ( 0,0,1 ) superstructure reflections . the solid lines are simulations for the charge order scenario while the dashed line is related to the homogeneously mixed - valence scenario . ( reprinted from@xcite . copyright 2008 , american physical society.),width=7 ] based on previous interpretations of the fe @xmath216 absorption spectra and on simulations it was possible to decompose the absorption structure and to assign the different peaks to specific fe sites [ see figure[feo_schlappa ] ( a ) ] . the low - energy features ( red solid line ) were assigned to fe@xmath285 . remarkably , the ( 0,0,@xmath402 ) shows a resonance exactly at these energies which means that the ( 0,0,@xmath402 ) peak is due to an order , which involves only @xmath400-site fe@xmath285 ions . this implies further that there is an orbital order of the @xmath403 electrons that distinguishes different @xmath400-site fe@xmath285 ions . this experimental result on the orbital ordering of the @xmath403 electrons of the fe@xmath285 ions agrees with the theoretical predictions on the basis of lda+@xmath310 calculations@xcite . the ( 0,0,1 ) superstructure reflection shoes maxima at two energies corresponding to the @xmath400-site fe@xmath285 and the @xmath400-site fe@xmath278 ions . this is exactly expected for a charge order involving the two @xmath400-site fe ions . in figure[feo_schlappa](c ) the measured resonance profile is compared with simulations for a charge ordered state ( solid line ) and a homogeneous fe@xmath404 configuration ( dashed ) . the very good agreement with the charge order state and the disagreement with the homogeneous state clearly supports charge order on the @xmath400 sites in magnetite . in this context , one should mention that the charge order of the fe @xmath221 electrons on the @xmath400-site is small and that one should rather speak of a modulation of the @xmath403 count which is partially screened by a charge transfer from the oxygen neighbors to the empty @xmath303 states@xcite . in transition metal compounds the degeneracy of partially occupied electronic states is very often lifted , resulting in a long - range orbital ordering of the occupied orbitals below a transition temperature . therefore orbital ordering phenomena play a fundamental role in determining the electronic and magnetic properties of many transition metal oxides . the origin of the orbital ordering is still under discussion . one scenario explains orbital ordering in terms of a purely electronic mechanism , i.e. , by a super - exchange between the transition metal ions@xcite . the other model explains orbital ordering by a lifting of the degeneracy of the atomic level by a cooperative jahn - teller effect , i.e. , by a net energy gain due to lowering of the electronic state and an energy loss by a local distortion of the ligands around the transition metal , similar to the peierls transition mentioned above . in this second scenario lattice effects are important . possibly in real systems a combination of the two models is realized . + + classical examples of orbitally ordered systems are kcuf@xmath139 @xcite and lamno@xmath139 . in case of manganites , in particular , orbital order has attracted considerable attention and controversy since they exhibit a wide diversity of ground states including phases with colossal magnetic resistance@xcite or charge and orbitally ordered ground states . in many cases small changes in some parameters such as the charge carrier doping or temperature can lead to transitions between disparate ground states . the origin of this rich physics is widely believed to be due to a complex interplay of charge , magnetic , orbital , and lattice degrees of freedom , which results in different strongly competing electronic phases . two prototypical examples are the manganites @xmath405mno@xmath139 and @xmath406mno@xmath334 ( @xmath284=trivalent rare earth ions , @xmath399= divalent alkaline - earth ions ) , which belong to the famous ruddlesden - popper series . in all these compounds each mn ion is surrounded by an o octahedron . the interaction with the o ions leads to a partial lifting of the degeneracy of the @xmath221 states into the lower @xmath403 states and the twofold degenerate @xmath303 states at higher energies . in the undoped case ( @xmath240=0 ) , the mn ions have a formal valence of 3 + with four electrons in the @xmath221 shell . since the hund s rule energy is much larger than the crystal field splitting , the mn @xmath407 is in a very stable high - spin state that corresponds to three spin - up electrons in the @xmath408 states and a single spin - up electron in the @xmath409 state , which results in a @xmath410=2 state of mn . the @xmath403 states , being less hybridised with the o ions than the @xmath411 states , are assumed to be more localised , also by strong correlation effects . the @xmath303 electrons are supposed to be more delocalised due to the stronger hybridisation with the o ions . on the other hand , for a cubic mno@xmath325 octahedron , a single electron in the degenerate @xmath411 levels can occupy any linear combination of the @xmath395 and @xmath412 orbitals . this is referred to as the orbital degree of freedom , which plays a very prominent role for the physics of manganites . the ground state of a cubic octahedron is therefore degenerate and , according to the theorem of jahn and teller , this degeneracy will always be lifted by a symmetry reduction , i.e. , a distortion of the octahedron . the mn @xmath407 is therefore strongly jahn - teller active causing local lattice distortions . when the filling of the @xmath303 states is close to 1 or close to a commensurate value , the distortions of the interconnected octahedra can occur in a cooperative way and one is then talking about a collective jahn - teller distortion . this collective ordering of jahn - teller distorted octahedra also corresponds to an orbital ordering . in other words the collective jahn - teller effect is one way to stabilize an orbitally ordered state . there is however another mechanism that can stabilize orbital order , even in the absence of electron - phonon coupling , which is given by the so - called kugel - khomskii exchange@xcite . it was discussed controversially which of the two mechanisms is the main driving force for orbital order in the manganites . a particular example for this controversy was given by the half - doped manganite la@xmath260sr@xmath315mno@xmath334 , which is composed of mno@xmath138 planes separated by a cubic ( lasr)o layer . upon hole doping , more and more mn ions with a formal valence of 4 + ( @xmath413 ) are created , which suppresses the collective jahn - teller effect and causes an increased itineracy of the @xmath303 electrons . the @xmath411 electrons thus play the role of conduction electrons coupled to the localised @xmath410=3/2 @xmath408 electrons . as discussed first by zener@xcite , the mobility of @xmath303 conduction electrons is strongly affected by the spin degrees of freedom : due to the so - called double exchange mechanism , the @xmath303 electrons can delocalise only for ferromagnetic alignment of neighbouring spins . upon increasing the temperature above the ferromagnetic transition temperature the configuration of the spins gets disordered , which causes a strong spin - charge scattering and thus an enhancement of the resistivity close to the curie temperature . applying an external magnetic field at this temperature can easily align the local spins leading to a strong magneto - resistance . this is a simple qualitative explanation of the colossal magneto - resistance observed e.g. in la@xmath260sr@xmath315mno@xmath139 ( x=0.3)@xcite which is not correct on a quantitative level , since it neglects the orbital and lattice degrees of freedom . the relevance of the latter becomes particular evident near @xmath240=0.5 where an electron ordering takes place which has attracted much attention and controversy . several studies have been performed on the compound la@xmath414sr@xmath415mno@xmath334 which is composed of mno@xmath138 planes separated by a cubic ( lasr)o layer . at room temperature the mn sites in this system are all crystallographically equivalent and have an average valency of + 3.5 . at @xmath416 240k a charge disproportionation occurs creating two inequivalent mn sites . originally the two sites were assigned to mn@xmath278 and mn@xmath291 ions , but later on it turned out that the charge difference of the two sites is much less than 1 . below @xmath274=120 k a long range antiferromagnetic ordering of the mn ions into a ce - type structure is realized , originally detected in the related compound la@xmath260ca@xmath315mno@xmath139 by neutron scattering@xcite . it is believed that this complex electronic order is stabilized by several interactions , but it was discussed controversially whether orbital order really exists in this system and , if it does , whether it is driven by a collective jahn - teller distortions or by superexchange interactions . synchrotron - based investigations of orbital order in the manganites were initiated by a seminal study of la@xmath414sr@xmath415mno@xmath334 using hard x - rays at the mn @xmath215 edge@xcite . superstructure reflections were detected , which could not be explained on the basis of the high - temperature crystal structure and were therefore explained in terms of an orbital ordering of the mn @xmath221 states . while there was consensus about the orbital origin of these reflections , the direct observation by resonant hard x - ray scattering was challenged , as the m @xmath215 edge is dominated by dipole - allowed transitions to the 4p states , which are not sensitive to orbital order but rather lattice distortions , as pointed out in a theoretical study @xcite . another theoretical paper@xcite thus suggested the use of rsxs experiments at the mn @xmath222 edges , which are directly related to virtual excitations into mn @xmath221 states and thus to orbital ordering . the calculations predicted different photon energy dependencies of the scattered intensity for orbital ordering being stabilized by superexchange interaction compared to jahn - teller effect . such rsxs studies on single - layered manganite la@xmath414sr@xmath415mno@xmath334 at the mn @xmath222 edges were subsequently reported in refs . @xcite . the first report was given by wilkins et al.@xcite , who studied the orbital and magnetic reflections at wave vectors of @xmath417 and @xmath418 , respectively . we note that the occurrence of orbital order causes the unit cell to quadruple in the @xmath419 plane . edges of la@xmath414sr@xmath415mno@xmath334 at the orbital - order superstructure reflection @xmath420 measured at 63k ( full black line ) compared with theoretical fits ( dashed red lines ) for ( a ) @xmath421 and ( b ) @xmath422 types of orbital ordering . in panel ( c ) a fit for an orthorhombic crystal field is presented . the inset shows the temperature dependence of the orbital order parameter . ( reprinted with permission from @xcite . copyright 2005 , american physical society.),width=7 ] figure[lsmoo_wilkins ] shows the energy dependence of the scattered intensity at the orbital order wave vector @xmath423 through the mn @xmath224 ( near 640 ev ) and @xmath226 edges ( near 650 ev ) which are related to virtual electric - dipole transitions between the mn @xmath225 and @xmath227 core levels to the unoccupied @xmath221 states . figure[lsmom_wilkins ] shows analogous data for the afm order reflection @xmath424 . it can be clearly observed in these figures that the orbital and magnetic reflection exhibit strong resonances at the mn @xmath216 edges . we note that these measurements provide the most direct prove of orbital ordering in these materials . in figures[lsmoo_wilkins ] and [ lsmom_wilkins ] the insets show the temperature dependences of the orbital and magnetic order parameters , respectively . the orbital order parameter decreases continuously with increasing temperature and disappears at @xmath425=230 k. the intensity of the antiferromagnetic reflection shows a similar behaviour but disappears at a lower temperature of @xmath274=120 k. edges of la@xmath414sr@xmath415mno@xmath334 at the magnetic superstructure reflection @xmath426 measured at 63 k ( full black line ) compared with theoretical fits ( dashed red lines ) for ( a ) @xmath421 and ( b ) @xmath422 types of orbital ordering . in panel ( c ) a fit for an orthorhombic crystal field is presented . the inset shows the temperature dependence of the magnetic order parameter . ( reprinted with permission from @xcite . copyright 2005 , american physical society.),width=7 ] in both figures the experimental data are compared with theoretical calculations of the rsxs spectra based on atomic multiplet calculations in a crystal field . in the calculations the crystal fields were modified in such a way as to fit the experimental data . in the panels ( a ) and ( b ) the cubic and the tetragonal crystal field parameters were adjusted for a @xmath421 and @xmath422 type of orbital ordering , respectively . in the panel ( c ) for the @xmath421 type ordering a small orthorhombic crystal field component was added . from the comparison of the experimental data with the theoretical calculations the authors concluded that the orbital ordering below @xmath425 is predominantly of @xmath421 type . the inclusion of a small orthorhombic crystal field component moderately improved the fits , although many details of the complicated experimental lineshape could not be reproduced by the calculation . notwithstanding these difficulties , calculations of the resonance profiles predicted a drastic reduction of the @xmath427 intensity ratio with decreasing tetragonal crystal field . this theoretical prediction was used for the interpretation of experimental results . the experimental temperature dependent rsxs data shown in figure[lsmot_wilkins ] ( a ) and ( b ) indicate that below the orbital ordering temperature @xmath425 the @xmath427 ratio increases , suggesting an increase of the tetragonal crystal field , i.e. , an increase of the jahn - teller distortion with cooling . approaching @xmath274 the ratio @xmath427 increases further and saturates below @xmath274 . edges of la@xmath414sr@xmath415mno@xmath334 . ( a ) integrated intensities of the main features of the orbital ordering @xmath420 reflection at at the @xmath224 edge ( triangles ) and at the @xmath226 edge ( inverted triangles ) . ( b ) intensity ratio of the integrated intensities @xmath427 . ( c ) temperature dependence of the integrated intensity of the magnetic @xmath426 reflection at 643 ev . ( reprinted with permission from @xcite . copyright 2005 , american physical society.),width=7 ] from these findings the authors concluded that there are two separate contributions causing the orbital ordering : a dominant mechanism related to the cooperative jahn - teller distortion of the o ions around the mn@xmath278 ions and the direct magnetic kugel - khomskii superexchange mechanism which further strengthens orbital ordering by short - range afm order and another increase by long - range afm order below @xmath274 . furthermore the authors concluded a strong interaction between orbital and magnetic correlations . nearly at the same time another group has performed similar rsxs experiments on the same compound la@xmath414sr@xmath415mno@xmath334@xcite . the diffraction profile at the mn @xmath216 edges for the the orbital order @xmath420 superstructure reflection were very close to those shown in figure[lsmoo_wilkins ] . in a following paper@xcite , the experimental data were compared to calculations of a small planar cluster consisting of a central active mn site with a first neighbor shell comprising o and mn sites . thus by allowing a hopping between the mn and the o ions an integer charge ordering of the mn ions could be abandoned . in addition to the jahn - teller distortion of the o ions , the spin magnetization of the inactive mn@xmath291 sites were explored as adjustable parameters . the calculations based on a @xmath428 orbital ordering of the @xmath411 electrons reproduced all spectral features of the resonance profile while calculations on a @xmath429 order did not , although the differences were not dramatic enough to provide a definite answer . a further result of the rsxs studies of dhesi et al.@xcite was the detection of a complicated temperature dependence of the individual features of the @xmath224 edge indicating that the complete @xmath224 edge intensities can not be assigned to one order parameter ( e.g. orbital ordering by a collective jahn - teller effect ) . these findings were supported by a following rsxs study of the same compound by staub et al.@xcite . they realized that the features observed in the resonance profiles of the magnetic reflection show the same temperature dependence while those of the orbital order reflection show a strong energy and temperature dependence . the high - energy features of the @xmath216 edges ( in figure[lsmoo_wilkins ] at 647 and 656 ev ) which were assigned to a jahn - teller ordering parameter saturate below @xmath274 . the other features assigned to a direct super - exchange ordering parameter still increases below @xmath274 . from this experimental result together with a polarisation dependence of the resonance profiles they came to the conclusion that there are two interactions leading to the orbital order . it was concluded that the orbital super - exchange interaction dominates over the jahn - teller distortion strain field interaction and drives the transitions . the rsxs results on the manganites described above , demonstrate that the lineshapes at the mn @xmath216 edges display rich spectral features , which contain a lot of information about the underlying order phenomena . to extract this information and to interpret the spectra is , however , challenging as it requires extensive theoretical modeling . here it should be mentioned that in addition various rsxs studies have been performed on double and multi layer manganites@xcite . although largely discussed in connection with 3d transition metal compounds , orbital order is a phenomenon that also occurs in other materials , such as @xmath284 compounds . here , the shielding of the @xmath229 shell by the outer @xmath349 electrons causes a weaker coupling of the electronic degrees of freedom of the @xmath229 electrons to the lattice and hence , allows studying orbital order mechanisms in a less complex situation . in a series of publications on orbital ordering of @xmath229 electrons in @xmath430 compounds mulders et al . could show that rsxs performed at the @xmath431 resonances is not just able to observe orbital order of the @xmath229 electrons but can in addition quantify the contributions of the different higher order multipole moments to the electronic ordering@xcite . in recent years , the interest in materials with coupled magnetic and ferroelectric order has seen an extraordinary revival@xcite . the incentive for numerous applied as well as fundamental studies of these materials is the perspective to manipulate electric polarisation by a magnetic field@xcite or magnetic order by an electric field@xcite . in this context , the notion multiferroics has been coined , which is , of course , more general and applies to a broader range of materials with coupled order parameters . the renewed interest in magnetoelectric materials was initiated by the discovery of strong coupling between magnetic and electric order in tbmno@xmath139 for which one can flip the electric polarisation by application of a magnetic field@xcite . the ferroelectric order in tbmno@xmath139 is connected to the occurrence of a cycloidal magnetic arrangement of the mn spins@xcite , a property that was also found for other members of the perovskite @xmath284mno@xmath139 series , where @xmath284 denotes a rare earth gd , tb , or dy@xcite . different theoretical models were used to explain the strong coupling of magnetic and electric order , however , arriving at the same expression that links the direction of the electric polarisation @xmath432 to the chirality of the spin structure@xcite : @xmath433 in this equation , @xmath434 and @xmath435 denote spins at the sites @xmath436 and @xmath437,and @xmath438 is the unit vector connecting the two sites . other rare earth manganites of the form @xmath284mn@xmath138o@xmath439 also exhibit multiferroic behaviour , with reversible switching of the ferroelectric polarisation by a magnetic field , as found in tbmn@xmath138o@xmath439@xcite . the coupling of magnetic and electric order in these latter materials is believed to be of a different origin , involving magnetoelastic coupling , ionic displacements or electronic ferroelectric polarisation . the majority of the recently studied magnetoelectrics are manganites , however , also cuprates , among them cupric oxide cuo itself have been found to be multiferroic@xcite , with a particularly high ordering temperature of @xmath440 k. as the multiferroic materials are transition metal oxides with complex order of the magnetic moments , neutrons have initially been the method of choice for their magnetic characterization@xcite . rsxs , on the other hand , is perfectly matching these materials as both the @xmath221 and @xmath229 electronic states of the transition metals as well as the @xmath218 states of oxygen can be addressed via resonant dipole transitions@xcite . thus , the method has provided substantial contributions to the understanding of the materials , also because circularly polarised x - rays are readily available at synchrotron radiation sources . by the magnetic structure factor , this provides direct access to a possible handedness , as present in helical@xcite or cycloidal arrangements@xcite . in addition , frustrated magnetic interactions result in rather long - period superstructures that can thus be accessed at soft x - ray wavelengths . [ scale=0.5 , trim= 0 0 0 0 , angle= 0 ] wilkins2009 - 1 tbmno@xmath139 is the most studied material of the @xmath284mno@xmath139 series . for this material , magnetic neutron diffraction provided the first experimental observation of a cycloidal magnetic structure in connection with ferroelectric polarisation@xcite . from the observation of components of the mn @xmath221 magnetization along both the @xmath362 and the @xmath367 crystallographic axis in the ferroelectric phase a non - collinear structure was concluded that together with a propagation along the @xmath362 axis has the proper chirality to induce a ferroelectric polarisation along the @xmath367 axis according to eq.[eqnpolarisation ] . the tb @xmath229 structure , on the contrary , remained collinear , as only a component along the @xmath441 axis was found , pointing out the essential role of the mn ordering for the ferrolectric properties of the material . the mn spin cycloid leads to substantial circular dichroism in x - ray scattering as first found in a non - resonant hard x - ray study using circularly polarised light , in this way directly linking to the handedness of the magnetic structure@xcite . while it was generally agreed that ferroelectricity is essentially due to a mn @xmath442 cycloid , further details of the magnetic structure could be elucidated taking advantage of the very high sensitivity of resonant magnetic soft x - ray diffraction . a refinement of the mn magnetic structure was obtained from resonant soft x - ray diffraction at the mn @xmath226 resonance@xcite . here , a magnetic f - type reflection along [ 0 k 0 ] was observed that would not be allowed for strictly a - type magnetic ordering in the perovskite structure of these materials@xcite . from modeling the azimuthal dependences of the diffraction intensities measured with linearly polarised x - rays ( figure[wilkins2009 - 1 ] ) it was concluded that a canted magnetic component along the @xmath367 axis must be present in all ordered phases of the material , rendering also the previously assumed sinusoidal phase non - collinear . [ scale=0.8 , trim= 0 0 0 0 , angle= 0 ] jang2011 - 1 further details of the magnetic structure were revealed by the application of circular polarised x - rays at the mn @xmath226 resonance ( see figure[jang2011 - 1 ] ) . the magnetic ( 0 @xmath443 0 ) reflection exhibits pronounced circular dichroism , thus representing an f - type magnetic cycloid that is coupled to the previously observed a - type cycloid@xcite . this clear dependence of the scattered intensity on the helicity of the x - rays is not to be confused with x - ray circular dichroism in absorption , as it is due to the magnetic structure factor@xcite and not an effect of the resonant form factor . the occurrence of an f - type magnetic cycloid is explained by antisymmetric exchange , the so - called dzyaloshinskii - moriya interaction that favours canting of the magnetic moments towards the @xmath367 axis@xcite . figure[jang2011 - 1 ] also indicates another application of resonant soft x - ray diffraction to multiferroic materials . with the help of an external electric field , the electric polarisation of tbmno@xmath139 is switched from pointing along -c to pointing along c. according to eq.[eqnpolarisation ] , this corresponds to magnetic cycloids of opposite chirality as depicted on the right panel of figure[jang2011 - 1 ] . therefore , the asymmetry in the scattering of circular polarised light changes sign , as can be inferred from the changing response to left ( @xmath222 ) and right ( @xmath444 ) circularly polarised light . hence , besides general refinements of the magnetic structures , rsxs can provide a useful contrast mechanism to identify and study microscopic details of the magnetic structure upon the application of external fields . the first demonstration of such an application was provided by bodenthin et al . for the case of ermn@xmath138o@xmath439@xcite . the contrast was obtained at the mn @xmath224 resonance using the commensurate ( 1/2 0 1/4 ) reflection , which is closely connected to the ferroelectric phase in this material . as shown in figure[bodenthin2008 - 1 ] ( upper panel ) , differences in the intensities of this peak are observed with an electric field applied compared to the situation without field . the differences are in fact small but significant , as shown in the inset , and the peak intensity reveals hysteretic behaviour , in this way demonstrating the manipulation of the magnetic structure by an external electric field@xcite . [ scale=1 , trim= 0 0 0 0 , angle= 0 ] bodenthin2008 - 1 the studies reported so far were mainly concerned with the magnetic order of the mn spins . however , also the ordering of the @xmath229 moments is expected to play a role for the ferroelectric properties of the @xmath284mno@xmath139 compounds . this may not be so important for tbmno@xmath139 , but the situation is definitely different in case of dymno@xmath139 . here , the ferroelectric polarisation is much larger than observed for tbmno@xmath139@xcite , and its temperature dependence is closely linked to @xmath229 ordering@xcite . by tuning the photon energy to the @xmath284 @xmath217 resonances , one takes advantage of the element - selectivity of the method that allows to study the @xmath229 magnetic ordering separately . using circularly polarised x - rays , it was shown that the @xmath229 moments in dymno@xmath139 themselves form a magnetic cycloid with the proper chirality to support ferroelectric polarisation along the @xmath367 axis according to eq.[eqnpolarisation ] , the same direction , as also promoted by the mn cycloid in the material@xcite . this is demonstrated in figure[schierle2010 - 1](c ) where the upper panels display the incommensurate ( @xmath445 ) diffraction peak , recorded at the dy @xmath231 resonance using left and right circularly polarised light . obviously , a large intensity difference is observed in the ferroelectric phase at 10 k , which is absent in the paraelectric phase at 20 k. this latter phase is characterized by a collinear sinusoidal magnetic modulation that exhibits no handedness . figure[schierle2010 - 1 ] also suggests an interesting approach to manipulate ferroelectric domains : as the photoelectric effect induces local charging of the sample surface in the highly insulating dymno@xmath139 , a radial electric field is induced that creates two regions of opposite electric polarisation along the crystallographic @xmath367 axis ( cf . figure[schierle2010 - 1](a ) ) . they are characterized by magnetic cycloids of opposite handedness . when the sample is scanned along @xmath242 , i.e. , parallel to the @xmath367 axis , the asymmetry of the magnetic diffraction signal , defined as @xmath446 , changes its sign as seen in figure[schierle2010 - 1](c ) . [ scale=0.4 , trim= 0 0 0 0 , angle= 0 ] schierle2010 - 1 the contrast obtained by the circular dichroic asymmetry in scattering could eventually be used to image multiferroic domains in the material as shown in figure[schierle2010 - 2 ] . the domain pattern was imprinted by the x - ray beam either while cooling through the paraelectric / ferroelectric phase transition ( position 1 ) or during heating close to the transition temperature in order of facilitate switching of the electric polarisation ( positions 2 and 3 ) . using the circular dichroic contrast of the incommensurate ( @xmath445 ) diffraction peak , the resulting domain pattern was imaged by scanning the sample@xcite . [ scale=0.4 , trim= 0 0 0 0 , angle= 0 ] schierle2010 - 2 while it was argued that the electric field generated by the local charging via the photoelectric effect@xcite is inducing the ferroelectric domains , later studies of ymn@xmath138o@xmath439 and ermn@xmath138o@xmath439 provided evidence that small currents flowing through the samples upon the application of the electric fields may also play a role@xcite . after essentially considering the @xmath221 and @xmath229 magnetic ordering in the multiferroic @xmath284 manganites , recent studies have eventually also elucidated the role of oxygen in multiferroics . as discussed before , an incommensurate magnetic cycloid was the driving force for ferroelectricity in @xmath284mno@xmath139 compounds . for tbmn@xmath138o@xmath139 , on the other hand , the magnetic ordering connected to the ferroelectric phase is commensurate@xcite , and almost collinear . nevertheless , strong magnetoelectric effects are observed@xcite . a possible mechanism involving charge transfer between mn@xmath278 and o was suggested to account for the large electric polarisation of the material@xcite . resonant soft x - ray diffraction at the o@xmath215 edge of tbmn@xmath138o@xmath439 provided clear evidence for a magnetic polarisation of the oxygen along with the commensurate antiferromagnetic ordering in the ferroelectric phase@xcite . figure[beale2010 - 1 ] displays the intensity change of the antiferromagnetic ( 1/2 0 1/4 ) reflection when the photon energy is scanned across the o @xmath215 edge , revealing resonant behaviour at energies that correspond to unoccupied oxygen states characterized by hybridisation with mn 3@xmath253 states . model calculations of this resonance behaviour with and without mn spin supported the interpretation that in fact a magnetic polarisation at the o site is observed that is antiferromagnetically correlated@xcite . these results support the above mentioned theoretical models@xcite that the o magnetic polarisation is key to the magnetoelectric coupling mechanism . [ scale=0.8 , trim= 0 0 0 0 , angle= 0 ] beale2010 - 1 a similar result was obtained by partzsch et al.@xcite , who compared the temperature dependence of the ( 1/2 0 1/4 ) reflection in the isostructural compound ymn@xmath138o@xmath439 both at the mn @xmath224 resonance and the o @xmath215 edge ( see figure[partzsch2011 - 1 ] ) . while the behaviour at the mn @xmath224 resonance closely resembles the magnetic order parameter as measured by neutron diffraction ( top ) , the temperature dependence at the o@xmath215 edge is rather different and follows the spontaneous polarisation in the material , i.e. , tracking the ferroelectric order parameter ( bottom ) . this provides evidence that the covalency of the mn and o atoms plays a central role for the ferroelectric polarisation in ymn@xmath138o@xmath439 . density - functional calculations indeed show that the spin order drives a redistribution of the valence band electrons resulting in a purely electronic contribution to the ferroelectric polarisation@xcite . [ scale=0.8 , trim= 0 0 0 0 , angle= 0 ] partzsch2011 - 1 a further result in this context is to be mentioned here : a study of tbmn@xmath138o@xmath439 revealed the existence of an internal field defined by @xmath447 . a quantity that is related to that expression and that was measured by resonant diffraction follows the same temperature dependence as the ferroelectric polarisation@xcite . [ scale=0.8 , trim= 0 0 0 0 , angle= 0 ] wu2010 - 1 the majority of multiferroic materials studied by rsxs are manganites . the recent discovery that cupric oxide cuo is a multiferroic even at comparably high temperatures around 230k@xcite , has triggered magnetic rsxs studies of this compound . rsxs has been used to identify ferroelectric nanoregions by diffuse scattering@xcite ( see figure[wu2010 - 1 ] ) . an anomalous memory effect for the direction of the electric polarisation in the commensurate - incommensurate magnetic transition has been detected that coincides with the ferroelectric transition . from the rsxs results , incorporated with simulations of diffuse scattering , it was proposed that a preserved spin handedness in the multiferroic nanoregions is responsible for this memory effect in the magnetically induced ferroelectric properties of cuo . copper oxide was also recently studied by scagnoli et al.@xcite . rather than the ferroelectric properties , this investigation was focused on the low - temperature phase of cuo . polarisation analysis as well as particular dependencies of diffracted intensities on incident circular polarisation revealed the existence of so - called orbital currents in the material . the increasing number of studies using rsxs is closely connected with the high intensity and the particular properties of the x - ray radiation available at modern synchrotron radiation sources . the control of the incident x - ray polarisation and its use was already discussed throughout this article . another characteristic is the high transverse coherence of the radiation that can be provided by undulator sources . together with the longitudinal coherence given by the high energy resolution , synchrotrons can provide fully coherent x - ray beams for correlation spectroscopy as well as imaging . and finally , synchrotron radiation has a well - defined time structure that can be used for time - resolved experiments . recent development have pushed the resolution into the femtosecond region , which allows to access the time scales involved in the ordering phenomena discussed throughout this article . both , high coherence and femtosecond time resolution is eventually provided by free - electron lasers , which will permit both spatially and time - resolved studies . as the scope of this account is to provide a more general overview of rsxs , it can not discuss these latter developments in depth . coherent scattering using soft x - rays , in particular , is already well - developed , and a complete description of this field is beyond the scope of this article . time - resolved experiments using rsxs , on the other hand , are scarce , even though studies related to the physics of complex order in correlated materials have been carried out with femtosecond lasers already at length . but also here , the instrumental development is on the way , and many exciting result can be expected in the near future . while a complete description of these fields can not be accomplished in this article , still , the last section is devoted to a short outlook to what can be achieved using the high sensitivity to magnetic and electronic ordering in the soft x - ray regime , starting with a classical method of x - ray diffraction that only recently was extended to the soft x - ray regime . resonant magnetic x - ray diffraction is a method essentially applied to single - crystalline material as discussed throughout this article . and in fact , magnetic x - ray diffraction from powders is very difficult because of the weak signal intensity compared to charge scattering and fluorescence . for the determination of magnetic structures , magnetic neutron powder diffraction is the method of choice , because the available momentum transfer allows to measure a large number of reflections required for structural analysis and refinement . nevertheless , resonant x - ray diffraction may be useful in cases , where the synthesis of new materials does not yield single crystals , even if the full potential of the method like azimuth dependent studies can not be used . [ scale=0.8 , trim= 0 0 0 0 , angle= 0 ] garcia - fernandez2008 - 1 even in the hard x - ray regime , resonant magnetic powder diffraction studies are scarce . a recent investigation of gdni@xmath138ge@xmath138 at the comparably weak gd @xmath224 edge used polarisation analysis and fluorescence background suppression to prepare the magnetic signal @xcite . already a decade earlier , resonant magnetic diffraction was demonstrated for the case of uo@xmath138 @xcite , exploiting the huge enhancement at the u @xmath233 resonance , a situation comparable to soft x - ray resonances discussed in this article . however , these two studies essentially demonstrated the feasibility of resonant magnetic x - ray powder diffraction , rather than providing new insights into the microscopic properties of the materials . the situation has changed with recent rsxs studies @xcite that were carried out at the swiss light source . figure [ garcia - fernandez2008 - 1 ] ( left panel ) shows a two - dimensional image of the diffraction pattern obtained from a powder sample of smbamn@xmath138o@xmath325 . the data were obtained at the mn @xmath224 resonance using an in - vacuum x - ray ccd camera and represent a section of the debye - scherrer ring corresponding to orbital order in the material . vertical integration yields a clearly discernible ( 1/4 1/4 0 ) diffraction peak . diffraction from a powder , of course , provides an average over all azimuthal directions . therefore , an important input for the study of the type of order by measuring azimuthal dependences is not available . but the energy dependence of the diffraction peak allows to further characterize the orbital order in this layered manganite material as @xmath448 type , compared to @xmath449 present in single - layer manganites @xcite . further , the temperature dependence of the diffraction peak was studied ( see figure [ garcia - fernandez2008 - 1 ] , right panel ) , which provides a measure of the orbital order parameter . below the ordering temperature of @xmath450k , a second orbital phase transition around 210k is observed that is also reflected in resistivity and magnetization data . the intensity increase of the orbital reflection is quantitatively reproduced by a structure factor calculation that provides a scenario involving a change of the orbital stacking at the second transition @xcite . [ scale=0.8 , trim= 0 0 0 0 , angle= 0 ] bodenthin2011 - 1 another example of resonant powder diffraction was given by a study of rare - earth nickelates @xmath284nio@xmath139 . the interest in these perovskite nickelates stems from metal - insulator transitions with transition temperatures that can be tuned by the ionic radius of the rare - earth ion and that are accompanied by magnetic and charge order . figure [ bodenthin2011 - 1 ] displays the intensity of the ( 1/2 0 1/2 ) magnetic diffraction peak across the ni @xmath216 resonance for a series of @xmath284nio@xmath139 compounds , with the ionic radius of the @xmath284 decreasing from pr to y. again , data were obtained with an x - ray ccd camera and subsequent integration across the debye - scherrer ring . the energy dependencies shown in fig . [ bodenthin2011 - 1 ] reveal a striking similarity , with no energy shifts and little variation of the spectral shape . this shows that the electronic structure of the materials essentially stays the same across the series . more specifically , model calculations of the energy dependencies revealed that the change in the charge disproportionation between inequivalent ni sites connected with charge order in the material can not be larger than 0.05 electron charges despite the substantial change in the metal - insulator transition temperature @xcite . while the application of resonant soft x - ray diffraction to powder materials is still rare , these examples show the usefulness of the method beyond a mere demonstration of feasibility . conventional x - ray diffraction investigates predominantly periodic structures . the diffraction patterns are transformed into real - space atomic maps . the determination of non - periodic nanoscale structures by x - ray scattering is much more difficult since the inversion of the intensity profile suffers from the intrinsic loss of phase information . in standard experiments the diffuse scattering intensity around a bragg peak yields some information on the disorder of a system , as also discussed in section[sec : casestudies ] in connection with magnetic roughness . by limiting the diffraction volume to the coherence volume , extra information about the structure is available . in this case the measured intensity profile has strong fringes and which represent a specific microscopic configuration of the sample , which are called `` speckles '' . speckle patterns can be recorded as a function of time , in this way probing the time - dependence of correlations in a material . in holography , on the other hand , a reference beam is used to interfere with the light scattered from the sample . this allows retrieval of the scattering phase and hence reconstruction of the scattering object by a simple fourier transform . scattering of coherent light is a technique that is well established in the optical regime , as optical lasers providing highly coherent light are available for a long time . recent years have seen a transfer of the method to the x - ray range@xcite , which has become possible with the advent of undulator radiation sources at third - generation synchrotrons that provide x - ray beams with sufficient degree of coherence . a high degree of longitudinal coherence of x - rays is anyway provided by monochromatization , but only undulators with their high brilliance provide sufficient transverse coherence as well , resulting in a highly coherent x - ray beam . the full power of coherent x - ray scattering will be exploited after the installation of x - ray free electron lasers . in the meantime , both , correlation spectroscopy as well as holographic imaging were successfully carried out using soft x - rays , exploiting the magnetic sensitivity at resonance . a first experiment that transferred the method of coherent scattering into the soft x - ray regime was carried out by price et al . @xcite , who studied fluctuations in smectic liquid crystals . these experiments , however , did not exploit electronic resonances in the materials . the power of coherent scattering in rsxs for the imaging of magnetic domains has been demonstrated in a landmark experiment by eisebitt et al.@xcite in which the meandering maze - like magnetic domains in co / pt multilayers with alternating up and down magnetization perpendicular to the layer surface were investigated . the magnetic contrast has been achieved by exploiting the magnetic dichroism in resonance at the co @xmath216 edges using circularly polarised photons . a schematic picture of the experimental set - up is shown in figure[fig : hellwig2006 ] . the coherent part of the undulator radiation is selected by an aperture of micrometer diameter and hits a specially designed sample structure : co / pt multilayers are deposited on a silicon nitride membrane that is transparent at soft x - ray energies . the imaging pinhole is of micrometer diameter . the sample also comprises the nanometer - sized reference pinhole that defines a reference beam . the x - rays scattered by the magnetic structure interfere with the reference beam and form a holographic interference pattern , i.e. , the speckle pattern , which encodes the image of the magnetic nanostructure . the final image derived by a fourier transform and shown in the inset in the upper right corner of figure[fig : hellwig2006 ] demonstrates the detection of the labyrinth stripe domain pattern which is characteristic for magnetic systems with perpendicular anisotropy . magnetic field dependent studies revealed a transformation of the labyrinth domain into isolated domains and , finally near saturation , into isolated bubble domains with increasing field . following this first demonstration of lens - less imaging with soft x - rays , further experiments improved the efficiency of the method @xcite and demonstrated that the resolution of the method can be substantially enhanced beyond the fabrication limit of the reference structure @xcite . it was also shown that lens - less imaging can even be accomplished without reference beam by oversampling methods @xcite . these exploring experiments demonstrated that with photon sources with shorter pulses and higher brightness , such as free electron lasers , there will be a pathway to study dynamics of magnetic structures in real time to foster progress in rapid magnetic reading and writing processes . the availability of free electron lasers in the soft x - ray region with a spectacular increase of some nine order of magnitude in peak brilliance in combination with femtosecond time resolution and high coherence@xcite allow now ground breaking , completely new experiments on the dynamics of magnetic nanoscaled structures . [ width=7cm , angle=90 ] hellwig2006 as one of the first examples , we present the investigation of meandering maze - like domains in co / pt multilayers with alternating up and down magnetization perpendicular to the layer surface ( see above ) . speckle patterns recorded at the free electron laser flash with a time - resolution of 30 fs are depicted in figure[gutt2010 - 1]@xcite . magnetic contrast for the domains was obtained by using photons at the co@xmath451 edge and a geometry which maximizes the second term in eq.[eqn : hannon ] . information on the mean value of the size of the domains could be obtained from a single 30 fs free electron laser pulse . in this work it was also demonstrated that the speckle signal for subsequent pulses provided information on the magnetization dynamics as a function of time . these pioneering results point to the possibility of ultrafast magnetization studies down to the 100 fs timescale . [ scale=0.73 , trim= 0 0 0 0 , angle= 0 ] gutt2010 - 1 all these experiments are carried out in transmission geometry , with sample and reference beam scattered from the same substrate , which inherently provides a rigid coupling between sample and reference . in these cases , samples require a special design with particularly thin films in the nanometer range that remain transparent at resonance energies in the soft x - ray regime . in reflection geometry , the scope of coherent scattering can be substantially broadened towards application to surfaces of bulk materials as discussed throughout this paper . while this is more demanding , a successful demonstration at 500 ev photon energy using a test structure was achieved @xcite . [ scale=1 , trim= 0 0 0 0 , angle= 0 ] konings2011 - 1 another example of coherent scattering carried out in reflection geometry is given by correlation spectroscopy across the phase transition of helical magnetic ordering in a ho thin film@xcite as shown in figure[konings2011 - 1 ] . the helical magnetic structure of ho leads to a pronounced magnetic diffraction peak that can be probed by x - rays at the ho @xmath231 resonance@xcite ( cf . [ sec : long - periodafm ] ) . at 52k , well below the ordering temperature of the film ( @xmath452k ) , the speckle pattern connected with this diffraction peak is static , as shown on the top panel . it shows that magnetic fluctuations at this temperature are practically absent . upon heating to 70k close to the ordering temperature , the speckle patterns change with time , indicating the development of magnetic fluctuations at the phase transition . a close inspection of the speckle patterns as well as the time - averaged intensity distribution on the bottom of figure[konings2011 - 1 ] shows that some parts still remain fixed . hence , fluctuations at the phase transition do not affect the whole sample , revealing a non - ergodic behaviour of the system . an analogous experiment on the orbital order of the half - doped manganite pr@xmath414ca@xmath414mno@xmath139 revealed similar behaviour near the phase transition , with both pinned and slowly fluctuating orbital domains @xcite , the examples of coherent scattering described here open a broad field of applications to the materials and their various electronic ordering phenomena discussed throughout this paper . the development of lasers with pulse duration in the femtosecond range opened a new field to study ultrafast processes . magnetic and other complex ordering phenomena can be studied using photons in the optical range , as changes of the symmetry in a system upon ordering are often reflected in the optical properties of the material , particularly involving second - order processes that are excited using intense laser light . thus , optical wavelengths provide contrast to study the ultrafast dynamics connected with the decay of ferromagnetic or more complex ordering phenomena in transition metal oxides@xcite . however , as soft x - ray resonances provide highest sensitivity to spin , orbital , and charge order in combination with element selectivity , the development of free electron lasers in this energy range represents a large step in this field . in addition to free electron lasers , also slicing facilities at third - generation synchrotron sources provide the ultrashort photon pulses required for time - resolved x - ray diffraction studies with femtosecond time resolution . a setup installed at the swiss light source provides ultrashort x - ray pulses for hard x - ray diffraction . using structural superlattice reflections associated with charge and orbital order , it was possible to study the dynamics in la@xmath453ca@xmath454mno@xmath139@xcite . [ scale= 0.57 , trim= 0 0 0 0 , angle= 0 ] holldack2010 - 1 a slicing beamline covering the soft x - ray region and providing linear as well as circularly polarised light is installed at bessy ii at the helmholtz - zentrum berlin@xcite . here , studies of demagnetization dynamics could be extended to the soft x - ray region , with the particular advantage of element selectivity and the possibility to access spin and orbital moment separately . first experiments were concerned with x - ray absorption only@xcite , but recently , also rsxs experiments with fs time resolution were carried out successfully . for that purpose , eute films were used that were already discussed in section[magneticdepth ] , as they exhibit magnetic bragg peaks of unprecedented quality at the eu - m@xmath439 resonance@xcite . time resolved experiments were carried out using a pump - probe scheme as illustrated in figure[holldack2010 - 1 ] . the system was excited by a laser using a wavelength of 400 nm that is sufficiently small for an excitation across the band gap of the material and corresponds approximately to a @xmath455 excitation . the photon energy of the probe pulse was tuned to the eu @xmath231 resonance and the scattering geometry was chosen to correspond to the ( @xmath456 ) afm bragg peak that occurs almost at a scattering angle of @xmath457 . [ scale=0.73 , trim= 0 0 0 0 , angle= 0 ] holldack2010 - 2 figure[holldack2010 - 2 ] displays the intensity of the afm diffraction peak as a function of the time delay between the optical pump and the x - ray probe pulse . interestingly , a very fast decay of the afm order is observed , which is faster than ( @xmath458 ) fs . this is quite remarkable for a semiconductor , as the fast demagnetization mechanisms involving conduction electrons known from metals are not available . figure[holldack2010 - 2 ] also displays a rocking scan through the afm diffraction peak recorded with a 100 fs probe pulse without laser pumping . this demonstrates that in this type of experiments the capabilities of diffraction beyond recording the mere peak intensity can be exploited to study the temporal evolution of spatial correlations in the ordered phase . while experimental facilities at the free - electron laser in hamburg ( flash ) at desy ( deutsches elektronen - synchrotron ) and the linac coherent light source ( lcls ) at the slac national accelerator laboratory have paved the way to ultrafast studies , time - resolved scattering experiments for the study of correlated materials using soft x - rays are presently still scarce . an experiment recently carried out at flash is concerned with the decay of orbital order in the prototypical correlated material magnetite fe@xmath139o@xmath334@xcite(cf . section[magnetite ] ) . in figure[pontius2011 - 1 ] the time - dependent intensity of the ( 0 0 1/2 ) reflection , measured with photons at the o@xmath215 edge , is shown as a function of time after a femtosecond laser pump pulse , for several pump fluences . since the intensity of the peak in resonance is directly related to the unoccupied density of states , this intensity provides direct information about the electronic order and hence of the melting of charge / orbital order induced with the infrared pulse . an ultrafast melting of the charge / orbital order has been found in the rapid decrease of the intensity of the superstructure peak . the persistent intensity of the peak after 200 ps even for pump fluences that correspond to sample heating above the verwey transition ( inset of figure[pontius2011 - 1 ] ) was interpreted in terms of a transient phase characterized by the existence of partial charge / orbital order which has not been observed in equilibrium . in this way the time - dependent rsxs experiment have demonstrated that important information on the dynamics of phase transitions in correlated systems can be obtained . [ scale=0.4 , trim= 0 0 0 0 , angle= 0 ] pontius2011 - 1 a third time - resolved experiment , albeit only with picosecond resolution , was reported recently from diamond light source@xcite . time resolution was achieved here by gating the synchrotron radiation pulses synchronized with the laser pulses . the material studied was la@xmath414sr@xmath415mno@xmath334 , which is characterized by spin and orbital order ( cf . section[sec : manganites ] ) . these two types of order are represented by the ( @xmath459 ) afm bragg peak , and the ( @xmath460 ) orbital diffraction peak , respectively . figure[ehrke-1 ] displays the evolution upon laser excitation of the two diffraction peaks , revealing that spin order can be quickly destroyed , while the orbital order persists . the results were interpreted in terms of an intermediate transient phase in which the spin order is completely removed by the photoexcitation while the orbital order is only weakly perturbed . in this way the time - dependent rsxs experiments have demonstrated that it is possible to separate the spin dynamics determined by the electronic structure from jahn - teller contributions which are more related to the lattice dynamics . [ scale=0.2 , trim= 0 0 0 0 , angle= 0 ] ehrke2011 - 1 during the last two decades rsxs has established itself as a powerful tool in modern solid state physics to investigate magnetic , orbital and charge ordering phenomena associated with electronic degrees of freedom . since the technique is coupled to specific core excitations it is element specific . the techniques combines diffraction methods with x - ray absorption spectroscopy and therefore delivers structural nanoscale information on the modulation of the valence band electron density , on the bond orientation , i.e. , orbital ordering , and on the spin density . in particular rsxs has provided important structural and spectroscopic information on 3@xmath253 transition metal compounds and 4@xmath100 systems which comprise interesting solids such as high-@xmath305 superconductors , charge density compounds , giant and colossal magneto - resistance systems which may be important for spintronics . since rsxs has a finite probing depth , the method is not only suited for the investigation of bulk properties : rsxs provides also important structural information on small crystals , ultra thin films , on multilayer systems and interfaces , and on all kind of nanostructures . the future of the technique will strongly depend on the instrumental development of the photon sources and the diffractometers . variable polarisation is already realized at various beamlines at synchrotron radiation facilities . fast switching of the polarisation will help to detect small polarisation effects . in several rsxs diffractometers the application of suitable polarisation analyzers will allow to extend investigations of magnetic and orbital ordering in a large class of interesting materials . arrays of channel plate detectors or charge coupled devices ( ccd ) will enable to perform faster diffraction experiments . sample environment will play an important role in future rsxs experiments : the possibility to apply high magnetic fields and low temperatures to the sample will offer interesting studies on new ground states in correlated systems . furthermore , the development of intense coherent soft x - ray sources by the development of fourth generation synchrotron radiation facilities ( energy recovery linacs and free electron lasers ) will allow to reveal , element specific and with nanometer spacial resolution , complicated non - periodic lattice , magnetic , and charge structures . here important issues will be phenomena related to fluctuation near phase transitions , chemical reactions , defects and precipitations in materials etc . finally , with the advent of free - electron laser photon sources in the soft x - ray region , interesting time - dependent pump - probe studies on the structural evolution of photo - excited solids will be feasible on a large scale . possibly movies of chemical reactions , studies on matter under extreme conditions , watching magnetic spin flips in real time , and unraveling the functional dynamics of biological materials will be feasible . the future of rsxs will be not only related to bulk studies of the charge , orbital , and spin order in transition metal compounds . rather the focus will be also on similar studies in thin films , all kind of artificial nanostructures and in particular on the structure and the electronic properties of interfaces . therefore those rsxs experiments will strongly depend on the development of preparation stations for artificial nanostructures , e.g. by laser deposition or molecular beam epitaxy . one focus of future rsxs experiments will still be related to complex bulk properties of correlated materials as a function of composition , temperature , and magnetic field . there , mainly charge , spin , and orbital ordering as well as jahn - teller distortions will be studied in correlated transition metal and rare earth compounds . new bulk experiments could be considered in the fields of vortex lattices of superconductors since the charge inside and outside of vortexes is different . since similar to the o @xmath215 edges in tm oxides , the c @xmath215 edges in soft matter are strongly dependent on the chemical bonding of c atoms , rsxs will develop to a great tool for investigations of nanoscale structures and spacial distributions in polymer blends , multiblock copolymers , polymer solutions , lipid membranes , colloids , micelles , and polymeric biological composite particles . furthermore , using coherent light sources , disordered systems will be studied applying speckle methods . finally rsxs will be extended to the time domain via setting up pump - probe experiments to study the dynamics of phase transitions . the other focus of future rsxs will be certainly related in the analysis of structural and electronic properties of artificial structures on the nanometer scale which have great promise for novel functionality . in particular , rsxs will help to resolve the problem to characterize the electronic structure of oxide hetero - epitaxial devices in which , as already discussed above , at the interfaces between two band insulators metallicity and even superconductivity has been detected . at present it is not decided whether this observation is due to the polar nature of the layers or due to o vacancies . future rsxs experiments will certainly help to resolve open questions in this field . thus one can be confident that rsxs will have a great impact on a vast variety of scientific fields ranging from solid state physics across materials science and chemistry to biology and medicine . the authors thank j. hill , u. staub , and s. wilkins for helpful suggestions . 100 d. g. hawthorn , k. m. shen , j. geck , d. c. peets , h. wadati , j. okamoto , s .- w . huang , d. j. huang , h .- j . lin , j. d. denlinger , ruixing liang , d. a. bonn , w. n. hardy , and g. a. sawatzky . resonant elastic soft x - ray scattering in oxygen - ordered @xmath464 . , 84:075125 , 2011 . m. w. haverkort , n. hollmann , i. p. krug , and a. tanaka . symmetry analysis of magneto - 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since the mid-1990s we built visual interactive maps of bibliographic and database information at strasbourg astronomical observatory , and some of these , with references , are available at murtagh [ 2006d ] . the automated annotation of such maps is not easy . at the time of writing zdnet and the bbc ( british broadcasting corporation ) use interactive annotated maps to support information navigation . in zdnet s case , some prominent terms are graphically presented and can be used to carry out a local search ; and in the bbc case , terms relating to downloadable radio programs are displayed in moving sizes and locations . in the work described in this article , we adopt a different approach : we select the terms of interest in a manual or semi - automated way . this not only represents expert user judgement but also allows for inclusion of rare or very frequent terms . in one of our three case studies , we use an automated way to select such terms . for selected terms , we use their inter - relationships to build a hierarchy and use this as a central device for summarizing information and supporting navigation . `` ontologies are often equated with taxonomic hierarchies of classes ... but ontologies need not be limited to such a form '' [ gruber 2001 ] . gruber is cited in gmez - prez et al . [ 2004 ] as characterizing an ontology as `` an explicit specification of a conceptualization '' . in wache et al . [ 2001 ] , ontologies are motivated by semantic heterogeneity of distributed data stores . this is also termed data heterogeneity and is counterposed to structural or schematic heterogeneity . ontologies are motivated by wache et al . [ 2001 ] `` for the explication of implicit and hidden knowledge '' , as `` a possible approach to overcome the problem of semantic heterogeneity '' . so , ontologies may help with integration of diverse , but related , data ; or they may help with clarifying or disambiguating distinctions in the heterogeneous data . ontologies are likely to be of immediate help in supporting querying . for example , the query model may be based on the ontology ( or ontologies ) used . there is extensive activity on standards and software , relating more to the above - mentioned schematic rather than semantic heterogeneity , and a useful survey of this area is denny [ 2004 ] . denny takes an ontology in a broad - ranging view as a knowledge - representation scheme . a short review of some recent approaches in this area follows . ahmad and gillam [ 2005 ] develop a semi - automated approach using text with no markup . multiword expressions are determined , and frequency of occurrence information is used to point to term or phrase importance . a stop list is used to avoid irrelevant words . part of speech analysis is not used . a semantic net is formed to allow development of the ontology elements . abou assali and zanghi [ 2006 ] use syntactic part of speech tagging to determine the nouns . these authors retain sufficiently frequent nouns . they apply the notion of weak subsumption : if for the most part a word is in a text that another is in , and not vice versa , then this leads to a hierarchical relationship . chuang and chien [ 2005 ] assert that multiway trees are appropriate for concept hierarchies , whereas binary trees are built using hierarchical clustering algorithms . hence they modify the latter to provide more appropriate output . ( a formal approach for mapping a binary hierarchical classification tree onto a multiway hierarchy is described in murtagh [ 2006b ] . ) a hierarchical clustering has often been used to represent an ontology . note that this is usually not a concept hierarchy . a concept hierarchy is based on a subsumption relationship between terms , whereas a hierarchical clustering is an embedded set of clusters of the term set . later in this article ( section [ secorientree ] ) , we show a way to derive a concept hierarchy , involving subsumption of terms , from a hierarchic clustering . a hierarchic clustering is typically a binary , rooted , terminal labeled , ranked tree , and a concept hierarchy is typically a multiway , rooted , terminal and non - terminal labeled , ranked tree . by starting with the former ( binary ) tree representation , we have an extensive theoretical and formal arsenal at our disposal , to represent the main lines of what we need to do , and to help to avoid ad hoc , user parameter - based , `` engineering '' approaches . as seen later in this work , we start by laying the foundations of our perspective by basing this on binary trees , and later proceed to the multiway tree . an alternative approach can be found in ganesan et al . [ 2003 ] , where similarities or distances on trees are redefined and re - axiomatized for the case of multiway trees . an alternative representation for an ontology is a lattice , and formal concept analysis ( fca ) is a methodology for the analysis of such lattices . if we have a set of documents or texts , @xmath0 , characterized by an index term set @xmath1 , then as janowitz [ 2005 ] shows , hierarchical clustering and fca are loosely related . hierarchical clustering is based on pairwise distances or dissimilarities , @xmath2 ( @xmath3 is the set of non - negative reals ) . fca is based on partially ordered sets ( posets ) such that there is a dissimilarity @xmath4 ( @xmath5 is the power set of the index terms , @xmath1 ) . other approaches ( rule - based ; machine learning approaches , etc . ; layered , engineering , approaches with maintenance management see maedche [ 2006 ] ) are also available . one difficulty with such `` engineering '' approaches is that there is an ad hoc understanding of the problem area , and often there is dependence on somewhat arbitrary threshold and selection criteria that do not generalize well . our approach formalizes the problem area the information space in terms of its local or global topology . where we do have selection criteria , such user interaction is at the application goal level . visualization is often an important way to elucidate semantic heterogeneity for the user . visual user interfaces for ontological elucidation are discussed in murtagh et al . [ 2003 ] , with examples that include interactive , responsive information maps based on the kohonen self - organizing feature map ; and semantic network graphs . a study is presented in murtagh et al . [ 2003 ] of client - side visualization of concept hierarchies relating to an economics information space . the use of `` semantic road maps '' to support information retrieval goes back to doyle [ 1961 ] . motivation , following murtagh et al . [ 2003 ] , includes the following : ( i ) visualization of the semantic structure and content of the data store allows the user to have some idea before submitting a query as to what type of outcome is possible . hence visualization is used to summarize the contents of the database or data collection ( i.e. , information space ) . ( ii ) the user s information requirements are often fuzzily and ambiguously defined at the outset of the information search . hence visualization is used to help the user in his / her information navigation , by signaling the relationships between concepts . ( iii ) ontology visualization therefore helps the user before the user interacts with the information space , and during this interaction . it is a natural enough progression that the visualization becomes the user interface . this article is organised as follows . to begin with , in section [ sect2 ] , we table the issue of whether or not there is inherent hierarchical structure in a text , or a collection of texts . in section [ sect3 ] we show how we can rigorously determine the extent of inherent hierarchical structure in a text . this quantifying of inherent hierarchical structure is then used in subsections [ sect41 ] , [ sect42 ] , [ sect52 ] and [ sect7 ] . a text provides both global and linear semantics , and how we can process these two different perspectives on a given text is discussed in section [ sect4 ] . a central aspect of our approach is a new distance or metric , which we have recently introduced and exemplified on another data analysis problem . this new distance is described in subsection [ sect5 ] . in section [ sect6 ] we apply what we have described in earlier sections to the selection of salient and characteristic pairs and triplets of terms , and also the selection of pertinent terms . our motivation is not just the traditional view of phrase counting ( even though we incorporate this view ) but rather the characterization of text content using its internal ( local hierarchical ) structure . a natural approach to defining a concept hierarchy lies in use of a hierarchical clustering algorithm . however , the latter forms an embedded sequence of clusters , so that a hierarchy of concepts must somehow be derived from it . in section [ secorientree ] we first of all show that `` converting '' any hierarchical clustering into a hierarchy of concepts is relatively straightforward . however we do have to face the problem of a unique , and beyond that best , solution . we show how we can admirably address this need for a unique solution . our innovative approach is based on the foundations laid in sections [ sect2 ] and [ sect4 ] of this article . we analyze three different data sets in this work : firstly a set of documents , with some degree of heterogeneity , to illustrate our key goal ; secondly a homogeneous text , partitioned into successive textual segments ; and thirdly a small homogeneous text , partitioned at the sentence level , proxied by lines of text . we select terms , indeed nouns , in a partially automated way , since this crucial aspect of ontology design may benefit from being user - driven , and may have scalability advantages . in later sections we address the issue of finding and presenting structure in text . we link such structure with the textual content . consequently a key , initial question is to know whether or not there is structure present , and to what extent . a first problem to be addressed is whether or not the document has any hierarchical structure to begin with . as input , we have possibly a fully tagged document ( based , e.g. , on part - of - speech tagging , schmid [ 1994 ] ) . however in this work , we start with free text , because it is the most generally available and applicable framework . additional information provided by part - of - speech tagging can be of use to us , as we will show later . next we consider the issue of whether or not a document has sufficient inherent hierarchical structure to warrant further investigation . we could approach this problem by fitting a hierarchy , and there are many algorithms for doing so ( such as any hierarchical clustering algorithm ; de soete [ 1986 ] describes a least squares optimal fitting approach ) . however departure from inherent hierarchical structure is not easily pinpointed . after all , we have an output induced structure , and we are told , let s say , that the fit is 80% ( defined as @xmath6 where @xmath7 is input dissimilarity , @xmath8 is tree or ultrametric distance read off the output , and the sums are over all pairs ) , which is not very revealing nor useful . an alternative `` bottom - up '' approach is pursued here , which allows easy assessment of inherent structure , and also pinpointing where this occurs or does not occur . a formal definition of hierarchical structure is provided by ultrametric topology ( in turn , related closely to p - adic number theory ) . the triangular inequality holds for a metric space : @xmath9 for any triplet of points @xmath10 . in addition the properties of symmetry and positive definiteness are respected . the `` strong triangular inequality '' or ultrametric inequality is : @xmath11 for any triplet @xmath10 . an ultrametric space implies respect for a range of stringent properties . for example , the triangle formed by any triplet is necessarily isosceles , with the two large sides equal ; or is equilateral . in an ultrametric space ( i.e. , a space endowed with an ultrametric , or an ultrametric topology ) , one `` lives '' , so to speak , in a tree . all `` moves '' between one location and another are as if one descended the tree to a common tree node , and then reclimbed to the target point . topologically , an ultrametric goes a lot further : all points in a circle or sphere are centers , for example ; or the radius of a sphere is identical to its diameter . the triangle property respected by any triplet of points in an ultrametric space affords a useful way to quantify extent of hierarchical structure . we will describe our `` extent of hierarchical structure '' , on a scale of 0 ( no respect for ultrametricity ) to 1 ( everywhere , respect for the ultrametric or tree distance ) algorithmically . we examine triplets of points ( exhaustively if possible , or otherwise through sampling ) , and determine the three angles formed by the associated triangle . we select the smallest angle formed by the triplet points . then we check if the other two remaining angles are approximately equal . if they are equal then our triangle is isosceles with small base , or equilateral ( when all triangles are equal ) . the approximation to equality is given by 2 degrees ( 0.0349 radians ) . our motivation for the approximate ( `` fuzzy '' ) equality is that it makes our approach robust and independent of measurement precision . this approach works very well in practice [ murtagh 2004 ; 2006a ] . we may note our one assumption for our data when we look at triangles in this way : scalar products define angles so that by assuming our data are in a hilbert space ( a complete normed vector space with a scalar product ) we may proceed with this analysis . this hilbert space assumption is very straightforward in practice . when finite ( as is always the case for us , in practice ) , we are using a euclidean space . often in practice , for arbitrary euclidean data , there is very little ultrametricity as quantified by the proportion of triangles satisfying the ultrametric requirement . but recoding the data can be of great help in dramatically increasing the proportion of such ultrametricity - respecting triangles [ murtagh 2004 ; 2005a ] . if we recode our data such that each pairwise distance or dissimilarity is mapped onto one element of the set @xmath12 , then as seen in subsection [ sect5 ] below the triangular inequality becomes particularly easy to assess for existence of , or non - existence of , a locally ultrametric relationship . in our use of free text , we have already noted how a mapping into a euclidean space gives us the capability to define distance in a simple and versatile way . in correspondence analysis [ murtagh 2005b ] , the texts we are using provide the rows , and the set of terms used comprise the column set . in the output , euclidean factor coordinate space , each text is located as a weighted average of the set of terms ; and each term is located as a weighted average of the set of texts . ( this simultaneous display is sometimes termed a biplot . ) so texts and terms are both mapped into the same , output coordinate space . this can be of use in understanding a text through its closest terms , or vice versa . a commonly used methodology for studying a set of texts , or a set of parts of a text ( which is what we will describe below ) , is to characterize each text with numbers of terms appearing in the text , for a set of terms . the @xmath13 distance is an appropriate weighted euclidean distance for use with such data [ benzcri 1979 ; murtagh 2005b ] . consider texts @xmath14 and @xmath15 crossed by words @xmath16 . let @xmath17 be the number of occurrences of word @xmath16 in text @xmath14 . then , omitting a constant , the @xmath13 distance between texts @xmath14 and @xmath15 is given by @xmath18 . the weighting term is @xmath19 . the weighted euclidean distance is between the _ profile _ of text @xmath14 , viz . @xmath20 for all @xmath16 , and the analogous _ profile _ of text @xmath15 . ( our discussion is to within a constant because we actually work on _ frequencies _ defined from the numbers of occurrences . ) correspondence analysis allows us to project the space of documents ( we could equally well explore the terms in the _ same _ projected space ) into a euclidean space . it maps the all - pairs @xmath13 distance into the corresponding euclidean distance . for a term , we use the ( full rank ) projections on factors resulting from correspondence analysis . as noted , this factor space is endowed with the ( unweighted ) euclidean distance . we will also take into consideration the strongest `` given '' in regard to any classical text : its linearity ( or total ) order . a text is read from start to finish , and consequently is linearly ordered . a text endowed with this linear order is analogous to a time series . if we use the correspondence analysis ( full dimensionality ) factor coordinates for each term , then the textual time series we are dealing with is seen to be a multivariate time series . just as the way we code our input data plays a crucial role in the resulting analysis , so also the recoding of pairwise distances can influence the analysis greatly . in murtagh [ 2005a ] we introduced a new distance , which we will term the `` change versus no change '' , cvnc , metric , and showed its benefits on a wide range of ( financial , biomedical , meteorological , telecoms , chaotic , and random ) time series . motivation for using this new metric is that it greatly increases the ultrametricity of the data . the cvnc metric is defined in the following way . take the euclidean distance squared , @xmath21 for all @xmath22 , where we have terms @xmath23 in the factor space with coordinates @xmath24 . it will be noted below in this section how this assumption of euclidean distance squared has worked well but is not in itself important : in principle any dissimilarity can be used . we enforce sparseness on our given squared distance values , @xmath25 . we do this by approximating each value @xmath26 , in the range @xmath27 , by an integer in @xmath28 . the value of @xmath29 must be specified . in our work we set @xmath30 . the recoding of distance squared is with reference to the mean distance squared : values less than or equal to this will be mapped to 1 ; and values greater than this threshold will be mapped to 2 . thus far , the recoded value , @xmath31 is not necessarily a distance . with the extra requirement that @xmath32 whenever @xmath33 it can be shown that @xmath31 is a metric [ murtagh 2005a ] : * theorem : * _ the recoded pairwise measure , @xmath34 , defined as described above from any dissimilarity , is a distance , satisfying the properties of : symmetry , positive definiteness , and triangular inequality . _ to summarize , in our coding , a small pairwise dissimilarity is mapped onto a value of 1 ; and a large pairwise dissimilarity is mapped onto a value of 2 . identical values are unchanged : they are mapped onto 0 . this coding can be considered as encoding pairwise relationships as `` change '' , i.e. 2 , versus `` no change '' , i.e. 1 , relationships . then , based on these new distances , we use the ultrametric triangle properties to assess conformity to ultrametricity . the proportion of ultrametric triangles allows us to fingerprint our data . for any given triplet ( of terms , with pairwise cvnc distances ) , if the triplet is to be compatible with the ultrametric inequality , each set of three recoded distances is necessarily of one of the following patterns : trivial : : : at least one ( recoded ) distance is 0 , in which case we do not consider it . equilateral : : : recoded distances in the triplet are 1,1,1 or 2,2,2 , defining an equilateral triangle . ultrametric isosceles : : : recoded distances in the triplet are 1,2,2 in any order , defining an isosceles triangle with small base . non - ultrametric : : : recoded distances in the triplet are 1,1,2 in any order . the non - ultrametric case here is seen to be an isosceles triangle with large base . we could `` intervene '' and change one of the values to make it ultrametric . if we change the 2-value to a 1-value , this will produce an equilateral triangle , which is ultrametric . in this case , we are approximating our three values optimally from below , and the resulting ultrametric is termed the subdominant , or maximally inferior , ultrametric . the associated stepwise algorithm for constructing a hierarchy is known as the single link hierarchical clustering algorithm . on the other hand , we could change one of the 1-values to a 2 . this is not unique , since we could change either of the 1-values . the resulting hierarchy is termed the minimally superior ultrametric . the associated stepwise algorithm for constructing a hierarchy is known as the complete link hierarchical clustering algorithm . all of this is very clear from the case considered here . the recoding into the cvnc metric is a particular example of symbolic coding . see murtagh [ 2006c ] . in the next section , we will show the usefulness of this cvnc metric for quantifying inherent hierarchical structure . in this section we first describe the data set used . next , based on the foundation of the previous section , we quantify inherent hierarchical structure in our data . this justifies going further , to harness and exploit this structure . we use 14 texts taken from wikipedia ( mid-2006 ) , and coverted to straight text from html . table [ tabprop ] shows the numbers of words in each . .properties of texts used . [ cols= " < , < , > , > , > " , ] to the extent that our data satisfies , globally and throughout , the ultrametric inequality , we can adopt any of the widely used hierarchical clustering algorithms ( single , complete , average linkage ; minimum variance , median , centroid ) to induce an identical , unique hierarchy . but when we find our data to be , say , 86% ultrametric , as is not untypically the case in practice , then we must consider carefully what our aim is . if we wished to look at each and every isosceles triangle , then in the case of the artificial intelligence text this means , out of a total of 2,027,795 triplets ( i.e. , @xmath35 ) we must consider 1,007,597 . what we will do instead is return to taking our text as a time series . we have 231 unique nouns in the artificial intelligence text . in the text , these nouns are used , in total , on 405 occasions . so our text is a time series of 405 values . for successive nouns in this textual time series , the cvnc metric has an evident meaning : _ we are noting semantic change versus lack of change as we read through the text_. we examine successive triplets in the textual time series . for the artificial intelligence text , we find 45% of the triplets to be equilateral ; 37% of the triplets are isosceles ; and 18% of the triplets are non - ultrametric . the isosceles triplets point to a dominance or subsumption relationship that will be of use for us in a concept hierarchy . say we have a triplet @xmath36 . say , further , that the cvnc distance between @xmath37 and @xmath38 is 1 , so therefore there is no change in progressing from use of term @xmath37 to use of term @xmath38 . however both @xmath37 and @xmath38 are at cvnc distance 2 to term @xmath39 , and this betokens a semantic change . so the relationship is simply represented as @xmath40 . the term @xmath39 dominates or subsumes @xmath37 and @xmath38 . the following results hold . firstly , say that a successive triplet of values , in any order , is found as @xmath36 , and later in the text , again , this triplet is found in any order . then the relationship between the three recoded distances in both cases will be identical . _ for a given triplet , in any order within the triplet , the relationship is unique . _ secondly , consider any other term , @xmath41 , such that some or all of the terms @xmath36 are found to have a relationship with @xmath41 . as an example , we meet with @xmath42 at one point in the text , and later we meet with @xmath43 . then there is no influence by @xmath41 on the relationship ensuing from the @xmath43 triplet , vis - - vis the relationship ensuing from the earlier @xmath42 triplet . _ we have locality of the relationship in any given triplet , from successive terms . the relationship is strictly local to the given triplet . _ among the isosceles triangles in the artificial intelligence text , we find the following relationships . .... ( computer science ) branch ( home computer ) world ( analysis systems ) formalism ( analysis systems ) reasoning ( expert system ) conclusion ( expert system ) amounts ( example networks ) reasoning ( networks learning ) reasoning ( pattern recognition ) capabilities ( control systems ) computation ( consciousness systems ) logic ( medicine computer ) commentators ( computer technology ) commentators ( application feature ) os ( application feature ) languages ( libraries systems ) specialist ( libraries systems ) programmers ( software engineering ) development ( software engineering ) practices ( programs example ) logic ( example type ) logic ( projects publications ) life ( publications bayesian ) life ( bayesian networks ) life ( cybernetics systems ) agents ( systems control ) agents ( wiki web ) website ( wiki web ) category ( algorithm implementations ) projects ( algorithm implementations ) demonstrations ( implementations research ) demonstrations ( research group ) demonstrations .... however there are other isosceles triplets that are less self - evident . for this reason therefore we take all texts . for the 14 texts , we have 6439 nouns , and 1470 unique nouns . with our cvnc metric on all pairs of nouns , the complete link hierarchical clustering method gives 21 clusters in all . while one application of the foregoing is to deriving common pairs and triplets of terms , in practice it would be better to combine all relationships into a `` bigger picture '' . we will address this below in section [ secorientree ] . presenting a result with around 1500 terms does not lend itself to convenient display . we ask therefore what the most useful perhaps the most discriminating terms are . in correspondence analysis both texts and their characterizing terms are projected into the same factor space . see figure [ caout ] . so , from the factor coordinates , we can easily find the closest term(s ) to a given text . we do this for each of the 14 texts , and find the closest terms , respectively , as follows : .... bayesian automaton brain captcha psychologists image maps logic topologies databases agents representation game games .... a hierarchical clustering of these is shown in figure [ den15 ] . the ward minimum variance method is used , as being appropriate for structuring data well ( see murtagh [ 1984b ] ) and also having an agglomerative criterion that is appropriate for the prior euclidean embedding ( viz . , inertia - based in both cases ) . the data clustered are exactly those illustrated in the best planar projection of figure [ caout ] : these are 14 texts in a 4048-dimensional term space . due to centering in the dual spaces , the inherent dimensionality of both text and term spaces are : min@xmath44 = 13 . based on the dual spaces , we carry out the eigen - reduction in the space of smaller original dimensionality ( viz . , the space of the terms , which are in a 14-dimensional space ) , and then subsequently project into the 4048-dimensional space . proceeding further , the 5 closest terms to any given text , based on the full inherent dimensionality of this data ( viz . , smaller of dimensionality of texts , and dimensionality of terms ) , are as follows . .... text and set of 5 closest characterizing terms : artificial intelligence bayesian intelligence consciousness brains chatterbots artificial life automaton automata biology chemical allelomimesis artificial neural networks brain prediction forecasting aircraft epitomes captcha captcha captchas robot intelligence chemistry computational linguistics psychologists logics morphology pragmatics logicians computer vision image images diagnosis dimensionality dimensions evolutionary computation maps intelligence robot biology chemistry fuzzy logic logic mapping animals brakes armies genetic algorithms topologies communications music finance representations machine translation databases chemistry database memory robot multi - agent system agents agent robotics cybernetics robot semantic network representation database map namespaces robot turing test game chatterbot consciousness memory intelligence virtual world games gameplay topography communication representations .... we have noted in the introduction how a hierarchical clustering may be the starting point for creating a concept hierarchy , but the two representations differ . in this section we show how we can move from an embedded set of clusters , to an oriented tree . orientation in the latter case aims at expressing subsumption . consider the dendrogram shown in figure [ den1 ] , which represents an embedded set of clusters relating to the 8 terms . we will consider first such a strictly 2-way hierarchy , where we assume that no two agglomerations take place at precisely the same level . in the later case study , in subsection [ sect7 ] , we will consider the practical case where agglomerations take place at the same level . rather than the 14 texts used in section [ sect6 ] , to clarify the presentation in this section we will take just one text . we took aristotle s _ categories _ , which consisted of 14,483 individual words . we broke the text into 24 files , in order to base the textual analysis on the sequential properties of the argument developed . in these 24 files there were 1269 unique words . we selected 66 nouns of particular interest . a sample ( with frequencies of occurrence ) : man ( 104 ) , contrary ( 72 ) , same ( 71 ) , subject ( 60 ) , substance ( 58 ) , ... no stemming or other preprocessing was applied . for the hierarchical clustering , we further restricted the set of nouns to just 8 . ( these will be seen in the figures to be discussed below . ) the data array was doubled [ murtagh 2005b ] to produce an @xmath45 array , which with removing 0-valued text segments ( since , in one text segment , none of our selected 8 nouns appeared ) gave an @xmath46 array , thereby enforcing equal weighting of ( equal masses for ) the nouns used . the spaces of the 8 nouns , and of the 23 text segments ( together with the complements of the 23 text segments , on account of the data doubling ) are characterized at the start of the correspondence analysis in terms of their frequencies of occurrence , on which the @xmath13 metric is used . the correspondence analysis then `` euclideanizes '' both nouns and text segments . we used a 7-dimensional ( corresponding to the number of non - zero eigenvalues found ) euclidean embedding , furnished by the projections onto the factors . a hierarchical clustering of the 8 nouns , characterized by their 7-dimensional ( euclidean ) factor projections , was carried out : figure [ den1 ] . the ward minimum variance agglomerative criterion was used , with equal weighting of the 8 nouns . figure [ den2 ] shows a canonical representation of the dendrogram in figure [ den1 ] . both trees are isomorphic to one another . figure [ den2 ] is shown such that the sequence of agglomerations is portrayed from left to right ( and of course from bottom to top ) . we say that figure [ den2 ] is a canonical representation of the dendrogram , implying that figure [ den1 ] is not in canonical form . in figure [ den3 ] , the canonical representation has its non - terminal nodes labeled . next , figure [ den4 ] shows a superimposed oriented binary rooted tree , on @xmath47 nodes , which is isomorphic to the dendrogram on @xmath48 terminal nodes . this oriented binary tree is an inorder traversal of the dendrogram . sibson s [ 1973 ] `` packed representation '' of a dendrogram uses just such an oriented binary rooted tree , in order to define a permutation representation of the dendrogram . from our example , the packed representation permutation can be read off as : @xmath49 : for any terminal node indexed by @xmath14 , with the exception of the rightmost which will always be @xmath48 , define @xmath50 as the rank at which the terminal node is first united with some terminal node to its right . discussion of combinatorial properties of dendrograms , as related to such oriented binary rooted trees , and associated down - up and up - down permutations , can be found in murtagh [ 1984a ] . , but now with successively _ later _ agglomerations always represented by _ right _ child node . apart from the labels of the initial pairwise agglomerations , this is otherwise a unique representation of the dendrogram ( hence : `` existence '' and `` object '' can be interchanged ; so can `` disposition '' and `` fact '' ; and finally `` name '' and `` disposition '' ) . in the discussion we refer to this representation , with later agglomerations always parked to the right , as our canonical representation of the dendrogram.,width=529 ] , with non - terminal nodes numbered in sequence . these will form the nodes of the oriented binary tree . we may consider one further node for completeness , 8 or @xmath51 , located at an arbitrary location in the upper right.,width=529 ] finally , in figure [ den5 ] , we `` promote '' terminal node labels to the nodes of the oriented tree . we will use exactly the procedure used above for defining a permutation representation of the oriented tree . first the left terminal label is promoted to its non - terminal node . next , the right terminal label is promoted as far up the tree as is necessary in order to find an unlabeled non - terminal node . this procedure is carried out for all non - terminal labels , working in sequence from left to right ( i.e. , consistent with our canonical representation of the dendrogram ) . the rightmost label is not shown : it is at an arbitrary location in the upper right hand side , with a tree arc oriented towards the top non - terminal node of the dendrogram , now labeled as `` motion '' . in this section , we have specified a consistent procedure for labeling the nodes of an oriented tree , starting from the labels associated with the terminal nodes of a dendrogram . we start therefore with embedded clusters , and end up with terms and directed links between these terms . there is some non - uniqueness : any two labels associated with terminal nodes that are left and right child nodes of one non - terminal node can be interchanged . this clearly leads to a different label promotion outcome . our promotion procedure was motivated by the permutation representation of an oriented binary tree , as described above . here too we do not claim uniqueness of permutation representation . but we do claim optimality in the sense of parsimony , and well - definedness . in the case of a multiway tree with very few distinct levels , the promotion procedure becomes very simple , but continues to be non - unique . in the previous subsection , we discussed an algorithm which takes a hierarchical clustering , and hence a dendrogram , into an inorder tree traversal , and hence a permutation of the set of terms used . the formal procedure discussed in the previous subsection suffers from non - uniqueness : alternative permutations could be defined . this leads us to question the relationship of subsumption ( or direction in the oriented tree ) . in this section we will develop another approach which is even more closely associated with the data that we are analyzing . we have already seen that triangle properties between triplets of points , or data objects , are fundamental to ultrametricity and hence to tree representation . a dendrogram , representing a hierarchical clustering , allows us to read off , for all triplets of points , either ( i ) isosceles triangles , with small base , or ( ii ) equilateral triangles , and ( iii ) no other triangle configuration . the reason for the last condition is simply that non - isosceles , or isosceles with large base , triangles are incompatible with the ultrametric , or tree , metric . we will leave aside for the present the equilateral triangle case . firstly , it implies that all 3 points are _ ex aequo _ in the same cluster . secondly , therefore we will treat them altogether as a concept cluster . thirdly , the equilateral case does not arise in the example we will now explore . and [ den3 ] , with clusters indicated by ellipses . shown here are ellipses covering the clusters at nodes 7 , 6 , 5 , and 3.,width=529 ] in figure [ den6 ] , cluster number 3 indicates the following isosceles triangle with small base : ( ( existence , object ) position ) . our notation is : ( ( x , y ) z ) , such that triplet x , y , z has small base x , y , and the side lengths x , z and y , z are equal . this is necessarily implied by relationships represented in figure [ den6 ] . so , motivated by this triangle view of the cluster number 3 part of the dendrogram we will promote `` position '' to the cluster number 3 node . similarly we will promote `` motion '' to the cluster number 5 node . note the consistency of our perspective on the cluster number 3 and 5 nodes relative to how the associated terms here form an isosceles triangle with small base . we will straight away generalize this definition . in any case of a node in the form of nodes 3 or 5 , where we have a 2-term left subtree , and a 1-term right subtree , where left and right are necessarily labeled in this way given the canonical representation of the dendrogram , then : _ the left subtree is dominated by the right subtree_. we will next look at cluster number 6 ( remaining with figure [ den6 ] ) . as always for such trees , the node corresponding to this cluster has two subtrees , one to the left ( here : 3 ) and one to the right ( here : 5 ) . since our dendrogram is in canonical form , any such node has a subtree with smallest non - terminal node level to the left ; and the subtree which was more recently formed in the sequence of agglomerations to the right . based on either or both of these criteria which serve to define what are the left and right subtrees we define the ordering relationship : _ the left subtree is dominated by the right subtree_. figure not available : see pdf version of paper at www.cs.rhul.ac.uk/home/fionn/papers/auto_onto.pdf figure [ concepts ] summarizes the concept relations that we can derive in a similar way from any dendrogram . figure [ aristontoex ] indicates how the concept relations , shown in figure [ concepts ] , are to be used . firstly the term set is summarized , using our selection of terms . scaling to large data sets is addressed in this way . secondly , in our interactive implementation ( web address : + thames.cs.rhul.ac.uk/@xmath52dimitri/textmap ) , we allow the terms shown to continually move in a limited way , to get around the occlusion problem , and we also allow magnification of the display area for this same reason . thirdly , terms other than those shown are highlighted when a cursor is passed over them . next , double clicking on any term gives a ranked list of text segment names , ordered by frequency of occurrence by this term . clicking on the text segment gives the actual text at the bottom of the display area . are shown in decreasing size ( and in rainbow colors , from red ) , with other terms ( in all , 66 ) displayed with a dash , and all text segments ( in all , 24 ) represented by an asterisk . the principal factor plane of a correspondence analysis ( based on the 24 text segments @xmath53 66 terms frequencies of occurence ) output is used.,width=529 ] we proceed now to a third case study of this work , where we have a multiway hieararchy ( and not a binary hierarchy ) from the start . we require a frequency of occurrence matrix which crosses the terms of interest with parts of a free text document . for the latter we could well take documentary segments like paragraphs . oneill [ 2006 ] is a 660-word discussion of ubiquitous computing from the perspective of human computing interaction . with this short document we used individual lines ( as proxies for the sequence of sentences ) as the component parts of the document . there were 65 lines . this facilitates retrieval of a small segment of such a single document . we chose this text to work with because it is a very small text ( a single text compared to the data used in section [ sect6 ] , and a far smaller text compared to that used in section [ secorientree ] ) . based on a list of nouns and substantives furnished by the part - of - speech tagger ( schmid , 1994 ) , we focused on the following 30 nouns : support = @xmath54 `` agents '' , `` algorithms '' , `` aspects '' , `` attempts '' , `` behaviours '' , `` concepts '' , `` criteria '' , `` disciplines '' , `` engineers '' , `` factors '' , `` goals '' , `` interactions '' , `` kinds '' , `` meanings '' , `` methods '' , `` models '' , `` notions '' , `` others '' , `` parts '' , `` people '' , `` perceptions '' , `` perspectives '' , `` principles '' , `` systems '' , `` techniques '' , `` terms '' , `` theories '' , `` tools '' , `` trusts '' , `` users '' @xmath55 . this set of 30 terms was used to characterize through presence / absence the 65 successive lines of text , leading to correspondence analysis of the @xmath56 presence / absence matrix . this yielded then the definition of the 30 terms in a factor space . in principle the rank of this space ( taking account of the trivial first factor in correspondence analysis , relating to the centering of the cloud of points ) is min ( @xmath57 ) . however , given the existence of zero - valued rows and/or columns , the actual rank was 25 . therefore the full rank projection of the terms into the factor space gave rise to 25-dimensional vectors for each term , and these vectors are endowed with the euclidean metric . define this set of 30 terms as the _ support _ of the document . based on their occurrences in the document , we generated the following _ reduced _ version of the document , defined on this support , which consists of the following ordered set of 52 terms : reduced document = `` goals '' `` techniques '' `` goals '' `` disciplines '' `` meanings '' `` terms '' `` others '' `` systems '' `` attempts '' `` parts '' `` trusts '' `` trusts '' `` people '' `` concepts '' `` agents '' `` notions '' `` systems '' `` people '' `` kinds '' `` behaviours '' `` people '' `` factors '' `` behaviours '' `` perspectives '' `` goals '' `` perspectives '' `` principles '' `` aspects '' `` engineers '' `` tools '' `` goals '' `` perspectives '' `` methods '' `` techniques '' `` criteria '' `` criteria '' `` perspectives '' `` methods '' `` techniques '' `` principles '' `` concepts '' `` models '' `` theories '' `` goals '' `` tools '' `` techniques '' `` systems '' `` interactions '' `` interactions '' `` users '' `` perceptions '' `` algorithms '' this reduced document is just the `` time series '' of the nouns of interest to us , as they are used in traversing the document from start to finish . each noun in the sequence of 52 nouns is represented by its 25-dimensional factor space vector . out of 43 unique triplets , with self - distances removed , we found 31 to respect the ultrametric inequality , i.e. 72% . our measure of ultrametricity of this document , based on the support used , was thus 0.72 . for a concept hierarchy we need an overall fit to the data . using the euclidean space perspective on the data , furnished by correspondence analysis , we can easily define a terms @xmath53 terms distance matrix ; and then hierarchically cluster that . consistent with our analysis we recode all these distances , using the cvnc mapping onto @xmath58 for unique pairs of terms . now approximating a global ultrametric from below , achieved by the single linkage agglomerative hierarchical clustering method ( and this best fit from below , termed the subdominant or maximal inferior ultrametric , is optimal ) , and an approximation from above , achieved by the complete linkage agglomerative hierarchical clustering method ( and this best fit from above , termed a minimal superior ultrametric , is non - unique and hence is one of a number of best fits from above ) , will be identical if the data is fully ultrametric - embeddable . if we had an ultrametricity coefficient equal to 1 we found it to be 0.72 for this data then it would not matter what agglomerative hierarchical clustering algorithm ( among the usual lance - williams methods ) that we select . in fact , we found , with an ultrametricity coefficient equal to 0.72 , that the single and complete linkage methods gave an identical result . this result is shown in figure [ fig1 ] . a convenient label promotion procedure to apply here is first to re - represent the terminal labels from left to right as : @xmath54 `` users '' , `` trusts '' , @xmath59 , `` agents '' , `` algorithms '' @xmath55 ; @xmath54 `` goals '' , `` perspectives '' @xmath55 ; @xmath54 `` tools '' @xmath55 ; @xmath54 `` techniques '' @xmath55 ; and @xmath54 `` methods '' @xmath55 . this is the canonical form , with ordering of left and right subtrees now extended to all subtrees . next , we must in fairness take the nodes at level 2 as being _ ex aequo _ , @xmath54 `` tools '' @xmath55 ; @xmath54 `` techniques '' @xmath55 ; and @xmath54 `` methods '' @xmath55 . similarly at level 1 , we also have two clusters that are _ ex aequo _ : @xmath54 `` users '' , `` trusts '' , @xmath59 , `` agents '' , `` algorithms '' @xmath55 ; and @xmath54 `` goals '' , `` perspectives '' @xmath55 . figure not available : see pdf version of paper at www.cs.rhul.ac.uk/home/fionn/papers/auto_onto.pdf figure [ ubiq ] shows our resulting scheme where level 1 clusters dominate level 2 clusters . this provides our ontology . the granularity of this one document is , as mentioned above , line - based , and there are 65 lines in all . hence retrieval of one or more of these document snippets is supported , and the ontology is based on a 30-noun document support . having first appraised text collections in terms of their local hierarchical structure , we then proceeded in this work to show how this new methodology could be employed for a wide range of tasks that include : * finding salient pairs and triplets of terms , which are not necessarily in sequence ; * permitting us to consider any given text as a whole with all pairwise relationships between terms , or alternatively as a time series with relationships restricted to terms that are successive in sequence ; * passing seamlessly from the exploration of local hierarchical structure to global hierarchical structure ; * especially when global hierarchical structure is manifest , being able to use any of a wide range of agglomerative clustering criteria to furnish the same resultant hierarchy ; * determining a hierarchy of concepts from the embedded , partially ordered subsets provided by a hierarchical clustering ; * obtaining unique results when given a 2-way hierarchical clustering tree , and then readily generalizing this to the practical case of multiway trees ; * exemplifying an efficient and an effective textual data processing pipeline ; and * through the measurement of local hierarchical structure , having available an approach to validating the appropriateness of any data for this data analysis pipeline . by analysis of text through local hierarchical relationships between terms we determine extensive internal textual structure , without being stifled by the more traditional approach of fitting some global structure , such as a hierarchy , to the text . a ( local ) hierarchical structure is a powerful one : it includes peer as well as subsumption types of relationships . we stress that we can find very pronounced hierarchical structures of this sort if we encode the text is novel ways . an example is to start with a euclidean spatial embedding of the terms and documents ( or segments of a document ) , which is quite traditional ; and then look at interrelationships between terms using `` relatively close / similar '' versus `` relatively distant / new '' ( and this alone can be shown to have metric properties ) . another example of an encoding - related strategy is not to take into consideration all interrelationships between terms , but only between successive terms , and thereby view the text as a particular type of time series . user interactivity with the system is to select the terms of interest ( people s personal names , industrial product names , location or venue names , etc . ) . the interrelationships between these terms are then explored through their local hierarchical links . our general application targeted is , as stated in murtagh et al . [ 2003 ] , to have readily available a self - description of data , as a basis for visually - based interactive and responsive querying of , retrieval from , and navigation of data collections . this work was carried out in the context of the european union sixth framework project , `` ws - talk , web services communicating in the language of their community '' , 20042006 . pedro contreras and dimitri zervas contributed to this work . the textmap demonstrator was developed by dimitri zervas . : : abou assali , a. and zanghi , h. 2006 . automated metadata hierarchy derivation . in _ proc . ieee ictta06 _ ( damascus , syria ) . : : ahmad , k. and gillam , l. ( 2005 ) . automatic ontology extraction from unstructured texts . in _ odbase 2005_. : : benzcri , j.p . _ lanalyse des donnes , tome i taxinomie , tome ii correspondances _ , 2nd ed . ( dunod , paris ) . : : chuang shui - lung and chien lee - feng ( 2005 ) . taxonomy generation for text segments : a practical web - based approach , acm transactions on information systems . 23 , 363396 . : : de soete , g. ( 1986 ) . a least squares algorithm for fitting an ultrametric tree to a dissimilarity matrix . _ pattern recognition letters _ , 2 , 133137 . : : denny , m. ( 2004 ) . ontology tools survey , revisited , at + http://www.xml.com/pub/a/2004/07/14/onto.html , july 2004 . : : doyle , l.b . semantic road maps for literature searches . _ journal of the acm_. 8 , 553578 . : : ganesan , p. , garcia - molina , h. and widom , j. ( 2003 ) . exploiting hierarchical domain structure to compute similarity _ acm transactions on information systems_. 21 , 6493 . : : gmez - prez , a. , fernndez - lpez , m. and corcho , o. ( 2004 ) . _ ontological engineering ( with examples from the areas of knowledge management , e - commerce and the semantic web ) _ ( springer , berlin ) . : : gruber , t. ( 2001 ) . what is an ontology ? , http://www-ksl.stanford.edu/kst/what-is-an-ontology.html , sept . : : janowitz , m.f . cluster analysis based on abstract posets , preprint . also presentation , enst de bretagne , 30 oct . 2004 ; dimacs , 9 mar . 2005 ; and sfc , 31 may 2005 . : : maedche , a. ( 2006 ) . _ ontology learning for the semantic web _ ( kluwer , dordrecht ) . : : murtagh , f. ( 1984a ) . counting dendrograms : a survey . _ discrete applied mathematics _ , 7 , 191199 . : : murtagh , f. ( 1984b ) . structures of hierarchic clusterings : implications for information retrieval and for multivariate data analysis . _ information processing and management _ , 20 , 611617 . : : murtagh , f. ( 2004 ) . on ultrametricity , data coding , and computation , _ journal of classification , 21 , 167184 . : : murtagh , f. ( 2005a ) . identifying the ultrametricity of time series , _ european physical journal b _ , 43 , 573579 . : : murtagh , f. ( 2005b ) . _ correspondence analysis and data coding with r and java _ ( chapman & hall / crc ) . : : murtagh , f. ( 2006a ) . ultrametricity in data : identifying and exploiting local and global hierarchical structure , arxiv : math.st/0605555v1 19 may 2006 . : : murtagh , f. ( 2006b ) . the haar wavelet transform of a dendrogram , + arxiv : cs.ir/0608107v1 28 aug 2006 , forthcoming in _ journal of classification_. : : murtagh , f. ( 2006c ) . symbolic dynamics in text : application to automated construction of concept hierarchies , _ festschrift for edwin diday _ , in press . available at : www.cs.rhul.ac.uk/home/fionn/papers : : murtagh . f. ( 2006d ) . visual user interfaces , interactive maps , references , theses , http://astro.u-strasbg.fr/@xmath52fmurtagh/inform : : murtagh , f. , taskaya , t. , contreras , p. , mothe , j. and englmeier , k. ( 2003 ) . interactive visual user interfaces : a survey , _ artificial intelligence review _ , 19 , 263283 . : : oneill , e. ( 2006 ) . understanding ubiquitous computing : a view from hci , in discussion following r. milner , ubiquitous computing : how will we understand it ? , _ computer journal _ , 49 , 390399 . : : schmid , h. ( 1994 ) . probabilistic part - of - speech tagging using decision trees . in _ proc . new methods in language processing _ ( see treetagger site ) . : : treetagger , www.ims.uni-stuttgart.de/projekte/corplex/treetagger/ decisiontreetagger.html : : sibson , r. ( 1973 ) . slink : an optimally efficient algorithm for the single - link cluster method , _ computer journal _ , 16 , 3034 . : : wache , h. , vgele , t. , visser , u. , stuckenschmidt , h. , schuster , g. , neumann , h. and hbner , s. ( 2001 ) . ontology - based integration of information a survey of existing approaches . in _ proceedings of the ijcai-01 workshop : ontologies and information sharing_.

tremendous progress has been realized in the last decades concerning the theoretical foundation of quantum optics in dielectric media . while the historical approach proposed by jauch and watson @xcite was already based on the standard canonical quantization formalism for fields , it neglected dispersion and dissipation which are intrinsic properties of any causal dielectric media satisfying kramers krnig relations . since then several important studies were devoted to the extension of the method to inhomogeneous and artificially structured media which are central issues in modern micro and nano photonics @xcite . furthermore , theoretical approaches adapted to transparent but dispersive media with negligible losses have been also developed based on different techniques such as the slowly varying envelope approximation @xcite or the quasi modal expansion method which is valid near resonance for polaritons @xcite . more recently , losses were included in the theory by adding phenomenologically some optical dissipation channels in the light propagation path @xcite . such a method was successfully used for the modeling of casimir forces in dissipative media @xcite and surface plasmon polaritons @xcite . + moreover , the most fundamental progress was probably done when huttner and barnett , and others @xcite proposed a self - consistent canonical quantization procedure for an homogeneous and causal dielectric medium by coupling photonic degrees of freedom with mechanical oscillator variables acting as thermal baths . the method , based on the pioneer works by fano and hopfield @xcite ( see also ref . @xcite ) , was subsequently extended to several inhomogeneous systems including anisotropic and magnetic properties @xcite . in parallel to these theoretical works based on the standard canonical quantization method , a different and powerful axis of research appeared after the work by gruner and welsch @xcite ( see also ref.@xcite ) based on the quantum langevin noise approach used in cavity qed ( i.e. , quantum electrodynamics ) @xcite . the method is also known as the green tensor method @xcite since it relies on efficient green dyadic techniques used nowadays in nano - photonics and plasmonics @xcite . this ` langevin - noise ' approach , which actually extends earlier ` semi classical ' researches based on the fluctuation - dissipation theorem by lifshitz and many others in the context of casimir and optical forces @xcite , was successfully applied in the recent years to many issues concerning photonics @xcite and nano - plasmonics where dissipation can not be neglected @xcite . in this context the relationship between the huttner - barnett approach on the one side and the langevin noise method on the other side has attracted much attention in the last years , and several works attempted to demonstrate the validity of the langevin noise method from a rigorous hamiltonian perspective which is more in a agreement with the canonical huttner - barnett approach @xcite . + the aim of this work is to revisit these derivations of the equivalence between the langevin noise and hamitonian method and to show that some unphysical assumptions actually limit the domain of validity of the previous attempts . more precisely , as we will show in this work , the analysis and derivations always included some hypothesis concerning causality and boundary conditions which actually lead to circularity in the deductions and are not applicable to the most general inhomogeneous systems used in nano - optics . specifically , these derivations , like the fluctuation - dissipation reasoning in lifshitz and rytov works @xcite , give too much emphasis on the material origin of quantum fluctuations for explaining macroscopic quantum electrodynamics in continuous media . however , as it was already pointed out in the 1970 s @xcite , one must include with an equal footings both field and matter fluctuations in a self consistent qed hamiltonian in order to preserve rigorously unitarity and causality @xcite . while this does nt impact too much the homogeneous medium case considered by huttner and barnett @xcite it is crucial to analyze further the inhomogeneous medium problem in order to give a rigorous foundation to the gruner and welsch theory @xcite based on fluctuating currents . this is the central issue tackled in the present work . + the layout of this paper is as follows : in section ii we review the lagrangian method developed in our previous work @xcite based on an alternative dual formalism for describing the huttner - barnett model . in this section we summarize the essential elements of the general lagrangian and hamiltonian model necessary for the present study . in particular we present the fundamental issue about the correct definition of hamiltonian which will be discussed at length in this article . in section iii we provide a quantitative discussion of the huttner - barnett model for an homogeneous dielectric medium . we discuss a modal expansion into plane waves and separate explicitly the electromagnetic field into classical eigenmodes and noise related langevin s modes . we show that both contributions are necessary for preserving unitarity and time symmetry . we consider limit cases such as the ideal hopfield - fano polaritons @xcite without dissipation and the weakly dissipative polariton modes considered by milonni and others @xcite . we discuss the physical interpretation of the hamiltonian of the whole system and interpret the various contributions with respect to the langevin noise method and to the loss - less hopfield - fano limit . in section iv we generalize our analysis to the inhomogeneous medium case by using a green dyadic formalism in both the frequency and time domain . we demonstrate that in general it is necessary to keep both pure photonic and material fluctuations to preserve the unitarity and time symmetry of the quantum evolution . we conclude with a discussion about the physical meaning of the hamiltonian in presence of inhomogeneities and interpret the various terms associated with photonic and material modes . in ref . @xcite we developed a new lagrangian formalism adapted to qed in dielectric media without magnetic properties . here we will use this model to derive our approach but a standard treatment based on the minimal coupling scheme @xcite or the power - zienau @xcite transformation would lead to similar results . we start with the dual lagrangian density : @xmath0 where @xmath1 and @xmath2 are the magnetic and displacement fields respectively . in this formalism the usual magnetic potential @xmath3 , defined such as @xmath4 , is replaced by the dual electric potential @xmath5 ( in the ` coulomb ' gauge @xmath6 ) defined by @xmath7 implying @xmath8 the material part @xmath9 of the lagrangian density in eq . [ 1 ] reads @xmath10 with @xmath11 the material oscillator fields describing the huttner - barnett bath coupled to the electromagnetic field . the coupling depends on the polarization density which is defined by @xmath12 where the coupling function @xmath13 defines the conductivity of the medium at the harmonic pulsation @xmath14 . from eq . [ 1 ] and euler - lagrange equations we deduce the dynamical laws for the electromagnetic field @xmath15 with the electric field @xmath16 . similarly for the material oscillators we have : @xmath17we point out that the lagrangian density in eq . [ 1 ] includes a term @xmath18 which is necessary for the derivation of the dynamical laws for the material fields @xmath19 @xcite . furthermore , to complete the qed canonical quantization procedure of the material field we introduce the lowering @xmath20 and rising @xmath21 operators for the bosonic material field from the relation @xmath22 . as explained in ref . @xcite by using the equal time commutation relations between the canonical variables @xmath11 and @xmath23 , we deduce the fundamental rules @xmath24=\delta(\omega-\omega')\delta^3(\mathbf{x}-\mathbf{x'})\textbf{i}. \label{33}\end{aligned}\ ] ] ( with @xmath25 the unit dyad ) and @xmath26=[\mathbf{f}^\dagger_\omega(\mathbf{x},t),\mathbf{f}^\dagger_{\omega'}(\mathbf{x}',t)]=0 $ ] allowing a clear interpretation of @xmath20 and @xmath27 as lowering and rising operators for the bosonic states associated with the matter oscillators . + moreover , eqs . [ 5],[8 ] can be formally integrated leading to @xmath28 where @xmath29 is an initial time and where @xmath30 is a fluctuating dipole density distribution defined by : @xmath31 \nonumber\\ \label{35}\end{aligned}\ ] ] with @xmath32 and where by definition @xmath33 . we therefore have @xmath34 which is reminiscent of the general linear response theory used in thermodynamics @xcite . we point out that the term @xmath35 can be seen as an induced dipole density . however , as we will show in the next section the electric field itself is decomposed into a purely fluctuating term @xmath36 and a scattered field @xmath37 which depends on the density @xmath38 . therefore , the contribution @xmath35 to @xmath39 is also decomposed into a pure photon - fluctuation term @xmath40 and an induced term @xmath41 related to material fluctuations @xmath38 . + importantly , the linear susceptibility @xmath42 which is defined by @xmath43 characterizes completely the dispersive and dissipative dielectric medium . we can show that the permittivity @xmath44 is an analytical function in the upper part of the complex plane @xmath45 , i.e. , @xmath46 , provided @xmath42 is finite for any time @xmath47 . from this we deduce the symmetry @xmath48 and it is possible to derive the general kramers - krnig relations existing between the real part @xmath49\equiv\widetilde{\varepsilon}'(\mathbf{x},\omega)$ ] and the imaginary part @xmath50\equiv\widetilde{\varepsilon}''(\mathbf{x},\omega)$ ] of the dielectric permittivity . therefore , the huttner - barnett model characterized by the conductivity @xmath51 is fully causal and can be applied to describe any inhomogeneous dielectric media in the linear regime . + the central issue of the present paper concerns the definition of the hamiltonian @xmath52 in the huttner - barnett model . we remind that in ref . @xcite we derived the result : @xmath53 with @xmath54 where @xmath55:$ ] means , as usually , a normally - ordered product for removing the infinite zero - point energy . inserting the definition for @xmath20 obtained earlier we get for the material part @xmath56 which has the standard structure for oscillators ( i.e. , without the infinite zero - point energy ) . + however , hutner and barnett @xcite after diagonalizing their hamiltonian found that the total evolution is described in the homogeneous medium case by @xmath57 while as we will see this is actually a correct description ` for all practical purpose ' in a homogeneous dissipative medium for large class of physical boundary conditions , this is in general not acceptable in order to preserve time - symmetry and unitarity in the full hilbert space for interacting matter and light . the general method based on langevin forces and noises avoided quite generally mentioning that difficult point . we emphasize that while the conclusions presented in refs . @xcite is accepted by more or less all authors on the subject @xcite they have been some some few dissident views ( see refs . @xcite ) claiming , that in the context of an input - output formalism , the langevin noise formalism is not complete unless we consider as well fluctuations of the free photon modes ( see also the replies with an opposite perspective in refs . @xcite ) . in the present work we will generalize and give a rigrous qed like hamiltonian foundations to the prescriptions of refs . @xcite and we will show that it is actually necessary to include a full description of photonic and material quantum excitations in order to preserve unitarity . in order to appreciate this fact further we will first consider the problem associated with quantization of the electromagnetic field . we first introduce the paradigmatic homogeneous medium case considered initially by huttner and barnett @xcite , i.e. , with @xmath58 . we start with faraday s law : @xmath59 rewritten according to eq . [ 21 ] as : @xmath60 inserting eq . [ 4 ] and using the coulomb ( transverse ) gauge condition we get : @xmath61 we use the modal expansion method developed in ref . @xcite and write @xmath62 with @xmath63 a generic label for the wave vector @xmath64 , @xmath65 ( here we consider as it is usually done the periodical ` box ' born - von karman expansion in the rectangular box of volume @xmath66 ) , @xmath67 or 2 , labels the two transverse polarization states with unit vectors @xmath68 , and @xmath69 ( conventions and details are given in appendix a of ref . @xcite ) . inserting eq . [ 40 ] into eq . [ 91 ] we obtain the dynamical equation : @xmath70 with the time dependent source term @xmath71 to solve this equation we use the laplace transform of the fields which is defined below . + we are interested in the evolution for @xmath72 of a field @xmath73 which fourier transform is not necessary well mathematically defined since the field is not going to zero fast enough for @xmath74 ( e.g. , a fluctuating current or field ) . the method followed here is to consider the forward laplace transform of the different evolution equations ( such an approach was also used by suttorp by mixing both forward and backward laplace s transforms @xcite ) . to deal with this problem we first change the time @xmath75 variable in @xmath76 and define @xmath77 . we define the ( forward ) laplace transform of @xmath78 as : @xmath79 with @xmath80 ( @xmath14 a real number and @xmath81 ) . the presence of the term @xmath82 ensures the convergence . we will not here introduce the backward laplace transform @xmath83 since the time @xmath29 is arbitrary and can be sent into the remote past if needed . + as it is well known , the ( forward ) laplace transform is connected to the usual fourier transform since we have @xmath84 where @xmath85 is the fourier transform of @xmath86 with respect to @xmath75 . + now for the specific problem considered here we obtain the separation @xmath87 . the @xmath88 contribution corresponds to what classically we call a sum of eigenmodes supported by the medium ( i.e. with @xmath89 ) while the @xmath90 term is the fluctuating field generated by the langevin source @xmath91 . more explicitly , we have for the source term : @xmath92 where we use eq . [ 93 ] . the propagator function @xmath93 is expressed as a bromwich contour : @xmath94 where @xmath95 is defined as @xmath96 . we remind that a bromwich integral by definition will vanish for @xmath97 so that on the left side we actually mean @xmath98 . + the function @xmath93 has some remarkable properties which should be emphasized here . we first introduce the ` zeros ' of @xmath99 , i.e. , the set of roots @xmath100 solutions of @xmath101 . from the causal properties of @xmath102 we have @xmath103 and therefore we deduce @xmath104 implying that the ` + ' and ` - ' roots are not independent . the important fact is that the roots are located in the lower complex plane associated with a negative imaginary part of the frequency @xcite ( this is proven in appendix a ) . now , as shown in appendix b the integral in eq . [ 95 ] can be computed by contour integration in the complex plane after closing the contour with a semi - circle in the lower plane and using the cauchy residue theorem . we get : @xmath105 we also get @xmath106 for @xmath107 ( after integration in the upper plane and considering in detail the case @xmath108 ) . clearly , the function @xmath93 is here expanded into a sum of modes which define the polaritons of the problem ( this is shown explicitly in the next subsection ) . here , the normal mode frequency @xmath109 are complex numbers ensuring the damped nature of the waves in the future direction . we emphasize that in our knowledge this kind of formula has never been discussed before . the expansion in eq . [ 98 ] is however rigorous and generalizes the quasimodal approximations used in the weak dissipation regime and discussed in subsection iii(f ) . + similarly the source - free term @xmath110 reads : @xmath111 with the new propagator function @xmath112 like for @xmath113 we get @xmath114 for @xmath107 . additionally , the boundary condition at @xmath115 ( i.e. , @xmath116 ) imposes @xmath117 ( see appendix b ) . + the electromagnetic field can be calculated using the expansion eq . [ 94 ] or [ 100 ] . first , the mathematical and physical structure of the free - field is seen by using the modal expansion : @xmath118\boldsymbol{\hat{\epsilon}}_{\alpha , j}\phi_\alpha(\mathbf{x})\nonumber\\ = \sum_{\alpha , j , m}\frac{-\tilde{\varepsilon}(\omega_{\alpha , m}^{(-)})}{2i\omega_\alpha}\frac{e^{-i\omega_{\alpha , m}^{(-)}(t - t_0)}}{\frac{\partial(\omega\sqrt{\tilde{\varepsilon}(\omega)})}{\partial \omega}|_{\omega_{\alpha , m}^{(-)}}}[\dot{q}_{\alpha , j}(t_0)\nonumber\\-i\omega_{\alpha , m}^{(-)}q_{\alpha , j}(t_0)]\boldsymbol{\hat{\epsilon}}_{\alpha , j}\phi_\alpha(\mathbf{x } ) + hcc.\label{117}\end{aligned}\ ] ] where we used the symmetries of the modal expansion @xcite together with @xmath119 . from this we directly obtain : @xmath120\boldsymbol{\hat{k}}_{\alpha}\times\boldsymbol{\hat{\epsilon}}_{\alpha , j}\phi_\alpha(\mathbf{x})\nonumber\\ = \sum_{\alpha , j , m}\frac{-\tilde{\varepsilon}(\omega_{\alpha , m}^{(-)})}{2c}\frac{e^{-i\omega_{\alpha , m}^{(-)}(t - t_0)}}{\frac{\partial(\omega\sqrt{\tilde{\varepsilon}(\omega)})}{\partial \omega}|_{\omega_{\alpha , m}^{(-)}}}[\dot{q}_{\alpha , j}(t_0)\nonumber\\-i\omega_{\alpha , m}^{(-)}q_{\alpha , j}(t_0)]\boldsymbol{\hat{k}}_{\alpha}\times\boldsymbol{\hat{\epsilon}}_{\alpha , j}\phi_\alpha(\mathbf{x } ) + hcc . \label{118}\end{aligned}\ ] ] and similarly for the magnetic field : @xmath121\boldsymbol{\hat{\epsilon}}_{\alpha , j}\phi_\alpha(\mathbf{x})\nonumber\\ = \sum_{\alpha , j , m}\frac{\omega_{\alpha , m}^{(-)}\tilde{\varepsilon}(\omega_{\alpha , m}^{(-)})}{2c\omega_\alpha}\frac{e^{-i\omega_{\alpha , m}^{(-)}(t - t_0)}}{\frac{\partial(\omega\sqrt{\tilde{\varepsilon}(\omega)})}{\partial \omega}|_{\omega_{\alpha , m}^{(-)}}}[\dot{q}_{\alpha , j}(t_0)\nonumber\\-i\omega_{\alpha , m}^{(-)}q_{\alpha , j}(t_0)]\boldsymbol{\hat{\epsilon}}_{\alpha , j}\phi_\alpha(\mathbf{x } ) + hcc . \label{119}\end{aligned}\ ] ] the ( transverse ) electric field associated with this free solutions can also be obtained from the definition @xmath122 . we have thus @xmath123 and considering the laplace transform in eqs . [ 100 ] and [ 101 ] we can express the electric field as a function of @xmath93 , i.e. , @xmath124\boldsymbol{\hat{k}}_{\alpha}\times\boldsymbol{\hat{\epsilon}}_{\alpha , j}\phi_\alpha(\mathbf{x})\nonumber\\ = \sum_{\alpha , j , m}\frac{-1}{2c}\frac{e^{-i\omega_{\alpha , m}^{(-)}(t - t_0)}}{\frac{\partial(\omega\sqrt{\tilde{\varepsilon}(\omega)})}{\partial \omega}|_{\omega_{\alpha , m}^{(-)}}}[\dot{q}_{\alpha , j}(t_0)\nonumber\\-i\omega_{\alpha , m}^{(-)}q_{\alpha , j}(t_0)]\boldsymbol{\hat{k}}_{\alpha}\times\boldsymbol{\hat{\epsilon}}_{\alpha , j}\phi_\alpha(\mathbf{x } ) + hcc . \label{120}\end{aligned}\ ] ] from eqs . [ 119 ] and [ 120 ] we see that @xmath125 in agreement with maxwell s equation for a free - field ( the other maxwell s equations are also automatically fulfilled by definition ) . + importantly , in the vacuum case where @xmath126 we have @xmath127 and we find that the vacuum fields are given by @xmath128 with @xmath129 as expected . in the general case however causality imposes that the imaginary part of @xmath109 is negative . therefore the optical modes labeled by @xmath63 , @xmath130 and @xmath131 are damped in time ( the only exception being of course the vacuum case where the only contribution to the field arises from the source - free term @xmath88 since the @xmath90 terms vanishes together with @xmath38 ) . as a consequence , if @xmath132 the free terms vanish asymptotically . in particular , if @xmath133 ( corresponding to initial conditions fixed in the infinite remote past ) we can omit for all practical purposes the contribution of @xmath134 , @xmath135 , and @xmath136 to the field observed at any finite time @xmath75 ( unless we are in the vacuum ) . this is indeed what was implicitly done by huttner and barnet @xcite and gruner and welsch @xcite and that is why for all calculational needs they completely omitted the discussion of the @xmath134 , @xmath135 , and @xmath136 fields . however , for preserving the unitarity of the full evolution one must necessarily include both @xmath88 and @xmath90 terms . while this problem is apparently only technical we will see in the following its importance for inhomogeneous systems . + the previous discussion concerning the omission of the @xmath88 source free terms is very important since it explains the mechanism at work in the huttner barnet model @xcite . to clarify that point further we now express the scattered field @xmath90 using a green tensor formalism ( in appendix c we introduce alternative descriptions based on vectorial and scalar potentials ) . we first observe that from @xmath137 we obtain after integration by parts : @xmath138 with the dyadic propagator @xcite : @xmath139 the meaning of the tensor @xmath140 , which depends on @xmath141 , becomes more clear if we introduce the green tensor @xmath142 solution of @xmath143 we observe that if we separate the tensor into a transverse and longitudinal part we get @xmath144 with for the transverse dyadic green function @xmath145 and for the longitudinal part @xmath146 with the unit longitudinal dyadic distribution @xmath147 . we have also the important relations between the tensor @xmath140 and @xmath142 : @xmath148 now if we write @xmath149 and introduce the relation @xmath150 with @xmath151 the free mode given by eq . [ 120 ] and @xmath152 the source field , induced by @xmath153 , which is given by @xmath154 we get @xmath155 , i.e. , eq . [ rehh ] as it could be checked directly after comparing eq . [ s ] with eqs . [ gper ] and [ gpar ] . others important relations between the dyadic formalism and scalar green function are given in appendix d. + importantly , we can write all scattered field as a function of the rising and lowering operators @xmath156 , @xmath157 . in order to give explicit expressions we use the fourier transform notations ( i.e. , with @xmath80 with @xmath158 ) and the relation @xcite @xmath159\nonumber\\ \label{36}\end{aligned}\ ] ] we obtain : @xmath160 @xmath161where @xmath162 is a lowering operator associated with the transverse fluctuating field and defined as @xmath163 such that from eq . [ 33 ] we get the mode commutator @xmath164 and the harmonic time evolution @xmath165 if we impose the initial condition @xmath166 . + for the longitudinal electric field we deduce from eq . [ gpar ] and eq . [ cool2 ] @xmath167 ( in agreement with the definition @xmath168 ) and therefore : @xmath169 we have @xmath170 and the commutator @xmath171=\delta_{\alpha,\beta}\delta_{j , k}\delta(\omega-\omega')$ ] and the time evolution @xmath172 with similar initial condition as for the transverse field . regrouping these definition we have obviously @xmath173 furthermore , we can easily show that we have also @xmath174 leading to @xmath175 the description of the scattered field @xmath90 given here corresponds exactly to what huttner and barnett @xcite called the quantized field obtained after generalizing the diagonalization procedure of fano and hopfield@xcite . here we justify these modes by using the laplace transform method and by taking the limit @xmath176 explicitly . this means that we neglect the contribution of the @xmath88 transverse field which is infinitely damped at time @xmath75 . importantly @xmath30 does nt vanish since the time evolution of @xmath177 in eq . [ 35 ] is harmonic . it is important to further comment about causality and on the structure of the total field as a sum of @xmath88 and @xmath90 modes . the @xmath88 ( classical polariton ) modes are indeed exponentially damped in the future direction meaning that a privileged temporal direction apparently holds in this model . this would mean that we somehow break the time symmetry of the problem . however , since the evolution equations are fundamentally time symmetric this should clearly not be possible . similarly , propagators such as @xmath142 are also spatially damped at large distance , see eq . [ 103 ] , since we have terms like @xmath178 . this also seems to imply a privileged time direction and would lead to a kind of paradox . however , we should remind that only the sum @xmath88 + @xmath90 has a physical meaning and this sum must preserves time symmetry . indeed , we remind that time reversal applied to electrodynamics implies that if @xmath179 , @xmath180 , and @xmath181 is a solution of the coupled set of equations given in section ii then the time reversed solutions @xcite @xmath182 , @xmath183 , and @xmath184 is also defining a solution of the same equations ( we have also @xmath185 and @xmath186 ) . now , considering the dipole density evolution we get from eq . [ 18 ] after some manipulations : @xmath187 where @xmath188 is defined as @xmath38 ( see the definition eq . [ 35 ] ) but with @xmath189 replaced by @xmath190 . the presence of the time @xmath191 everywhere has a clear meaning . indeed , from special relativity choosing a time reference frame with @xmath192 ( passive time reversal transformation ) implies that the causal evolution of @xmath39 defined for @xmath72 will become an anticausal evolution defined for @xmath193 . going back to the active time reversal transformation eq . [ 18bis ] we see that the new linear susceptibility @xmath194 is given by @xmath195 ( @xmath196 is explicitly written in order to emphasize the anticausal structure . however , since we have @xmath197 and @xmath198 we have @xmath199 . this means that the new permittivity ( with poles in the upper half complex frequency space ) is now associated with growing anticausal modes since @xmath200 . the ( active ) time reversed evolution is defined for @xmath201 so that we indeed get modes decaying into the past direction while growing into the future direction . this becomes even more clear for the time reversal evolution of the electromagnetic field given by the separation in @xmath88 and @xmath90 modes . the time reversal applied on the @xmath88 field such as @xmath202 corresponding to classical polaritons is now involving frequency @xmath203 instead of @xmath109 . this leads to reversed temporal evolution such as @xmath204 and @xmath205 associated with growing waves in the future direction ( since the new poles are now in the upper half complex frequency space ) . more generally , we have shown ( see appendix c ) that the full evolution of either @xmath88 or @xmath90 fields is completely defined by the knowledge of the green function and propagators such as @xmath206 ( see eq . [ 106 ] ) with the causal function @xmath93 given by the bromwich fourier integral @xmath207 . now , like for the susceptibility @xmath208 time reversal leads to a new propagator @xmath209 and therefore by a reasoning equivalent to the previous one to @xmath210 involving the complex conjugate @xmath211 with a fourier expansion @xmath212 which again involves the anticausal permittivity @xmath213 . this naturally leads to growing waves in the future direction and to green function spatially growing as@xmath214 . the full equivalence between the two representation of the total field @xmath215 is completed if first we observe that the initial conditions at @xmath29 ( i.e. , @xmath216 , @xmath217 , etc ... ) are now replaced by final boundary condition at @xmath218 ( i.e. , @xmath219 , @xmath220 , etc ... ) . second , since the @xmath191 value is arbitrary we can send it into the remote future if we want and we will have thus a evolution expressed in term of growing modes for all times @xmath221 . the value of the field at such a boundary is of course arbitrary , so that if we want the two representation can describe the same field if for the final field at @xmath222 we take the field evolving from @xmath29 using the usual causal evolution from past to future . this equivalence is of course reminiscent from the dual representations obtained using either retarded or advanced waves @xcite . indeed , the total electric field @xmath223 can always be separated into @xmath224 or equivalently @xmath225 where @xmath226 and @xmath227 label the retarded and advanced source fields respectively and @xmath228 and @xmath229 label the homogeneous ` free ' fields coming from the past and from the future respectively . this implies that only a specific choice of boundary conditions in the past or future can lead to a completely causal evolution and therefore time symmetry in not broken in the evolution equations but only through a specification of the boundary conditions . in other worlds , in agreement with the famous loschmidt and poincar objections to boltzmann , strictly deriving time irreversibility from an intrinsically time reversible dynamics is obviously impossible without additional postulates . this point was fully recognized already by boltzmann long ago and was the basis for his statistical interpretation of the second law of thermodynamics @xcite . + in order to further understand the implication of the huttner - barnett model @xcite we should discuss how the full hamiltionan finally reads in this formalism . this is central since the langevin noise equations developed by gruner and welsch @xcite use only the @xmath230 hamitonian . in the huttner - barnett model the full dynamics is determined by the complete knowledge of the dipole density @xmath231 so that the results of gruner and welsch @xcite should be in principle justifiable . however , if we are not careful , this will ultimately break unitary and time - symmetry . in order to understand the physical mechanism at work one should first clarify the relation existing between the hutner - barnett model , using @xmath231 as a fundamental field , and the historical hopfield - fano approach @xcite for defining discrete polariton modes as normal coordinate solutions of the full hamiltonian . + we remind that the historical method for dealing with polariton is indeed based on the pioneer work by fano and hopfield @xcite for diagonalizing the full hamiltonian . the procedure is actually reminiscent of the classical problem of finding normal coordinates and normal ( real valued ) eigen frequencies associated with free vibrations of a system of linearly coupled harmonic oscillators . the classical diagonalization method @xcite relies on the resolution of secular equations already studied by laplace . actually , the first ` semiclassical ' treatment made by born and huang @xcite is mathematically correct and equivalent to the one made later by fano and hopfield even though the relation between those formalisms is somehow hidden behind the mathematical symbols . the full strategy for finding normal coordinates and frequencies becomes clear if we use fourier transforms of the various fields for the problem under study . indeed , a fourier expansion of a field component @xmath232 will obviously define the needed harmonic expansion . in the problems considered by born , huang , fano , and hopfield , the fourier spectrum @xmath233 is a sum of dirac distributions @xmath234 peaked on the real valued eigenvibrations @xmath235 . this specific situation needs a complete discussion since the hopfield - fano model @xcite leads to an exact diagonalization of the full hamiltonian @xmath52 . this will in turn makes clear some fundamental relations with the laplace transform formalism used in the previous subsections . + we start with the fourier transformed equations @xmath236 @xmath237 and @xmath238 with the constraint @xmath239 , @xmath240 coming from the real valued nature of the fields . from eq . [ b ] we get using the properties of distributions : @xmath241\sqrt{\frac{2\sigma_\omega}{\pi}}\widetilde{\textbf{e}}(\omega)+\widetilde{\textbf{x}_\omega}^{(sym)}(\omega)\label{d}\end{aligned}\ ] ] with @xmath242\nonumber\\ \label{e}\end{aligned}\ ] ] and where we used the reality constraint and introduced constants of motions @xmath243 , @xmath244 which will becomes annihilation and creation operators in the second quantized formalism . the principal value can be conveniently written @xmath245=\frac{1}{\omega^2-(\omega+i0^+)^2}-\frac{i\pi}{2\omega}[\delta(\omega-\omega)-\delta(\omega+\omega)]$ ] or equivalently @xmath245=\frac{1}{\omega^2-(\omega - i0^+)^2}+\frac{i\pi}{2\omega}[\delta(\omega-\omega)-\delta(\omega+\omega)]$ ] . this leads to two different representations of eq . [ d ] : @xmath246 where @xmath247 and @xmath248 can also be written like eq . [ e ] , i.e. , respectively as @xmath249 $ ] or @xmath250 $ ] . this discussion is reminiscent of the different representation given in section iii d involving retarded , advanced or time symmetric modes . of course the usual causal representation is @xmath251 which is obtained from the definition of @xmath252 given in section iii at the limit @xmath133 . however , all the descriptions are rigorously equivalent . it is also easy to show that we have @xmath253 which justifies why we called this field symmetrical . it corresponds to a representation of the problem mixing boundary conditions in the future and the past in a symmetrical way like it was used for instance by feynman and wheeler in their description of electrodynamics @xcite as discussed in section iii d. + now after inserting the causal representation of eq . [ f ] in eq . [ c ] and then into eq . [ a ] we get @xmath254 with @xmath255 and @xmath256 this causal permittivity @xmath257 being given by the huttner - barnett model @xcite the secular equations @xmath258 for transverse modes have no root in the upper complex frequency half - plane and in particular on the real frequency axis ( the longitudinal term is discussed below ) . this means that , unlike eq . [ d ] , eq . [ g ] has in general no dirac term corresponding to independent eigenmodes . the electric field is thus represented by a source term @xmath259 obtained like in the previous subsection using a causal green function . the absence of free normal modes for the electric field is of course reminiscent from the rapid decay of the free modes @xmath88 when @xmath133 as discussed before . the representation given here does nt distinguish between transverse and longitudinal fields but this should naturally occur since we have the constraint @xmath260 which implies @xmath261 . together with eq . [ a ] we thus get @xmath262 eq . [ long ] is actually reminiscent of the charge screening by @xmath257 . here we used the fact that @xmath257 has no root on the real axis otherwise the imaginary part of the permittivity should vanish and the the medium would be lossless at the frequency @xmath263 a fact which is prohibited by physical consideration about irreversibility @xcite . this reasoning is rigorously not valid at @xmath264 since the imaginary part of the permittivity is a odd function on the real axis . but then in general to have a root at @xmath264 it would require that the real part of the permittivity vanishes as well and this is not allowed from usual permittivity model ( see eq . [ b4 ] ) which makes therefore this possibility very improbable . + the previous reasoning is clearly formally equivalent to the ones obtained in the previous subsection and in both case the field @xmath265 or @xmath231 completely determine the electromagnetic evolution . still , there are exceptions for instance in the drude lorentz model with @xmath266 which forms the basis for the hopfield @xcite polariton model . this model is rigorously not completely lossless since we have @xmath267+\frac{i\pi\omega_p^2}{2\omega_0}(\delta(\omega-\omega_0)-\delta(\omega+\omega_0))$ ] corresponding to a singular absorption peak . + moreover , in this hopfield model @xcite , which is a limit case of the huttner barnett model @xcite , we get the exact evolution equation @xmath268 which in the case of the longitudinal modes leads to solving the secular equation @xmath269 . it has the solution @xmath270 where @xmath271 isthe longitudinal plasmon frequency . however relating this result to eq . [ long ] requires careful calculations since @xmath257 is here a highly singular distribution . indeed , applying eq . [ long ] will lead to find solutions of @xmath272 with @xmath273 with @xmath274 a longitudinal vector field . this singular @xmath251 field at @xmath275 seems to imply that polaritons are resonant at such frequency in contradiction with the result leading to @xmath276 for such polariton modes . now , due to the presence of absorption peaks at @xmath275 we have near this singular points @xmath277 where @xmath278 is supposed to be regular @xcite . moreover , outside this narrow absorption band the medium is effectively lossless and instead of eq . [ long ] we have @xmath279 which has the singular solution @xmath280 with @xmath281 a longitudinal vector field . importantly , from the hypothesis of regularity at @xmath282 we have @xmath283 and therefore @xmath284 which means that @xmath285 everywhere . the lorentz - drude model leads therefore to genuine longitudinal polaritons eigenfrequencies @xmath286 solutions of @xmath287 . we emphasize that the same result could be obtained using the laplace transform method . we have indeed @xmath288 with @xmath289 . therefore we can rewrite the longitudinal scattered field as @xmath290 . moreover since there is no longitudinal @xmath88 electric field and since @xmath291 we have @xmath292 which shows that the genuine longitudinal polariton oscillates at the frequency @xmath293 as expected . + the transverse polariton modes of the hopfield model are obtained in a similar way from eq . [ g ] with @xmath294 with @xmath295 a transverse vector field . as explained in the appendix g solving the problem with a plane wave expansion labeled by @xmath63 and @xmath130 leads again to a secular equation @xmath296 for the two transverse modes ( @xmath297 ) giving a quartic dispersion relation @xmath298 with two usual hopfield solutions @xmath299}}{\sqrt{2}}.\nonumber\\ \end{aligned}\ ] ] the two - modes field have now the structure @xmath300 with @xmath301 with @xmath302 an amplitude coefficient for the mode ( see appendix e and @xcite for a derivation ) . + one of the most important issue in the context of the hopfield model concerns the hamiltonian . indeed , in this model the full hamiltonian eq . [ 10 ] @xmath303 reads @xmath304.\label{polaritonhopfield}\end{aligned}\ ] ] if we isolate first the longitudinal term we get @xmath305\nonumber\\=\int d^3\mathbf{x}[\frac{(\partial_t\mathbf{p}_{||})^2+\omega_l^2\mathbf{p}_{||}^2}{2\omega_p^2}]=2\frac{\omega_l^2}{\omega_p^2}\int d^3\mathbf{x}\boldsymbol{\beta}^\ast\boldsymbol{\beta}\end{aligned}\ ] ] we can of course introduce a fourier transform of the dipole field as @xmath306 and new polariton fields amplitudes @xmath307 . we thus have @xmath308 . this expression of the hamiltonian is standard for normal coordinates expansion in linearly coupled harmonic oscillators . + furthermore , using commutators like eq . [ 33 ] one deduce @xmath309=i\hbar\delta^3(\mathbf{x}'-\mathbf{x})\textbf{i}$ ] and other similar ones . in the fourier space we thus obtain @xmath310=i\hbar\delta_{\alpha,\beta}$ ] which lead after straightforward transformation to the commutators @xmath311=\delta_{\alpha,\beta}$ ] , @xmath312=[f_{\alpha,||}^\dagger(t),f_{\beta,||}^\dagger(t)]=0.$ ] with @xmath313 ( the time dependence in the heisenberg picture means @xmath314 ) . this naturally leads to the hopfield - fano hamiltonian expansion for longitudinal polaritons : @xmath315 a similar analysis can be handled for the transverse polaritons modes but the calculation is a bit longer ( see appendix e ) . to summarize this calculation in few words : using a fourier expansion of the different primary transverse field operators in eq . [ polaritonhopfield ] we get after some manipulations the hopfield - fano expansion @xcite @xmath316 with the operator @xmath317 obeying the usual commutation rules for rising and lowering operators . again this result is expected in a modal expansion using normal coordinates and again the same result could be alternatively obtained using the laplace transform method . to summarize , the approach developed previously using the laplace transform formalism agrees with the normal coordinates methods based on the fourier expansion in the frequency domain . both approaches lead to the conclusion that for an homogeneous medium the various electromagnetic and material fields are completely determined by the knowledge of the matter oscillating dipole density @xmath318 ( fourier s method ) or @xmath231 ( laplace s method ) . in the limit @xmath133 both approach are equivalent and there is no contribution of the free field in a homogenous dissipative medium ( the residual @xmath319,@xmath320 is exponentially damped in the regime @xmath133 ) . we also showed that if losses in the huttner barnett model are sharply confined in the frequency domain we can find exact polaritonic modes which agree with the historical method developed by fano and hopfield @xcite . these modes fully diagonalize the hamitonian @xmath52 . while we focused our study on the particular drude lorentz model the result is actually generalizable @xcite to homogenous media with conductivity @xmath321 which lead to a permittivity @xmath322+i\frac{\sigma(\omega)}{\omega}.\label{dangerous}\end{aligned}\ ] ] in particular the hamiltonian can in these special cases be written as a sum of harmonic oscillator terms corresponding to the different longitudinal and transversal polariton modes . we thus write @xmath323 and @xmath324 where @xmath131 and @xmath325 label the discrete longitudinal and transverse polaritons modes . however , for a more general huttner - barnett model where the permittivity is only constrained by kramers - kronig relations such a simple interpretation is not possible and the hamiltonian is not fully diagonalized . the additional physical requirement @xcite imposing that the imaginary part of the permittivity should be rigorously positive valued , i.e. , @xmath326 , also prohibits these exceptional cases which should therefore only appear as ideal limits with loss confined in infinitely narrow absorption bands . however as we will show in the next subsection the lossless idealization represents a good approximation for a quite general class of medium with weak dissipation . the previous results of hopfield and fano have still a physical meaning and are for example used with success for the description of planar cavity polaritons @xcite . + it is particularly relevant to consider what happens in the huttner barnett approach if we relax a bit the demanding constraints of the original hopfield - fano model based on eq . [ dangerous ] . for this we consider a medium with low loss such as the medium can be considered with a good approximation as transparent in a given spectral band where the field is supposed to be limited . this approach was introduced by milonni @xcite and is based on the hamiltonian obtained long ago by brillouin @xcite and later by landau and lifschitz @xcite for dispersive but slowly absorbing media . the main idea is to replace the electromagnetic energy density @xmath327 in the full hamiltonian by a term like @xmath328 where @xmath329 is the approximately real valued permittivity of the field at the central pulsation @xmath330 with which the wave - packet propagates . since this approach has been successfully applied to quantize polaritons @xcite or surface plasmons @xcite it is particularly interesting to justify it in the context of the more rigorous huttner - barnett approach developed here . in the mean time this will justify the use of hopfield - fano approach as an effective method applicable for the low loss regime which is a good assumption in most dielectric ( excluding metals supporting lossy plasmon modes ) . + from poynting s theorem , it is usual in macroscopic electromagnetism to isolate the work density @xmath331 such as the energy conservation reads @xmath332 by direct integration we thus get the usual formula for the time derivative of the total energy @xmath333 such as : @xmath334 which cancels if the fields decay sufficiently at spatial infinity ( assumption which will be done in the following ) as it can be proven after using the poynting vector divergence and stokes theorem . we now consider a temporal integration window @xmath335 to compute the average derivative @xmath336 where @xmath337 means the magnetic energy variation during the time @xmath335 and @xmath338=\int_{t}^{t+\delta t}dt'[ ... ]$ ] is an integration domain from an initial time @xmath75 to a final time @xmath339 . the next step is to fourier expand the field in the frequency domain and we write @xmath340 where the positive frequency part of the field is defined as @xmath341 ( the negative frequency part is then @xmath342^\dagger$ ] ) . we use similar notation for the displacement field and we introduce the fourier field @xmath343 . in order to achieve the integration eq . [ energybri ] the temporal window will be supposed sufficiently large compared to the typical period @xmath344 of the light pulse . this allows us to simplify the calculation and most contributions cancel out during the integration @xcite . additionally to perform the calculation we assume that we have @xmath345 . this is a usual formula in classical physics where the term @xmath346 is supposed equal to zero , but here we are dealing with a quantized theory and we can not omit this term . furthermore , we showed that in the huttner - barnett model when @xmath133 only the @xmath90 contributions discussed in section iii c remain @xcite . assuming therefore this regime the transverse field @xmath347 , and @xmath348 are fully expressed as a function of operators @xmath162 . [ rehhd ] and [ rehhe ] allow us to define the fourier fields : @xmath349 @xmath350 for @xmath351 ( for @xmath352 we have @xmath353 where @xmath354 is given by eq . [ glopglop ] at the positive frequency @xmath355 . similar symmetries and properties hold for the electric and magnetic fields ) . we thus see that the relation @xmath356 is approximately fulfilled if we consider only frequencies near a resonance at @xmath357 ( the spectral distribution in eq . [ glopglop ] is thus extremely peaked since losses are weak ) . this dispersion relation occurs for transverse polaritons modes @xmath358 ( neglecting the imaginary part ) and if the wave packet of spectral extension @xmath359 is centered on such a wavelength we can replace with a good approximation @xmath360 by @xmath361 in the numerator of eq . [ glopglop ] leading thus to @xmath356 . after this assumption the calculation can be done like in classical textbooks @xcite and eq . [ energybri ] becomes ( this usual calculation will not be repeated here ) : @xmath362)=0 , \nonumber\\ \label{energytri}\end{aligned}\ ] ] where @xmath330 denotes now the transverse polariton frequency @xmath363 $ ] for the homogeneous medium considered here . in this formula the imaginary part of @xmath329 is systematically neglected in agreement with the reasoning discussed for instance in ref . alternatively eq . [ energytri ] could be rewritten using the time average @xmath364 in order to get the classical brillouin formula for the electric energy density in the medium but this will not be useful here . moreover , we introduce the mode operators : @xmath365 where @xmath366 is a frequency window centered on the polariton pulsation @xmath363:=\omega'_{\alpha , m}$ ] . now , if we suppose that the electromagnetic field is given by a sum of such transverse modes ( without overlap of the frequency domains @xmath366 ) then eq . [ energytri ] reads : @xmath367e_{\alpha , j , m}^{(-)}e_{\alpha , j , m}^{(+)})=0 , \label{energyquadri}\end{aligned}\ ] ] where the contribution @xmath368 arises from a modal expansion of the magnetic field and from using the resonance condition in the numerator of eq . [ rehhb ] ( which involves @xmath369 ) . + what is also fundamental here is that we have the commutators ( the derivation in the complex plane in given in appendix f ) : @xmath370=\delta_{\alpha,\beta}\delta_{j , l}\delta_{m , n}\frac{\hbar\omega_{\alpha , m}}{2}\frac{d\omega_{\alpha , m}^2}{d\omega_\alpha^2}.\nonumber\\ \label{window}\end{aligned}\ ] ] and @xmath371=[e_{\alpha , j , m}^{(-)}(t),e_{\beta , l , n}^{(-)}(t)]=0 $ ] . these relation imply the existence of effective rising and lowering operators @xmath372 for polaritons defined by @xmath373 . + these relations were phenomenologically obtained by milonni @xcite after quantizing brillouin s energy formula . here we justify this result from the ground using the huttner - barnett formalism . importantly , after defining the optical index of the polariton mode @xmath374 we can rewrite @xmath375 as @xmath376 where @xmath377 is the group velocity of the mode defined by @xmath378 . this allows us to rewrite the operators as @xmath379 ( since @xmath380 ) and finally to have : @xmath381 the total energy is thus defined as @xmath382 which is a constant of motion defined up to an arbitrary additive constant . this formula involves only the transverse modes so that actually it gives the energy @xmath383 associated with the transverse polariton modes in weekly dissipative medium and represents a generalization of hopfield - fano results as an effective but approximative model . + few remarks are here necessary . first , the model proposed here relies on the assumption that the fields is a sum of wave packets spectrally non overlapping . this hypothesis which was also made by garrison and chio @xcite was then called the ` quasi multimonochromatic ' approximation . this assumption is certainly not necessary since milonni s model includes as a limit the rigorous hopfield- fano model @xcite which does nt rely on such an assumption . in order to justify further milonni s approach @xcite and relax the hypothesis made it is enough to observe first , that in eq . [ glopglop ] the approximation @xmath356 is quite robust even if the fields is spectrally very broad . indeed , since losses are here supposed to be very weak the resonance will practically cancel out if @xmath14 differs significantly of a values where the condition @xmath357 occurs . second , if we insert formally eqs . [ glopglop ] and eq . [ reglop ] with the previous assumption into eq . [ energytri ] then instead of the term @xmath384 in eq . [ energytri ] we get a term @xmath385 . this contribution is in general more complicated because @xmath386 depends on @xmath14 . however , using explicitly eqs . [ glopglop ] and eq . [ reglop ] and specially the fourier expansion in plane waves we see that for the specific fields considered here eq . [ energyquadri ] still holds . this means that we can again introduce polariton operators @xmath387 , and @xmath388 defined by eq . [ modepolar ] . as previously these operators depend on a frequency window @xmath366 and here these are introduced quite formally for taking into account the fact that the resonance @xmath389 is extremely peaked near the different polariton frequencies @xmath390 . a product like @xmath391 occurring in the integration will thus not contribute unless the frequency @xmath392 and @xmath14 are in a given window @xmath366 . [ energyquadri ] thus results as a very good practical approximation . + remarkably , as observed in eq . [ window ] ( and explained in appendix e ) the commutator does not depend explicitly on the size of these windows ( which are only supposed to be small compared to the separation between the different mode frequencies and large enough to include the resonance peaks as explained in appendix e ) . therefore , this allows us to renormalize these operators as before by introducing the same rising and lowering polariton operators @xmath393 such as eq . [ energy5 ] and @xmath394 hold identically . we thus have completed the justification of the milonni s approach for dielectric medium with weak absorption @xcite . + an other remark concerns the longitudinal electric field which was omitted here since it does not play an active role in pulse propagation through the medium . we have indeed @xmath395 so that the reasoning was only done on the transverse modes . however , this was not necessary and one could have kept the longitudinal electric field all along the reasoning . since the transverse part is a constant as we showed before , this should be the case for the longitudinal part as well since @xmath396 is also an integral of motion . now , a reasoning similar to the previous one for transverse waves will lead to the brillouin formula for the longitudinal electric energy : @xmath397 where the longitudinal polariton modes are defined by : @xmath398 in this formalism the longitudinal polariton frequencies @xmath399 are the solutions of @xmath400 ( where losses are again supposed to be weak ) . the commutator can be defined using a method equivalent to eq . [ window ] and we get : @xmath401=\delta_{\alpha,\beta}\delta_{m , n}\frac{\hbar}{|m_{\alpha , m}|}. \label{windowlong}\end{aligned}\ ] ] with @xmath402 . after defining the lowering polariton operator as @xmath403 we thus obtain : @xmath404 as it should be . s approach @xcite leads therefore to an effective justification of longitudinal polaritons as well and this includes the hopfield - fano @xcite model as a limiting case when losses are vanishing outside infinitely narrow absorption bands . + a final important remark should be done since it concerns the general significance of the scattered field ( s ) in the lossless limit . indeed , we see from eq . [ modepolar ] that the transverse mode operators @xmath405 and @xmath405 rigorously vanish in the limit @xmath406 ( this is not true for longitudinal operators . [ modepolarlong ] which are physically linked to bound and coulombian fields ) . in agreement with the sub - section iii c we thus conclude that in the vacuum limit one should consider the @xmath88 fields as the only surviving contribution . however , we also see that for all practical needs if the losses are weak but not equal to zero then by imposing @xmath407 the @xmath88 terms should cancel and only will survive a scattered term which will formally looks as a free photon in a bulk medium with optical index @xmath408 . therefore , we justify the formal canonical quantization procedure used by milonni and others @xcite which reduces to the historical quantization methods in the ( quasi- ) non dispersive limit @xcite . however , this can only be considered as an approximation and therefore the original claim presented in ref . @xcite that the scattered field @xmath90 is sufficient for justifying the exact limit @xmath406 without the @xmath88 term was actually unfounded . as we will see this will become specially relevant when we will generalize the langevin noise approach to an inhomogeneous medium . the central issue in this work is to interpret the physical meaning of quantized polariton modes in the general huttner barnett framework of section ii and this will go far beyond the limiting hopfield - fano @xcite or milonni s approaches @xcite which are valid in restricted conditions when losses and/or dispersion are weak enough . for the present purpose we will focus on the homogeneous medium case ( the most general inhomogeneous medium case is analyzed in the next section ) . it is fundamental to compare the mode structure of eqs . [ rehhd ] , [ rehhe ] , [ rehhepar ] , and [ rehhb ] on the one side and the mode structure of eqs . [ 118 ] , [ 119 ] and [ 120 ] on the other side which are associated respectively with the free modes ` ( 0 ) ' and the scattered modes ` ( s ) ' . the ` ( 0 ) ' modes are the eigenstates of the classical propagation problem when we can cancel the fluctuating term @xmath38 . this is however not allowed in qed since we are now considering operators in the hilbert space and one can not omit these terms without breaking unitarity . inversely the scattered modes are the modes which were considered by huttner and barnett @xcite . moreover only these ` ( s ) ' modes survive here if the initial time @xmath29 is sent into the remote past , i.e. , if @xmath407 . for all operational needs it is therefore justified to omit altogether the ` ( 0 ) ' mode contribution in the homogeneous medium case . clearly , this was the choice made by gruner and welsch @xcite and later by more or less all authors working on the subject ( see however refs . @xcite ) which accepted this rule even for non - homogeneous media . if we accept this axiom then the hamiltonian of the problem seems to reduce to @xmath409 , i.e. in the homogeneous medium case , to @xmath410 which depends only on the fluctuating operators @xmath411 , @xmath412 in agreement with the langevin force / noise approach of gruner and welsch @xcite . + now , there is apparently a paradox : the complete hamiltonian of the system is in agreement with eq . [ 10 ] given by @xmath413 can we show that @xmath52 is actually equivalent to @xmath230 ? this is indeed the case for all practical purposes at least for the homogeneous medium case treated by huttner and barnett@xcite that we analyzed in details before . to see that remember that @xmath52 is actually a constant of motion therefore we should have @xmath414 . this equality reads also @xmath415 . now , the central point here is to use the boundary condition at time @xmath29 which imply @xmath416 . furthermore , since the time evolution of @xmath417 is harmonic we have @xmath418 . altogether these relations imply @xmath419 so that we have : @xmath420 in other words we get a description in which the fock number states associated with the fluctuating operators @xmath177 or equivalently @xmath162 , @xmath411 diagonalize not the full hamiltonian but only a part that we noted @xmath230 . however , this is not problematic since the remaining term @xmath421 is also clearly by definition a constant of motion since it only depends on fields at time @xmath29 . how can we interpret this constant of motion ? we can clearly rewrite it as @xmath422 . therefore , in the @xmath423 potential vector formalism defined in section ii , this constant depends on both @xmath424 and @xmath425 , i. e. , lowering and rising operators associated with the transverse @xmath426 and @xmath427 fields , and it also depends on the operators @xmath428 , @xmath429 which are associated with the fluctuating dipole distribution at the initial time @xmath29 . furthermore since we have also @xmath430 , @xmath431 , @xmath432 and since @xmath433 , @xmath434 we can alternatively write : @xmath435 this means that the remaining term @xmath436 depends on the knowledge of @xmath437 and @xmath438 electromagnetic free field . since in the limit @xmath133 these @xmath88 electromagnetic terms vanish at any finite time @xmath75 this would justify to consider @xmath436 as an inoperative constant . however , we have also the contribution of @xmath439 which play a fundamental role in the determination of @xmath440 , @xmath441 and @xmath442 at the finite time @xmath75 . therefore , while @xmath436 is a constant of motion it nevertheless contains quantities which will affect the evolution of the surviving @xmath90 fields at time @xmath75 . it is for this reason that we can say that for ` all practical purposes only ' @xmath436 is unnecessary and that @xmath230 is sufficient for describing the energy problem . + there are however many remarks to be done here concerning this analysis . first , while the hamiltonian @xmath230 gives a good view of the energy up to an additive inoperative constant it is the full hamiltonian which is necessary for deriving the equations of motions from hamilton s equations or equivalently from heisenberg s evolution like @xmath443 $ ] . it is also only with @xmath52 that time symmetry is fully preserved . in particular do not forget that in deriving the fano - hopfield @xcite formalism we introduced ( see eq . [ g ] ) the evolution @xmath444 which depends on the causal ` in ' field @xmath445 and on the causal permittivity @xmath257 . since the knowledge of @xmath446 is equivalent to @xmath91 in the limit @xmath447 the ( forward ) laplace transform method is leading to the same result that the usual fano method and this fits as well with the gruner and welsch langevin s equations @xcite . however , instead of eq . [ g ] we could equivalently use the anticausal equation @xmath448 where @xmath449 replaces @xmath449 and where the causal permittivity @xmath257 becomes now @xmath450 which is associated with amplification instead of the usual dissipation . the green integral equation now becomes @xmath451 where @xmath208 replaces @xmath452 ( see eq . [ 20bis ] ) and is associated with the anticausal dynamics which is connected to the time - reversed evolution . both formalism developed with either ` in ' or ` out ' fields are completely equivalent but this shows that we have the freedom to express the scattered field in term of @xmath453 , and @xmath454 or in term of @xmath455 , and @xmath456 . since there is in general no fully propagative ` free ' electromagnetic modes ( if we exclude the lossless medium limit considered by hopfield and fano @xcite ) then the surviving fields at finite time @xmath75 obtained either with @xmath133 or @xmath457 correspond to decaying or growing waves in agreement with the results discussed for the laplace transform methods . the full hamiltonian @xmath52 is thus expressed equivalently either as @xmath458 ( with @xmath459 the fluctuating field defined in section ii ) or as @xmath460 with @xmath461 the fluctuating field using using a final boundary condition at time @xmath222 . in the limits @xmath133 , @xmath457 this leads to @xmath462 or @xmath463 where the remaining terms ( see eq . [ rema ] ) @xmath464 and @xmath465 are two different integrals of motion . therefore , it shows that the representation chosen by gruner and welsch in ref . @xcite is not univocal and that one could reformulate all the theory in terms of @xmath449 instead of @xmath445 to respect time symmetry . furthermore , we emphasize that while the two hamiltonians @xmath466 and @xmath467 have formally the same mathematical structure they are not associated with the same physical electromagnetic fields since eq . [ glopiglopa ] and eq . [ outoftime ] corresponds respectively to decaying and growing radiated fields . causality therefore requires to make a choice between two different representations . it is only the choice on a boundary condition in the remote past or future together with thermodynamical considerations which allow us to favor the decaying regime given by eq . [ glopiglopa ] . + a different but related point to be discussed here concerns the hopfield - fano limit @xcite for which the general conductivity like @xmath468 leads to quasi - lossless permittivity ( see eq . [ dangerous ] ) . in this regime we found that an exact diagonalization procedure can be handled out leading to genuine transverse and longitudinal polaritons . these exceptions also fit with the time - symmetry considerations discussed previously since the quasi - absence of absorption makes the problem much more time symmetrical that in the cases where the only viable representations involve either @xmath449 or @xmath445 . of course the hopfield - fano model is an idealization which , in the context of the huttner - barnett framework , gets a clear physical interpretation only as an approximation for low - loss media , as explained in section iii g. + this leads us to a new problem , which is certainly the most important in the present work : considering the vacuum limit @xmath469 discussed after eq . [ 120 ] we saw that only the @xmath88 electromagnetic modes survive in this regime and that the @xmath90 modes are killed together with @xmath470 . these vacuum modes are completely decoupled from the undamped mechanical oscillators motion @xmath471 . in this limit time symmetry is of course respected and we see that the full set of eigenmodes diagonalizing the hamiltonian corresponds to the uncoupled free photons and free mechanical oscillator motions . an other way to see that is to use again the fourier formalism instead of the laplace transform method . from eq . [ g ] we see that in the vacuum limit it is not the scattered field defined by eq . [ glopiglopa ] which survives ( since @xmath470 ) but an additional term corresponding to free space photon modes . while this should be clear after our discussion this point has tremendous consequences if we want to generalize properly the huttner - barnett approach to an inhomogeneous medium . in such a medium the permittivity @xmath472 is position dependent . now , generally speaking in nanophotonics we consider problems where a dissipative object like a metal particle is confined in a finite region of space surrounded by vacuum . the susceptibility @xmath42 therefore vanishes outside the object and we expect electromagnetic vacuum modes associated with free space photon to play a important role in the final analysis . this should contrast with the huttner - barnett case for homogenous medium which supposes an unphysical infinite dissipative medium supporting bulk polaritons or plasmons . the inhomogeneous polariton case will be treated in the next section . in order to deal with the most general situation of polaritons in inhomogeneous media we need first to consider the formal separation between source fields and free - space photon mode . the separation used here is clearly different from the one developed in the previous section since we now consider on the one side as source term the total polarization @xmath473 ( see eq . [ 18 ] ) which includes both the fluctuating term @xmath30 but also the induced polarization @xmath35 and on the other side as source - free terms some general photon modes solutions of maxwell equations in vacuum . in order not to get confused with the previous notations we now label @xmath474 the vacuum modes and @xmath475 the modes induced by the total dipolar distribution @xmath473 . + we start with the second order dynamical equation @xmath476 which can be solved using the method developed for eq . [ 90 ] . indeed , by imposing @xmath477 in eq . [ 90 ] and by replacing @xmath38 by @xmath39 , @xmath134 by @xmath478 , @xmath479 by @xmath480 and so on in the calculations of section iii we can easily obtain the formalism needed . consider first the vacuum fields @xmath478 , @xmath481 and @xmath482 . from eqs . [ 118 ] , [ 119 ] in the limit @xmath126 we get a plane - wave modal expansion for the free photon field which we write in analogy with eq . [ 57b ] as @xmath483 with the modal expansion coefficients @xmath484 showing the harmonic structure of the fields . of course this transverse vacuum field satisfies maxwell s equations without source terms and in particular @xmath485 with @xmath486 . + we now study the scattered - fields @xmath487 , @xmath488 and @xmath489 . like for the calculations presented in section iii for the homogeneous medium case it appears convenient to use the green dyadic formalism which is well adapted for nanophotonics studies in particular for numerical computation of fields in complex dielectric environment where no obvious spatial symmetry are visible . starting with this formalism we get for the @xmath475 electric field : @xmath490 equivalently , by using the inverse laplace transform ( see appendix d for details ) @xmath491 , we obtain in the time domain @xmath492 this analysis implies @xmath493 and therefore we have @xmath494 . + moreover , most studies consider instead of the propagator @xmath495 the dyadic green function @xmath496 which is a solution of @xmath497 and which is actually connected to @xmath495 by @xmath498 . this dyadic green function is very convenient since practical calculations very often involve not the displacement field @xmath426 but the electric field @xmath499 . we obtain : @xmath500 alternatively in the time domain we have for the scattered field @xmath501 : @xmath502 this can rewritten by using the propagator @xmath503 together with eq . [ definiti ] as : @xmath504 finally , we can also use the dyadic formalism to represent the magnetic field @xmath505 . starting from eq . [ 6 ] which yields @xmath506 and therefore @xmath507 where we introduced the definition @xmath508 ( here we used the condition @xmath509 ) . in the time domain we thus directly obtain @xmath510 which yields @xmath511 as expected . + we emphasize here once again the fundamental role played by the boundary conditions at @xmath29 . what we showed is that at the initial time @xmath29 we have @xmath493 and thus @xmath512 which means , by definition of our vacuum modes , @xmath513 . in other words , the electric field associated with vacuum modes equals the total displacement fields at the initial time . this is interesting since it also implies @xmath514 ( which means @xmath515 ) . this can be written after separation into transverse and longitudinal parts as @xmath516 for the transverse ( solenoidal or divergence- free ) components and @xmath517 for the longitudinal ( irrotational or curl - free ) components . . [ gourbi ] is well known in qed since it rigorously agrees with the definition of the longitudinal field obtained in usual coulomb gauge using the @xmath3 potential instead of @xmath423 . untill now we did nt specify the form of the dipole density @xmath473 . the separation between source and free terms for the field was therefore analyzed from a microscopic perspective where the diffracted fields @xmath518 and @xmath505 was generated by the full microscopic current . in order to generalize the description given in section iii for the homogeneous medium we will now use the separation eq . [ 18 ] of @xmath473 into a fluctuating term @xmath30 and an induced contribution @xmath35 of essentially classical origin . using the laplace transform we get @xmath519 . now from the previous section we have therefore for the laplace transform of the electric field the following lippman - schwinger integral equation @xmath520.\nonumber\\ \label{149}\end{aligned}\ ] ] in order to get a meaningful separation of the total field we here define @xmath521 and @xmath522 we have clearly @xmath523 and @xmath524 this allows us to interpret @xmath142 as the green function of the inhomogeneous dielectric medium while @xmath525 is a free solution of maxwell s equation in the dielectric medium in absence of the fluctuating source @xmath153 . by direct replacement of eqs . [ 150 ] and [ 151 ] into [ 149 ] one get @xmath526 meaning that the total field can be seen as the sum of the free solution @xmath525 and of scattering contribution @xmath527 induced by the fluctuating source @xmath153 . + in the time domain we get for the electric field @xmath528 and for the magnetic field @xmath529 which are completed by the constitutive relation : @xmath530 . similarly we obtain for the new propagator : @xmath531\nonumber\\ -\int_{0}^{t}d\tau'\chi(\mathbf{x},\tau')\mathbf{u}_\chi(t-\tau',\mathbf{x},\mathbf{x''})\nonumber\\ \label{156}\end{aligned}\ ] ] we have the important properties @xmath532 , @xmath533 . the total electromagnetic field in the time domain is thus expressed as : @xmath534 @xmath535 with @xmath536 , @xmath537 finally , the constitutive relation for the scattered displacement field @xmath538 reads : @xmath539 . the boundary conditions at @xmath29 together with the field equations determine the full evolution and we have : @xmath540 we point out that the description of the longitudinal field should be treated independently in this formalism since at any time @xmath75 we have the constraint @xmath541 which shows that fluctuating current and field are not independent . more precisely , if we insert the constraint @xmath541 into the lagrangian formalism developed in section ii we get a new effective lagrangian density for the longitudinal field which reads : @xmath542 from eq . [ 2bb ] we get the euler - lagrange equation : @xmath543 with @xmath544 and which agrees with eq . [ 8 ] if the constraint @xmath541 is used . now , the formal solution of eq . [ 8b ] is obtained from eq . [ 17bb ] @xmath545 and allows for a separation between a fluctuating term @xmath546 and a source term @xmath547 . from this we naturally deduce @xmath548 integral eqs . [ 17bb ] or [ dde ] could in principle be solved iteratively in order to find expressions @xmath549 and @xmath550 which are linear functionals of @xmath551 and @xmath552 . alternatively , this can be done self - consistently using the laplace transform of eq . [ dde ] which reads @xmath553 and leads to : @xmath554 in the time domain we have thus @xmath555 with the effective susceptibility defined as@xmath556 we have equivalently for the effective susceptibility @xmath557 . after closing the contour in the complex plane we get @xmath558 where the sum is taken over the longitudinal modes solutions of @xmath559 ( with @xmath560 and @xmath561 by definition ) . the frequencies considered here are in general spatially dependent since the medium is inhomogeneous and are therefore very often difficult to find . in the limit of the homogeneous lossless medium we obtain the hopfield - fano @xcite model . + we emphasize that the present description of the polariton field contrast with the integral solution of eq . [ g ] @xmath444 which was obtained for the homogeneous medium : @xmath562 and which included only a scattering contribution @xmath90 due to the cancellation of the @xmath88 term for @xmath133 . here , we can not neglect or cancel the @xmath88 mode solutions since in general the medium is not necessary lossy at spatial infinity . this will be in particular the case for all scattering problems involving a localized system such as a metal or dielectric antenna supporting plasmon - polariton localized modes . however , mostly all studies , inspirited by the success of the huttner - barnett model @xcite for the homogeneous lossy medium , and following the langevin - noise method proposed gruner and welsch @xcite , neglected or often even completely omitted the contribution of the @xmath88 modes . still , these @xmath88 modes are crucial for preserving the unitarity of the full matter field dynamics and can not be rigorously omitted . only in those case where absorption is present at infinity we can omit the @xmath88 modes . + to clarify this point further consider a medium made of a spatially homogeneous background susceptibility @xmath563 and of a localized susceptibility @xmath564 such as @xmath565 at spatial infinity . the electromagnetic field propagating into the medium with total permittivity @xmath566 can be thus formally developed using the lippeman - schwinger equation as : @xmath567.\nonumber\\ = \overline{\mathbf{e}'}^{(1)}(\mathbf{x},p ) -\int d^3\mathbf{x ' } \frac{p^2}{c^2}\mathbf{g}_{\chi_1}(\mathbf{x},\mathbf{x'},ip)\nonumber\\ \cdot [ \bar{\chi_2}(\mathbf{x}',p)\overline{\mathbf{e}'}(\mathbf{x},p)+\overline{\mathbf{p}'}^{(0)}(\mathbf{x'},p)].\nonumber\\\label{154bb}\end{aligned}\ ] ] where we have defined a new background free field @xmath568 and a new green dyadic tensor for the background medium @xmath569 we have naturally @xmath570 and similarly @xmath571 . + importantly , in the time domain we can write : @xmath572\nonumber\\ -\mathbf{p}^{(0)}(\mathbf{x},t)+\int_{0}^{t}d\tau\chi_2(\mathbf{x},\tau)\textbf{e}(\mathbf{x},t-\tau).\nonumber\\ \label{15bbb}\end{aligned}\ ] ] we can check that we have @xmath573 . moreover , since from eq . [ 150b ] ( written in the time domain ) we have @xmath574 and since @xmath575 , we deduce that at the initial time @xmath29 @xmath576 as it should be to agree with the general formalism presented in section iv a. + now , since the background dissipative medium is not spatially bound the @xmath577 field associated with damped modes will vanish if @xmath133 as explained before . we could therefore be tempted @xcite to eliminate from the start @xmath577 and thus get in the laplace transform language the effective formula : @xmath578\nonumber\\ = -\int d^3\mathbf{x ' } \frac{p^2}{c^2}\mathbf{g}_{\chi}(\mathbf{x},\mathbf{x'},ip)\cdot\overline{\mathbf{p}'}^{(0)}(\mathbf{x'},p ) ] . \label{154bbb}\end{aligned}\ ] ] with the total green dyadic function @xmath579 obeying to eq . [ 152 ] with the total permittivity @xmath566 . it is straightforward to check that @xmath580 satisfies also eq . [ 151 ] so that it is the same green function . + however , removing @xmath577 from the start in eq . [ 15bbb ] would mean that the boundary conditions at the initial time @xmath29 have been obliviated since we should now necessarily have @xmath581 . this corresponds to a very specific boundary condition which is certainly allowed in classical physics ( where we can put @xmath582 ) but which in the quantum formalism means that we break the unitarity of the evolution . to say it differently , it means that in the langevin noise formalism @xcite the photon field @xmath423 is not anymore an independent canonical contribution to the evolution since all electromagnetic fields are induced by the material part . the green formalism presented by gruner and welsch @xcite , and abundantly used since @xcite , represents therefore an alternative theory which rigorously speaking is not equivalent , contrary to the claim in refs . @xcite , to the lagrangian formalism discussed in section ii for the general huttner - barnett model @xcite . our analysis , as already explained in the introduction , agrees with the general studies made in the 1970 s and 1980 s in qed @xcite since one must include with an equal footings both the field and matter fluctuations in a self consistent qed in order to preserve rigorously unitarity and causality . + two issues are important to emphasize here . first , observe that in the limit where the background susceptibility @xmath583 vanishes then the term @xmath584 in general does not cancel at any time , and therefore the coupling to photonic modes can not be omitted even in practice from the evolution at finite time @xmath75 . this is is particularly important in nanophotonics where an incident exiting photon field interact with a localized nano - antenna . it is therefore crucial to analyze further the impact of our findings on the quantum dynamics of polaritons in presence of sources such as quantum fluorescent emitters . this will the subject of a subsequent article . + the second issue concerns the hamiltonian definition in the new formalism . indeed , the definition of the full system hamiltonian @xmath52 was previously given for the homogeneous medium case in section iii g. we showed ( see eq . [ 158o ] ) that @xmath52 is given by @xmath585 where @xmath230 is the material hamiltonian defined in eq . [ 30c ] and which depends only on the free mode operators @xmath156 , @xmath157 . this @xmath586 is the hamiltonian considered by the noise langevin approach and the remaining term ( see eq . [ rema ] ) @xmath587 is an additional constant of motion . this constant proved to be irrelevant for all practical purposes since the only surviving electromagnetic fields ( i.e. if @xmath133 ) are the induced @xmath90 modes which are generated by the fluctuating dipole density @xmath30 ( see however the different remarks concerning time symmetry at the end of section iii g ) . now , for the inhomogeneous problem the complete reasoning leading to eq . [ 158ob ] is still rigorously valid . the main difference being that in general the constant of motion @xmath587 is not irrelevant at all since the @xmath88 electromagnetic modes are not in general vanishing even if @xmath133 . + in order to clarify this point we should now physically interpret the term @xmath436 . we first start with the less relevant term in optics : the longitudinal polariton . indeed , the longitudinal field is here decoupled from the rest and evolves independently using the lagrangian density @xmath588 defined in eq . [ 2bb ] ( this should not be necessarily true if the polaritons are coupled to external sources such as fluorescent emitters ) . we thus get the following hamiltonian @xmath589:.\nonumber\\ \label{2c}\end{aligned}\ ] ] @xmath590 is a constant of motion and can used ( with the hamiltonian formalism ) to deduce the evolution equation ( see eq . [ 8b ] ) and the solution eq . [ 17bb ] . since @xmath590 is a constant of motion we have @xmath591 and from the form of the solution we obtain the equivalent formula @xmath592 . \label{2d}\end{aligned}\ ] ] where we have clearly by definition @xmath593and thus @xmath594 ( with @xmath595 ) . while @xmath596 is a constant of motion it is not irrelevant here since the equivalence of eq . [ 2c ] and eq . [ 2d ] leads to the complete solution eq . [ 17bb ] for @xmath597 . oppositely , taking @xmath598 and omitting @xmath596 would lead to the free solution @xmath599 in contradiction with the dynamical law . this again stresses the importance of keeping all contributions in the evolution and hamiltonian . + in order to analyze the transverse field hamiltonian we should comment further on the difference between the description using the @xmath423 potential used in this work and the most traditional treatment using the * a * potential ( in the coulomb gauge ) . indeed , by analogy with the separation between @xmath475 and @xmath474 modes discussed in section vi a we can using the * a * potential vector representation get a separation between free - space modes @xmath600 and source field @xmath601 . here , we label these modes by an additional prime for reasons which will become clear below . first , the source field contribution @xmath602 is given by @xmath603 and with by definition @xmath604 . importantly we have @xmath605 meaning also @xmath606 . the free - space modes @xmath607 are easily obtained using a plane wave expansion as @xmath608 where @xmath609 . using the @xmath3 potential we therefore get for the free electromagnetic fields : @xmath610 these transverse fields satisfy maxwell s equation in vacuum like the free - fields given by eq . [ 121 ] do as well . however , it should now be clear that this two sets of free - fields given either by eq . [ 140nnn ] or eq . [ 121 ] are not equivalent . to see that we must express the source field @xmath611 . the longitudinal contribution @xmath612 is given by the usual instantaneous coulomb field @xmath613 since there is no other longitudinal contribution this leads to @xmath614 which is eq . [ gourbi ] . as mentioned already this is the usual result . however the most important term here is the transverse source field : @xmath615 . we get for this term @xmath616 which is also written as : @xmath617 equivalently we have in the frequency domain : @xmath618 \nonumber\\ \label{143}\end{aligned}\ ] ] this transverse scattered field vanishes at @xmath29 : @xmath619 and therefore at this initial time it differs by an amount @xmath620 from the scattered field given by eq . we thus get @xmath621 . importantly , by comparing eq . [ 136 ] and eq . [ 144 ] we obtain a relation for the free - space modes in the two representations using either the @xmath423 or @xmath3 potential vectors : @xmath622 this is reminiscent from the relation @xmath623 . it shows that while the two fields @xmath624 and @xmath625 are solutions of the same maxwell s equations in vacuum they are not defined by the same initial conditions . we must therefore be extremely careful when dealing with the modes in order not to get confused with the solutions chosen . we also mention that the scattered magnetic field @xmath626 is given by @xmath627 or equivalently by @xmath628 in the frequency domain this gives : @xmath629 a comparison with the formulas obtained in the subsection vi a shows that @xmath630 differs from @xmath505 but that at time @xmath29 both vanish so that @xmath631 . it implies that @xmath632 so that while @xmath633 differs from @xmath634 for @xmath635 they become equal at the initial time @xmath29 . again , this stresses the difference between the representations based either on @xmath423 or @xmath3 . + now , this description using @xmath3 leads to a clear interpretation of @xmath636 . indeed , at time @xmath29 only the @xmath607 solution survives for the transverse part of the field . importantly , the set of free - space solutions @xmath607 actually depends on lowering and rising operators @xmath637 , @xmath638 defined such that @xmath639=\delta_{\alpha,\beta}\delta_{j , k}$ ] and @xmath640 with @xmath641 . therefore , by a reasoning similar to the one leading to eq . [ 158o ] we deduce @xmath642 we clearly here get a physical interpretation of the remaining term @xmath643 as a energy sum over the transverse photon modes propagating in free space . these free photons are calculated using the @xmath3 potential vector . from eq . [ 145 ] we know that these modes differ in general from those in the @xmath423 potential vector since @xmath644 is not identical to @xmath645 unless the polarization density @xmath646 cancels ( which is the case in vacuum ) . + now , in classical physics the meaning of expansion eq . [ energytransverse ] is clear : it corresponds to a diagonalization of the hamiltonian in term of normal coordinates , i.e. , like for classical mechanics @xcite , and similarly to the huang , fano , hopfield procedure for polaritons @xcite . in qed the problem is different since , as explained in details in ref . @xcite fields like @xmath647 and @xmath648 ( and their hermitian conjugate variables ) do not commute , unlike it is for @xmath649 and @xmath648 . it is thus not possible to find common eigenstates of the @xmath650 operators for photons ( in the usual representation ) and for @xmath651 associated with the material fluctuations . this is not true for the representation using @xmath652 and @xmath651 operators but now the full hamiltonian is not fully diagonalized as seen from eq . [ 158ob ] . only if one neglect the remaining term @xmath436 , like it was done in refs . @xcite , can we diagonalize the hamiltonian . however , then we get the troubles concerning unitarity , causality and time symmetry discussed along this manuscript . the general formalism discussed in this article using the @xmath423 potential provide a natural way for dealing with qed in dispersive and dissipative media . it is based on a canonical quantization procedure generalizing the early work of huttner and barnett @xcite for polaritons in homogeneous media . the method is unambiguous as far as we conserve all terms associated with free photons @xmath653 and material fluctuations @xmath346 for describing the quantum evolution . in particular , in order to preserve the full unitarity and the time symmetry of the coupled system of equations we have to include in the evolution terms associated with fluctuating electromagnetic modes @xmath319 , @xmath320 which have a classical interpretation as polariton eigenmodes and can not in general be omitted if the medium is spatially localized in vacuum . we also discussed alternative representation based on the potential @xmath3 instead of @xmath423 . at the end both representations are clearly equivalent and could be used for generalizing the present theory to other linear media including tensorial anisotropy , magnetic properties , and constitutive equations coupling @xmath654 and @xmath655 ( magneto / electric media ) . moreover , the most important finding of the present article concerns the comparisons between the generalized huttner - barnett approach , advocated here , which involves both photonic and material independent degrees of freedom , and the langevin - noise method proposed initially by gruner and welsch @xcite which involves only the material degrees of freedom associated with fluctuating currents . we showed that rigorously speaking the langevin - noise method is not equivalent to the full hamiltonian qed evolution coupling photonic and material fields . only in the regime where the dissipation of the bulk surrounding medium is non vanishing at spatial infinity could we , i.e. , for all practical purposes , identify the two theories . however , even with such assumptions the langevin noise model is breaking time - symmetry since it considers only decaying modes while the full hamiltonian theory used in our work accepts also growing waves associated with anti - thermodynamic processes . we claim that this is crucial in nanophotonics / plasmonics where quantum emitters , spatially localized , are coupled to photonic and material modes available in the complex environment , e.g. , near nano - antennas in vacuum ( i.e. in a spatial domain where losses are vanishing at infinity ) . since most studies consider the interaction between molecules or quantum - dots and plasmon / polaritons using the langevin noise approach we think that it is urgent to clarify and clean up the problem by analyzing the coupling regime using the full hamiltonian evolution advocated in the present work . finally , we suggest that this work could impact the interpretation and discussion of pure qed effects such as the casimir force or the lamb shift which are strongly impacted by polariton and plasmon modes . all this will be the subject for future works and therefore the present detailed analysis is expected to play an important role in nanophotonics and plasmonics for both the classical and quantum regimes . this work was supported by agence nationale de la recherche ( anr ) , france , through the sinphonie ( anr-12-nano-0019 ) and placore ( anr-13-bs10 - 0007 ) grants . the author gratefully acknowledges several discussions with g. bachelier . the relation @xmath656 admits zeros @xmath100 as postulated in the text . writing @xmath45 one of such zero and @xmath657 the condition eq . [ b1 ] means : @xmath658 from which we deduce after eliminating @xmath659 @xmath660 therefore a necessary but not sufficient condition for having zeros is that if @xmath661 for such a zero then @xmath662 while if @xmath663 then @xmath662 . actually we also see from eq . [ b2 ] that the zero are allowed to be located along the real or imaginary axis of the complex @xmath14 plane if @xmath664 along these axes . this is in general not possible for a large class of permittivity function . consider for example the quite general causal permittivity , i.e. , satisfying the kramers - krnig relation , defined by : @xmath665 with @xmath666 . then we have also @xmath667 clearly , @xmath668 if @xmath669 and @xmath670 in contradiction with the necessary condition for zeros existence mentioned before . this reasoning is valid in one quarter plane but now , if @xmath671 is a zero @xmath672 is also a zero . therefore , the absence of a zero in the quarter plane @xmath669 , @xmath670 implies the absence of zero in the second quarter plane @xmath673 , @xmath670 and therefore eq . [ b1 ] do not admit any zero in the upper half plane for a very usual permittivity like eq . [ b3 ] . actually , the case @xmath674 should be handled separately . we find from eq . [ b4 ] that for such value @xmath664 . this is acceptable in order to have a zero existence in agreement with eq . however , from eq . [ b2 ] we find also that if a zero exists along the axis @xmath674 then we should have as well @xmath675 . this is in contradiction with eq . [ b4 ] which implies @xmath676 . this completes the proof for the permittivity given by eq . + the question concerning the generality of the proof is however still handling . huttner and barnett mentioned the existence of such a proof in the landau and lifschitz text - book @xcite but has no solution in the upper frequency half - plane ( see refs . @xcite).,width=309 ] it is relevant to detail the missing proof here . in order to get the complete result we will use a method used by landau and lifshitz ( see ref . @xcite p. 380 ) . in the complex @xmath14-plane we define @xmath677 . from the properties of @xmath678 we deduce @xmath679 . this implies that @xmath680 is real along the imaginary axis and that @xmath681 and @xmath682 along the real axis . furthermore due to causality we have @xmath683 and therefore @xmath684 if @xmath669 along the real axis . now we consider the closed contour integral ( see fig . 1 ) @xmath685 along @xmath686 made of the real axis and of the semi circle @xmath687 of infinite radius @xmath688 in the upper half plane . however , @xmath689 is a real number and since @xmath680 is not real along the real axis there is no pole along @xmath686 unless @xmath689 is infinite or null . therefore , since @xmath680 is analytical in the upper half plane eq . [ b5 ] gives the numbers of zeros of @xmath690 in this half space . + we then rewrite eq . [ b5 ] as an integral in the complex @xmath691 : @xmath692 where @xmath693 is the image of @xmath686 along the mapping @xmath694 . in particular the origin @xmath695 is mapped on it self while the semi circle of radius @xmath696 is mapped onto the circle of radius @xmath697 . the half real axis @xmath698 corresponding to @xmath351 is mapped onto a complex curve located in the upper half plane of the complex @xmath680-space ( since @xmath699 along this half line ) . similarly , the second half axis @xmath700 is mapped in the lower half plane . as shown on the figure if @xmath701 then the contour integral omits the point @xmath702 and there is no pole involved in the integral which therefore vanishes . we thus deduce that eq . [ b1 ] has no solution in the upper half - plane in the @xmath14 space which is the proof needed . the calculation of @xmath93 defined by the bromwich integral eq . [ 95 ] for @xmath703 : @xmath704 can be handled after closing the contour integral in the lower plane . however , since @xmath389 is not analytical in such lower plane we must include the poles @xmath100 ( all located in the lower plane see appendix a ) , i.e. the residues , in the integral . we use the separation : @xmath705\nonumber\\ \label{96}\end{aligned}\ ] ] and express it as a function of @xmath706 near each poles ( i.e , in the limit @xmath707 ) . we get : @xmath708 from the condition @xmath104 and the equality @xmath709 we then get for @xmath703 after integration in the lower plane eq . [ 98 ] and @xmath106 for @xmath97 ( after integration in the upper plane where no pole are present ) . the value at @xmath108 deserves some careful analysis . indeed , if @xmath108 the integration along the semicircle do not vanish exponentially with it radius @xmath696 and if we choose to integrate in the upper half plane ( where there is no pole ) we get @xmath710 if @xmath688 since @xmath711 in this limit in the upper half plane . therefore we have indeed @xmath712 and the function is continuous at @xmath108 . of course the null value for negative time @xmath713 have no meaning since the laplace transform is only interested in the evolution for positive time . + this leads to the sum rule : @xmath714=0.\label{101}\end{aligned}\]]this leads to the sum rule : @xmath714=0.\label{99}\end{aligned}\ ] ] the free term @xmath110 is defined as @xmath715e^{p(t - t_0)}}{\omega_\alpha^2+(1+\bar{\chi}(p))p^2}\nonumber\\ = u_\alpha(t - t_0)\dot{q}_{\alpha , j}(t_0)+\dot{u}_\alpha(t - t_0)q_{\alpha , j}(t_0)\nonumber\\ \label{102}\end{aligned}\ ] ] with @xmath716 like for @xmath113 we get @xmath717 . the last line of eq . [ 102 ] is therefore justified from the fact that the laplace transform of @xmath718 is @xmath719 . now , the boundary condition at @xmath115 imposes @xmath117 . therefore , from eq . [ 103 ] we deduce the second sum rule : @xmath720=1.\label{104}\end{aligned}\ ] ] the value at @xmath108 is not defined univocally since @xmath718 defined through the bromwich integral of @xmath721 is discontinuous . we point out that considering a direct integration at @xmath108 could lead to contradictions since the integration along @xmath722 do not vanish . if we choose to integrate in the upper half plane ( where there is no pole ) we get @xmath723 if @xmath688 since @xmath711 in this limit in the upper half plane . we would get @xmath724 ( a similar calculation could be done in the lower space including poles and residues and we would obtain once again @xmath725 ) . here we considered carefully the limit to prevent us from such a contradiction . the source field can be written as : @xmath726 . after some algebras we get : @xmath727 where we introduced the green function:@xmath728 computed by contour integration in the complex plane and solution of @xmath729 . of course , along the real axis @xmath730 we get @xmath731 which is the usual green function for an homogenous medium . we can also write this field without introducing @xmath14 by using the green propagator @xmath732 which leads to : @xmath733 we have also @xmath734 which represent the generalization of retarded propagator expansion for a lossy and dispersive medium . the role of causality is here crucial since the modes are always damped when the time is growing in the future direction as expected from pure thermodynamical considerations . this means in particular that @xmath735 tends to vanish exponentially as @xmath75 goes to infinity . importantly , in the vacuum limit ( @xmath126 ) we get naturally @xmath736 and in the limit @xmath407 we obtain the retarded potential @xmath737 however , in the vacuum limit we have also : @xmath738 therefore @xmath739 actually vanishes and we get in this limit @xmath740 as it should be . now , from eq . [ 105 ] and from the field definition we easily get the integral formulas for @xmath741 and @xmath742 : @xmath743 these equations have a clear interpretation in term of generalized hertz potentials . in particular , taking the limit @xmath176 and using the properties of convolutions , together with the fact that @xmath744 , we get @xmath745 with @xmath746 by rewriting @xmath747 in eq . [ cool ] we get after some rearrangements : @xmath748}{p^2(1+\bar{\chi}(p))}\label{137}\end{aligned}\ ] ] which involves the scalar green function defined in eq . [ 104 ] and from @xmath749 @xmath750\nonumber\\+ \mathbf{i}\delta^{3}(\mathbf{x}-\mathbf{x'})\label{138}\end{aligned}\ ] ] these formulas must be taken carefully since they are not actually valid at the source location , i.e. , if @xmath751 due to the bad convergence of the series defining the dyadic green function . after regularization we can obtain the result @xmath752\nonumber\\ + \frac{c^2}{p^2(1+\bar{\chi}(p))}\mathbf{l}\delta^{3}(\mathbf{x}-\mathbf{x'})\label{139}\end{aligned}\ ] ] and @xmath753\label{140}\end{aligned}\ ] ] with @xmath754 a dyadic term depending on the way we define the principal value @xcite : @xmath755 for a small exclusion spherical volume surrounding the point @xmath756 we get @xmath757 , i.e. , the depolarization field predicted by the clausius - mosotti formula @xcite . + in the particular case @xmath477 we write @xmath758 and similarly for other green functions . we also consider the time evolution which in vacuum relies on the propagators @xmath759 explicit calculations lead to : @xmath760 and similarly for the transverse part of @xmath761 : @xmath762 while for the longitudinal part we get : @xmath763 we deduce automatically the boundary conditions @xmath764 . we also obtain @xmath765 and @xmath766 from which we obtain the boundary condition : @xmath767 . we thus obtain : @xmath768 finally , from eq . [ 139 ] we find explicitly for the time dependent @xmath769 field : @xmath770.\label{a16}\end{aligned}\ ] ] from eq . [ b ] we deduce in the hopfield model @xmath771 this leads to the solution @xmath772 and therefore to @xmath773 where the permittivity is given by the lossless lorentz - drude formula @xmath774 . for the transverse fields we expand the different fields as @xmath775 with @xmath776 and @xmath777 . now , we write @xmath294 with @xmath295 a transverse vector field . we thus have @xmath778 with @xmath779 . we thus get the equation @xmath780 near the singularities @xmath781 we get : @xmath782\widetilde{e}_{\alpha , j}(\omega)=\omega_0 ^ 2\gamma_{\alpha , j}\delta(\omega-\omega_0)\end{aligned}\ ] ] and therefore supposing the regularity ( as for the longitudinal case ) we have @xmath783 . the secular equation @xmath784\widetilde{e}_{\alpha , j,\pm}(\omega)=0 $ ] associated with the transverse modes can be easily solved and this indeed leads to @xmath785 with @xmath786 with @xmath302 an amplitude coefficient for the transverse polariton mode . importantly we again get from the regularity condition @xmath787 and thus @xmath788 like for the longitudinal mode . this implies @xmath789 and from eq . [ reeu ] we get @xmath790 . at the end of the day we obtain the following fields amplitudes @xmath791 and @xmath792 ( using definitions similar to eq . [ defi ] ) . these define the hopfield transformation between the old variables @xmath793 , @xmath794 , @xmath795 , @xmath796 and the new polaritonic variables representing the normal coordinates of the problem @xmath302 and @xmath797 . up to a normalization this is equivalent to the work by hopfield . we start with the amplitude for the transverse polariton field @xmath798 where @xmath366 is a frequency window centered on the polariton pulsation @xmath363:=\omega'_{\alpha , m}$ ] . this field is equivalently written as @xmath799 where @xmath800 is a window function such as @xmath801 if @xmath14 belongs to the interval @xmath366 and @xmath802 otherwise . from the commuting properties of @xmath162 ( see eqs . [ 33 ] and [ commuting ] ) we deduce the commutator : @xmath370=\frac{\hbar}{\pi}\delta_{\alpha,\beta}\delta_{j , l}\nonumber\\ \cdot\int_0^{+\infty } \frac{d\omega\omega^4\tilde{\varepsilon}''(\omega)}{|\omega_\alpha^2-\tilde{\varepsilon}(\omega)\omega^2|^2 } f_{\alpha , m}(\omega)f_{\beta , n}(\omega)\end{aligned}\ ] ] with @xmath803 . now consider the integral : @xmath804 since for weak losses the integrand is extremely peaked on the value @xmath390 we can write @xmath805 as @xmath806 we use the approximation @xmath807 which is valid near the pole where the conditions @xmath808 approximately holds for transverse polaritons . we thus get @xmath809 with @xmath810 and @xmath811 . and where we used the integral @xmath812 which is easily calculated in the complex plane . from this we finally obtain the commutator of eq . [ window ] . we point out that the result does not explicitly depends on the extension of the frequency windows @xmath366 if losses and dispersion are weak enough to have @xmath813 . + for the longitudinal polariton field we have a similar calculation . starting with the definition : @xmath814 we can calculate the commutator @xmath815 $ ] . we have @xmath401=\frac{\hbar}{\pi}\delta_{\alpha,\beta}\nonumber\\ \cdot \int_0^{+\infty } d\omega\frac{\tilde{\varepsilon}''(\omega)}{|\tilde{\varepsilon}(\omega)|^2 } f_{\alpha , m}(\omega)f_{\beta , n}(\omega)\end{aligned}\ ] ] with again @xmath803 . we have to evaluate the integral @xmath816 which like for the transverse modes in the limit of weak losses and dispersion leads after straightforward calculations to @xmath817from this we deduce the commutator given in eq . 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access to large data sets on human activities and interactions has long been limited by the difficulty and cost of gathering such information . recently , the ever increasing availability of digital traces of human actions is widely enabling the representation and the analysis of massive amounts of information on human behavior . the representation of this information in terms of complex networks @xcite has led to many research efforts because of the naturally interlinked nature of these new data sources . tracing human behavior in a variety of contexts has become possible at very different spatial and temporal scales : from mobility of individuals inside a city @xcite and between cities @xcite , to mobility and transportation in an entire country @xcite , all the way to planetary - scale travel @xcite . mobile devices such as cell phones make it possible to investigate mobility patterns and their predictability @xcite . on - line interactions occurring between individuals can be monitored by logging instant messaging or email exchange @xcite . recent technological advances further support mining real - world interactions by means of mobile devices and wearable sensors , opening up new avenues for gathering data on human and social interactions . bluetooth and wifi technologies give access to proximity patterns @xcite , and even face - to - face presence can be resolved with high spatial and temporal resolution @xcite . the combination of these technological advances and of heterogeneous data sources allow researchers to gather longitudinal data that have been traditionally scarce in social network analysis @xcite . a dynamical perspective on interaction networks paves the way to investigating interesting problems such as the interplay of the network dynamics with dynamical processes taking place on these networks . in this paper , we capitalize on recent efforts @xcite that made possible to mine behavioral networks of face - to - face interactions between individuals , in a variety of real - world settings and in a time - resolved fashion . we present an in - depth analysis of the data we collected at two widely different events . the first event was the infectious exhibition @xcite held at the science gallery in dublin , ireland , from april @xmath0 to july @xmath0 , 2009 . the second event was the acm hypertext 2009 conference @xcite hosted by the institute for scientific interchange foundation in turin , italy , from june @xmath1 to july @xmath2 , 2009 . in the following , we will refer to these events as sg and ht09 , respectively . intuitively , interactions among conference participants differ from interactions among museum visitors , and the concerned individuals have very different goals in both settings . the study of the corresponding networks of proximity and interactions , both static and dynamic , reveals indeed strong differences but also interesting similarities . we take advantage of the availability of time - resolved data to show how dynamical processes that can unfold on the close proximity network such as the propagation of a piece of information or the spreading of an infectious agent unfold in very different ways in the investigated settings . in the epidemiological literature , traditionally , processes of this kind have been studied using either aggregated data or under assumptions of stationarity for the interaction networks : here we leverage the time - resolved nature of our data to assess the role of network dynamics on the outcome of spreading processes . at a more fundamental level , simulating simple spreading processes over the recorded interaction networks allows us to expose several properties of their dynamical structure as well as to probe their causal structure . the paper is organized as follows : first , we briefly describe the data collection platform and our data sets in section [ data ] ; in section [ static - properties ] we discuss the salient features of the networks of interactions aggregated on time windows of one day . these networks are static objects , carrying only information about the cumulative time that daily each pair of individuals has spent in face - to - face proximity . section [ dynamic - properties ] analyzes the dynamical properties of face - to - face interactions between conference participants and museum visitors . section [ resilience ] further characterizes the aggregated network structures by investigating the effect of incremental link removal . finally , section [ information - diffusion ] investigates the role played by causality in information spreading along the proximity network , and section [ conclusions ] concludes the paper and defines a number of open questions . the data collection infrastructure uses active radio - frequency identification devices ( rfid ) embedded in conference badges to mine face - to - face proximity relations of persons wearing the badges . rfid devices exchange ultra - low power radio packets in a peer - to - peer fashion , as described in refs . exchange of radio packets between badges is only possible when two persons are at close range ( @xmath3 to @xmath4 m ) and facing each other , as the human body acts as a rf shield at the carrier frequency used for communication . the operating parameters of the devices are programmed so that the face - to - face proximity of two individuals wearing the rfid tags can be assessed with a probability in excess of @xmath5 over an interval of @xmath6 seconds , which is a fine enough time scale to resolve human mobility and proximity at social gatherings . false positives are exceedingly unlikely , as the ultra - low power radio packets used for proximity sensing can not propagate farther than @xmath4-@xmath7 m , and a sustained excess of packets is needed in order to signal a proximity event . when a relation of face - to - face proximity ( or `` contact '' , as we will refer to it in the following ) is detected , the rfid devices report this information to receivers installed in the environment ( rfid readers ) . the readers are connected to a central computer system by means of a local area network . once a contact has been established , it is considered ongoing as long as the involved devices continue to exchange at least one radio packet for every subsequent interval of @xmath6 seconds . conversely , a contact is considered terminated if an interval of @xmath6 seconds elapses with no packets exchanged . for a detailed description of the sensing platform and some of its deployments , see refs.@xcite . the deployments at the science gallery in dublin @xcite and at the ht09 conference in turin @xcite involved vastly different numbers of individuals and stretched along different time scales . the former lasted for about three months and recorded the interactions of more than @xmath8 visitors ( more than @xmath9 face - to - face contacts recorded ) , whereas the latter took place over the course of three days and involved about @xmath10 conference participants ( about @xmath11 contacts ) . behaviors are also very different : in a museum , visitors typically spend a limited amount of time on site , well below the maximum duration permitted by the museum opening hours , they are not likely to return , and they follow a rather pre - defined path , touching different locations that host the exhibits . in a conference setting , on the other hand , most attendees stay on - site for the entire duration of the conference ( a few days ) , and move at will between different areas such as conference room , areas for coffee breaks and so on . the coverage of the community was different in both settings . at the science gallery , visitors were equipped with a rfid tag upon entering the venue , as part of an interactive exhibit , and therefore almost the totality of them were tracked . on the other hand , at ht09 , about 75% of the participants volunteered to being tracked . this sampling may introduce some biases in the results . sampling issues are also commonly encountered in the study of static complex networks @xcite . reference @xcite has shown that for a broad variety of real - world deployments of the rfid proximity - sensing platform used in this study , the behavior of the statistical distributions of quantities such as contact durations is not altered by unbiased sampling of individuals . on the other hand , we can not completely rule out that a systematic bias is introduced by the selection of volunteers , if volunteers and non - volunteers have different behavioral patterns . accurately checking this point would require monitoring an independent data source for face - to - face contacts , and because of scalability issues this would be feasible only for small control groups . issues regarding the effect of missing data and incomplete sampling on the properties of dynamical processes unfolding on the networks also deserve attention and will be the subject of future investigations . we start by analyzing aggregated networks of interaction obtained by aggregating the raw proximity data over one day . this aggregation yields a social graph where nodes represent individuals , and an edge is drawn between two nodes if at least one contact was detected between those nodes during the interval of aggregation . therefore , every edge is naturally weighted by the total duration of the contact events that occurred between the tags involved , i.e. , by the total time during which the corresponding individuals have been in face - to - face proximity . the choice of daily time windows seems quite natural in our settings . it would represent , for instance , a typical time scale for a description of articulated social networks based on surveys , in which each participant would ( ideally ) declare who s / he has encountered during the course of the day . such a choice for the duration of the time - window , albeit natural , is by no means unique @xcite . for instance , it is possible to aggregate the data over longer periods of time ( weeks or months ) to investigate the stationarity of the collected data @xcite . shorter aggregation times of the order of a few minutes are also useful , for instance , to resolve circadian activity patterns at the venue under investigation . figure [ aggregated - networks ] displays the aggregated contact networks for june 30@xmath12 at the ht09 conference ( top left ) , and for three representative days for the sg museum deployment . despite the large variation in the number of daily museum visitors , ranging from about @xmath13 to @xmath14 , the chosen days illustrate many features of the sg aggregated networks , in particular the presence of either a single or two large connected components ( cc ) in the network . days with smaller numbers of visitors can also give rise to aggregated networks made of a larger number of small isolated clusters . as shown in fig . [ nclusters ] , depending on the number of visitors the number of cc can in fact vary substantially . for a large number of visitors , typically only one cc is observed . for a low number of visitors , on the other hand , many clusters are formed . overall one also notices that the network diameter ( highlighted in all the plots of fig . [ aggregated - networks ] ) is considerably longer for sg than for ht09 aggregated networks , reflecting the different behavioral patterns in these settings . the small - world nature or lack thereof of the aggregated networks can be investigated statistically by introducing a proper null model . to this end , we construct a randomized network using the rewiring procedure described in ref . the procedure consists in taking random pairs of links @xmath15 and @xmath16 involving @xmath17 distinct nodes , and rewiring them as @xmath18 and @xmath19 . this procedure preserves the degree of each node and the degree distribution @xmath20 , while destroying the degree correlations between neighboring nodes , as well as any other correlations linked to node properties . the procedure is carried out so that initially distinct ccs do not get merged . since the rewiring procedure can not be implemented for the rare ccs with less than four nodes , these small ccs are removed from the aggregated networks before rewiring . figure [ rewired - networks ] displays a single realization of the null model for the networks in the top row of fig . [ aggregated - networks ] . we notice that the rewired version of the aggregated ht09 network is very similar to the original version , whereas the null model for the aggregated network of the sg data on july 14@xmath12 is more `` compact '' than the original network and exhibits a much shorter diameter . similar considerations hold for the other aggregated networks of the sg deployment . . left : ht09 deployment , june 30@xmath12 . right : sg deployment , july 14@xmath12 . the network diameters are highlighted as in fig . [ aggregated - networks ] . in the sg case , the randomized network is much more `` compact '' than the original one , with a much shorter diameter.,title="fig : " ] . left : ht09 deployment , june 30@xmath12 . right : sg deployment , july 14@xmath12 . the network diameters are highlighted as in fig . [ aggregated - networks ] . in the sg case , the randomized network is much more `` compact '' than the original one , with a much shorter diameter.,title="fig : " ] more quantitatively , we measure the mean number of nodes one can reach from a randomly chosen node by making @xmath21 steps on the network , a quantity hereafter called @xmath22 . for a network consisting of a single connected component , the definition of @xmath22 implies that @xmath23 where @xmath24 is the average node degree , @xmath25 is the total number of nodes in the network and @xmath26 the saturation value of @xmath27 on the network . the saturation value @xmath26 is reached when @xmath21 is equal to the length of the network diameter , and may vary for different realizations of the random networks . for a network consisting of several ccs one has to take into account the probability @xmath28 that the chosen node belongs to a given cc , where @xmath29 is the number of nodes in the @xmath30-th cc . as a consequence , eq . ( [ m - l - single - component ] ) generalizes to @xmath31 where @xmath32 is the average node degree on the @xmath30-th cc . this ensures that the quantity @xmath33 , regardless on the number of cc , assumes the same value when @xmath34 , and saturates to unity for both the aggregated and rewired network . figure [ m - l - plots ] displays @xmath33 for the aggregated networks on the top row of fig . [ aggregated - networks ] , as well as its value averaged on @xmath10 randomized networks ( the average value of @xmath22 converges rapidly already when calculated on a few tens of randomized networks ) . we notice the striking similarity between the results for the ht09 original and randomized networks , where about @xmath35 of the individuals lie , in both cases , within two degrees of separation . in the sg case , conversely , the same @xmath35 percentage is reached with six degrees of separation for the original network , but with only three degrees of separation on the corresponding randomized networks . the same calculation , performed on other aggregated sg networks , yields qualitatively similar results , always exposing a dramatic difference from the null model . steps on the network , @xmath22 , divided by its saturation limit @xmath26 , for daily aggregated networks ( circles ) and their randomized versions ( triangles ) . for the randomized case , data are averaged on @xmath10 realizations . left : network aggregated on june 30@xmath12 for the ht09 case . right : sg deployment , july 14@xmath12 . the solid lines are only guides for the eye.,title="fig : " ] steps on the network , @xmath22 , divided by its saturation limit @xmath26 , for daily aggregated networks ( circles ) and their randomized versions ( triangles ) . for the randomized case , data are averaged on @xmath10 realizations . left : network aggregated on june 30@xmath12 for the ht09 case . right : sg deployment , july 14@xmath12 . the solid lines are only guides for the eye.,title="fig : " ] averaged over all daily aggregated networks , for the ht09 ( left ) and the sg ( right ) cases . , title="fig : " ] averaged over all daily aggregated networks , for the ht09 ( left ) and the sg ( right ) cases . , title="fig : " ] one of the standard observables used to characterize a network topology is the degree distribution @xmath20 , i.e. , the probability that a randomly chosen node has @xmath36 neighbors . figure [ p - k ] reports the degree distributions of the daily aggregated networks , averaged over the whole duration of the ht09 deployment ( left ) and sg deployment ( right ) . for the sg case , we left out the few isolated nodes that contribute to the degree distribution for @xmath37 only . the @xmath20 distributions are short - tailed in all cases : @xmath20 decreases exponentially in the sg case , and even faster for ht09 . we notice that the ht09 degree distribution exhibits a peak at @xmath36 around @xmath38 , pointing to a characteristic number of contacts established during the conference . moreover , the average degree in the ht09 case , @xmath39 , close to @xmath6 , is more than twice as high as that for the sg networks , @xmath40 which is close to @xmath41 . this represents another clear indication of the behavioral difference of conference participants versus museum visitors ( the fact that the average degree is high for conference attendees can be regarded as a goal of the conference itself ) . finally , we observe that a large fraction of the recorded contacts are sustained for a short time : for instance , removing all the contacts with a cumulated duration below one minute yields @xmath40 about @xmath42 for ht09 and @xmath40 about @xmath43 for sg . the availability of time - resolved data allows one to gain much more insight into the salient features of the social interactions taking place during the deployments than what could be possible by the only knowledge of `` who has been in face - to - face proximity of whom '' . we first investigated the presence duration distribution in both settings . for the conference case , the distribution is rather trivial , as it essentially counts the number of conference participants spending one , two or three days at the conference . the visit duration distribution for the museum , instead , can be fitted to a lognormal distribution ( see fig . [ p - visits ] ) , with geometric mean around @xmath44 minutes . this shows that , unlike the case of the conference , here one can meaningfully introduce the concept of a characteristic visit duration that turns out to be well below the cutoff imposed by museum opening hours . the existence of a characteristic visit duration sheds light on the elongated aspect of the aggregated networks of visitor interactions ( see fig . [ aggregated - networks ] ) . indeed museum visitors are unlikely to interact directly with other visitors entering the venue more than one hour after them , thus preventing the aggregated network from exhibiting small - world properties . figure [ node - color - code ] reports the sg aggregated networks for two different days , where the network diameter is highlighted and each node is colored according to the arrival time of the corresponding visitor . one notices that , as expected from the aforementioned properties of the visit duration distribution , there is limited interaction among visitors entering the museum at different times . furthermore , the network diameter clearly defines a path connecting visitors that enter the venue at subsequent times , mirroring the longitudinal dimension of the network . these findings show that aggregated network topology and longitudinal / temporal properties are deeply interwoven . let us now focus on the temporal properties of social interactions . at the most detailed level , each contact between two individuals is characterized by its duration . the corresponding distributions are shown in fig . [ p - contact ] . as noted before , in both the ht09 and sg cases most of the recorded interactions amount to shortly - sustained contacts lasting less than one minute . however , both distributions show broad tails they decay only slightly faster than a power law . this behavior does not come as a surprise , as it has been observed in social sciences in a variety of context ranging from human mobility to email or mobile phone calls networks @xcite . more interestingly , the distributions are very close ( except in the noisy tail , due to the different number of contributing events ) , showing that the statistics of contact durations are robust across two very different settings . this robustness has been observed in ref . @xcite across different scientific conferences , but the museum setting corresponds to a situation in which a flux of individuals follows a predefined path , and this strong similarity between distributions was therefore not expected a priori . at a coarser level , aggregated networks are characterized by weights on the links , that quantify for how long two individuals have been in face - to - face proximity during the aggregation interval . figure [ p - w ] displays the distributions of these weights @xmath45 . these distributions are very broad @xcite : while most links correspond to very short contacts , some correspond to very long cumulated durations , and all time scales are represented , that is , no characteristic interaction timescale ( except for obvious cutoffs ) can be determined . we note that at this coarser level of analysis the distributions are again very similar . in the ht09 ( left ) and sg ( right ) aggregated networks ( data for all daily networks ) . the strength of a node quantifies the cumulated time of interaction of the corresponding individual with other individuals . , title="fig : " ] in the ht09 ( left ) and sg ( right ) aggregated networks ( data for all daily networks ) . the strength of a node quantifies the cumulated time of interaction of the corresponding individual with other individuals . , title="fig : " ] of nodes of degree @xmath36 . the figures show @xmath46 ( circles ) , for the ht09 ( left ) and sg ( right ) deployments ( the solid line is only a guide for the eye ) . the dashed lines stand for a linear fit and a power law fit to the data for the ht09 and sg deployments , respectively . distinct increasing and decreasing trends are respectively observed . the inset for the ht09 deployment shows a distribution of linear coefficients @xmath47 calculated for 4000 reshufflings of the network weights and the fitted value from the data collected at ht09 ( vertical line ) . the inset for the sg deployment shows @xmath48 on a doubly logarithmic scale ( circles ) together with the power law fit to the data ( dashed line ) . , title="fig : " ] of nodes of degree @xmath36 . the figures show @xmath46 ( circles ) , for the ht09 ( left ) and sg ( right ) deployments ( the solid line is only a guide for the eye ) . the dashed lines stand for a linear fit and a power law fit to the data for the ht09 and sg deployments , respectively . distinct increasing and decreasing trends are respectively observed . the inset for the ht09 deployment shows a distribution of linear coefficients @xmath47 calculated for 4000 reshufflings of the network weights and the fitted value from the data collected at ht09 ( vertical line ) . the inset for the sg deployment shows @xmath48 on a doubly logarithmic scale ( circles ) together with the power law fit to the data ( dashed line ) . , title="fig : " ] for each individual , the cumulated time of interaction with other individuals is moreover given by the strength @xmath49 of the corresponding node @xcite , i.e. , by the sum of the weights of all links inciding on it . the strength distributions @xmath50 are displayed in fig . [ p - s ] for the aggregated networks of the ht09 conference ( left ) and of the sg museum case ( right ) . unlike @xmath36 , the node strength @xmath49 spans several orders of magnitude , ranging from a few tens of seconds to well above one hour . the node strength @xmath49 can be correlated with the node degree @xmath36 by computing the average strength @xmath51 of nodes of degree @xmath36 @xcite . while a completely random assignment of weights yields a linear dependency with @xmath51 proportional to @xmath52 , where @xmath53 is the average link weight , super - linear or sub - linear behaviors have been observed in various contexts @xcite . a super - linear dependence such as the one observed in some conference settings @xcite hints at the presence of super - spreader nodes that play a prominent role in processes such as information diffusion @xcite . on the other hand , the sub - linear dependence observed for large - scale phone call networks @xcite corresponds to the fact that more active individuals spend on average less time in each call . figure [ s - over - k ] displays the ratio @xmath54 for the sg and ht09 daily aggregated networks . two different trends appear despite the large fluctuations : a slightly increasing trend in the conference setting , and a clearly decreasing one in the museum setting . in particular , the behavior of @xmath55 for the ht09 case ( left plot in figure [ s - over - k ] ) can be fitted linearly yielding a linear coefficient @xmath56 ( p - value @xmath57 ) . by reshuffling @xmath58 times the weights of the network links and performing the same linear fit for each reshuffling , we obtain a distribution of linear coefficients @xmath47 . such distribution , whose mean is zero , is shown in the inset of the left plot in figure [ s - over - k ] together with the value of @xmath47 from the ht09 daily aggregated networks ( vertical line ) . the observed value of @xmath47 at the ht09 is an outlier of the distribution ( @xmath59 percentile ) , thus showing that the observed behavior of @xmath55 can hardly arise by a random assignment of link weights . on the other hand , the observed behavior of @xmath55 at the sg can be fitted to a power law with a negative exponent i.e. it decreases linearly on a double logarithmic scale such as the one shown in the inset of the right plot in figure [ s - over - k ] . these results indicate that individuals who encountered the same number of distinct persons can have different spreading potentials , depending on the setting . it also gives a warning about characterizing spreading by only measuring the number of encounters , which can yield a rather misleading view . the issue of network vulnerability to successive node removal has attracted a lot of interest in recent years starting from the pioneering works of refs . @xcite , that have shown how complex networks typically retain their integrity when nodes are removed randomly , while they are very fragile with respect to targeted removal of the most connected nodes . while the concepts of node failures and targeted attacks are pertinent for infrastructure networks , successive removals of nodes or links is more generally a way to study network structures @xcite . for instance , detecting efficient strategies for dismantling the network sheds light on the network community structure , as it amounts to finding the links that act as bridges between different communities @xcite . moreover , in the context of information or disease spreading , the size of the largest connected component gives an upper bound on the number of nodes affected by the spreading . identifying ways to reduce this size , by removing particular links , in order to break and disconnect the network as much as possible , is analogous in terms of disease spreading to finding efficient intervention and containment strategies . in order to test different link removal strategies , we consider different definitions of weight for a link connecting nodes @xmath30 and @xmath60 in the aggregated contact network : + the simplest definition of link weight is given by the cumulated contact duration @xmath61 between @xmath30 and @xmath60 . in the following , we will refer to this weight as `` contact weight '' . + the topological overlap @xmath62 , introduced in ref . @xcite , is defined as @xmath63 \ , , \ ] ] where @xmath64 is the degree of node @xmath65 and @xmath66 measures the number of neighbors shared by nodes @xmath30 and @xmath60 . this measure is reminiscent of the edge clustering coefficient @xcite , and evaluates the ratio of the number of triangles leaning upon the @xmath67 edge with the maximum possible number of such triangles given that @xmath30 and @xmath60 have degrees @xmath68 and @xmath69 , respectively . edges between different communities are expected to have a low number of common neighbors , hence a low value of @xmath62 . + finally , the structural similarity of two nodes is defined as the cosine similarity @xmath70 \ , , \ ] ] where @xmath71 is the set of neighbors shared by nodes @xmath30 and @xmath60 , and the sums at the denominator are computed over all the neighbors of @xmath30 and @xmath60 . cosine similarity , which is one of the simplest similarity measures used in the field of information retrieval @xcite , takes into account not only the number of shared neighbors of @xmath30 and @xmath60 , but also the similarity of the corresponding edge strengths , i.e. the similarity of individuals in terms of the time they spent with their neighbors . once again , edges connecting different communities are expected to have a low value of @xmath72 . based on these three weight definitions , we consider four different strategies for link removal , namely : removing the links in increasing / decreasing order of contact weight , in increasing order of topological overlap , and in increasing order of cosine similarity . the former two strategies are the simplest one can devise , as they do not consider the neighborhoods topology . the latter two strategies were implemented in an incremental fashion , by recomputing the lists of links ranked in order of increasing overlap or cosine similarity whenever a link was removed , and then removing the links in the updated list order ) and ( [ eq : cos - simil ] ) , leads to sub - optimal results . the deviation from the updating strategy becomes apparent only when more than @xmath73 of links have been removed since @xmath62 and @xmath72 deal with local quantities only . as a consequence , each link removal amounts to a local perturbation of the network , contrary to what happens with non - local quantities such as the betweenness centrality @xcite . ] . an issue also arises from the fact that all the generalized weights mentioned above produce a certain amount of link degeneracy ( in particular when using the contact weight ) : for instance , many links may have the same ( small ) value @xmath61 , or exactly @xmath74 overlap or similarity . each link removal procedure carries therefore a certain ambiguity , and the results may depend on which links , among those with the same contact weight / overlap / similarity , are removed first . of the links on the sg museum daily aggregated network of july 14@xmath75 . clockwise from top : links removed in decreasing contact weight order , increasing contact weight order , increasing topological overlap order and increasing cosine similarity order . the largest cc is highlighted in each case.,title="fig : " ] of the links on the sg museum daily aggregated network of july 14@xmath75 . clockwise from top : links removed in decreasing contact weight order , increasing contact weight order , increasing topological overlap order and increasing cosine similarity order . the largest cc is highlighted in each case.,title="fig : " ] of the links on the sg museum daily aggregated network of july 14@xmath75 . clockwise from top : links removed in decreasing contact weight order , increasing contact weight order , increasing topological overlap order and increasing cosine similarity order . the largest cc is highlighted in each case.,title="fig : " ] of the links on the sg museum daily aggregated network of july 14@xmath75 . clockwise from top : links removed in decreasing contact weight order , increasing contact weight order , increasing topological overlap order and increasing cosine similarity order . the largest cc is highlighted in each case.,title="fig : " ] the impact of link removal on network fragmentation can be measured by monitoring the variations of the size of the largest cc , hereafter called @xmath76 , as a function of link removal . if the network is initially divided into two ccs , labeled @xmath77 and @xmath78 , of similar initial sizes @xmath79 , we call @xmath76 the size of the largest cc surviving in the network ( which does not need to be a subnetwork of @xmath77 ) . we used the apex `` @xmath74 '' to denote quantities expressed for the original network , before any link removal . in order to alleviate the problems arising from link degeneracy , we averaged @xmath76 on @xmath10 different link orderings ( i.e. we reshuffled the list of links of equal generalized weight before removing them ) . an example of a single realization of the removal strategies for the sg aggregated network of july 14@xmath12 is shown in fig . [ dismantled - networks ] . we observe that a removal of @xmath80 of the network links has a far deeper impact on the network when the removal is based on the topological overlap ( the size of the largest cc is @xmath81 ) or cosine similarity ( @xmath82 ) rather than on decreasing ( increasing ) contact weight ( @xmath83 @xmath84 ) . more quantitatively , fig . [ dismantled - networks - delta - n1 ] shows that removing links according to their topological overlap is the most efficient strategy . this is in agreement with previous results @xcite that have shown that topological criteria detect efficiently the links that act as bridges between communities . due to their high degeneracy , removing first the links with small contact weights approximates a random removal strategy that is far from optimal . despite this limitation , removing the links with small contact weights can outperform the removal of links with high contact weight since the latter are usually found within dense communities , while links between communities have typically small contact weights . of the largest cc as a function of the fraction of removed links , for several removal strategies , and for different daily aggregated networks in the ht09 and sg deployments . for all networks , removing links in increasing topological overlap order and increasing cosine similarity order have the most disruptive effects . the ht09 aggregated network is in all cases more resilient than the sg aggregated networks.,title="fig : " ] of the largest cc as a function of the fraction of removed links , for several removal strategies , and for different daily aggregated networks in the ht09 and sg deployments . for all networks , removing links in increasing topological overlap order and increasing cosine similarity order have the most disruptive effects . the ht09 aggregated network is in all cases more resilient than the sg aggregated networks.,title="fig : " ] of the largest cc as a function of the fraction of removed links , for several removal strategies , and for different daily aggregated networks in the ht09 and sg deployments . for all networks , removing links in increasing topological overlap order and increasing cosine similarity order have the most disruptive effects . the ht09 aggregated network is in all cases more resilient than the sg aggregated networks.,title="fig : " ] of the largest cc as a function of the fraction of removed links , for several removal strategies , and for different daily aggregated networks in the ht09 and sg deployments . for all networks , removing links in increasing topological overlap order and increasing cosine similarity order have the most disruptive effects . the ht09 aggregated network is in all cases more resilient than the sg aggregated networks.,title="fig : " ] the strategy based on link topological overlap proves slightly more effective than the strategy based on link similarity : the information on the link contact weights incorporated in the definition of @xmath72 ( eq . [ eq : cos - simil ] ) does not enhance the decrease of @xmath76 . this can be explained through the following argument : topological overlap link ranking usually leads to a higher degeneracy with respect to similarity - based link ranking . as a consequence , for a network with similar values of @xmath85 and @xmath86 , a strategy based on topological overlap is more likely to dismantle in parallel both @xmath77 and @xmath78 than a similarity - based strategy , as it has no bias towards a specific component . the opposite strategy of a complete dismantling of @xmath77 that leaves @xmath78 intact would result in @xmath87 even after the complete disintegration of @xmath77 . this effect is illustrated in fig . [ dismantled - networks - delta - n1 ] for the sg aggregated networks of may @xmath88 , which are indeed composed of two large ccs ( see fig . [ aggregated - networks ] ) . interestingly , and as expected from the previous comparisons , rather different results are obtained for the ht09 and sg aggregated networks . the conference network is more resilient to all strategies , and significant levels of disaggregation are reached only by removing large fractions ( @xmath89 ) of the links , sorted by their topological overlap . for the sg aggregated networks , on the other hand , targeting links with small topological overlap or cosine similarity is a quite effective strategy , which can be intuitively related to the modular structure visible in fig . [ aggregated - networks ] . aggregated networks often represent the most detailed information that is available on social interactions . in the present case , they would correspond to information obtained through ideal surveys in which respondents remember every single person they encountered and the overall duration of the contacts they had with that person . while such a static representation is already informative , it lacks information about the time ordering of events , and it is unable to encode causality . the data from our measurements do not suffer from this limitation , as they comprise temporal information about every single contact . therefore , these data can be used to investigate the unfolding of dynamical processes . they also allow to study the role of causality in diffusion processes , such as the spreading of an infectious agent or of a piece of information on the encounter networks of individuals . in the following we will mainly use an epidemiological terminology , but we may equally imagine that the rfid devices are able to exchange some information whenever a contact is established . individuals will be divided into two categories , susceptible individuals ( s ) or infected ones ( i ) : susceptible individuals have not caught the `` disease '' ( or have not received the information ) , while infected ones carry the disease ( or have received the information ) and can propagate it to other individuals . in order to focus on the structure of the dynamical network itself , we consider in the following a deterministic snowball si model @xcite : every contact between a susceptible individual and an infected one , no matter how short , results in a transmission event in which the susceptible becomes infected , according to @xmath90 . in this model , individuals , once infected , do not recover . such a deterministic model allows to isolate the role played in the spreading process by the structure of the dynamical network ( e.g. its causality ) . its role would otherwise be entangled with the stochasticity of the transmission process and the corresponding interplay of timescales . of course , any realistic epidemiological model should include a stochastic description of the infection process , since the transmission from an infected individual to a susceptible one is a random event that depends on their cumulative interaction time . the resulting dynamics would depend on the interplay between contact and propagation times . we leave the study of this interesting type of interplay to future investigations . in our numerical experiments , for each day we select a single `` seed '' , i.e. , an individual who first introduces the infection into the network . all the other individuals are susceptible and the infection spreads deterministically as described above . by varying the choice of the seed over individuals , we obtain the distribution of the number of infected individuals at the end of each day . the transmission events can be used to define the network along which the infection spreads ( i.e. , the network whose edges are given by @xmath91 contacts ) , hereafter called the _ transmission network_. due to causality , the infection can only reach individuals present at the venue after the entry of the seed . as a consequence , in the following we will use the term _ partially aggregated network _ to indicate the network aggregated from the time the seed enters the museum / conference to the end of the day . we note that the partially aggregated network defined in this way can be radically different from ( much smaller than ) the network aggregated along the whole day . at the sg museum , for two different choices of the seed ( blue node at the bottom ) . transparent nodes and light gray edges represent individuals not infected and contacts not spreading the infection , respectively . red nodes and dark gray links represent infected individuals and contacts spreading the infection , respectively . the diameter of the transmission network and of the partially aggregated networks are shown respectively with blue and orange links . the black node represents the last infected individual . , title="fig : " ] at the sg museum , for two different choices of the seed ( blue node at the bottom ) . transparent nodes and light gray edges represent individuals not infected and contacts not spreading the infection , respectively . red nodes and dark gray links represent infected individuals and contacts spreading the infection , respectively . the diameter of the transmission network and of the partially aggregated networks are shown respectively with blue and orange links . the black node represents the last infected individual . , title="fig : " ] figure [ infection - network ] shows two partially aggregated networks for july 14@xmath12 at the sg museum , for two different choices of the seed ( blue node ) , and the corresponding transmission networks . the transmission network is of course a subnetwork of the partially aggregated network : not all individuals entering the premises after the seed can be reached from the seed by a _ causal _ path , and not all links are used for transmission events . in order to emphasize the branching nature of infection spreading , we represent the transmission network with successively infected nodes arranged from the bottom to the top of the figure . we notice that the diameter of both the transmission and the partially aggregated network may not include the seed and/or the last infected individual . from the seed to all the infected individuals calculated over the transmission network ( circles ) and the partially aggregated networks ( triangles ) . the distributions are computed , for each day , by varying the choice of the seed over all individuals . , title="fig : " ] from the seed to all the infected individuals calculated over the transmission network ( circles ) and the partially aggregated networks ( triangles ) . the distributions are computed , for each day , by varying the choice of the seed over all individuals . , title="fig : " ] from the seed to all the infected individuals calculated over the transmission network ( circles ) and the partially aggregated networks ( triangles ) . the distributions are computed , for each day , by varying the choice of the seed over all individuals . , title="fig : " ] from the seed to all the infected individuals calculated over the transmission network ( circles ) and the partially aggregated networks ( triangles ) . the distributions are computed , for each day , by varying the choice of the seed over all individuals . , title="fig : " ] the presence of a few triangles in the transmission network is due to the finite time resolution of the measurements . let us consider , for instance , the case of an infected visitor @xmath92 who infects @xmath93 , followed by a simultaneous contact of @xmath92 and @xmath93 with the susceptible @xmath94 . in this case it is impossible to attribute the infection of @xmath94 to either @xmath92 or @xmath93 , and both the @xmath95 and the @xmath96 links are highlighted in the transmission network as admissible transmission events . as a consequence , we slightly overestimate the number of links in the transmission network of fig . [ infection - network ] . in the case of fig . [ infection - network ] , the number of links is between @xmath3 and @xmath97 larger than for a tree with the same number of nodes . at finer time resolutions , some of the diffusion paths of fig . [ infection - network ] would actually be forbidden by causality . a general feature exemplified by fig . [ infection - network ] is that the diameter of the transmission network ( blue path ) is longer than the diameter of the partially aggregated network ( orange path ) , a first signature of the fact that the fastest paths between two individuals , which are the ones followed by the spreading process , do not coincide with the shortest path over the partially aggregated network @xcite . ) along the transmission network , versus the total duration of the epidemics ( time interval from the entry of the seed to the last infection event).,title="fig : " ] ) along the transmission network , versus the total duration of the epidemics ( time interval from the entry of the seed to the last infection event).,title="fig : " ] ) along the transmission network , versus the total duration of the epidemics ( time interval from the entry of the seed to the last infection event).,title="fig : " ] ) along the transmission network , versus the total duration of the epidemics ( time interval from the entry of the seed to the last infection event).,title="fig : " ] the difference between the fastest and the shortest paths for a spreading process can be quantitatively investigated . figure [ path - length - distr ] reports the distribution of the network distances @xmath98 between the seed and every other infected individual along both the transmission networks and the aggregated networks . when calculated on the partially aggregated network , @xmath98 measures the length of the _ shortest _ seed - to - infected - individual path , whereas it yields the length of the _ fastest _ seed - to - infected - individual path when calculated on the transmission network . we observe that the length distribution of fastest paths , i.e. , the @xmath99 distribution for the transmission network , always turns out to be broader and shifted toward higher values of @xmath98 than the corresponding shortest path distribution , i.e. , @xmath99 for the partially aggregated network . the difference is particularly noticeable in the case of may @xmath100 and july @xmath101 for the sg deployment , and june @xmath102 for the ht09 conference , where the longest paths on the transmission network are about twice as long as the longest paths along the partially aggregated network . these results clearly underline that in order to understand realistic dynamical processes on contact networks , information about the time ordering of the contact events turns out to be essential : the information carried by the aggregated network may lead to erroneous conclusions on the spreading paths . it is also possible to study the length of the path connecting the first ( seed ) to the last infected individual along the transmission network . we measure the fastest seed - to - last - infected - individual path ( a quantity hereafter called `` transmission @xmath98 '' ) as a function of the duration of the spreading process , defined as the time between the entry of the seed and the last transmission event . as shown by fig . [ time - length - distr ] , a clear correlation is observed between the transmission @xmath98 and the duration of the spreading process for the sg case ( pearson coefficients @xmath103 for may @xmath100 and may @xmath104 , and @xmath105 for july @xmath101 ) . no significant correlation is instead observed for the ht09 conference . this highlights the importance of the longitudinal dimension in the sg data , and gives a first indication of the strong differences in the spreading patterns , that we further explore in the following . and @xmath106 quantile of the distribution of infected individuals at the end of each day , and the red lines correspond to the median ( @xmath107 quantile ) . the @xmath108 and @xmath109 are also shown ( black horizontal lines ) . , title="fig : " ] and @xmath106 quantile of the distribution of infected individuals at the end of each day , and the red lines correspond to the median ( @xmath107 quantile ) . the @xmath108 and @xmath109 are also shown ( black horizontal lines ) . , title="fig : " ] and @xmath106 quantile of the distribution of infected individuals at the end of each day , and the red lines correspond to the median ( @xmath107 quantile ) . the @xmath108 and @xmath109 are also shown ( black horizontal lines ) . , title="fig : " ] and @xmath106 quantile of the distribution of infected individuals at the end of each day , and the red lines correspond to the median ( @xmath107 quantile ) . the @xmath108 and @xmath109 are also shown ( black horizontal lines ) . , title="fig : " ] let us now consider some other quantitative properties of the spreading process , in particular the number of individuals reached by the infection / information at the end of one day . in the sg case fig . [ boxplot - number - infected ] shows the distributions for each day , as boxplots , displaying the median together with the @xmath108 , @xmath110 , @xmath106 and @xmath109 percentiles . days are arranged horizontally from left to right , in increasing number of visitors . a high degree of heterogeneity is visible . the blue line corresponds to the number of daily visitors , that is the maximum number of individuals who can potentially be infected . we observe that the number of infected individuals is usually well below this limit . the number of reached individuals also depends on the number of cc in the aggregated network , as the spreading process can not propagate from one cc to another . in fact , the limit for which all visitors are infected can be reached only if the aggregated network is globally connected , that occurs only when the global number of visitors is large enough . these results hint at the high intrinsic variability of the final outcome of an epidemic - like process in a situation where individuals stream through a building . a totally different picture emerges for the ht09 conference , where the infection is almost always able to reach all the participants . as mentioned previously , the spreading process can not reach individuals who have left the venue before the seed enters , or the individuals who belong to a cc different from that of the seed . therefore , we consider the ratio of the final number of infected individuals , @xmath111 to the number @xmath112 of individuals who can be potentially reached through causal transmission paths starting at the seed . the distributions of this ratio is reported in fig . [ ratio - ninf - nsus ] . we observe that in the case of ht09 ( left ) almost all the potentially infected individuals will be infected by the end of the day , whereas the distribution of @xmath113 is broader in the sg case ( right ) . we notice that a static network description would inevitably lead to all individuals in the seed s cc catching the infection , a fact that can be a severe ( and misleading ) approximation of reality . for each day the chosen seed generates a deterministic spreading process for which we can compute the cumulative number of infected individuals as a function of time , a quantity hereafter referred to as an incidence curve . figure [ epidemics - within - day ] shows the results for a selected day of the ht09 conference and for three different days of the sg data . for the ht09 ( left ) and the sg ( right ) data , averaged over all potential seeds . @xmath111 is the final number of infected individuals at the end of one day , while @xmath112 is the number of individuals that could potentially be reached by a causal transmission path starting at the seed . @xmath112 is given by the number of individuals visiting the premises in the same day , from the time the seed enters the premises , and belonging to the same cc as the seed . , title="fig : " ] for the ht09 ( left ) and the sg ( right ) data , averaged over all potential seeds . @xmath111 is the final number of infected individuals at the end of one day , while @xmath112 is the number of individuals that could potentially be reached by a causal transmission path starting at the seed . @xmath112 is given by the number of individuals visiting the premises in the same day , from the time the seed enters the premises , and belonging to the same cc as the seed . , title="fig : " ] ( aggregated network consisting of a single cc with @xmath114 individuals ) ; sg network for july 14@xmath12 ( one cc , @xmath115 individuals ) , may 19@xmath12 ( two ccs , @xmath116 individuals ) and may 20@xmath12 ( two ccs , @xmath117 individuals ) . each curve corresponds to a different seed , and is color - coded according to the starting time of the spreading . , title="fig : " ] ( aggregated network consisting of a single cc with @xmath114 individuals ) ; sg network for july 14@xmath12 ( one cc , @xmath115 individuals ) , may 19@xmath12 ( two ccs , @xmath116 individuals ) and may 20@xmath12 ( two ccs , @xmath117 individuals ) . each curve corresponds to a different seed , and is color - coded according to the starting time of the spreading . , title="fig : " ] ( aggregated network consisting of a single cc with @xmath114 individuals ) ; sg network for july 14@xmath12 ( one cc , @xmath115 individuals ) , may 19@xmath12 ( two ccs , @xmath116 individuals ) and may 20@xmath12 ( two ccs , @xmath117 individuals ) . each curve corresponds to a different seed , and is color - coded according to the starting time of the spreading . , title="fig : " ] ( aggregated network consisting of a single cc with @xmath114 individuals ) ; sg network for july 14@xmath12 ( one cc , @xmath115 individuals ) , may 19@xmath12 ( two ccs , @xmath116 individuals ) and may 20@xmath12 ( two ccs , @xmath117 individuals ) . each curve corresponds to a different seed , and is color - coded according to the starting time of the spreading . , title="fig : " ] in the case of the ht09 conference , the earliest possible seeds are the conference organizers , but little happens until conference participants gather for the coffee break and/or meet up at the end of the first talk , between @xmath118:@xmath119 and @xmath120:@xmath119 . a strong increase in the number of infected individuals is then observed , and a second strong increase occurs during the lunch break . due to the concentration in time of transmission events , spreading processes reach very similar ( and high ) incidence levels after a few hours , regardless of the initial seed or its arriving time . even processes started after @xmath121:@xmath119 can reach about @xmath122 of the conference participants . thus , the crucial point for the spreading process does not consist in knowing where and when the epidemic trajectory has started , but whether the seed or any other subsequently infected individual attend the coffee break or not . a different picture is obtained in the sg case : first , in order to reach almost all participants the epidemics must spread on a globally connected network and start early ( black curves for july @xmath123 data ) . even in such a favorable setting for spreading , the incidence curves do not present sharp gradients , and later epidemics are unable to infect a large fraction of daily visitors . the incidence curves for may @xmath124 and @xmath125 of fig . [ epidemics - within - day ] show that different scenarios can also occur : due to the fragmented nature of the network , the final fraction of infected individuals can fluctuate greatly , and sharp increases of the incidence can be observed when dense groups such as those visible in fig . [ aggregated - networks ] are reached . in this paper we have shown that the analysis of time - resolved network data can unveil interesting properties of behavioral networks of face - to - face interaction between individuals . we considered data collected in two very different settings , representative of two types of social gatherings : the ht09 conference is a `` closed '' systems in which a group of individuals gathers and interacts in a repeated fashion , while the sg museum deployment is an `` open '' environment with a flux of individuals streaming through the premises . we took advantage of the accurate time - resolved nature of our data sources to build dynamically evolving behavioral networks . we analyzed aggregated networks , constructed by aggregating the face - to - face interactions during time intervals of one day , and provided a comparison of their properties in both settings . we assessed the role of network dynamics on the outcome of dynamical processes such as spreading processes of informations or of an infectious agent . our analysis shows that the behavioral networks of individuals in conferences and in a museum setting exhibit both similarities and important differences . the topologies of the aggregated networks are widely different : the conference networks are rather dense small - worlds , while the sg networks have a larger diameter and are possibly made of several connected components they do not form small - worlds , and their `` elongated '' shape can be put in relation with the fact that individuals enter the premises at different times and remain there only for a limited amount of time . the networks differences are also unveiled by a percolation analysis , which reveals how the sg aggregated networks can easily be dismantled by removing links that act as `` bridges '' between groups of individuals ; on the contrary , aggregated networks at a conference are more `` robust '' , even with respect to targeted link removal . interestingly , some important similarities are also observed : the degree distributions of aggregated networks , for example , are short - tailed in both cases . moreover , despite the higher social activity at a conference , both the distribution of the contact event durations and the distribution of the total time spent in face - to - face interactions by two individuals are very similar . the study of simple spreading processes unfolding on the dynamical networks of interaction between individuals allowed us to delve deeper into the time - resolved nature of our data . comparison of the spreading dynamics on the time - dependent networks with the corresponding dynamics on the aggregated networks shows that the latter easily yields erroneous conclusions . in particular , our results highlight the strong impact of causality in the structure of transmission chains , that can differ significantly from those obtained on a static network . the temporal properties of the contacts are crucial in determining the spreading patterns and their properties . studies about the role of the initial seed and its properties on the spreading patterns , or the determination of the most crucial nodes for propagation , can be misleading if only the static aggregated network is considered . in more realistic dynamics , the fastest path is typically not the shortest path of the aggregated network , and the role of causality is clearly visible in the analysis of the seed - to - last - infected paths . spreading phenomena unfold in very different ways in the two settings we investigated : at a conference , people interact repeatedly and with bursts of activity , so that transmission events also occur in a bursty fashion , and most individuals are reached at the end of the day ; in a streaming situation , instead , the fraction of reached individuals can be very small due to either the lack of global connectivity or the late start of the spreading process . detailed information on the temporal ordering of contacts is therefore crucial . we also note that in more realistic settings with non - deterministic spreading , information about the duration of contacts , and not only their temporal ordering , would also turn out to be very relevant and lead to an interesting interplay between the contact timescale and the propagation timescale @xcite . future work will also address the issue of sampling effects : the fact that not all the conference attendees participated to the data collection may lead to an underestimation of spreading , since spreading paths between sampled attendees involving unobserved persons may have existed , but are not taken into account . we close by stressing that as the data sources on person - to - person interactions become richer and ever more pervasive , the task of analyzing networks of interactions is unavoidably shifting away from statics towards dynamics , and a pressing need is building up for theoretical frameworks that can appropriately deal with streamed graph data and large scales . at the same time , we have shown that access to these data sources challenges a number of assumptions and poses new questions on how well - known dynamical processes unfold on dynamic graphs . data collections of the scale reported in this manuscript are only possible with the collaboration and support of many dedicated individuals . we gratefully thank the science gallery in dublin for inspiring ideas and for hosting our deployment . special thanks go to michael john gorman , don pohlman , lynn scarff , derek williams , and all the staff members and facilitators who helped to communicate the experiment and engage the public . we thank the organizers of the acm hypertext 2009 conference and acknowledge the help of ezio borzani , vittoria colizza , daniela paolotti , corrado gioannini and the staff members of the isi foundation , as well as the help of several hypertext 2009 volunteers . we also thank harith alani , martin szomszor and gianluca correndo of the live social semantics team . we warmly thank bitmanufaktur and the openbeacon project , and acknowledge technical support from milosch meriac and brita meriac . we acknowledge stimulating discussions with alessandro vespignani and vittoria colizza . this study was partially supported by the fet - open project dynanets ( grant no . 233847 ) funded by the european commission . finally , we are grateful for the valuable feedback , the patience and the support of the tens of thousands of volunteers who participated in the deployments . d. watts , connections a twenty - first century science , nature 445 ( 2007 ) 489 . g. chowell , j. m. hyman , s. eubank , c. castillo - chavez , scaling laws for the movement of people between locations in a large city , phys . e 68 ( 2003 ) 066102 . d. balcan , v. colizza , b. gonalves , h. hu , j.j . ramasco , a. vespignani , multiscale mobility networks and the spatial spreading of infectious diseases , proc . natl . acad . usa 106 ( 2009 ) 21484 - 21489 . s. golder , d. wilkinson , b. huberman , rhythms of social interaction : messaging within a massive online network , _ communities and technologies 2007 : proceedings of the third communities and technologies conference _ michigan state university , 2007 . p. hui , a. chaintreau , j. scott , r. gass , j. crowcroft , c. diot , pocket switched networks and human mobility in conference environments , proceedings of the 2005 acm sigcomm workshop on delay - tolerant networking , philadelphia , pennsylvania , usa ( 2005 ) 244 - 251 . n. eagle , a. pentland , reality mining : sensing complex social systems , personal and ubiquitous computing 10 ( 2006 ) 255 - 268 . e. oneill , v. kostakos , t. kindberg , a. fatah gen . schieck , a. penn , d. stanton fraser , t. jones , instrumenting the city : developing methods for observing and understanding the digital cityscape , lecture notes in computer science 4206 ( 2006 ) 315 - 332 . a. pentland , honest signals : how they shape our world , mit press , cambridge ma , 2008 . h. alani , m. szomsor , c. cattuto , w. van den broeck , g. correndo , a. barrat , live social semantics , 8@xmath12 international semantic web conference iswc2009 , lecture notes in computer science 5823 ( 2009 ) 698 - 714 , http://dx.doi.org/10.1007/978-3-642-04930-9_44 . w. van den broeck , c. cattuto , a. barrat , m. szomsor , g. correndo , h. alani , the live social semantics application : a platform for integrating face - to - face presence with on - line social networking , first international workshop on communication , collaboration and social networking in pervasive computing environments ( percol 2010 ) , proceedings of the 8th annual ieee international conference on pervasive computing and communications , mannheim , germany ( 2010 ) 226 - 231 . onnela , j. saramki , j. hyvonen , g. szab , m. argollo de menezes , k. kaski , a .- l- barabsi , j. kertsz , analysis of a large - scale weighted network of one - to - one human communication , new j. phys 9 ( 2007 ) 179 . onnela , j. saramki , j. hyvnen , g. szab , d. lazer , k. kaski , j. kertsz , a .- barabsi , structure and tie strengths in mobile communication networks , proc . natl . acad . usa 104 ( 2007 ) 7332 - 7336 . g. kossinets , j. kleinberg , d. watts . the structure of information pathways in a social communication network , proceedings of the 14@xmath126 acm sigkdd international conference on knowledge discovery and data mining ( 2008 ) . j. stehl , n. voirin , a. barrat , c. cattuto , v. colizza , l. isella , c. rgis , j .- f . pinton , n. khanafer , w. van den broeck , p. vanhems , infectious disease spread on a data - driven dynamic contact network , submitted . g. csardi and t. nepusz , the igraph software package for complex network research , interjournal complex systems ( 2006 ) 1695 .

the magnetic field evolution in plasmas such as the solar corona and the earth s magnetosphere often follows an ideal evolution in which all topological features of the field remain unchanged over time . this invariance provides a motivation to investigate topological properties of the magnetic field such as magnetic null points , flux surfaces and periodic field lines . some of these topological features , particularly magnetic null points , are surprisingly stable and turn out to be conserved not only under an ideal plasma dynamics but also for a wide range of non - ideal flows . mathematically this is reflected in the existence of an index theorem ( greene , 1992 ) for magnetic null points which allows only a small number of bifurcation processes to either generate or destroy magnetic null points . this underlines the importance of magnetic null points and their properties for the structure and dynamics of astrophysical plasmas . generic magnetic null points ( i.e. null points where all eigenvalues of the linearisation of the magnetic field at the null have non - zero real part ) are associated with certain distinguished field lines and flux surfaces , namely spines ( or @xmath0 lines in the terminology of lau & finn , 1990 ) and fan surfaces ( or @xmath1 surfaces ) as well as , sometimes , magnetic separators , the intersection lines of two fan surfaces . combined , these features are sometimes termed the magnetic skeleton ( e.g. priest & titov , 1996 ) . in dynamical systems terminology , null points are hyperbolic fixed points of a volume preserving ( divergence - free ) vector field and the spine and fan of a null are stable or unstable invariant manifolds . a null whose fan surface corresponds to a stable manifold and the spine line to an unstable manifold is designated as type a while the converse is known as type b. the magnetic separator is a heteroclinic intersection of fan surfaces , lying in the fan plane of both a type a and type b null as shown in figure [ fig : initialstate2 ] . conservation of topology has important implications for the plasma evolution , limiting the evolutions that may occur and the amount of free magnetic energy bound in the configuration . however , even for plasmas which are largely ideal , the frozen - in condition can be violated in localised regions ( such as current sheets ) where non - ideal effects become important . the key requirement for such a process to change the topology is to have a non - zero electric field component in the direction parallel to the magnetic field ( hesse & schindler , 1988 ) . such a component ( for further details see sections 4 and 5 ) enables a change in connectivity of the magnetic field lines and so a change of topology , the process being known as magnetic reconnection . reconnection may allow for significant magnetic energy release , both locally at the reconnection site and globally in the previously topologically bound energy . historically , models of magnetic reconnection were solely two - dimensional . theoretical considerations show that in two - dimensions reconnection can only occur at a hyperbolic ( x - type ) null - point of the field ( for details see e.g. hornig , 2007 ) . such null - points , lying at the intersection of four topologically distinct flux domains , are also likely sites of current sheet formation . the resultant reconnection is now fairly well understood ( for reviews see , e.g. biskamp 2000 , priest & forbes 2000 ) , although modelling still required to fully determine the importance of the complete physics including kinetic effects ( birn & priest 2007 ) . in three - dimensions , the typical situation in astrophysical plasmas , there are no fundamental restrictions on where reconnection occurs , the only requirement being the presence of a localised non - ideal term in ohm s law ( hesse & schindler , 2007 ; schindler _ et al . _ 1988 ) . accordingly , reconnection may take place wherever such non - ideal terms become important . three - dimensional null - points are thought to be one such location ( klapper _ et al . _ various forms of reconnection can occur at 3d nulls ( for a review see priest & pontin 2009 ) and the nature of the reconnection has been analysed in some detail ( e.g. pontin _ et al . _ 2004 , 2005 ; galsgaard & pontin 2011 ) . current sheets may also form away from null points ( e.g. titov _ et al . _ 2002 ; browning _ et al . _ 2008 ; wilmot - smith _ et al . _ models for the local reconnection process here ( e.g. hornig & priest 2003 , wilmot - smith _ et al . _ 2006 , 2009 ) show that , just as in the 3d null case , there are many distinct 3d characteristics of reconnection , some of which we discuss below . one topologically distinguished location where current sheets may form away from nulls is at separators which , like the 2d null point , lie at the intersection of four flux domains . it is this characteristic that has been used to argue why separators should be prone to current sheet formation ( lau & finn 1990 ) . the minimum energy state for certain field configurations has the current lying along separators ( longcope 1996 , 2001 ) and current sheet formation at separators has been observed in numerical simulations ( e.g. galsgaard & nordlund , 1997 ; haynes _ et al . _ 2007 ) . reconnection taking place at separators is known as separator reconnection . a recent series of papers ( including galsgaard & parnell 2005 ; haynes _ et al . _ 2007 ; parnell _ et al . _ 2008 ; parnell _ et al . _ 2010a ) examine the reconnection taking place in a ` fly - by ' experiment where two magnetic flux patches on the photosphere are moved past each other and the resultant evolution in the corona is followed . the authors find strong current concentrations and therefore reconnection around each of a number of separators ( up to five in some time frames , haynes _ et al . _ these separators are created in the reconnection process and connect the same two nulls ( see also longcope & cowley , 1996 ) . magnetic flux is found to evolve in a complex manner , being reconnected multiple times through these distinct reconnection sites ( parnell _ et al . _ 2008 ) . separators were also found to be important in a simulation of magnetic - flux emergence into the corona ( parnell _ et al . _ 2010b ) where current sheets between the emerging and pre - existing coronal field were shown to be threaded by numerous separators , a feature of complex magnetic fields suggested by albright ( 1999 ) . et al . _ ( 2010b ) found the regions of highest integrated parallel electric field to be at or in association with the separators , so arguing that reconnection occurs at the separators themselves . separator reconnection also appears to be a key process at the magnetopause where magnetic nulls or even clusters of nulls appear in the northern and southern polar cusp regions ( dorelli _ et al . _ 2007 these nulls are linked by a separator along the dayside magnetopause which will therefore be involved in reconnection with the interplanetary magnetic field . indeed reconnection in such a configuration has been implied by observations from the cluster spacecraft ( xiao _ et al . _ 2007 these studies partially motivate the present work where we want to investigate how and why multiple separators can be involved in reconnection . although separator configurations have been analysed for some time ( e.g. chance _ et al . _ 1992 ; craig _ et al . _ 1999 ) , there remain several unanswered basic questions surrounding the nature of separator reconnection , particularly involving the way magnetic flux evolves , i.e. exactly how the changes in field line connectivity occur . there have been suggestions that the reconnection involves a ` cut and paste ' of field lines at the separator itself in a manner similar to the 2d null case ( e.g. lau & finn 1990 ; priest & titov 1996 ; longcope _ et al . _ however , previous detailed investigations into the nature of individual 3d reconnection events both at and away from null points have shown the flux evolution to be quite distinct from the 2d picture ( hornig & priest 2003 ; pontin _ et al . _ 2004 , 2005 ) and , in the light of these models , such a simple flux evolution would be surprising . indeed a number of key features of 3d reconnection are now known which show many of its features are quite distinct to the 2d case . accordingly it would appear useful to re - examine the fundamental nature of separator reconnection and consider whether it has its own particular distinguishing characteristics . in doing so we also wish to address a conclusion of parnell et al . ( 2010a ) who suggested that separator reconnection does not appear to involve the nulls that lie at both ends of the separator as well as the recent findings ( parnell _ et al . _ 2010b ) of very large numbers of separators sometimes appearing in numerical mhd experiments . accordingly the aim of the present work is to describe in detail the nature of an isolated 3d reconnection event in the vicinity of a separator . we do so using a simple analytical model which is described in detail in section [ ref : themodel ] . the model allows us to consider typical magnetic field connectivities resulting from separator reconnection in section [ sec : bif ] and the nature of the magnetic flux evolution in section [ sec : evolve ] . we consider how reconnection rates may be determined in separator configurations in section [ sec : rates ] before discussing our findings and concluding in sections [ sec : discussion ] & [ sec : conc ] . a fully self - consistent model for reconnection must incorporate a dynamic evolution which generates current sheet(s ) as well as the reconnection that takes place at those current sheets and changes the magnetic field topology . an example in the solar corona is the emergence of a magnetic flux tube from the convection zone and reconnection with the pre - existing coronal magnetic field . inherent in such events is an enormous separation of scales between the global dynamic process and the local reconnection events . accordingly , a typical approach to model reconnection itself is to start with a local magnetic field configuration that is considered susceptible to current sheet formation ( such as , in two dimensions , an x - type null point of the field ) . in simulations the magnetic field is then confined to a finite region and the boundaries driven in such a manner as to initiate a reconnection event which can then be studied in detail . determining physically realistic boundary conditions is just one of the obstacles in this modelling technique . here we also take a simplified approach to consider the local reconnection process , aiming to model the effect of reconnection on the field topology in a three - dimensional magnetic separator configuration . as discussed in section 1 , such configurations are thought to be likely sites for current sheet formation and associated reconnection in the solar atmosphere . to construct our model we take advantage of a generic feature of reconnection , namely , the presence of an electric field with a component parallel to the magnetic field in a localised region . such an electric field will , from faraday s law , generate a magnetic flux ring around it . the flux ring will grow in strength until the reconnection ceases . this situation is illustrated in figure [ fig : addstwist ] and can be further motivated by considering the equation for the evolution of magnetic helicity . expressing the electric field , @xmath2 , as @xmath3 ( where @xmath4 is the vector potential for the magnetic field @xmath5 , @xmath6 is a potential and @xmath7 represents time ) , we have that the helicity density evolves as @xmath8 in general three - dimensional situations the condition for magnetic reconnection to occur is the existence of isolated regions at which @xmath9 . such regions act as source terms for magnetic helicity ( see the right - hand side of equation [ eq : helicityevolution ] ) thus imparting a localised twist to the configuration at a reconnection site . we may investigate the effect that reconnection has on a certain magnetic field topology by determining the effect of additional localised twist within the configuration . adding any new field component in a manner consistent with maxwell s equations provides a basis for realistic field modifications . although the question of which states could be accessed in a dynamic evolution is outwith the scope of the model , comparison of results with large - scale numerical simulations may help to determine the plausibility of results ( see section [ sec : discussion ] for a discussion of this point ) . moving on to the specific details of our model , we take as a basic ( pre - reconnection ) state a potential magnetic field configuration in which two magnetic null points at @xmath10 are connected by a separator and is given by @xmath11 here @xmath12 determines the location of the two nulls along the @xmath13-axis while @xmath14 and @xmath15 give the characteristic field strength and length scale , respectively . here we set @xmath16 , @xmath17 , and @xmath18 . this configuration is exactly that illustrated in figure [ fig : initialstate2 ] . the spine of the upper null point ( itself located at @xmath19 @xmath20 ) is the ( one - dimensional ) unstable manifold of the null and lies along the line @xmath21 . the fan surface of the upper null , the ( two - dimensional ) stable manifold lies in the plane @xmath22 and is bounded below by the spine of the lower null - point . the spine of this lower null point ( itself located at @xmath19 , @xmath23 ) is the ( one - dimensional ) stable manifold of the null and lies along the line @xmath24 while the fan surface is the ( two - dimensional ) unstable manifold . it lies in the plane @xmath25 and is bounded above by the spine of the upper null point . the two null points are connected by a separator at the intersection of the two fan surfaces , @xmath26 , @xmath27 $ ] . throughout we consider the magnetic field over the spatial domain @xmath28 $ ] and @xmath29 $ ] . to simulate the topological effect of reconnection in this configuration we add a magnetic flux ring of the form @xmath30 here the flux ring is centred at @xmath31 , the parameter @xmath32 relates to the radius of the ring , @xmath33 to the height and @xmath34 to the field strength . in the primary model considered here we choose to centre the flux ring along the separator ( at @xmath35 ) and take the parameters @xmath36 , @xmath37 and @xmath38 giving the particular flux ring @xmath39 we add this flux ring to the potential field in a smooth manner , taking a time evolution satisfying faraday s law , @xmath40 specifically we set @xmath41 with @xmath42 so that the time evolution takes place in @xmath43 . ( note that the gradient of a scalar could also be added to the electric field which could allow for the superimposition of a stationary ideal flow . we have , for simplicity , neglected this possibility . ) similar evolutions can be obtained for the addition of the more general flux ring to the potential field . the addition of the magnetic flux ring creates a localised region of twist in the centre of the domain and we now wish to determine whether and how the magnetic field topology is changed . in order to do so first notice that the flux ring is sufficiently localised that the magnetic field at the null points remains unchanged during the time evolution . accordingly for each magnetic null point we may trace magnetic field lines in the fan surface in the neighbourhood of the null point and out into the volume . this method allows us to determine how the fan surfaces are deformed by the reconnection and to locate any intersections of these surfaces , i.e. separators . we describe our findings on the field topology in the following section . in order to follow the evolution of magnetic field topology during the reconnection we begin by showing in figure [ fig : fansz0 ] the intersection of the fan surfaces with the @xmath44 plane . the strength of the flux ring increases linearly in time up to the final state which we have normalised as @xmath45 . we show the intersections by tracing field lines from each fan surface in the close neighbourhood of the corresponding null which is possible since the disturbance is localised near the centre of the domain and so the eigenvalues associated with the nulls do not change in time . in the images the fan surface of the lower null is coloured blue and that of the upper null in orange . we first note that in the initial phase of the process the angle between the fan surfaces decreases . if the reconnection is weak the process can stop in this phase without leading to any change in the magnetic skeleton . however , a stronger reconnection event can lead to a further closing of the angle between the fan surfaces until they intersect ( at about @xmath46 in our model ) . recall that crossings of the fan surfaces give the location of magnetic separators in that particular plane . hence the process creates two new separators and correspondingly two new magnetic flux domains . in order to properly identify the various flux domains we label each of the distinct topological regions with the numbers i vi , as shown in the lower - right hand image of figure [ fig : fansz0 ] . we shall examine the nature of these flux domains later in this same section , but at this point it is already possible to make some general statements about this type of bifurcation . obviously the manner in which the fan planes can fold and intersect leads to the process always creating ( or the reverse process annihilating ) separators in pairs . there is no way in which we can create a single new separator as long as the reconnection is localised . ( this excludes that the whole domain under consideration is non - ideal and that separators enter or leave the domain across the boundary ) . also shown in figure [ fig : fansz0 ] are components of the vector field @xmath47 in the @xmath44 plane which is the vector field perpendicular to the central separator at @xmath48 . this field structure is initially hyperbolic but becomes elliptic as the reconnection continues . additionally a separator typically has both elliptic and hyperbolic field regions along its length . these findings coincide with those of parnell _ et al . _ ( 2010 ) who examined the local magnetic field structure along separators in a three - dimensional mhd simulation and found both elliptic and hyperbolic perpendicular field components . a three - dimensional view of the fan surfaces at @xmath49 ( an instant when three separators are present ) is shown in figure [ fig : fans07 ] . since each separator begins and ends at the null points , the surfaces must fold in a complex way such that the three intersections merge and coincide at the nulls . this folding takes place as the fan surface of one null approaches the spine of the other null and creates ` pockets ' of magnetic flux running parallel to the spine , as shown in figure [ fig : fans07 ] where the flux passing through the @xmath50 boundary can be seen . the new flux domains alone are shown separately in the right - hand image of figure [ fig : fans07 ] and correspond to the two new flux domains ( types v and vi ) that appear in the first bifurcation as shown in figure [ fig : fansz0 ] . we now examine in more detail the connectivity of the magnetic flux in the topologically distinct flux domains i vi ( figure [ fig : fansz0 ] ) , focusing on the flux that passes through the central plane , @xmath44 ( flux above the upper null and below the lower null is not of interest here ) . the flux through the central plane can be distinguished in two ways . firstly we can follow field lines in the negative direction to determine on which side of the fan plane of the lower null they end ( @xmath51 or @xmath52 ) and secondly we can follow the field lines in positive direction to determine on which side of the fan plane of the upper null they end ( @xmath53 or @xmath54 ) . combined this technique shows the nature of the field line connectivity in each of the four different flux domains . flux in the domain labelled i enters the domain through the boundary @xmath55 and leaves through the boundary @xmath56 . flux in domain ii enters through @xmath57 and leaves through @xmath56 , that in domain iii enters through @xmath57 and leaves through @xmath50 while that in domain iv enters through @xmath55 and leaves through @xmath50 . note that this distinction of fluxes is based on topological features of the null points and is independent of the particular choice of our boundary . following the first bifurcation at @xmath58 the two new flux domains v and vi are created . field lines in flux domain v have the same basic connectivity type as those in flux domain iv ( i.e. they enter through through the @xmath55 boundary and leave through @xmath50 boundary ) . field lines in flux domain vi have the same basic connectivity type as those in flux domain ii ( i.e. they enter through the @xmath57 boundary and leave through @xmath56 boundary ) . although these two basic connectivity types existed prior to the bifurcation , the flux is enclosed within topologically distinct regions as it threads the new ` pockets ' previously described ( see figure [ fig : fans07 ] ) . as the reconnection continues , further distortion of the surfaces gives another bifurcation at @xmath59 generating an additional pair of separators , as shown in figure [ fig : fansend ] . thus at the end of the reconnection considered here five separators are present in the configuration , each connecting the same pair of nulls . figure [ fig : fansend ] also shows a colour coding of the various flux domains according to connectivity type . this is shown at the end of the reconnection and gives a summary of the way the flux domains relate to each other ( i.e. field lines in flux domains shown in the same colour have the same connectivity type ) . we envisage this type of reconnection as occurring , for example , at the dayside magnetopause due to interaction with the interplanetary magnetic field . accordingly , a crucial question is exactly how the changes in flux connectivity occur and how flux is transferred between topologically distinct domains . in the magnetopause example this would have implications for how the solar wind plasma can interact with that in the magnetosphere . we begin to address this question in the following section . in order to track the magnetic flux in time and determine how the changes in connectivity occur we examine some particular _ magnetic flux velocities _ and so begin by briefly discussing their motivation . recall that an ideal evolution of the magnetic field is one satisfying @xmath60 ( where @xmath61 is the plasma velocity ) the curl of which gives @xmath62 in such a situation the magnetic field and a line element have same evolution equation and so flux is ` frozen - in ' to the plasma and the magnetic topology is conserved . a real plasma evolution has @xmath63 where @xmath64 represents some non - ideal term ( such as @xmath65 in a resistive mhd evolution where @xmath66 is the resistivity and @xmath67 the electric current ) which is typically localised to some region of space . in this paper the effect of the non - ideal term is modelled by the addition of a magnetic flux ring . even with the inclusion of the non - ideal term we may sometimes still find a velocity @xmath68 with respect to which the magnetic flux is frozen - in if can be written as @xmath69 where @xmath70 is an arbitrary function , since taking the curl again and using faraday s law gives an equation of the form . in three - dimensional magnetic reconnection we have localised regions where @xmath71 and , as a result , a unique flux velocity @xmath68 can not be found ( hornig & priest 2003 ) . instead we may consider field lines as they enter and leave the non - ideal ( @xmath71 ) region , fixing them on one of these sides . we choose a transversal surface that lies below the non - ideal region and integrate along magnetic field lines into and out of the non - ideal region . parameterising a magnetic field line by @xmath72 and starting the integration from the point @xmath73 ( with @xmath74 ) on the transversal surface we may determine @xmath70 as @xmath75 and subsequently @xmath68 as @xmath76 carrying out the integration for @xmath70 until field lines meet a boundary and leave the domain and noting that @xmath77 at these boundaries we can find a particular flux velocity @xmath78 and visualise the effect of reconnection on the magnetic flux by showing its component perpendicular to the boundaries . this procedure can only be carried out in the presence of a single magnetic separator ; multiple separators lead to ambiguities in the potential @xmath70 . note that in the example considered here the electric field which is responsible for the non - ideal evolution decays exponentially away from its centre . at @xmath79 the electric field has fallen to a value of @xmath80 of the maximum in the domain , sufficiently small to be considered an ideal environment and so we choose this as our transversal surface for the calculation of @xmath70 . in the cartoon in figure [ fig : sephft2 ] ( left - hand image ) the direction of the flux velocity @xmath81 is shown for a surface of field lines bounding the non - ideal region . the situation considered here can be compared with that without null points ( hornig & priest 2003 ) which is indicated in figure [ fig : sephft2 ] ( right - hand image ) . in the non - null case the flux velocity has the well - known feature of counter - rotating flows . with the addition of null points , the cross - section of the flux tube which bounds the non - ideal region is split into two separate domains connected by a singular line , the spine , for each of the two nulls . along the spine the flux velocity becomes infinite . this feature can be also found in figure [ fig : fluxvel ] , which shows the component of @xmath82 perpendicular to the @xmath50 boundary at @xmath83 . the left - hand image shows streamlines of @xmath82 ( grey lines ) and the intersection of the fan surface of the lower null with the boundary ( black line ) . the flow direction is counter - clockwise . the magnitude of @xmath84 is shown in the right - hand image . a logarithmic scale is taken and the flow becomes infinite at the point of intersection of the upper null s spine ( @xmath85 ) . the flux velocity shown in figure [ fig : fluxvel ] is typical of that in the early evolution when only one separator is present . a rotational flux velocity is found within the flux tube passing through the non - ideal region . additionally , the fan surface behaves as if advected by this flux velocity . with these considerations in mind we may consider the nature of the reconnection to be as follows . for exactness in the discussion we assume the field lines are fixed below the non - ideal region although a symmetric situation occurs in the alternative case with field lines fixed above the non - ideal region . a magnetic field line threading the non - ideal region leaves the domain through a side boundary , @xmath50 , say ( that for which the flux velocity is illustrated in figure [ fig : fluxvel ] ) . there are two distinct ways the field line topology changes , illustrative phases of which are shown in figure [ fig : recntypes ] : 1 . the first situation is shown in the upper panel of figure [ fig : recntypes ] . the field line initially lies to the right of the fan surface of the lower null ( flux domain iv ) . the motion of this unanchored end is towards the spine of the upper null . it reaches the spine in a singular moment at which the flux velocity is infinite . here the field line is connected to the upper null , lying in the fan surface of that null ( this is a dynamic situation with the fan surfaces moving in time ; the fan sweeps across the anchored end of the field line ) . the field line then moves through the spine and is connected to the opposite side boundary as it leaves the domain ( flux domain i ) . in this process reconnection is occurring continuously while the field line is connected to the non - ideal region . the ` flipping ' of of the magnetic field line and change in its topology occurs at a singular moment during the reconnection . the second situation is shown in the lower panel of figure [ fig : recntypes ] . the field line initially lies to the left of the fan surface of the lower null ( flux domain iii ) . the motion of the unanchored end is again towards the spine of the upper null . the fan surface of the upper null is moving away from the anchored end of the field line while order to change its global connectivity ( pass through the upper null ) the entire field line must be lying in the fan plane of that null . accordingly a ` flipping ' of the field line can only occur at or after the first bifurcation of separators . in this bifurcation a fold ( pocket ) in the fan surface is created which sweeps up and over the anchored end of the field line . in the moment at which the global field line connectivity changes the field line is connected to the ( rising ) pocket of the fan and the upper null . in the following instant the unanchored end of the field line moves to the opposite side boundary ( @xmath56 ) and the field connectivity is the new type vi . while the separator reconnection configuration is sometimes viewed as a three - dimensional analogue of two - dimensional reconnection case at an x - type null point , this example illustrates that the behaviour of the two is , in many ways , quite distinct . in the two - dimensional case an analysis of the magnetic flux velocities ( see , for example , hornig 2007 ) shows that the reconnection of field lines occurs _ at _ the magnetic null point where the magnetic flux velocity is infinite and the field lines are ` cut ' and rejoined . our analysis demonstrates that in the separator case a magnetic null is also key to the reconnection with field lines passing through the null at the moment their global connectivity changes , again when the flux velocity is infinite . however , this is a non - local process with the null itself far removed from the reconnection site ( indeed the null may lie in an ideal environment ) . the magnetic separator itself has only an indirect role in the process . we suggest that , physically , the locations of the singularity in the flux velocity may be associated with regions of strong particle acceleration in real separator reconnection events . having considered the way in which field line reconnection occurs in the separator configuration we proceed next to a quantitative analysis where we ask how to measure and interpret the rate of reconnection in the configuration with one or more separators . the question here is whether it is necessary to know the global field topology ( including the location and number of magnetic separators ) in order to determine the reconnection rate . in a two - dimensional configuration where reconnection takes place at an x - type null point of the field the reconnection rate is given by the value of the electric field at the null point and measures the rate at which magnetic flux is transferred between the four topologically distinct flux domains . in order to express the rate as a dimensionless quantity that electric field is normalised to a characteristic convective electric field and so the reconnection rate measured in terms ( fractions ) of the alfvn mach number . in three - dimensions we also have a measure for the rate of reconnection . this is given by the maximum integrated parallel electric field over all field lines that thread the non - ideal region ( schindler _ et al . _ 1988 ) : @xmath86 however , the interpretation of the ( maximum ) integral as a unique reconnection rate relies on the assumption that the topology of the magnetic field in the reconnection region is simple . this assumption is justified in our case only up to the time of the bifurcation of separators . the formulation is consistent with the two - dimensional measure of the electric field at the null with the two - dimensional reconnection rate being the three - dimensional reconnected flux per unit length in the invariant direction . note that the question of how or whether to normalise the three - dimensional reconnection rate has not been properly addressed . in contrast to the two - dimensional case a ` high ' reconnection rate can be obtained by having a strong value of the electric field or a long non - ideal region ( long path of the integral ) and the question of what constitutes ` typical ' convective electric field , or field strength to normalise it to , is not easily answered . to examine the reconnection rate in the model presented here , we first consider only the early stages of the evolution , @xmath87 $ ] , when a single separator is present ( along the @xmath19 line and with @xmath88 $ ] ) . during this time the field line with the field line with the maximum integrated parallel electric field along it is the separator field line itself . the linear increase in time of the flux ring ( simulating the reconnection ) implies that the electric field at this line is constant in time and so the reconnection rate of the configuration for @xmath87 $ ] is given by @xmath89 in this case the coincidence of the separator and the line of maximum parallel electric field gives the clear and intuitive interpretation of the reconnection rate as the rate at which flux is transferred between the topologically distinct flux domains ( from regions ii and iv and into regions i and iii ) . following the first bifurcation three separators are present in the domain ( and five following the second bifurcation ) ; we aim to determine how the reconnection rate should be measured and interpreted in this situation . we label the central separator @xmath90 and the two separators lying off the central axis following the first bifurcation @xmath91 and @xmath92 ( by symmetry the order is not important ) . the values of the quantities @xmath93 over time are shown in figure [ fig : recnratedata ] along with the associated cumulative reconnected fluxes ( time integrals of the reconnection rates ) . for @xmath94 $ ] a total of @xmath95 units of flux are reconnected through each of the separators @xmath91 and @xmath92 . the physical interpretation for the fluxes @xmath96 and @xmath97 comes from considering the difference in reconnected flux between the separators @xmath90 and @xmath91 ( or , equivalently , @xmath92 ) in the interval @xmath98 $ ] . this is given by @xmath99 this value is that of the magnetic flux passing through the surface bounded by the separators @xmath91 and @xmath90 at the end of the reconnection and so the flux contained in each of the new flux domains ( regions v and vi ) at the end of the reconnection ( @xmath45 ) . overall , the situation is illustrated in figure [ fig : recnrateinterpret ] where the flux transport between domains ( across fan surfaces ) is shown . in the figure the black arrows show flux transport in the situation where field lines are considered as as fixed from below the non - ideal region . the alternative , the flux transport obtained by considering field lines as fixed above the non - ideal region , is shown in grey . when one separator is present ( left - hand image ) the integrated parallel electric field along that separator gives information on flux transport between the four domains in the relatively simple way already discussed . when multiple separators are present ( right - hand image ) the separator with the highest integrated parallel electric field provides the primary transport of flux while the secondary separators provide additional information on the amount of flux in each domain . considering flux in domain vi for example ( the lower closed domain ) , the transport of flux into the domain ( thick grey and black arrows ) associated with the primary , central separator is faster than transport out of the domain ( thinner grey and black arrows ) associated with the secondary bounding separator . this results in a net transport of flux into this region . when three separators are present the reconnection rate with respect to the four basic flux connectivity types ( corresponding to flux domains i , ii / vi , iii , iv / v , as described in section [ sec : bif ] ) is given by @xmath100 for @xmath101 , the situation considered here , this rate is lower than @xmath102.= the reduction in the rate of change of flux between domains compared with the reconnection rate as given by the maximum integrated parallel electric field across the region ( @xmath102 ) is a result of recursive reconnection between the multiple separators . the recursive nature of the reconnection can be seen in figure [ fig : recnrateinterpret ] . as an example considering flux as fixed from below the non - ideal region ( i.e. with the grey arrows ) , the amount of flux in domain i is being reduced at a rate @xmath102 by reconnection into region vi but also increased at a rate @xmath103 from domain vi and rate @xmath104 from domain ii . note that the situation @xmath105 is also conceivable . in this case the flux in the regions @xmath106 and @xmath107 decreases in time and the additional flux coming from these regions can push the reconnection rate [ eq : threeseprate ] above the maximum of @xmath108 . both situations show that due to the non - trivial topology of the magnetic field the reconnection rate with respect to the flux domains can differ significantly from the reconnection rate [ eq:3drec ] . this is of particular relevance for the measurement and interpretation of reconnection since separators can be hard to detect in real systems ( and numerical simulations ) . we remark further that if one is , for example , interested in particle acceleration rather than flux evolution , it may not be important that the reconnection processes can cancel each other in a topological sense . in this case the sum of all the reconnection rates , @xmath109 would give a better measure for the efficiency of the reconnection process . the preceding analysis examined a rather particular situation in which a reconnection process was centred exactly on a magnetic separator that was identified as the reconnection line . a natural question arises : is a real reconnection event likely to be centred in this way or will it perhaps encompass a separator but with a different magnetic field line being the reconnection line ? it seems likely that both cases will occur in real reconnection events and so we now consider how our findings are altered by centring the reconnection away from the initial separator . such a modification to the model can easily be made by taking @xmath110 , @xmath111 , @xmath112 as not all zero in equation and making the appropriate adjustments to equation . a first question surrounds our finding that reconnection at a separator can create new separators . by choosing different locations ( @xmath113 ) at which to centre the reconnection region we can further address the question of whether reconnection in the vicinity of a separator tends to create new separators . we find that this is indeed the case so long as the reconnection region overlaps the separator to some degree . such an example is shown in figure [ fig : offaxis ] ( left ) . if the reconnection region lies on or about a fan surface but not including a separator then the effect of the reconnection is to twist or wrap up the fan surface ( figure [ fig : offaxis ] , right ) . then only a very particular combination of reconnection events that distort two separate fan surfaces could lead to an intersection of the surfaces and the creation of a new separator . we conclude that the creation of new separators by a reconnection event that includes a separator is a generic process . the finding may help to explain the large number of separators found in mhd reconnection simulations such as those of parnell _ et al . _ ( 2010b ) . next we consider the nature of the magnetic flux evolution where a single separator is present but the separator itself does not give the maximum integrated parallel electric field . figure [ fig : sephft2 ] shows the magnetic flux evolution in the case where the separator and the maximum integrated parallel electric field do coincide ( the reconnection region is centred at the separator ) . here a rotational flux evolution in the magnetic flux tube threading the non - ideal region splits into four ` wings ' from the presence of the nulls . a similar evolution occurs when the reconnection region is offset from the separator . in that case the flux tube is split into four wings of unequal sizes but , nevertheless , the primary features of a continuous change in field line connectivity , a flipping of magnetic field lines as their topology is changed and the non - local involvement of the null all remain . that is , it is not necessary for a separator to be the reconnection line in order for the the reconnection to have these distinguishing features of separator reconnection . in this way we can think of separator reconnection as a process that takes place when a magnetic separator passes through a reconnection region . the separator need not itself be associated with the maximum parallel electric field . separator reconnection is a ubiquitous process in astrophysical plasmas ( e.g. longcope _ et al . _ 2001 ; haynes _ et al . _ 2007 ; dorelli _ et al . _ 2007 ) but several fundamental details of how it takes place including how flux evolves in the process are not yet well understood . here we have introduced a simple model for reconnection at an isolated non - ideal region threaded by a magnetic separator in an effort to better understand the basics of the process . our first new finding is that reconnection events in the neighbourhood of a separator can ( and if sufficiently strong do ) create new separators . this occurs even though the null points themselves remain in an ideal region . such a bifurcation of separators has to occur in pairs ( from 1 to 3 , 5 , 7 , ... separators ) and the reverse process is , of course , also possible . the fan planes of the two null points divide the space into distinct regions which can be distinguished due to the connectivity of their field lines with respect to any surface enclosing the configuration . each bifurcation introduces a new pair of flux domains with a connectivity different from their neighbouring fluxes . the rate of change of flux between four neighbouring flux domains is usually found by the integral of the parallel electric field along their dividing separator . however , in cases where multiple separators thread the non - ideal region the change in flux between domains is more complicated , as expressed here by equation , and it will typically be less than the maximum of rates along the separators due to recursive reconnection between separators . furthermore , considerations of asymmetric cases show that the maximum of @xmath114 does not have to occur along a separator and hence even for a single separator and a single isolated reconnection region the maximum integrated parallel electric field may not correspond to the rate of change of flux between domains . in three dimensions the topology of the magnetic field must be known before a meaningful reconnection rate can be determined . finally , the nature of the magnetic flux evolution in a separator reconnection process shows several distinguishing characteristics including rotational flux velocities that are split along the separator . these characteristics are present regardless of whether the separator itself possesses the maximum integrated parallel electric field . this analytical model is necessarily limited , one such limitation being in not addressing the way in which reconnection could be initiated at a separator . simulations do show current layers forming in the neighbourhood of separators ( e.g. parnell _ et al . _ 2010b ) but the extent of the layers may be variable . for example , we have assumed here that the non - ideal region is localised to the central portion of the separator and the nulls remain in an ideal environment . if a non - ideal region were to extend further along the domain and to include the nulls then further bifurcation processes can occur due to bifurcations of null points and a considerably more complex topology can evolve . the authors would like to thank an anonymous referee for pointing out the possibility of having a reconnection rate with respect to the flux domains which is greater than any of the rates along separators . hornig , g. 2007 , ` fundamental concepts ' in ` reconnection of magnetic fields : magnetohydrodynamics and collisionless theory and observations ' , eds . birn , j. and priest , e.r . , cambridge university press . longcope , d. 2007 , ` separator reconnection ' in ` reconnection of magnetic fields : magnetohydrodynamics and collisionless theory and observations ' , eds . birn , j. and priest , e.r . , cambridge university press . plane at times indicated . superimposed is the vector field @xmath47 of horizontal magnetic field components at the relevant times . a folding of the fan surfaces creates new separators in pairs as the fan surfaces intersect . the lower - right hand image indicates our labelling of the topologically distinct regions.,title="fig:",scaledwidth=26.0% ] plane at times indicated . superimposed is the vector field @xmath47 of horizontal magnetic field components at the relevant times . a folding of the fan surfaces creates new separators in pairs as the fan surfaces intersect . the lower - right hand image indicates our labelling of the topologically distinct regions.,title="fig:",scaledwidth=26.0% ] plane at times indicated . superimposed is the vector field @xmath47 of horizontal magnetic field components at the relevant times . a folding of the fan surfaces creates new separators in pairs as the fan surfaces intersect . the lower - right hand image indicates our labelling of the topologically distinct regions.,title="fig:",scaledwidth=26.0% ] plane at times indicated . superimposed is the vector field @xmath47 of horizontal magnetic field components at the relevant times . a folding of the fan surfaces creates new separators in pairs as the fan surfaces intersect . the lower - right hand image indicates our labelling of the topologically distinct regions.,title="fig:",scaledwidth=26.0% ] plane at times indicated . superimposed is the vector field @xmath47 of horizontal magnetic field components at the relevant times . a folding of the fan surfaces creates new separators in pairs as the fan surfaces intersect . the lower - right hand image indicates our labelling of the topologically distinct regions.,title="fig:",scaledwidth=24.8% ] . three magnetic separators are present where the fan surfaces intersect . the location of the nulls is marked with black spheres . the folding and intersection of the fan surfaces has created two new flux domains . the corresponding flux tubes are also shown separately ( _ right _ ) where the three separators are marked with black lines ( the aspect ratio of the image has been further distorted to allow the main features to be identified).,title="fig:",scaledwidth=35.0% ] . three magnetic separators are present where the fan surfaces intersect . the location of the nulls is marked with black spheres . the folding and intersection of the fan surfaces has created two new flux domains . the corresponding flux tubes are also shown separately ( _ right _ ) where the three separators are marked with black lines ( the aspect ratio of the image has been further distorted to allow the main features to be identified).,title="fig:",scaledwidth=32.0%,height=377 ] plane at @xmath45 . superimposed is the vector field @xmath47 . five separators are present at this time . were the reconnection process to continue , further folding of the fan surfaces would continue to generate separators in pairs in this manner . ( _ right _ ) at @xmath45 the various topologically flux domains are coloured according to their boundary connectivity types ( for the main regions compare with figure 3 ) showing the relation between connectivities of the various domains.,title="fig:",scaledwidth=42.5% ] plane at @xmath45 . superimposed is the vector field @xmath47 . five separators are present at this time . were the reconnection process to continue , further folding of the fan surfaces would continue to generate separators in pairs in this manner . ( _ right _ ) at @xmath45 the various topologically flux domains are coloured according to their boundary connectivity types ( for the main regions compare with figure 3 ) showing the relation between connectivities of the various domains.,title="fig:",scaledwidth=40.0% ] boundary at @xmath83 . the left - hand image shows streamlines of the flux velocity ( grey lines ) and the location of the fan surface of the lower null ( black line ) . the right - hand image indicates the magnitude of the flow which becomes singular at the spine ( @xmath85 ) . the direction of the flow is anti - clockwise . , scaledwidth=70.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( upper panel ) and @xmath115 ( lower panel ) . , title="fig:",scaledwidth=19.0% ] ( solid black line ) and the separators @xmath91 , @xmath92 ( solid grey line ) that form in the first bifurcation . the cumulative reconnected flux across the separators is shown in the dashed lines of corresponding colour . the dotted vertical line indicates the moment of bifurcation of separators.,scaledwidth=99.0% ] . in the left - hand image the reconnection region is centred at @xmath116 and overlaps the initial separator . two more separators are created as the fan surfaces intersect . in the right - hand image the reconnection region is centred at @xmath117 on a fan surface but not including the separator . no new separator is created in this process.,title="fig:",scaledwidth=42.0% ] . in the left - hand image the reconnection region is centred at @xmath116 and overlaps the initial separator . two more separators are created as the fan surfaces intersect . in the right - hand image the reconnection region is centred at @xmath117 on a fan surface but not including the separator . no new separator is created in this process.,title="fig:",scaledwidth=45.1% ]

the aim of the present work is twofold . firstly , we study here the effect of introducing small pressure perturbations in an otherwise pressure - free gravitational collapse which was to terminate in a black hole final state . for such a purpose , spherically symmetric models of black hole and naked singularity formation for a general matter field are considered , which undergo a complete gravitational collapse under reasonable physical conditions while satisfying suitable energy conditions . secondly , we investigate the genericity and stability aspects of the occurrence of naked singularities and black holes as collapse endstates . the analysis of pressure perturbations in known collapse models , inhomogeneous but otherwise pressure - free , shows how collapse final states in terms of black hole or naked singularity are affected and altered . this allows us to examine in general how generic these outcomes are and we study in the initial data space the set of conditions that lead the collapse to a naked singularity and investigate how ` abundant ' these are . while it is known now for some time that both black holes and naked singularities do arise as collapse endstates under reasonable physical conditions , this helps us understand and analyze in a clear manner the genericity aspects of occurrence of these objects in a complete gravitational collapse of a massive matter cloud in general relativity . the physics that is accepted today as the backbone of the general mechanism describing the formation of black holes as the endstate of collapse relies on the very simple and widely studied oppenheimer - snyder - datt ( osd ) dust model , which describes the collapse of a spherical cloud of homogeneous dust @xcite , @xcite . in the osd case , all matter falls into the singularity at the same comoving time while an horizon forms earlier than the singularity , thus covering it . a black hole results as the endstate of collapse . still , homogeneous dust is a highly idealized and unphysical model of matter . taking into account inhomogeneities in the initial density profile it is possible to show that the behaviour of the horizon can change drastically , thus leaving two different outcomes as the possible result of generic dust collapse : the black hole , in which the horizon forms at a time anteceding the singularity , and the naked singularity , in which the horizon is delayed thus allowing null geodesics to escape the central singularity where the density and curvatures diverge , to reach faraway observers @xcite-@xcite . it is known now that naked singularities do arise as a general feature in general relativity under a wide variety of circumstances . many examples of singular spacetimes can be found , but their relevance in models describing physically viable scenarios has been a matter of much debate since the first formulation of the cosmic censorship hypothesis ( cch ) @xcite . in particular , the formation of naked singularities in dynamical collapse solutions of einstein field equations remains a much discussed problem of contemporary relativity . the cch , which states that any singularity occurring in the universe must be hidden within an event horizon and therefore not visible to faraway observers , has remained at the stage of a conjecture for more than four decades now . this is also because of the difficulties lying in a concrete and definitive formulation of the conjecture itself . while no proof or any mathematically rigorous formulation of the same exists in the context of dynamical gravitational collapse ( while some proofs exist for particular classes of spacetimes that do not describe gravitational collapse , as in @xcite and @xcite ) , many counterexamples have been found over the past couple of decades @xcite-@xcite . many of these collapse scenarios are restricted by some simplifying assumptions such as the absence of pressures ( dust models ) or the presence of only tangential pressures @xcite-@xcite . it is well - known that the pressures can not be neglected in realistic models describing stars in equilibrium . it seems natural therefore that if one wishes to study analytically what happens during the last stages of the life of a massive star , when its core collapses under its own gravity thus forming a compact object as a remnant , pressures must be taken into account . therefore , further to early works that showed the occurrence of naked singularities in dust collapse , much effort has been devoted to understanding the role played by pressures @xcite-@xcite . the presence of pressures is a crucial element towards the description of realistic sources as we know that stars and compact objects are generally sustained by matter with strong stresses ( either isotropic or anisotropic ) . at first it was believed that the naked singularity scenario could be removed by the introduction of pressures , thus implying that more realistic matter models would lead only to the formation of a black hole . we now know that this is not the case . the final outcome of collapse with pressure is entirely decided by its initial configuration and allowed dynamical evolutions and it can be either a black hole or a naked singularity . furthermore it is now clear that within spherical collapse models ( be it dust , tangential pressure or others ) the data set leading to naked singularities is not a subset of ` zero measure ' of the set of all possible initial data . despite all this work we can still say that much more is to be understood about the role that general pressures play during the final stages of collapse . perfect fluid collapse has been studied mostly under some simplifying assumptions and restrictions in order to gain an understanding , but a general formalism for perfect fluids described by a physically valid equation of state is still lacking due to the intrinsic difficulties arising from einstein equations . considering both radial and tangential pressures is a fundamental step in order to better understand what happens in the ultra - dense regions that forms at the center of the collapsing cloud prior to the formation of the singularity . for this reason , perfect fluids appear as a natural choice since these are the models that are commonly used to describe gravitating stars in equilibrium and since it can be shown that near the center of the cloud regularity implies that matter must behave like a perfect fluid . in the present paper we use a general formalism to analyze the structure of collapse in the presence of perfect fluid pressures . this helps to understand better realistic collapse scenarios and their outcomes and brings out clearly the role played by pressures towards the formation of black holes or naked singularities as the endstate of collapse . we examine what are the key features that determine the final outcome of collapse in terms of a black hole or a naked singularity when perfect fluids , without any restriction imposed by the choice of an equation of state , are considered . the reason we do not assume an explicit equation of state here is that the behavior of matter in ultra - dense states in the final stages of collapse is unknown . on the other hand , having regularity and energy conditions satisfied provides a physically reasonable framework to study collapse endstates . we find that not only naked singularities are not ruled out in perfect fluid collapse scenarios but also that the separation between the black hole region and the naked singularity region in the space of all possible evolutions has some interesting features . in particular , we show that the introduction of small pressures can drastically change the final fate of the well - known pressureless models . for example , we see that adding a small pressure perturbation to an inhomogeneous dust model leading to a black hole can be enough to change the outcome of collapse to a naked singularity , and viceversa . further to this , we investigate here the space of initial data and collapse evolutions in generality , in order to examine the genericity of naked singularities in collapse . to study small pressure perturbations as well as the genericity and stability aspects , we use the general formalism for spherically symmetric collapse developed earlier @xcite , @xcite in order to address the basic problem of how generic is a given collapse scenario which leads to the formation of naked singularities . given the existence of an increasing number of models describing collapse leading to a naked singularity , the issue of genericity and stability of such models in the space of initial data has become the crucial ingredient in order to decide whether the cosmic censorship hypothesis in its present form should be conserved , modified or dropped altogether . it should be noted , however , that the concepts such as genericity and stability are far from well - defined in a unique manner in general relativity , as opposed to the newtonian gravity . a major difficulty towards such a task is the non - uniqueness of topology , or the concept of ` nearness ' itself in a given spacetime geometry @xcite . one could define topology on a space of spacetime metrics by requiring that the metric component values are ` nearby ' or also additionally requiring that their @xmath0-th derivatives are also nearby , and in each case the resulting topologies will be different . this is in fact connected in a way with the basic problem in arriving at a well - formulated statement of the cosmic censorship itself . there have been attempts in the past to examine the genericity and stability of naked singularities in special cases . for example , in @xcite it was shown that for certain classes of massless scalar field collapse the initial data leading to naked singularity has , in a certain sense , a positive codimension , and so the occurrence of naked singularity is unstable in that sense . on the other hand , it was shown in @xcite , @xcite that naked singularity occurrence is stable in the sense of the data sets leading to the same being open in the space of initial data . but these need not be dense in this space and so ` non - generic ' if we use the definition of ` genericity ' in the sense given in the dynamical systems theory ( where a set of initial data leading to a certain outcome is said to be generic if it is open and dense within the set of all initial data ) . in that case , however , both black hole and naked singularity final states turn out to be ` non - generic ' . therefore , in the following we adopt a more physical definition of ` genericity ' in the sense of ` abundance ' , and we call generic an initial data set that has a non - zero measure , and which is open ( though not necessarily dense ) in the set of all initial data . with this definition , the results obtained in @xcite and @xcite would mean that both black hole and naked singularity are generic collapse endstates . we note that we do not deal here with the self - similar models , or scalar fields , which is a somewhat special case . therefore , the issue of genericity and stability of naked singularities in collapse remains wide open , for spherically symmetric as well as non - spherical models and for different forms of matter fields . our consideration here treats in this respect a wide variety of physically reasonable matter fields for spherically symmetric gravitational collapse . in section [ einstein ] , the general structure for einstein equations to study spherical collapse is reviewed and we describe how the equations can be integrated thus obtaining the equation of motion for the system . in section [ perturbation ] we examine the structure of the initial data sets of collapse leading to black hole and naked singularity to gain an insight on genericity of such outcomes for some special models and effect of introducing small pressure perturbations is investigated . section [ genericity ] then considers the genericity aspects of the outcomes of collapse with respect to initial data sets . we prove that the initial data sets leading to black holes and naked singularities in the space of all initial data sets for perfect fluid collapse are both generic . section [ eos ] is devoted to a brief discussion on equations of state . finally , in section [ remarks ] we outline the key features of the above approach and its advantages , and point to possible future uses of the same for astrophysical and numerical applications . in this section we summarize and review the key features on spherical gravitational collapse analysis , and also reformulate some of the key quantities and equations , especially those relating to the nature and behaviour of the final singularity curve . this will be useful in a later section in analyzing the small pressure perturbations in a given collapse scenario , and subsequently towards a general analysis of the genericity aspects of the occurrence of naked singularities and black holes as collapse final states . the regularity conditions and energy conditions that give physically reasonable models are discussed here . the final stages of collapse are discussed , evaluating key elements that determine when the outcome will be a black hole or a naked singularity . we shall find a function , related to the tangent of outgoing geodesics at the singularity whose sign solely determines the time of formation of trapped surfaces in relation with the time of formation of the singularity . we also analyze here the occurrence of trapped surfaces during collapse and the possibility that radial null geodesics do escape thus making it visible . we see how both features are related to the sign of the above mentioned function , thus obtaining a necessary and sufficient condition for the visibility of the singularity . the most general spacetime describing a spherically symmetric collapsing cloud in comoving coordinates @xmath1 and @xmath2 depends upon three functions @xmath3 , @xmath4 and @xmath5 , and takes the form , @xmath6 the energy momentum tensor reads , @xmath7 where @xmath8 is the energy density and @xmath9 and @xmath10 are the radial and tangential stresses . the metric functions @xmath11 , @xmath12 and @xmath13 are related to the energy - momentum tensor via the einstein equations that can be written in the form : @xmath14 where @xmath15 is the misner - sharp mass of the system ( representing the amount of matter enclosed in the comoving shell labeled by @xmath1 at the time @xmath2 ) and for convenience we have defined the functions @xmath16 and @xmath17 as , @xmath18 the collapse scenario is obtained by requiring @xmath19 and the central ` shell - focusing ' singularity is achieved for @xmath20 , where the density and spacetime curvatures blow up . divergence of @xmath8 is obtained also whenever @xmath21 , thus indicating the presence of a ` shell - crossing ' singularity . such singularities are generally believed to be gravitationally weak and do not correspond to divergence of curvature scalars , therefore indicating that they are removable by a suitable change of coordinates @xcite,@xcite . for this reason in the forthcoming discussion we will be concerned only with the shell - focusing singularity thus assuming @xmath22 . since there is a scale invariance degree of freedom we can choose the initial time in such a way so that @xmath23 . therefore we can introduce the scaling function @xmath24 defined by , @xmath25 with @xmath26 , so that the collapse will be described by @xmath27 and the singularity will be reached at @xmath28 . we see that this is a better definition for the singularity since at @xmath29 the energy density does not diverge anywhere on the spacelike surfaces , including at the center @xmath30 . this is seen immediately from the regular behaviour of the mass function near the center that imposes that @xmath15 must go like @xmath31 close to @xmath30 ( as it will be shown later ) . such a regularity was not clear from equation , especially along the central curve @xmath30 , where we have @xmath20 , without the introduction of @xmath32 . in this manner the divergence of @xmath8 is only reached at the singularity . we note that @xmath32 acts like a label for successive events near the singularity since it is monotonically decreasing in @xmath2 and therefore can be used as a ` time ' coordinate in the place of @xmath2 itself near the singularity . we shall consider the misner - sharp mass to be in general a function of the comoving radius @xmath1 and the comoving time @xmath2 , expressed via the ` temporal ' label @xmath32 as @xmath33 near the center of the cloud this is just equivalent to a rewriting @xmath34 . it can be shown that vanishing of the pressure gradients near the center of the cloud imply that the radial and tangential stresses must assume the same value in the limit of approach to the center @xcite . this requirement comes from the fact that the metric functions should be at least @xmath35 at the center of the cloud and is a consequence of the fact that the einstein equation for a general fluid contains a term in @xmath36 that therefore must vanish at @xmath30 . since we are interested in the final stages of collapse of the core of a star it is therefore straightforward to take that the cloud behaves like a perfect fluid in proximity of @xmath30 . we shall then take , @xmath37 in such a case , we are then left with five equations in the six unknowns @xmath8 , @xmath38 , @xmath15 , @xmath11 , @xmath12 and @xmath32 . the system becomes closed once an equation of state for the fluid matter is defined or assumed , but in general it is possible to study physically valid dynamics ( namely those satisfying regularity and energy conditions ) without assuming a priori an equation of state . in fact it is reasonable to suppose that any equation of state that holds at the departure from equilibrium , when the gravitational collapse commences , will not continue to hold in the extreme regimes achieved when approaching the singularity . in this case , we are then left with the freedom to choose one of the functions . if we take @xmath15 as the free function then from einstein equations and , @xmath38 and @xmath8 will follow immediately and they can be evaluated explicitly once we know @xmath32 and its derivatives . further , from the requirement for perfect fluid collapse we can write equation as @xmath39 equation can be integrated once we define a suitable function @xmath40 from @xmath41 then we get , @xmath42 where the integration function @xmath43 can be interpreted in analogy with the dust models and is seen to be related to the velocity of the infalling matter shells . finally , from equation we can write the equation of motion for @xmath13 in the form of an effective potential as @xmath44 this allows us to study the dynamics of the collapsing system in analogy of the usual phase - space tools of the classical mechanics type models . equation can be expressed in terms of the scaling factor @xmath32 as , @xmath45 where the minus sign has been considered in order to study the collapse . we see that in order to have a solution we must have @xmath46 , we can call this a ` reality condition ' that is necessary for the collapse dynamics to occur . solving the equation solves the set of einstein equations . einstein equations provide the relations between the spacetime geometry and the matter distribution within it , however , they do not give any statement about the type of matter that is responsible for the geometry . on a physical ground , not every type of matter distribution is allowed , and therefore some restrictions on the possible matter models must be made based on considerations of physical reasonableness . this usually comes in the form of energy conditions ensuring the positivity of mass - energy density . further , regularity conditions must be imposed in order for the matter fields to be well - behaved at the initial epoch from which the collapse evolves and at the center of the cloud . firstly , the finiteness of the energy density at all times anteceding the singularity and regularity of the misner - sharp mass in @xmath30 imply that in general we must have , @xmath47 where @xmath48 is a regular function going to a finite value @xmath49 in the limit of approach to the center . if @xmath15 does not go as @xmath31 or higher power in the limit of approach to the center @xmath30 , we immediately see from the einstein equation for @xmath8 that there would be a singularity at the center at the initial epoch , which is not allowed by the regularity conditions as we are interested in collapse from regular initial configurations . also , requiring that the energy density has no cusps at the center is reflected in the condition , @xmath50 as seen before , the behaviour of the pressure gradients near @xmath30 suggests that the tangential and radial pressures become equal in limit of approach to the center , thus justifying our assumption of a perfect fluid type of matter . further since the gradient of the pressures must vanish at @xmath30 , we see that @xmath51 near @xmath30 which for the metric function @xmath11 implies that near the center , @xmath52 where the function @xmath53 can be absorbed in a redefinition of the time coordinate @xmath2 . from the above , via equation we can write @xmath54 from the analogy with the lemaitre - tolman - bondi ( ltb ) models we can evaluate the regularity requirements for the velocity profile @xmath43 . since near the center @xmath55 can be written as , @xmath56 we now see how to interpret the free function @xmath43 in relation with the known ltb dust models . in fact the cases with @xmath57 constant are equivalent to the bound ( @xmath58 ) , unbound ( @xmath59 ) and marginally bound ( @xmath60 ) ltb collapse models . also , the condition that there be no shell - crossing singularities may imply some further restrictions on @xmath55 . as is known , matter models describing physically realistic sources must be constrained by some energy conditions . the weak energy condition in our case implies @xmath61 the first one is achieved whenever @xmath62 . in fact from equation we see that positivity of @xmath8 is compatible with positivity of @xmath63 only if @xmath62 . therefore we must have @xmath64 that close to the center will be satisfied whenever @xmath65 . now from @xmath66 we can rewrite @xmath8 as @xmath67 from which we see that the second weak energy condition is satisfied whenever @xmath68 finally the choice of a mass profile satisfying the above equation allows us to rewrite the condition as @xmath69 which is obviously satisfied if the pressure is positive ( since it implies @xmath70 ) but can be satisfied also by some choice of negative pressure profiles . the dynamical evolution of collapse is entirely determined once the initial data set is given @xcite-@xcite . specifying the initial conditions consists in prescribing the values of the three metric functions and of the density and pressure profiles as functions of @xmath1 on the initial time surface given by @xmath71 . this reduces to defining the following functions : @xmath72 at the initial time the choice of the scale function @xmath32 is such that @xmath73 , furthermore , from @xmath74 we get @xmath75 . since the initial data must obey einstein equations it follows that not all of the initial value functions can be chosen arbitrarily . in fact the choice of the mass profile together with einstein equations is enough to specify the four remaining functions . from equations and , writing @xmath76 we get @xmath77 while from equation we get , @xmath78 from equation we can write , @xmath79 with @xmath80 related to @xmath81 by @xmath82 in turn , @xmath83 can be related to the function @xmath84 , defined by equation , at the initial time via equation , @xmath85 finally , the initial condition for @xmath12 can be related to the initial value of the function @xmath86 from equation and equation , @xmath87 since we are studying the final stages of collapse and the formation of black holes and naked singularities , we must require the initial configuration to be not trapped . this will allow for the formation of trapped surfaces during collapse and therefore we must require @xmath88 from which we see how the choice of the initial matter configuration @xmath81 is related to the initial boundary of the collapsing cloud . some restrictions on the choices of the radial boundary must be made in order not to have trapped surfaces at the initial time . this condition is reflected on the initial configuration for @xmath17 and @xmath16 since @xmath89 and this condition also gives some constraints on the initial velocity . in fact to avoid trapped surfaces at the initial time the velocity of the infalling shells must satisfy @xmath90 we see that the initial velocity of the cloud must always be positive and that the case of equilibrium configuration where @xmath91 can be taken only in the limit . the consideration of a perfect fluid matter model implies that the misner - sharp mass is in general not conserved during collapse . therefore the matching with an exterior spherically symmetric solution leads to consider the generalized vaidya spacetime . it can be proven that matching to a generalized vaidya exterior is always possible when the collapsing cloud is taken to have compact support within the boundary taken at @xmath92 , and the pressure of the matter is assumed to vanish at the boundary @xcite-@xcite . matching conditions imply continuity of the metric and its first derivatives across the boundary surface . such a matching is in principle always possible but it should be noted that matching conditions together with regularity and energy conditions , might impose some further restrictions on the allowed initial configurations . in the following we are interested in the local visibility of the central singularity occurring at the end of the collapse . therefore we will restrict our attention to a neighborhood of the central line @xmath30 . in this case it is easy to see that there always exist a neighborhood for which no shell crossing singularities occur . this is seen by the fact that @xmath93 and therefore , since @xmath94 and shell crossing singularities are defined by @xmath21 , we can always fulfill @xmath95 in the vicinity of the center . furthermore , as it was mentioned before , matter behaves like a perfect fluid close to the center . this can be seen from the fact that regularity of the metric functions at the initial time requires that @xmath96 does not blow up at the regular center . this in turn implies that from equation we must have @xmath97 , and the condition holds for any time @xmath2 before the singularity . we study now the possible outcomes of collapse evolution . it is known that in general the final fate of the complete collapse of the matter cloud will be either a black hole or a naked singularity , depending on the choice of the initial data and the dynamical evolutions as allowed by the einstein equations . in order to analyze the final outcome of collapse we shall perform a change of coordinates from @xmath98 to @xmath99 , thus considering @xmath100 . as mentioned earlier this is always possible near the center of the cloud due to the monotonic behaviour of @xmath32 . in this case the derivative of @xmath32 with respect to @xmath1 in the @xmath98 coordinates shall be considered as a function of the new coordinates , @xmath101 . we see that regularity at the center of the cloud implies @xmath102 as @xmath103 . we can then consider the metric function @xmath104 which is given by equation , which now becomes , @xmath105 where now @xmath106 and @xmath107 denotes derivative with respect to @xmath1 in the @xmath99 coordinates . this implies , @xmath108 for the sake of clarity , we may assume here that near the center the mass function @xmath109 can be written as a series as , @xmath110 as a regularity condition , we take @xmath111 and @xmath112 . the function @xmath84 can then be written as an expansion and it takes the form , @xmath113 with the first terms @xmath114 given by @xmath115 if we restrict our analysis to constant @xmath32 surfaces then in equation we can put @xmath116 and its derivatives to be zero . on the other hand if we approach the singularity along a generic curve we can not neglect the terms in @xmath116 . the equation of motion takes the form @xmath117 which can be inverted to give the function @xmath118 that represents the time at which the comoving shell labeled @xmath1 reaches the event @xmath32 , @xmath119 then the time at which the shell labeled by @xmath1 becomes singular can be written as a singularity curve as @xmath120 regularity ensures that , in general , @xmath118 is at least @xmath35 near the singularity and therefore can be expanded as , @xmath121 with @xmath122 and @xmath123 and @xmath124 . of course the situations with discontinuities can be analyzed as well with minor technical modifications to the above formalism . in our case , assuming that @xmath118 can be expanded implies that the first two terms in the expansion of @xmath40 must vanish . as seen before vanishing of first term is consistent with the regularity condition for @xmath11 that follows from the pressure gradients at the center , while vanishing of second term implies that @xmath125 must be a constant , which gives , in accordance with the requirement that the energy density has no cusps at the center , that @xmath111 . the singularity curve then takes the form @xmath126 where @xmath127 is the time at which the central shell becomes singular . by a simple calculation , retaining for the sake of completeness all the terms in the expansions of @xmath48 and expanding @xmath11 as @xmath128 , we obtain @xmath129 and @xmath130dv \ ; , \ ] ] where we have defined @xmath131 . the apparent horizon is the boundary of trapped surfaces which in general is given by , @xmath132 in the case of spherical collapse the above equation reduces to @xmath133 , which together with equation gives , @xmath134 this describes a curve @xmath135 given by @xmath136 inversely , the apparent horizon curve can be expressed as the curve @xmath137 which gives the time at which the shell labeled by @xmath1 becomes trapped . in the dust case , the condition @xmath138 implies that approaching the singularity the radius of the apparent horizon must shrink to zero thus leaving @xmath139 as the only point of the singularity curve that can in principle be visible to far away observers . on the other hand , in the perfect fluid case we note that models where the mass profile has different dependence on @xmath32 will lead to totally different structures for the apparent horizon and the trapped region . indeed in full generality there can be cases where non - central singularities become visible . this is possible in the case in which @xmath109 goes to zero as @xmath32 goes to zero , leaving @xmath140 bounded ( see e.g. @xcite ) . presently , we are interested in the case where only the central singularity would be visible . in order to understand what are the features relevant towards determining the visibility of the singularity to external observers we can evaluate the time curve of the apparent horizon in such cases as @xmath141 where @xmath142 is the singularity time curve , whose initial point is @xmath143 . near @xmath30 , equation can be written in the form , @xmath144 from which we see how the presence of pressures affects the time of the formation of the apparent horizon . in fact , all the initial configurations that cause @xmath145 ( or @xmath146 in case that @xmath145 vanishes ) to be positive will cause the apparent horizon curve to be increasing , and trapped surfaces to form at a later stage than the singularity , thus leaving the possibility that null geodesics escape from the central singularity . by studying the equation for outgoing radial null geodesics it is possible to determine that whenever the apparent horizon is increasing in time at the singularity there will be families of outgoing future directed null geodesics that reach outside observers from the central singularity , at least locally . it can be shown that positivity of the first non - null coefficient @xmath147 is a necessary and sufficient condition for the visibility of the central singularity @xcite . nevertheless the scenario of collapse of a cloud composed of perfect fluid offers some more intriguing possibilities . in fact we can see from equation that whenever the mass function @xmath15 goes to zero as collapse evolves it is possible to delay the formation of trapped surfaces in such a way that a portion of the singularity curve @xmath142 becomes timelike . this in turn leads to the possibility that non - central shells are visible when they become singular @xcite , thus introducing a new scenario that is not possible for dust collapse . it is easy to verify that in order for the mass function to be radiated away during the evolution the pressure of the fluid must be negative at some point before the formation of the singularity . despite this seemingly artificial feature negative pressures are worth investigating as they could point to a breakdown of classical gravity and could describe the occurrence of quantum effects close to the formation of the singularity @xcite . we will now use the general formalism developed above to study how the outcomes of gravitational collapse , either in terms of a black hole or naked singularity , are altered once an arbitrarily small pressure perturbation in the initial data set is introduced . the lemaitre - tolman - bondi ( ltb ) model ( @xcite-@xcite ) for inhomogeneous dust and homogeneous perfect fluid is reviewed describing necessary conditions for the ltb collapse scenario . then certain perfect fluid models are given , using the treatment above , by making specific choices for the free function so that it reduces to the ltb case for some values of the parameters . we show how the choice of these parameters or introduction of small pressure perturbations is enough to change the final outcome of collapse of the inhomogeneous dust . from equations we see that if we account for regularity and physically valid density and pressure profiles ( typically including only quadratic terms in @xmath1 ) we have @xmath148 . then the final outcome of collapse will be decided by @xmath146 as written in equation . we see immediately that once the matter model is fixed globally , thus specifying @xmath48 , the sign of @xmath146 depends continuously on the values at the initial time taken by the parameters @xmath149 , @xmath150 and @xmath151 ( with @xmath152 ) . by continuity then we can say that , away from the critical surface for which @xmath153 , if a certain initial configuration leads to a black hole ( thus having @xmath154 ) , then changing slightly the values of the initial parameters @xmath48 , @xmath155 , @xmath84 will not change the final outcome . the same result holds for naked singularities and leads us to conclude that every initial data set for which @xmath156 will have a small neighborhood leading to the same outcome @xcite . the same , however , can not be said for the surface separating these two possible outcomes of collapse , where @xmath153 . in this case it is the sign of the next non - vanishing @xmath157 that determines the final outcome and it is easy to see that the introduction of a small pressure such that @xmath158 becomes non - zero for some @xmath159 can change the final fate from black hole to naked singularity and viceversa . consider the scenario where the coefficients @xmath157 vanish for every @xmath160 . this critical surface represents the case of simultaneous collapse , or when @xmath161 , where a black hole forms at the end of collapse and it includes ( though it is not uniquely restricted to ) the oppenheimer - snyder - datt homogeneous dust collapse model . while this is the case for homogeneous dust , it is also easy to show that for inhomogeneous dust and for perfect fluid collapse also there are initial configurations that lead to simultaneous collapse , once inhomogeneities , velocity profile , and pressure are chosen suitably . in fact if we consider collapse of general type i matter fields , we can always suitably tune the parameters in order to have simultaneous collapse and therefore a black hole final outcome . nevertheless , in all these cases , the introduction of a small pressure can drastically change the final outcome by making some @xmath157 turn positive . of course in full generality there will also be regions in which the ` reality condition ' is not satisfied and therefore no final outcome is possible . but if we restrict our attention to a close neighborhood of the center we will always have a complete collapse of the inner shells thus leading to a black hole or a naked singularity . in this sense we can consider a small perturbation of any type i fluid collapse . we see that the initial data not lying on the critical surface will not change the outcome of collapse once a small perturbation in @xmath48 or @xmath38 or @xmath55 is introduced . on the other hand , those initial data sets that belong to the critical surface might indeed change outcome entirely as a result of the introduction of a small inhomogeneity , or a small pressure or small velocity . we shall now consider below some collapse models that can be obtained from the above formalism , and analyze these under the introduction of small pressure perturbations . the simplest model that can be studied for small pressure perturbations is the well - known lemaitre - tolman - bondi spacetime , where the matter form is dust with pressures assumed to be vanishing . it is interesting to know how the collapse outcome would change when small pressure perturbations are introduced in the cloud , which is a more realistic scenario compared to pressureless dust . the spacetime metric in this case takes the form , @xmath162 and it can be obtained from the above formalism not only if we impose the matter to be dust but also once we require homogeneity of the pressures ( namely imposing @xmath163 ) . in fact , if we take @xmath164 or @xmath165 , from einstein equations , together with the regularity condition for @xmath11 and @xmath43 we obtain @xmath166 and @xmath167 . the equation of motion becomes @xmath168 where @xmath48 is a function of @xmath2 only in the case of homogeneous pressure , and it is a function of @xmath1 only in the case of dust . in the case where @xmath169 is a constant we retrieve the oppenheimer - snyder - datt homogeneous dust model that , as it is known , develops a black hole at the end of collapse . the inhomogeneous dust model is obtained by requiring @xmath138 . in this case from equation follows @xmath165 and in general @xmath32 can be a function of @xmath1 and @xmath2 ( requiring @xmath170 is a necessary and sufficient condition to obtain the osd case ) . the final outcome of collapse is fully determined once the mass profile and the velocity profile are assigned ( see figure [ fig1 ] ) . and @xmath171 . in this case @xmath148 and @xmath146 determines the final outcome of collapse depending on the values of @xmath149 and @xmath150 . there are initial data sets that have a whole neighborhood leading to the same outcome in terms of either a black hole or naked singularity . the osd case lies on the critical surface separating the two outcomes which is defined by @xmath172 for all @xmath160 . ] for example , in the marginally bound case ( namely @xmath60 ) the singularity curve for inhomogeneous dust becomes @xmath173 and the apparent horizon curve is given by @xmath174 and in general collapse may lead to black hole or naked singularity depending on the behaviour of the mass profile @xmath175 . by writing @xmath175 near @xmath30 as a series @xmath176 we see that the lowest order non - vanishing @xmath81 ( with @xmath177 ) governs the final outcome of collapse . as expected in this case , we have @xmath178 in the perfect fluid ltb model ( corresponding to the frw cosmological models in case of expansion ) we have that requiring @xmath170 is a sufficient condition for having homogeneous collapse . in fact the following two statement can be easily proved : 1 . @xmath179 2 . @xmath180 the overall behaviour of the collapsing cloud is determined by the three functions @xmath181 , @xmath24 and @xmath182 ( which , as we have seen , are not independent from one another ) and the special cases of oppenheimer - snyder - datt metric and lemaitre - tolman - bondi perfect fluid metric can be summarized as follows . @xmath183 we consider now an example based on the above framework by introducing a small pressure perturbation to the inhomogeneous dust model described in the previous section . we consider in general @xmath184 and the mass function is chosen of the form @xmath185 where @xmath49 is a constant and the pressure perturbation is taken to be small in the sense that @xmath186 at all times ( in this way , as collapse progresses and the density diverges the model remains close to the ltb collapse as the pressure is always smaller than the density ) . we immediately see that setting @xmath187 reduces the model to that of inhomogeneous dust ( and @xmath188 further gives the osd homogeneous dust ) . we note that in this case no non - central singularities are visible since @xmath48 does not vanish at @xmath28 . therefore , just like in the dust case , only the central singularity at @xmath30 might eventually be visible . in this case we can integrate equation explicitly to obtain @xmath189 where @xmath190 now is the value taken by @xmath149 at the initial time when @xmath191 . therefore we can take the mass function in such a way that it corresponds to the ltb inhomogeneous dust at initial time with the pressure perturbation being triggered only at a later stage . we can therefore take , @xmath192 and the initial condition @xmath193 implies that @xmath194 ( remember that @xmath195 $ ] ) . by taking all the higher order terms to be vanishing we easily see that @xmath196 for @xmath197 and @xmath198 near @xmath30 . in this case the pressure and the energy density near @xmath30 become @xmath199 we therefore have two simple possibilities for the choice of the free function @xmath200 which determines @xmath48 : 1 . @xmath201 which implies @xmath202 and positive pressure . @xmath203 which implies @xmath204 and negative pressure . from the above , further assuming @xmath60 for simplicity in accordance with marginally bound ltb models , it follows immediately that @xmath205 and @xmath206 where we defined the function @xmath207 we see that @xmath146 is divided into two integrals . if we assume that the pressure perturbation is small ( thus considering @xmath49 to be big ) then the second integral can in principle be neglected . in fact for a suitable choice of @xmath49 it is not difficult to prove that the function at the denominator will be positive and monotonically increasing , and therefore it would not affect the sign of the integral , while the second integral will be small enough as compared to the first one . positivity of @xmath146 will then be decided by the sign of @xmath208 . , @xmath60 and @xmath209 . on the left the pressure is taken to be @xmath210 and it reduces to zero for @xmath211 . on the right the pressure perturbation is given by @xmath212 . the introduction of the pressure can uncover an otherwise clothed singularity depending on the values of @xmath213 , @xmath214 and @xmath190 . different choices of @xmath48 with @xmath215 will have a similar qualitative behaviour . ] , @xmath60 and @xmath209 . on the left the pressure is taken to be @xmath210 and it reduces to zero for @xmath211 . on the right the pressure perturbation is given by @xmath212 . the introduction of the pressure can uncover an otherwise clothed singularity depending on the values of @xmath213 , @xmath214 and @xmath190 . different choices of @xmath48 with @xmath215 will have a similar qualitative behaviour . ] in order to have naked singularity we must have @xmath216 . this is certainly the case whenever @xmath217 for any @xmath32 . therefore if we define @xmath218\}$ ] all the values of @xmath219 will lead to a naked singularity . on the other hand , values of @xmath220\}$ ] will lead to the formation of a black hole . for @xmath221 $ ] the explicit form of @xmath222 is what determines the sign of @xmath146 . it can be proven that we can have models in which positive values of @xmath190 lead to the formation of a naked singularity whereas the inhomogeneous dust case led to a black hole . we now analyze another case where the pressure perturbation introduced does not depend explicitly on @xmath32 . a similar situation was investigated by one of us earlier in @xcite . here we consider a pressure perturbation of ltb models of the form @xmath224 . if we impose @xmath225 in the equation , we can then solve for @xmath48 and explicitly evaluate @xmath17 . we obtain in that case , @xmath226 where @xmath227 and @xmath228 come from integration . here the case where @xmath229 reduces the system to dust . in this sense , if we keep @xmath230 we will consider this model to be a small perturbation of ltb in a similar way as was discussed in the previous example . then the radial derivatives of @xmath231 will correspond to the inhomogeneities in the ltb models while the pressure , or the function @xmath214 , can be taken to be either positive or negative ( with positive pressure corresponding to an increasing mass function @xmath48 and negative pressure corresponding to a decreasing mass function ) . in order to work on a specific model for the sake of clarity , we assume that @xmath214 and @xmath231 can be expanded near the center as , @xmath232 then the pressure and density become @xmath233 imposing regularity requires that @xmath234 , which implies @xmath235 . at the center of the cloud the pressure and density become @xmath236 and @xmath237 , and the energy conditions impose that @xmath238 and @xmath239 . then from equation we can integrate explicitly to obtain , @xmath240 where we have defined @xmath241 then we get , @xmath242 from the above expressions , we can easily obtain now @xmath145 and @xmath146 . for simplicity we consider here the case where @xmath243 . then @xmath148 and we get , @xmath244 } { \left(y_0v^3+z_0+\frac{4}{3}\frac{y_2}{z_0}v(1-v^3)\right ) ^{\frac{3}{2}}}\right]\sqrt{v}dv\ ] ] we see from here that the sign of @xmath146 is explicitly determined by the inhomogeneities ( @xmath245 ) and the pressure gradient ( @xmath246 ) ( see fig.[fig3 ] ) . once again taking @xmath247 and @xmath248 reduces the system to the osd collapse scenario and we see that the introduction of the slightest pressure can change drastically the outcome of collapse . on the other hand , taking only @xmath247 ( with @xmath249 ) we retrieve the ltb model and once again to change the final outcome of collapse we must choose @xmath246 suitably to balance the contribution to @xmath146 given by the inhomogeneities . therefore we see again that also within this perturbation model any collapse with initial data taken in a neighborhood of a model leading to a certain outcome and not lying on the critical surface will result in the same endstate . the equation for the apparent horizon curve can be easily written in this case and becomes @xmath250 which is a cubic equation in @xmath251 that admits in general three solutions in the case where @xmath252 . obviously this condition is satisfied near @xmath30 for positive pressures ( that correspond to negative @xmath214 ) and this indicates , as already stated , that the mass function is not vanishing at any time and therefore the central shell becomes trapped at the time of formation of the singularity . on the other hand , for negative pressures the central shell is not trapped and the formation of the apparent horizon can be shifted to some outer shells or removed altogether . as a function of @xmath246 and @xmath245 provides the phase space of initial data for the perfect fluid model with @xmath224 . here we have taken @xmath253 and @xmath254 . again , there are initial data sets that have a whole neighborhood leading to the same outcome which is either a black hole or naked singularity . the osd case lies on the critical surface separating the black holes from naked singularities . ] as we mentioned , the case of simultaneous collapse , which means that the final state of collapse is necessarily a black hole , need not be restricted to the oppenheimer - snyder - datt model only . in fact for different kinds of general type i matter fields there might be suitable choices of the parameters that lead the final state of collapse to be simultaneous . this is of course the case when the pressures are homogeneous , that is represented by the time reversal of the friedman - robertson - walker model , but more general matter models might also lead to the same behaviour . simultaneous collapse means that all matter shells terminate into the singularity at the same time . then we see from equation , which describes the singularity curve , that all coefficients @xmath147 must vanish , or equivalently that @xmath255 . from equation we can see that a sufficient condition for simultaneous collapse is @xmath256 this condition , for any given choice of the matter model , leads to a choice of the free function @xmath43 as @xmath257 it is easy to check that in the case of dust this reduces to @xmath258 and since the mass profile in this case is a function of @xmath1 only , we conclude that equation is satisfied for dust only by the case of homogeneous dust collapse ( where @xmath259 and @xmath260 ) . nevertheless , as we have said , this need not be the only case when the collapse is simultaneous . for example it s straightforward to see that when pressures are considered , the same condition as above holds for collapse of an homogeneous perfect fluid , where , in this case , @xmath261 . furthermore it is possible that the condition can be satisfied by some suitable function @xmath43 also for more general pressure profiles , since in general the mass function depends on both @xmath1 and @xmath2 , or for some other suitable choice of the velocity profile . in order to better understand the conditions under which we can have simultaneous collapse in full generality , let us now consider a general perfect fluid matter model given by a choice of @xmath109 ( which implies @xmath84 from equation ) , thus specifying all coefficients @xmath262 . firstly , we notice from equation that a suitable choice of @xmath263 is necessary in order for the reality condition to be fulfilled near the center . therefore , once we made this choice and carried out the integration for equation we see that @xmath264 depends linearly on @xmath265 only . in fact we can write @xmath266 from which we see that it will always be possible to choose @xmath265 suitably such that @xmath205 . the same reasoning can then be applied for all other coefficients @xmath147 that will depend linearly on @xmath267 as , @xmath268 and therefore for a given mass profile @xmath109 we can have simultaneous collapse if a suitable velocity profile @xmath43 given by @xmath269 exists . this means that the power series @xmath270 should converge to some function @xmath182 with a radius of convergence greater than the boundary of the cloud . this is certainly possible in the case of homogeneous pressures , where the condition that all @xmath147 vanish imposes that @xmath271 , and therefore @xmath272 . also , as we have seen in the examples above , this might be possible for other matter profiles as well . given any such model leading to simultaneous collapse ( and thus to the formation of a black hole ) , we have shown that the introduction of the slightest pressure perturbation in the initial data can turn the final outcome into a naked singularity . overall we have seen that the sign of @xmath146 and therefore the final outcome of collapse shares similar qualitative behaviour in different perfect fluid models as it is summarized in fig . [ fig4 ] . and with @xmath60 . the introduction of pressures @xmath38 and inhomogeneities @xmath190 can uncover an otherwise clothed singularity . also , different choices of @xmath48 can have the opposite effect , that is , a singularity that was naked can become covered . the initial data set @xmath273 has a whole neighborhood lying in the initial data space leading to naked singularity and can therefore be considered stable . the same holds for the initial data set @xmath274 leading to black hole . the ltb model ( obtained for @xmath165 and ) can lead to either of the outcomes and the osd scenario ( obtained when there are no inhomogeneities ) always lies on the critical surface separating the two outcomes and is therefore ` unstable ' . ] and with @xmath60 . the introduction of pressures @xmath38 and inhomogeneities @xmath190 can uncover an otherwise clothed singularity . also , different choices of @xmath48 can have the opposite effect , that is , a singularity that was naked can become covered . the initial data set @xmath273 has a whole neighborhood lying in the initial data space leading to naked singularity and can therefore be considered stable . the same holds for the initial data set @xmath274 leading to black hole . the ltb model ( obtained for @xmath165 and ) can lead to either of the outcomes and the osd scenario ( obtained when there are no inhomogeneities ) always lies on the critical surface separating the two outcomes and is therefore ` unstable ' . ] as we have seen , the final outcome of collapse depends upon the evolution of the pressure , the density and the velocity profiles . if the system is closed , as it is in the case where an equation of state describing the relation between @xmath38 and @xmath8 is given , then specifying the values of the above quantities at the initial time uniquely determines the final outcome of collapse . if the system is not closed , then we must further specify the behaviour of the free functions . once again for every possible choice of the free function(s ) the final outcome of collapse is decided by the initial values of @xmath38 , @xmath8 and @xmath55 . the genericity is defined here as every point in the initial data set leading to a naked singularity ( or a black hole ) has a neighborhood in the space of initial data for collapse whose points all lead to the same outcome . we show that the initial data set leading the collapse to a naked singularity forms an open subset of a suitable function space comprising of the initial data , with respect to an appropriate norm which makes the function space an infinite dimensional banach space . the measure theoretic aspects of this open set are considered and we argue that a suitable well - defined measure of this set must be strictly positive . this ensures genericity of initial data in a well - defined manner . at this point , the question of whether the given outcome is ` generic ' or not in a certain suitable sense yet to be defined , with respect to the allowed initial data sets , arises naturally . we shall therefore analyze the expression for the genericity of initial data leading the collapse to a naked singularity . similar conclusions apply to the case where @xmath145 vanishes and we must analyze equation and they can be used to investigate the genericity of the black hole formation scenario just as well . as is known the concept of ` genericity ' is not well - defined in general relativity . normally , by the word ` generic ' , one means ` in abundance ' or ` substantially big ' . this terminology has been used by many researcher working in relativity , and in gravitational collapse in particular ( see for example , @xcite-@xcite ) . in the theory of dynamical systems , however , the definition of ` genericity ' is given more tightly . there , one considers the class @xmath275 of all @xmath276 vector fields ( dynamical systems ) defined on a given manifold . a property @xmath277 satisfied by a vector field @xmath278 in @xmath275 is called generic if the set of all vector fields satisfying this property contains an open and dense subset of @xmath275 . this was the definition used by one of us in @xcite and @xcite . however , such a definition would render both black holes and naked singularities to be ` non - generic ' , as we remarked earlier . therefore in the present paper we have opted for a less stringent but physically more meaningful definition of genericity by requiring that the subset is open , and that it has a non - zero measure . the main reason for this comes from the fact that the ` denseness ' property depends on the topology used and the parent space used , and there are no unique and unambiguous definitions available in this regard as discussed earlier . hence the nomenclature of ` generic ' in the present paper is used in the sense in which most of the relativists use it , _ i.e. _ in the sense of abundance . this looks physically more satisfactory definition , allowing both black holes and naked singularities to be generic . in any case , the key point is that regardless of the definition used , both the collapse outcomes do share the same ` genericity ' properties , which is what our work here shows . first of all we note that the functions must satisfy the ` reality condition ' for the gravitational collapse to take place , namely @xmath279 where we have defined @xmath280 . we shall now prove that , given a mass function @xmath109 , and the function @xmath281 on the initial surface , there exists a large class of velocity distribution functions @xmath182 such that the final outcome is a naked singularity . we choose @xmath182 to satisfy the following differential equation on a constant @xmath32-surface , @xmath282^{\frac{3}{2 } } } = b(r , v ) \ ; , \ ] ] for @xmath283 , where @xmath284 is a continuous function defined on a domain @xmath285 \times [ 0 , 1 ] $ ] such that @xmath286^{\frac{3}{2 } } } < 0 \ ; , \ ] ] for all @xmath287 $ ] . it will then follow that @xmath288 as seen above , this condition ensures that central shell - focusing singularity will be naked . we prove the existence of @xmath289 as a solution of the differential equation with initial condition which @xmath284 will satisfy . for this purpose , we define @xmath290 then equation can be written as @xmath291 } { \left[m(r , v)+vb_{0}(r ) + 2va(r , v)\right]^{\frac{3}{2 } } } = \frac{1}{2 } \sqrt{v } \frac{\frac{dx}{dr } } { x^{\frac{3}{2 } } } = b(r , v ) \ ; , \ ] ] or @xmath292\equiv f(x , r ) \ ; , \ ] ] with the initial condition @xmath293 we now ensure the existence of a @xmath294-function @xmath295 as a solution of the above initial value problem defined throughout the cloud . the function @xmath296 is continuous in @xmath1 , with @xmath297 restricted to a bounded domain . with such domain of @xmath1 and @xmath297 , @xmath296 is also a @xmath294-function in @xmath297 which means @xmath296 is lipschitz continuous in @xmath297 . therefore , the differential equation has a unique solution satisfying the initial condition , provided @xmath296 satisfies a certain condition given below . further , we can ensure that the solution will be defined over the entire cloud , _ i.e. _ for all @xmath298 $ ] , by using the freedom in the choice of arbitrary function @xmath299 . for this , we consider the domain @xmath300 \times [ 0 , d]$ ] for some finite @xmath301 . let us take @xmath302 . then the differential equation has a unique solution defined over the entire cloud provided , @xmath303 this condition is to be satisfied according to usual existence theorems to guarantee existence of a unique solution ( see for example @xcite ) . equation implies @xmath304 _ i.e. _ , @xmath305\right|\le \frac{d}{r_{b } } \ ; .\ ] ] this , in turn , will be satisfied if @xmath306 for all @xmath307 $ ] . the collapsing cloud may start with @xmath308 small enough so that the expression @xmath309 which is always positive , satisfies the condition with @xmath297 restricted to a bounded domain . we then have infinitely many choices for the function @xmath299 , which is continuous and satisfies conditions and for each choice of @xmath32 . for each such @xmath299 , there will be a unique solution @xmath295 of the differential equation , satisfying initial condition , defined over the entire cloud , and in turn , there exists a unique function @xmath182 for each such choice of @xmath299 , that is given by the expression @xmath310 thus , we have shown the following : for a given constant @xmath32-surface and given initial data of mass function @xmath311 and @xmath312 satisfying physically reasonable conditions ( expressed on @xmath48 ) , there exists infinitely many choices for the function @xmath43 such that the condition is satisfied . the condition continues to hold as @xmath313 , because of continuity . hence , for all these configurations the central singularity developed in the collapse is a naked singularity . the above analysis shows that the initial data satisfying conditions , and lead the collapse to a naked singularity . if we change the sign in condition , call it condition @xmath314 , then above analysis apply and the initial data satisfying conditions , @xmath314 and lead the collapse to a black hole . in the cases discussed above , in addition to above conditions , energy conditions are also to be satisfied , and we have shown above that this is always possible for matter models leading to both possible outcomes . thus , from the above analysis , we get the following conditions which should be satisfied by the initial data in order that the end state of collapse is a naked singularity : 1 . energy conditions : @xmath315 , and @xmath316 . 2 . reality condition given by equation above . 3 . condition on @xmath284 : @xmath317 for naked singularity and @xmath318 for black hole . 4 . @xmath319 @xmath320 . for convenience , we denote the function @xmath321 then the reality condition ( 2 ) becomes @xmath322 . assuming this condition , condition ( 3 ) will be satisfied if and only if @xmath323 for naked singularity , and @xmath324 0 for a black hole . whenever @xmath325 is an increasing function of @xmath1 in the neighborhood of @xmath326 , we get its derivative positive , and so @xmath318 , and end state will be a black hole . on the other hand , if @xmath325 is a decreasing function of @xmath1 in the neighborhood of @xmath326 , we get its derivative negative , and so @xmath327 , and the endstate will be a naked singularity . regarding condition ( 4 ) , using the expression @xmath328 , it becomes @xmath329 which will be satisfied if @xmath32 and @xmath330 are sufficiently small . thus validity of all these conditions does not put any stringent restrictions on the initial data . the conclusion then is the following : if the initial data consisting of the mass function @xmath109 and function @xmath281 satisfies the above conditions , then there is a large class of velocity functions @xmath182 such that end state of collapse is either a black hole or a naked singularity , depending on the nature of function @xmath325 as explained above . we now show that the set of initial data @xmath331 satisfying the above conditions which lead the collapse to a naked singularity , is an open subset of @xmath332 , where @xmath278 is an infinite dimensional banach space of all @xmath333 or @xmath334 real - valued functions defined on @xmath335 \times [ 0 , d ] $ ] , endowed with the norms latexmath:[\[\begin{aligned } \parallel \textit{m}(r , v ) \parallel_{1 } & = & \sup_{{\mathcal{d } } } |\textit{m } | + \sup_{{\mathcal{d } } } |\textit{m}_{,r}| + \sup_{{\mathcal{d } } } |\textit{m}_{,v}| \ ; , \\ \parallel \textit{m}(r , v ) \parallel_{2 } & = & \parallel \textit{m}(r , v ) \parallel_{1 } + \sup_{{\mathcal{d } } } |\textit{m}_{,rr } | + \sup_{{\mathcal{d } } } |\textit{m}_{,rv}| + \sup_{{\mathcal{d } } } these norms are equivalent to the standard @xmath333 and @xmath334 norms , @xmath337 let @xmath338 be a subset of @xmath278 , where @xmath339 and @xmath340 . thus @xmath341 and @xmath342 are equivalent to energy conditions . we first show that @xmath343 is an open subset of @xmath278 . for simplicity , we use the @xmath333 norm , but a similar proof holds for the @xmath334 norm also . for @xmath344 , let us put @xmath345 , @xmath346 , @xmath347 , @xmath348 , @xmath349 and for @xmath1 varying in @xmath350 $ ] and @xmath351 $ ] , the functions involved herein are all continuous functions defined on a compact domain @xmath352 and hence , their maxima and minima exist . we define a positive real number @xmath353 let @xmath354 be @xmath333 in @xmath352 with @xmath355 . using above definition we get @xmath356 , @xmath357 , @xmath358 over @xmath352 . therefore , for choice of @xmath359 , the respective inequalities are @xmath360 that are satisfied on @xmath361 . the @xmath362 , @xmath363 and @xmath364 inequalities from above yield @xmath365 further , we can write @xmath366 - e_{1}| < e_{1 } $ ] where @xmath367 on @xmath352 . hence , @xmath368 on @xmath352 . using similar analysis for last four inequalities of equation , we obtain @xmath369 on @xmath352 . thus , @xmath370 , @xmath371 is @xmath333 , @xmath372 > 0 $ ] and @xmath373 > 0 $ ] on @xmath374 provided @xmath375 throughout @xmath361 . therefore , @xmath354 also lies in @xmath376 and hence , @xmath376 is an open subset of @xmath278 . using the similar argument we can show that the set @xmath377 is also an open subset of @xmath278 . thus the set of @xmath109 satisfying above conditions forms an open subset of @xmath278 , since intersection of finite number of open sets is open . similar arguments show that the set of @xmath281 and @xmath43 satisfying these conditions form separately open subsets of @xmath278 . hence using definition of product topology we see that the set @xmath378 defined above is an open subset of @xmath379 . thus initial data leading the collapse to a naked singularity forms an open subset of the banach space of all possible initial data , and therefore it is generic . we now discuss measure theoretic properties of the open set @xmath378 consisting of the initial data leading the collapse to a naked singularity . by referring to the relevant literature about measures on infinite dimensional separable banach spaces , we argue that this @xmath378 has strictly positive measure in an appropriate sense . for simplicity , we consider a single space @xmath278 and its open subset @xmath378 . we ask the question : does there exist a measure on @xmath278 which takes positive value on @xmath378 ? to answer this question , we note that @xmath278 is an infinite dimensional separable banach space , and it is a consequence of riesz lemma in functional analysis that every open ball in @xmath278 contains an infinite disjoint sequence of smaller open balls . so , if we want a translation invariant measure on @xmath278 then its value will be same on each of these balls . thus , if we demand that the surrounding ball has finite measure , then each of these smaller balls will have measure zero . otherwise sum of their measures would be infinite by countable additivity . in other words , for separable banach spaces , every open set has either measure zero or infinite under a translation invariant measure . so , if we wish to have a non - trivial measure on @xmath278 , then we have to discard the property of translation invariance . in that case we must shift to gaussian measures or wiener measures . under these measures , we can conclude that an open subset of @xmath278 will have a positive measure . for example , it is proved in @xcite ( theorem 2 on page 159 ) , that gaussian measure of an open ball in a separable banach space is positive . we can also use wiener measure on @xmath380)$]to get the same result ( see for example @xcite and @xcite ) . however , for all practical purposes in physics and astrophysics , physical functions could be assumed to be taylor expandable . thus assuming that our initial data is regular and taylor expandable , and again working for simplicity with a single function space , instead of product space , we can formulate our problem of measure as follows : let @xmath381 denote the space of taylor expandable functions defined on an interval @xmath382 $ ] . we consider initial data consisting of functions with first finite number of terms , say @xmath0 terms , which lead the collapse to a naked singularity . these functions will belong to a finite dimensional space isomorphic to @xmath383 . working with supremum norm as above and arguing similarly , we can prove that the initial data set satisfying conditions ( 1 ) to ( 4 ) above is an open subset of @xmath383 . now , we have a standard result ( see for example @xcite , prop . 4.3.4 , page 83 ) that a lebesgue measure of an open subset in @xmath383 is strictly positive . denoting this measure by @xmath384 and the open set by @xmath385 , we get @xmath386 . if , further , @xmath385 is bounded , then @xmath387 will be finite . hence normalized lebesgue measure of an open subset , and in particular , of an open ball in @xmath383 is also strictly positive , and in fact bounded . we ask the question : assuming that @xmath388 , can we get a measure on @xmath381 such that measure of @xmath378 is positive ? this is answered affirmatively by maxwell - poincar theorem which is stated as follows ( see for example @xcite ) : + consider the sequence of the normalized lebesgue measures on the euclidean spheres @xmath389 of radius @xmath390 and the limit of spaces @xmath391 then the weak limit of these measures is the standard gaussian measure @xmath359 which is the infinite product of the identical gaussian measures on the line with zero mean and variance @xmath392 . thus the limit exists and is positive on an open subset @xmath378 . it is also possible to give other approaches which answers affirmatively the existence of such limits which are termed as infinite products or in general ` projective limits ' . we describe briefly one such approach as described by yamasaki @xcite ( for general concepts on measure theory , we refer to @xcite ) . + let @xmath393 denote the infinite product of real lines . let @xmath394 be the subspace of @xmath395 given by @xmath396 : there exists @xmath397 with @xmath398 for @xmath399 . then @xmath395 is the algebraic dual of @xmath394 and @xmath400 is the weak borel field of @xmath395 . members of @xmath401 are called weak borel subsets of @xmath395 . the space @xmath381 mentioned above can be seen isomorphic to a subspace of @xmath393 , and is isomorphic to @xmath394 if we consider a finite number of terms in the taylor expansion . let @xmath402 denote one dimensional lebesgue measure on @xmath403 . let @xmath404 be a sequence of borel sets of @xmath403 such that @xmath405 . we shall define two borel measures @xmath406 and @xmath407 by @xmath408 @xmath406 is @xmath409-finite , whereas @xmath407 is a probability measure on @xmath410 . consider the product measure @xmath411 then @xmath412 is @xmath409-finite on @xmath395 . + then we have the following theorem : for every weak borel set @xmath413 of @xmath395 , put @xmath414 this limit always exists and becomes a @xmath409-finite @xmath415-invariant measure on @xmath395 . then @xmath416 lies on @xmath417 where @xmath418 we note that the measure @xmath416 defined in this theorem is called the infinite dimensional lebesgue measure supported by @xmath419 . for any open bounded subset @xmath413 of @xmath395 , @xmath413 is a borel set and hence a weak borel set . moreover @xmath420 and both these factors are finite . thus the measure @xmath416 takes a non - zero value on an open bounded subset @xmath413 of @xmath395 . we can employ this measure instead of the gaussian measure mentioned in maxwell - poincar theorem to yield the desired result . in any case , use of probability measure is inevitable and we conclude that the space of initial data leading to a certain outcome ( be it black hole or naked singularity ) , within a specific collapse scenario has non zero measure with respect to the set of all possible initial data . as is known the presence of an equation of state introduces a differential relation for the previously considered free function that closes the system of einstein equations . examples of simple , astrophysically relevant , linear and polytropic equations of states are discussed below . in the scenario described above , the relation between the density and pressure could vary during collapse , as it is natural to assume in the case where we go from a nearly newtonian initial state to a final state where a very strong gravitational field is present . the equation relating @xmath38 to @xmath8 will therefore be represented by some function of @xmath1 and @xmath32 that is related to the choice of the free function @xmath48 . there are at present many indications that suggest how in the presence of high gravitational fields gravity can act repulsively and pressures can turn negative towards the end of collapse . therefore if that is the case then the equation of state relating pressure and density ( which is always positive ) must evolve in a non - trivial manner during collapse . typically we can expect an adiabatic behaviour with small adiabatic index at the beginning of collapse when the energy density is lower than the nuclear saturation energy . it is not unrealistic to suppose that the equation of state will have sharp transitions when matter passes from one regime to another , as is the case when the limit of the nuclear saturation energy is exceeded . towards the end of collapse repulsive forces become relevant thus giving rise to negative pressures and the speed of sound approaches the speed of light @xcite . nevertheless it is interesting to analyze the structure of collapse model within one specific regime once a fixed equation of state , of astrophysical relevance , is imposed . if we choose the equation of state to be linear barotropic or polytropic we can describe collapse of the star right after it departs from the equilibrium configuration where gravity was balanced by the nuclear reactions taking places at its center . from an astrophysical point of view , neglecting the energy coming from the nuclear reactions occurring at the interior is reasonable since we know that once the nuclear fuel of the star is exhausted the star is subject to its own gravity only and the departure from equilibrium occurs in a very short time . in this sense equilibrium models for stars ( such as the early models studied in the pioneering work by chandrasekhar @xcite ) constitute the initial configuration of our collapse model and the physical parameters used to construct those equilibrium models will translate into the initial conditions for density and pressure . as we mentioned before , introducing an equation of state is enough to ensure that the system of einstein equations is closed and so no freedom to specify any function remains . in fact a barotropic equation of state of the form , @xmath421 introduces a differential equation that must be satisfied by the mass function @xmath109 , thus providing the connection between equation and equation and making them dependent on @xmath422 and its derivatives only . the dynamics is entirely determined by the initial configuration and therefore we see how solving the equation of motion is enough to solve the whole system of equations . in this case solving the differential equation for @xmath48 might prove to be too complicated . nevertheless with the assumption that @xmath48 can be expanded in a power series as in equation we can obtain a series of differential equations for each order @xmath81 . expanding the pressure and density near the center we obtain explicitly the differential equations that , if they can be satisfied by all @xmath81 converging to a finite mass function @xmath48 , solve the problem , thus giving the explicit form of @xmath48 . from @xmath423 with @xmath424 and from einstein equations and we get @xmath425 with @xmath426 once again we see that without the knowledge of @xmath116 , which is related to @xmath427 , it is impossible to solve the set of differential equations in full generality . furthermore we can see that whenever pressures and density can be expanded in a power series near the center the behaviour close to @xmath30 approaches that of an homogeneous perfect fluid . there are a few equations of state that have been widely studied in equilibrium models for stars and that naturally translate into collapse models . the simplest one is a linear equation of state of the form @xmath428 where @xmath429 is a constant . this case was studied in @xcite where it was shown the existence of a solution of the differential equation for @xmath48 , which , from einstein equations and becomes @xmath430m_{,v}=0 \ ; .\ ] ] it was shown that both black holes and naked singularities are possible outcomes of collapse depending on the initial data and the velocity distribution of the particles . another possibility is given by a polytropic equation of state of the type @xmath431 such an equation of state is often used in models for stars at equilibrium and can describe the relation between @xmath38 and @xmath8 in the early stages of collapse . therefore the physical values for @xmath432 , @xmath433 , @xmath429 and @xmath434 at the initial time can be taken from such models at equilibrium and expressed in terms of the thermodynamical quantities of the system such as the temperature and the molecular weight of the gas . the pressure is typically divided in a matter part ( describing an ideal gas ) and a radiation part ( related to the temperature ) . the exponent @xmath434 is generally written as @xmath435 , where @xmath0 is called polytropic index of the system and is constrained by @xmath436 ( for @xmath437 the cloud has no boundary at equilibrium ) @xcite , @xcite . the formalism developed above can therefore be used to investigate such realistic scenarios for collapse of massive stars . we have studied here the general structure of complete gravitational collapse of a sphere composed of perfect fluid without a priori requiring an equation of state for the matter constituents , thus allowing for the freedom to choose the mass function arbitrarily , as long as physical reasonableness as imposed by regularity and energy conditions is satisfied . the interest of such an analysis comes from the fact that the class of perfect fluid models for matter is considered to be physically viable for the description of realistic objects in nature such as massive stars and their gravitational collapse . typically perfect fluids are considered to be physically more sound than models where matter is approximated by dust - like behaviour _ i.e. _ without pressures , or where matter is sustained by only tangential pressures ( though the ` einstein cluster ' describing a spherical cloud of counter - rotating particles has been shown to have some non - trivial physical validity @xcite-@xcite ) . what we have shown is that , within the class of perfect fluid collapses , both final outcomes , namely black holes and naked singularities , can be equally possible depending on the choice of the initial data and the free function @xmath15 . in fact our results show that naked singularities and black holes are both possible final states of collapse , much in the same way as it has already been proven in the simpler cases of inhomogeneous dust and matter exhibiting only tangential stresses . the sets of initial data leading to either of the outcomes share the same properties in terms of genericity and stability . the structure of initial data sets in the case of osd , ltb and pressure collapse and their inter - relationship is , however , a complicated issue . nevertheless , we can comment on this based on the studies in @xcite , @xcite , and the results proved in sections [ perturbation ] and [ genericity ] in this paper . in @xcite it was proven that the space of initial data @xmath438 leading ltb collapse to black hole or naked singularity forms an open subset of @xmath439 , where @xmath278 is the infinite dimensional banach space of real @xmath294 functions defined on the domain . as per the analogous results in the case of non - vanishing tangential pressures it follows that the initial data set leading the collapse to osd black holes is a non - generic subset of space of @xmath440 . the shortcoming of the tangential pressure case being that is not wholly physically satisfactory . therefore to investigate the perfect fluid case , we moved to a ` bigger ' space @xmath441 , since the initial data set comprises of @xmath442 . thus , in this space , the initial data set @xmath443 or @xmath444 or the union of both the sets , will become non - generic . mathematically speaking , this set is meager or nowhere dense in the space @xmath441 . this is proved by the study performed here . thus , the initial configurations for the end states in the case of ltb or osd models lie on the critical surface separating the two possible outcomes of collapse as discussed above . this analysis in fact shows how the structure of einstein equations is very rich and complex , and how the introduction of pressures in the collapsing cloud opens up a lot of new possibilities that , while showing many interesting dynamical behaviours , do not rule out either of the two possible final outcomes . there are physical reasons , however , for the perfect fluid model to be subdued to the choice of an equation of state and there is also increasing evidence that such an equation of state can not hold during the whole duration of the dynamical collapse . in fact there are indications that as the collapsing matter approaches the singularity large negative pressures arise , thus making the equation of state relating density to pressures depart from usual well - known equations of stellar equilibrium . nevertheless the study of similar scenarios with linear or polytropic equations of state can give insights in the initial stages of collapse of a star . all this is very important from astrophysical point of view where still little is known of the processes that happen towards the very end of the life of a star , when in a catastrophic supernova explosion the outer layers are expelled and the inner core collapses under its own gravity . as we mentioned , due to the intrinsic complexity of einstein equations for perfect fluid collapse , it is generally possible to solve the system of equations only under some simplifying assumptions ( like the choice of a specific mass function ) , and only close to the center of the cloud . the indications provided by the present analysis are then a first step towards a better understanding of what happens in the last stages of the complete gravitational collapse of a realistic massive body . furthermore the above formalism could possibly be used as the framework upon which to develop possible numerical simulations of gravitational collapse . as seen in the comoving frame , the positivity of @xmath145 ( or @xmath146 ) ) is the necessary and sufficient condition for the singularity to be visible , at least locally . numerical models of a collapsing star made of a perfect fluid with a polytropic equation of state ( or a varying equation of state that takes into account the phase transitions that occur in matter under strong gravitational fields ) , with the addition of rotation and possibly electromagnetic field might help us better understand whether the inner ultradense region that forms at the center of the collapsing cloud when the apparent horizon is delayed , might be visible globally and have some effects on the outside universe . many numerical models that describe dynamical evolutions leading to the formation of black holes exist both in 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with the recent appearance of the internet and the world - wide web , understanding the properties of growing networks with popularity - based construction rules has become an active and fruitful research area @xcite . in such models , newly - introduced nodes preferentially attach to pre - existing nodes of the network that are already `` popular '' . this leads to graphs whose structure is quite different from the well - known _ random graph _ @xcite in which links are created at random between nodes without regard to their popularity . this discovery of a new class of graph theory problems has fueled much effort to characterize their properties . one basic measure of the structure of such networks is the _ node degree _ @xmath5 defined as the number of nodes in the network that are linked to @xmath0 other nodes . in the case of the random graph , the node degree is simply a poisson distribution . in contrast , many popularity - driven growing networks have much broader degree distributions with a stretched exponential or a power - law tail . the latter form means that there is no characteristic scale for the node degree , a feature that typifies many networked systems @xcite . power laws , or more generally , distributions with highly skewed tails , characterize the degree distributions of many man - made and naturally occurring networks @xcite . for example , the degree distributions at the level of autonomous systems and at the router level exhibit highly skewed tails @xcite . other important internet - based graphs , such as the hyperlink graph of the world - wide web also appear to have a degree distribution with a power - law tail @xcite . these observations have spurred a flurry of recent work to understand the underlying mechanisms for these phenomena . a related example with interest to anyone who publishes , is the distribution of scientific citations @xcite . here one treats publications as nodes and citations as links in a citation graph . currently - available data suggests that the citation distribution has a power - law tail with an associated exponent close to @xmath6 @xcite . as we shall see , this exponent emerges naturally in the _ growing network _ ( gn ) model where the relative probability of linking from a new node to a previous node ( equivalent to citing an earlier paper ) is strictly proportional to the popularity of the target node . in this paper , we apply tools from statistical physics , especially the rate equation approach , to quantify the structure of growing networks and to elucidate the types of geometrical features that arise in networks with physically - motivated growth rules . the utility of the rate equations has been demonstrated in a diverse range of phenomena in non - equilibrium statistical physics , such as aggregation @xcite , coarsening @xcite , and epitaxial surface growth @xcite . we will attempt to convince the reader that the rate equations are also a simple yet powerful analysis tool to analyze growing network systems . in addition to providing comprehensive information about the node degree distribution , the rate equations can be easily adapted to analyze both heterogeneous and directed networks , the age distribution of nodes , correlations between node degrees , various global network properties , as well as the cluster size distribution in models that give rise to independently evolving sub - networks . thus the rate equation method appears to be better suited for probing the structure of growing networks compared to the classical approaches for analyzing random graphs , such as probabilistic @xcite or generating function @xcite techniques . in the next section , we introduce three basic models that will be the focus of this review . in the following three sections , we then present rate equation analyses to determine basic geometrical properties of these networks . we close with a brief summary . the models we study appear to embody many of the basic growth processes in web graphs and related systems . these include : * the _ growing network _ ( gn ) @xcite . nodes are added one at a time and a single link is established between the new node and a pre - existing node according to an attachment probability @xmath7 that depends only on the degree of the `` target '' node ( fig . [ network ] ) . + + [ network ] * the _ web graph _ ( wg ) . this represents an extension of the gn to incorporate link directionality @xcite and leads to independent , dynamically generated in - degree and out - degree distributions . the network growth occurs by two distinct processes @xcite that are meant to mimic how hyperlinks are created in the web ( fig . [ io - growth ] ) : * * with probability @xmath8 , a new node is introduced and it immediately attaches to an earlier target node . the attachment probability depends only on the in - degree of the target . * * with probability @xmath9 , a new link is created between already existing nodes . the choices of the originating and target nodes depend on the out - degree of the former and the in - degree of the latter . + * the _ multicomponent graph _ ( mg ) . nodes and links are introduced _ independently _ @xcite . ( i ) with probability @xmath8 , a new _ unlinked _ node is introduced , while ( ii ) with probability @xmath9 , a new link is created between existing nodes . as in the wg , the choices of the originating and target nodes depend on the out - degree of the former and the in - degree of the latter . step ( i ) allows for the formation of many clusters . because of its simplicity , we first study the structure of the gn @xcite . the basic approaches developed in this section will then be extended to the wg and mg models . we first focus on the node degree distribution @xmath5 . to determine its evolution , we shall write the rate equations that account for the change in the degree distribution after each node addition event . these equations contain complete information about the node degree , from which any measure of node degree ( such as moments ) can be easily extracted . for the gn growth process in which nodes are introduced one at a time , the rate equations for the degree distribution @xmath10 are @xcite @xmath11 the first term on the right , @xmath12 , accounts for processes in which a node with @xmath13 links is connected to the new node , thus increasing @xmath5 by one . since there are @xmath14 nodes of degree @xmath13 , the rate at which such processes occur is proportional to @xmath15 , and the factor @xmath16 converts this rate into a normalized probability . a corresponding role is played by the second ( loss ) term on the right - hand side ; @xmath17 is the probability that a node with @xmath0 links is connected to the new node , thus leading to a loss in @xmath5 . the last term accounts for the introduction of new nodes with no incoming links . we start by solving for the time dependence of the moments of the degree distribution defined via @xmath18 . this is a standard method of analysis of rate equations by which one can gain partial , but valuable , information about the time dependence of the system with minimal effort . by explicitly summing eqs . ( [ nk ] ) over all @xmath0 , we easily obtain @xmath19 , whose solution is @xmath20 . notice that by definition @xmath21 is just the total number of nodes in the network . it is clear by the nature of the growth process that this quantity simply grows as @xmath22 . in a similar fashion , the first moment of the degree distribution obeys @xmath23 with solution @xmath24 . this time evolution for @xmath25 can be understood either by explicitly summing the rate equations , or by observing that this first moment simply equals the total number of link endpoints . clearly , this quantity must grow as @xmath26 since the introduction of a single node introduces two link endpoints . thus we find the simple result that the first two moments are _ independent _ of the attachment kernel @xmath7 and grow _ linearly _ with time . on the other hand , higher moments and the degree distribution itself do depend in an essential way on the kernel @xmath7 . as a preview to the general behavior for the degree distribution , consider the strictly linear kernel @xcite , for which @xmath27 coincides with @xmath28 . in this case , we can solve eqs . ( [ nk ] ) for an arbitrary initial condition . however , since the long - time behavior is most interesting , we limit ourselves to the asymptotic regime ( @xmath29 ) where the initial condition is irrelevant . using therefore @xmath30 , we solve the first few of eqs . ( [ nk ] ) directly and obtain @xmath31 , @xmath32 , _ etc_. thus each of the @xmath5 grow linearly with time . accordingly , we substitute @xmath33 in eqs . ( [ nk ] ) to yield the simple recursion relation @xmath34 . solving for @xmath35 gives @xmath36 returning to the case of general attachment kernels , let us assume that the degree distribution and @xmath27 both grow linearly with time . this hypothesis can be easily verified numerically for attachment kernels that do not grow faster than linearly with @xmath0 . then substituting @xmath33 and @xmath37 into eqs . ( [ nk ] ) we obtain the recursion relation @xmath38 and @xmath39 . finally , solving for @xmath35 , we obtain the formal expression @xmath40 to complete the solution , we need the amplitude @xmath41 . using the definition @xmath42 in eq . ( [ nkgen ] ) , we obtain the implicit relation @xmath43 which shows that the amplitude @xmath41 depends on the entire attachment kernel . for the generic case @xmath44 , we substitute this form into eq . ( [ nkgen ] ) and then rewrite the product as the exponential of a sum of a logarithm . in the continuum limit , we convert this sum to an integral , expand the logarithm to lowest order , and then evaluate the integral to yield the following basic results : @xmath45 , & $ 0\leq\gamma<1$;\cr k^{-\nu } , \quad \nu>2 , & $ \gamma=1$;\cr { \rm best\ seller } & $ 1<\gamma<2$;\cr { \rm bible } & $ 2<\gamma$.}\end{aligned}\ ] ] thus the degree distribution decays exponentially for @xmath46 , as in the case of the random graph , while for all @xmath47 , the distribution exhibits robust stretched exponential behavior . the linear kernel is the case that has garnered much of the current research interest . as shown above , @xmath48}$ ] for the strictly linear kernel @xmath49 . one might anticipate that @xmath50 holds for all _ asymptotically _ linear kernels , @xmath51 . however , the situation is more delicate and the degree distribution exponent depends on microscopic details of @xmath7 . from eq . ( [ nkgen ] ) , we obtain @xmath52 , where the exponent @xmath53 can be tuned to _ any _ value larger than 2 @xcite . this non - universal behavior shows that one must be cautious in drawing general conclusions from the gn with a linear attachment kernel . , out - degree @xmath54 , and total degree 9.,scaledwidth=25.0% ] [ degrees ] as an illustrative example of the vagaries of asymptotically linear kernels , consider the shifted linear kernel @xmath55 . one way to motivate this kernel is to explicitly keep track of link directionality . in particular , the node degree for an undirected graph naturally generalizes to the in - degree and out - degree for a directed graph , the number of incoming and outgoing links at a node , respectively . thus the total degree @xmath0 in a directed graph is the sum of the in - degree @xmath56 and out - degree @xmath57 ( fig . [ degrees ] ) . ( more details on this model are given in the next section . ) the most general linear attachment kernel for a directed graph has the form @xmath58 . the gn corresponds to the case where the out - degree of any node equals one ; thus @xmath59 and @xmath60 . for this example the general linear attachment kernel reduces to @xmath61 . since the overall scale is irrelevant , we can re - write @xmath7 as the shifted linear kernel @xmath55 , with @xmath62 that can vary over the range @xmath63 . to determine the degree distribution for the shifted linear kernel , note that @xmath64 simply equals . using @xmath65 , @xmath66 and @xmath30 , we get @xmath67 and hence the relation @xmath53 from the previous paragraph becomes @xmath68 . thus a simple additive shift in the attachment kernel profoundly affects the asymptotic degree distribution . furthermore , from eq . ( [ nkgen ] ) we determine the entire degree distribution to be @xmath69 finally , we outline the intriguing behavior for super - linear kernels . in this case , there is a `` runaway '' or gelation - like phenomenon in which one node links to almost every other node . for @xmath70 , all but a finite number of nodes are linked to a _ node that has the rest of the links . we term such an overwhelmingly popular node as a `` bible '' . for @xmath71 , the number of nodes with a just a few links is no longer finite , but grows slower than linearly in time , and the remainder of the nodes are linked to an extremely popular node that we now term `` best seller '' . full details about this runaway behavior are given in @xcite . as a final parenthetical note , when the attachment kernel has the form @xmath72 , with @xmath73 , there is preferential attachment to poorly - connected sites . here , the degree distribution exhibits faster than exponential decay , @xmath74 . when @xmath75 , the propensity for avoiding popularity is so strong that there is a finite probability of forming a `` worm '' graph in which each node attaches only to its immediate predecessor . a practically - relevant generalization of the gn is to endow each node with an intrinsic and permanently defined `` attractiveness '' @xcite . this accounts for the obvious fact that not all nodes are equivalent , but that some are clearly more attractive than others at their inception . thus the subsequent attachment rate to a node should be a function of both its degree and its intrinsic attractiveness . for this generalization , the rate equation approach yields complete results with minimal additional effort beyond that needed to solve the homogeneous network . let us assign each node an attractiveness parameter @xmath76 , with arbitrary distribution , at its inception . this attractiveness modifies the node attachment rate as follows : for a node with degree @xmath0 and attractiveness @xmath77 , the attachment rate is simply @xmath78 . now we need to characterize nodes both by their degree and their attractiveness thus @xmath79 is the number of nodes with degree @xmath0 and attractiveness @xmath77 . this joint degree - attractiveness distribution obeys the rate equation , @xmath80 here @xmath81 is the probability that a newly - introduced node has attractiveness @xmath77 , and the normalization factor @xmath82 . following the same approach as that used to analyze eq . ( [ nk ] ) , we substitute @xmath65 and @xmath83 into eq . ( [ nk - het ] ) to obtain the recursion relation @xmath84 for concreteness , consider the linear attachment kernel @xmath85 . then applying the same analysis as in the homogeneous network , we find @xmath86 to determine the amplitude @xmath41 we substitute ( [ nk - het ] ) into the definition @xmath87 and use the identity @xcite @xmath88 to simplify the sum . this yields the implicit relation @xmath89 this condition on @xmath41 leads to two alternatives : if the support of @xmath77 is unbounded , then the integral diverges and there is no solution for @xmath41 . in this limit , the most attractive node is connected to a finite fraction of all links . conversely , if the support of @xmath77 is bounded , the resulting degree distribution is similar to that of the homogeneous network . for fixed @xmath77 , @xmath90 with an attractiveness - dependent decay exponent @xmath91 amusingly , the total degree distribution @xmath92 is no longer a strict power law @xcite . rather , the asymptotic behavior is governed by properties of the initial attractiveness distribution near the upper cutoff . in particular , if @xmath93 ( with @xmath94 to ensure normalization ) , the total degree distribution exhibits a logarithmic correction @xmath95 in addition to the degree distribution , we determine _ when _ connections occur . naively , we expect that older nodes will be better connected . we study this feature by resolving each node both by its degree and its age to provide a more complete understanding of the network evolution . thus define @xmath96 to be the average number of nodes of age @xmath97 that have @xmath13 incoming links at time @xmath22 . here age @xmath97 means that the node was introduced at time @xmath98 . the original degree distribution may be recovered from the joint age - degree distribution through @xmath99 . for simplicity , we consider only the case of the strictly linear kernel ; more general kernels were considered in ref . the joint age - degree distribution evolves according to the rate equation @xmath100 the second term on the left accounts for the aging of nodes . we assume here that the probability of linking to a given node again depends only on its degree and not on its age . finally , we again have used @xmath101 for the linear attachment kernel in the long - time limit . the homogeneous form of this equation implies that solution should be self - similar . thus we seek a solution as a function of the _ single _ variable @xmath102 rather than two separate variables . writing @xmath103 with @xmath104 , we convert eq . ( [ ck1 ] ) into the ordinary differential equation @xmath105 we omit the delta function term , since it merely provides the boundary condition @xmath106 , or @xmath107 . the solution to this boundary - value problem may be simplified by assuming the exponential solution @xmath108 ; this is consistent with the boundary condition , provided that @xmath109 and @xmath110 . this ansatz reduces the infinite set of rate equations ( [ fk1 ] ) into two elementary differential equations for @xmath111 and @xmath112 whose solutions are @xmath113 and @xmath114 . in terms of the original variables of @xmath97 and @xmath22 , the joint age - degree distribution is then @xmath115 thus the degree distribution for fixed - age nodes decays _ exponentially _ , with a characteristic degree that diverges as @xmath116 for @xmath117 . as expected , young nodes ( those with @xmath118 ) typically have a small degree while old nodes have large degree ( fig . [ age ] ) . it is the large characteristic degree of old nodes that ultimately leads to a _ power - law _ total degree distribution when the joint age - degree distribution is integrated over all ages . .,scaledwidth=40.0% ] the rate equation approach is sufficiently versatile that we can also obtain much deeper geometrical properties of growing networks . one such property is the correlation between degrees of connected nodes @xcite . these develop naturally because a node with large degree is likely to be old . thus its ancestor is also old and hence also has a large degree . in the context of the web , this correlation merely expresses that obvious fact that it is more likely that popular web sites have hyperlinks among each other rather than to marginal sites . to quantify the node degree correlation , we define @xmath119 as the number of nodes of degree @xmath0 that attach to an ancestor node of degree @xmath120 ( fig . [ corr - def ] ) . for example , in the network of fig . [ network ] , there are @xmath121 nodes of degree 1 , with @xmath122 . there are also @xmath123 nodes of degree 2 , with @xmath124 , and @xmath125 nodes of degree 3 , with @xmath126 . for the case @xmath127 and @xmath128.,scaledwidth=20.0% ] for simplicity , we again specialize to the case of the strictly linear attachment kernel . more general kernels can also be treated within our general framework @xcite . for the linear attachment kernel , the degree correlation @xmath119 evolves according to the rate equation @xmath129 the processes that gives rise to each term in this equation are illustrated in fig . [ corr - re ] . the first two terms on the right account for the change in @xmath130 due to the addition of a link onto a node of degree @xmath13 ( gain ) or @xmath0 ( loss ) respectively , while the second set of terms gives the change in @xmath130 due to the addition of a link onto the ancestor node . finally , the last term accounts for the gain in @xmath131 due to the addition of a new node . ) . the newly - added node and link are shown dashed.,scaledwidth=70.0% ] as in the case of the node degree , the time dependence can be separated as @xmath132 . this reduces eqs . ( [ nkl ] ) to the time - independent recursion relation , @xmath133 this can be further reduced to a constant - coefficient inhomogeneous recursion relation by the substitution @xmath134 to yield @xmath135 solving eqs . ( [ a ] ) for the first few @xmath0 yields the pattern of dependence on @xmath0 and @xmath120 from which one can then infer the solution @xmath136 from which we ultimately obtain @xmath137.\end{aligned}\ ] ] the important feature of this result is that the joint distribution does not factorize , that is , @xmath138 . this correlation between the degrees of connected nodes is an important distinction between the gn and classical random graphs . while the solution of eq . ( [ nkl - sol ] ) is unwieldy , it greatly simplifies in the scaling regime , @xmath139 and @xmath140 with @xmath141 finite . the scaled form of the solution is @xmath142 for fixed large @xmath0 , the distribution @xmath143 has a single maximum at @xmath144 . thus a node whose degree @xmath0 is large is typically linked to another node whose degree is also large ; the typical degree of the ancestor is 37% that of the daughter node . in general , when @xmath0 and @xmath120 are both large and their ratio is different from one , the limiting behaviors of @xmath143 are @xmath145 here we explicitly see the absence of factorization in the degree correlation : @xmath146 . in addition to elucidating the degree distribution and degree correlations , the rate equations can be applied to determine global properties . one useful example is the _ out - component _ with respect to a given node * x * this is the set of nodes that can be reached by following directed links that emanate from * x * ( fig . [ in - out ] ) . in the context of the web , this is the set of nodes that are reached by following hyperlinks that emanate from a fixed node to target nodes , and then iteratively following target nodes ad infinitum . in a similar vein , one may enumerate all nodes that refer to a fixed node , plus all nodes that refer these daughter nodes , _ etc_. this progeny comprises the in - component to node * x * the set from which * x * can be reached by following a path of directed links . [ in - out ] for simplicity , we study the in - component size distribution for the gn with a constant attachment kernel , @xmath147 . we consider this kernel because many results about network components are _ independent _ of the form of the kernel and thus it suffices to consider the simplest situation ; the extension to more general attachment kernels is discussed in @xcite . for the constant attachment kernel , the number @xmath148 of in - components with @xmath149 nodes satisfies the rate equation @xmath150 the loss term accounts for processes in which the attachment of a new node to an in - component of size @xmath149 increases its size by one . this gives a loss rate that is proportional to @xmath149 . if there is more than one in - component of size @xmath149 they must be disjoint , so that the total loss rate for @xmath148 is simply @xmath151 . a similar argument applies for the gain term . finally , dividing by @xmath152 converts these rates to normalized probabilities . for the constant attachment kernel , @xmath153 , so asymptotically @xmath154 . interestingly , eq . ( [ ik ] ) is almost identical to the rate equations for the degree distribution for the gn with linear attachment kernel , except that the prefactor equals @xmath155 rather than @xmath156 . this change in the normalization factor is responsible for shifting the exponent of the resulting distribution from @xmath6 to @xmath157 . to determine @xmath148 , we again note , by explicitly solving the first few of the rate equations , that each @xmath158 grows linearly in time . thus we substitute @xmath159 into eqs . ( [ ik ] ) to obtain @xmath160 and @xmath161 . this immediately gives @xmath162 this @xmath163 tail for the in - component distribution is a robust feature , _ independent _ of the form of the attachment kernel @xcite . this @xmath163 tail also agrees with recent measurements of the web @xcite . the complementary out - component from each node can be determined by constructing a mapping between the out - component and an underlying network `` genealogy '' . we build a genealogical tree for the gn by taking generation @xmath164 to be the initial node . nodes that attach to those in generation @xmath165 form generation @xmath166 ; the node index does not matter in this characterization . for example , in the network of fig . [ network ] , node 1 is the `` ancestor '' of 6 , while 10 is the `` descendant '' of 6 and there are 5 nodes in generation @xmath167 and 4 in @xmath168 . this leads to the genealogical tree of fig . [ genealogy ] . . the nodes indices indicate when each is introduced . the nodes are also arranged according to generation number.,scaledwidth=35.0% ] [ genealogy ] the genealogical tree provides a convenient way to characterize the out - component distribution . as one can directly verify from fig . [ genealogy ] , the number @xmath169 of out - components with @xmath149 nodes equals @xmath170 , the number of nodes in generation @xmath171 in the genealogical tree . we therefore compute @xmath172 , the size of generation @xmath165 at time @xmath22 . for this discussion , we again treat only the constant attachment kernel and refer the reader to ref . @xcite for more general attachment kernels . we determine @xmath172 by noting that @xmath172 increases when a new node attaches to a node in generation @xmath173 . this occurs with rate @xmath174 , where @xmath175 is the number of nodes . this gives the differential equation for @xmath176 with solution @xmath177 , where @xmath178 . thus the number @xmath169 of out - components with @xmath149 nodes equals @xmath179 note that the generation size @xmath172 grows with @xmath165 , when @xmath180 , and then decreases and becomes of order 1 when @xmath181 . the genealogical tree therefore contains approximately @xmath182 generations at time @xmath22 . this result allows us to determine the diameter of the network , since the maximum distance between any pair of nodes is twice the distance from the root to the last generation . therefore the diameter of the network scales as @xmath183 ; this is the same dependence on @xmath184 as in the random graph @xcite . more importantly , this result shows that the diameter of the gn is always small ranging from the order of @xmath185 for a constant attachment kernel , to the order of one for super - linear attachment kernels . in the world - wide web , link directionality is clearly relevant , as hyperlinks go _ from _ an issuing website _ to _ a target website but not vice versa . thus to characterize the local graph structure more fully , the node degree should be resolved into the _ in - degree _ the number of incoming links to a node , and the complementary _ out - degree _ [ degrees ] ) . measurements on the web indicate that these distributions are power laws with different exponents @xcite . these properties can be accounted for by the web graph ( wg ) model ( fig . [ io - growth ] ) and the rate equations provide an extremely convenient analysis tool . let us first determine the average node degrees ( in - degree , out - degree , and total degree ) of the wg . let @xmath186 be the total number of nodes , and @xmath187 and @xmath188 the in - degree and out - degree of the entire network , respectively . according to the elemental growth steps of the model , these degrees evolve by one of the following two possibilities : @xmath189 that is , with probability @xmath8 a new node and new directed link are created ( fig . [ io - growth ] ) so that the number of nodes and both the total in- and out - degrees increase by one . conversely , with probability @xmath190 a new directed link is created and the in- and out - degrees each increase by one , while the total number of nodes is unchanged . as a result , @xmath191 , and @xmath192 . thus the average in- and out - degrees , @xmath193 and @xmath194 , are both equal to @xmath195 . to determine the degree distributions , we need to specify : ( i ) the _ attachment rate _ @xmath196 , defined as the probability that a newly - introduced node links to an existing node with @xmath56 incoming and @xmath57 outgoing links , and ( ii ) the _ creation rate _ @xmath197 , defined as the probability of adding a new link from a @xmath198 node to a @xmath199 node . we will use rates that are expected to occur in the web . clearly , the attachment and creation rates should be non - decreasing in @xmath56 and @xmath57 . moreover , it seems intuitively plausible that the attachment rate depends only on the in - degree of the target node , @xmath200 ; _ i.e. _ , a website designer decides to create link to a target based only on the popularity of the latter . in the same spirit , we take the link creation rate to depend only on the out - degree of the issuing node and the in - degree of the target node , @xmath201 . the former property reflects the fact that the development rate of a site depends only on the number of outgoing links . the interesting situation of power - law degree distributions arises for asymptotically linear rates , and we therefore consider @xmath202 the parameters @xmath203 and @xmath204 must satisfy the constraint @xmath205 and @xmath206 to ensure that the rates are positive for all attainable in- and out - degree values , @xmath207 and @xmath208 . with these rates , the joint degree distribution , @xmath209 , defined as the average number of nodes with @xmath56 incoming and @xmath57 outgoing links , evolves according to @xmath210\\ & & \hskip 0.285 in + q\left[{(j-1+\lambda_{\rm out})n_{i , j-1 } -(j+\lambda_{\rm out})n_{ij}\over j+\lambda_{\rm out } n}\right ] + p\,\delta_{i0}\delta_{j1}.\nonumber\end{aligned}\ ] ] the first group of terms on the right accounts for the changes in the in - degree of target nodes by simultaneous creation of a new node and link ( probability @xmath8 ) or by creation of a new link only ( probability @xmath190 ) . for example , the creation of a link to a node with in - degree @xmath56 leads to a loss in the number of such nodes . this occurs with rate @xmath211 , divided by the appropriate normalization factor @xmath212 . the factor @xmath213 in eq . ( [ nij ] ) is explicitly written to make clear these two types of processes . similarly , the second group of terms account for out - degree changes . these occur due to the creation of new links between already existing nodes hence the prefactor @xmath190 . the last term accounts for the introduction of new nodes with no incoming links and one outgoing link . as a useful consistency check , one may verify that the total number of nodes , @xmath214 , grows according to @xmath215 , while the total in- and out - degrees , @xmath216 and @xmath217 , obey @xmath218 . by solving the first few of eqs . ( [ nij ] ) , it is again clear that the @xmath219 grow linearly with time . accordingly , we substitute @xmath220 , as well as @xmath221 and @xmath222 , into eqs . ( [ nij ] ) to yield a recursion relation for @xmath223 . using the shorthand notations , @xmath224 the recursion relation for @xmath223 is @xmath225n_{ij } = ( i-1+\lambda_{\rm in})n_{i-1,j}+a(j-1+\lambda_{\rm out})n_{i , j-1 } + p(1+p\lambda_{\rm in})\delta_{i0}\delta_{j1}.\end{aligned}\ ] ] the in - degree and out - degree distributions are straightforwardly expressed through the joint distribution : @xmath226 and @xmath227 . because of the linear time dependence of the node degrees , we write @xmath228 and @xmath229 . the densities @xmath230 and @xmath231 satisfy [ ii ] ( i+b)i_i & = & ( i-1+_in)i_i-1 + p(1+p_in)_i0 , + ( j+1q+_outq)o_j & = & ( j-1+_out)o_j-1+p1+p_outq_j1 , respectively . the solution to these recursion formulae may be expressed in terms of the following ratios of gamma functions [ i - sol ] i_i&=&i_0(i+_in)(b+1)(i+b+1)(_in ) , + [ o - sol ] o_j&=&o_1(j+_out ) ( 2+q^-1+_out q^-1 ) ( j+1+q^-1+_out q^-1)(1+_out ) , with @xmath232 and @xmath233 . from the asymptotics of the gamma function , the asymptotic behavior of the in- and out - degree distributions have the distinct power law forms @xcite , [ in ] i_i~i^-_in , _ in&=&2+p_in , + 0.7 in o_j~j^-_out , _ out&=&1+q^-1+_out pq^-1 , with @xmath234 and @xmath235 both necessarily greater than 2 . let us now compare these predictions with current data for the web @xcite . first , the value of @xmath8 is fixed by noting that @xmath236 equals the average degree of the entire network . current data for the web gives @xmath237 , and thus we set @xmath238 . now ( [ in ] ) contain two free parameters and by choosing them to be @xmath239 and @xmath240 we reproduced the observed exponents for the degree distributions of the web , @xmath241 and @xmath242 , respectively . the fact that the parameters @xmath203 and @xmath204 are of the order of one indicates that the model with linear rates of node attachment and bilinear rates of link creation is a viable description of the web . in addition to the degree distributions , current measurements indicate that the web consists of a `` giant '' component that contains approximately 91% of all nodes , and a large number of finite components @xcite . the models discussed thus far are unsuited to describe the number and size distribution of these components , since the growth rules necessarily produce only a single connected component . in this section , we outline a simple modification of the wg , the multicomponent graph ( mg ) , that naturally produces many components . in this example , the rate equations now provide a comprehensive characterization for the size distribution of the components . in the mg model , we simply separate node and link creation steps . namely , when a node is introduced it does not immediately attach to an earlier node , but rather , a new node begins its existence as isolated and joins the network only when a link creation event reaches the new node . for the average network degrees , this small modification already has a significant effect . the number of nodes and the total in- and out - degrees of the network , @xmath243 now increase with time as @xmath221 and @xmath244 . thus the in- and out - degrees of each node are time independent and equal to @xmath245 , while the total degree is @xmath246 . as in the case of the wg model , we study the case of a bilinear link creation rate given in eq . ( [ ac ] ) , with now @xmath247 to ensure that @xmath248 for all permissible in- and out - degrees , @xmath207 and @xmath249 . we study local characteristics by employing the same approach as in the wg model . we find that results differ only in minute details , _ e.g. _ , the in- and out - degree densities @xmath230 and @xmath231 are again the ratios of gamma functions , and the respective exponents are @xmath250 notice the decoupling the in - degree exponent is independent of @xmath204 , while @xmath235 is independent of @xmath251 . the expressions ( [ inout ] ) are neater than their wg counterparts , reflecting the fact that the governing rules of the mg model are more symmetric . to complement our discussion , we now outline the asymptotic behavior of the joint in- and out - degree distribution . although this distribution defies general analysis , we can obtain partial and useful information by fixing one index and letting the other index vary . an elementary but cumbersome analysis yields following limiting behaviors @xmath252 with @xmath253 we also can determine the joint degree distribution analytically in the subset of the parameter space where @xmath254 , _ i.e. _ , @xmath255 . in what follows , we therefore denote @xmath256 . the resulting recursion equation for the joint degree distribution is @xmath257 with @xmath258 . because the degrees @xmath56 and @xmath57 appear in eq . ( [ nij * ] ) with equal prefactors , the substitution @xmath259 reduces eqs . ( [ nij * ] ) into the constant - coefficient recursion relation @xmath260 we solve eq . ( [ m ] ) by employing the generating function technique . multiplying eq . ( [ m ] ) by @xmath261 and summing over all @xmath262 , we find that the generating function @xmath263 equals @xmath264 . expanding @xmath265 in @xmath266 yields @xmath267 which we then expand in @xmath268 by employing the identity @xmath269 . finally , we arrive at @xmath270 from which the joint degree distribution is @xmath271 thus again , the in- and out - degrees of a node are correlated : @xmath272 . let us now turn now to the distribution of connected components ( clusters , for brevity ) . for simplicity , we consider models with undirected links . let us first estimate the total number of clusters @xmath273 . at each time step , @xmath274 with probability @xmath8 , or @xmath275 with probability @xmath190 . this implies @xmath276 the gain rate of @xmath273 is exactly equal to @xmath8 , while in the loss term we ignore self - connections and tacitly assume that links are always created between different clusters . in the long - time limit , self - connections should be asymptotically negligible when the total number of clusters grows with time and no macroscopic clusters ( _ i.e. _ , components that contain a finite fraction of all nodes ) arise . this assumption of no self - connections greatly simplifies the description of the cluster merging process . consider two clusters ( labeled by @xmath277 ) with total in - degrees @xmath278 , out - degrees @xmath279 , and number of nodes @xmath280 . when these clusters merge , the combined cluster is characterized by @xmath281 thus starting with single - node clusters with @xmath282 , the above merging rule leads to clusters that always satisfy the constraint @xmath283 . thus the size @xmath0 characterizes both the in - degree and out - degree of clusters . to simplify formulae without sacrificing generality , we consider the link creation rate of eq . ( [ ac ] ) , with @xmath284 . then the merging rate @xmath285 of the two clusters is proportional to @xmath286 , or @xmath287 let @xmath288 denotes the number of clusters of mass @xmath0 . this distribution evolves according to @xmath289 the first set of terms account for the gain in @xmath288 due to the coalescence of clusters of size @xmath290 and @xmath291 , with @xmath292 . similarly , the second set of terms accounts for the loss in @xmath288 due to the coalescence of a cluster of size @xmath0 with any other cluster . the last term accounts for the input of unit - size clusters . these rate equations are similar to those of irreversible aggregation with product kernel @xcite . the primary difference is that we explicitly treat the number of clusters as finite . one can verify that the total number of nodes @xmath293 grows with rate @xmath8 and that the total number of clusters @xmath294 grows with rate @xmath295 , in agreement with eq . ( [ n ] ) . solving the first few eqs . ( [ comp ] ) shows again that @xmath288 grow linearly with time . accordingly , we substitute @xmath296 into eqs . ( [ comp ] ) to yield the time - independent recursion relation @xmath297 a giant component , _ i.e. _ , a cluster that contains a finite fraction of all the nodes , emerges when the link creation rate exceeds a threshold value . to determine this threshold , we study the moments of the cluster size distribution @xmath298 . we already know that the first two moments are @xmath299 and @xmath300 . we can obtain an equation for the second moment by multiplying eq . ( [ mk ] ) by @xmath301 and summing over @xmath302 to give @xmath303 . when this equation has a real solution , @xmath304 is finite . the solution is @xmath305 and gives , when @xmath306 , to a threshold value @xmath307 . for @xmath308 ( @xmath309 ) all clusters have finite size and the second moment is finite . in this steady - state regime , we can obtain the cluster size distribution by introducing the generating function @xmath310 to convert eq . ( [ mk ] ) into the differential equation @xmath311/q}.\ ] ] the asymptotic behavior of the cluster size distribution can now be read off from the behavior of the generating function in the @xmath312 limit . in particular , the power - law behavior @xmath313 implies that the corresponding generating function has the form @xmath314 here the asymptotic behavior is controlled by the dominant singular term @xmath315 . however , there are also subdominant singular terms and regular terms in the generating function . in eq . ( [ gen ] ) we explicitly included the three regular terms which ensure that the first three moments of the cluster - size distribution are correctly reproduced , namely , @xmath316 , @xmath317 , and @xmath318 . finally , substituting eq . ( [ gen ] ) into eq . ( [ cz ] ) we find that the dominant singular terms are of the order of @xmath319 . balancing all contributions of this order in the equation determines the exponent of the cluster size distribution to be @xmath320 this exponent satisfies the bound @xmath321 and thus justifies using the behavior of the second moment of the size distribution as the criterion to find the threshold value @xmath322 . for @xmath323 there is no giant cluster and the cluster size distribution has a power - law tail with @xmath324 given by eq . ( [ tau ] ) . intriguingly , the power - law form holds for any value @xmath309 . this is in stark contrast to all other percolation - type phenomena , where away from the threshold , there is an exponential tail in cluster size distributions @xcite . thus in contrast to ordinary critical phenomena , the entire range @xmath309 is critical . as a corollary to the power - law tail of the cluster size distribution for @xmath309 , we can estimate the size of the largest cluster @xmath325 to see how `` finite '' it really is . using the extreme statistics criterion @xmath326 we obtain @xmath327 , or @xmath328 this is very different from the corresponding behavior on the random graph , where below the percolation threshold the largest component scales logarithmically with the number of nodes . thus for the random graph , the dependence of @xmath329 changes from @xmath185 just below , to @xmath184 , just above the percolation threshold ; for the mg , the change is much more gentle : from @xmath330 to @xmath184 . these considerations suggest that the phase transition in the mg is dramatically different from the percolation transition . very recently , simplified versions of the mg were studied @xcite . numerical @xcite and analytical @xcite evidence suggest that the size of the giant component @xmath331 near the threshold scales as @xmath332 therefore , the phase transition of this dynamically grown network is of infinite order since all derivatives of @xmath331 vanish as @xmath333 . in contrast , static random graphs with any desired degree distribution @xcite exhibit a standard percolation transition @xcite . in this paper , we have presented a statistical physics viewpoint on growing network problems . this perspective is strongly influenced by the phenomenon of aggregation kinetics , where the rate equation approach has proved extremely useful . from the wide range of results that we were able to obtain for evolving networks , we hope that the reader appreciates both the simplicity and the power of the rate equation method for characterizing evolving networks . we quantified the degree distribution of the growing network model and found a diverse range of phenomenology that depends on the form of the attachment kernel . at the qualitative level , a stretched exponential form for the degree distribution should be regarded as `` generic '' , since it occurs for an attachment kernel that is sub - linear in node degree ( _ e.g. _ , @xmath44 with @xmath2 ) . on the other hand , a power - law degree distribution arises only for linear attachment kernels , @xmath51 . however , this result is `` non - generic '' as the degree distribution exponent now depends on the detailed form of the attachment kernel . we investigated extensions of the basic growing network to incorporate processes that naturally occur in the development in the web . in particular , by allowing for link directionality , the full degree distribution naturally resolves into independent in - degree and out - degree distributions . when the rates at which links are created are linear functions of the in- and out - degrees of the terminal nodes of the link , the in- and out - degree distributions are power laws with different exponents , @xmath234 and @xmath235 , that match with current measurements on the web with reasonable values for the model parameters . we also considered a model with independent node and link creation rates . this leads to a network with many independent components and now the size distribution of these components is an important characteristic . we have characterized basic aspects of this process by the rate equation approach and showed that the network is in a critical state even away from the percolation threshold . the rate equation approach also provides evidence of an unusual , infinite - order percolation transition . while statistical physics tools have fueled much progress in elucidating the structure of growing networks , there are still many open questions . one set is associated with understanding dynamical processes in such networks . for example , what is the nature of information transmission ? what governs the formation of traffic jams on the web ? another set is concerned with growth mechanisms . while we can make much progress in characterizing networks with idealized growth rules , it is important to understand the actual rules that govern the growth of the internet . these issues appear to be fruitful challenges for future research . it is a pleasure to thank francois leyvraz and geoff rodgers for collaborations that led to some of the work reported here . we also thank john byers and mark crovella for numerous informative discussions . finally , we are grateful to nsf grants int9600232 and dmr9978902 for financial support . recent reviews from the physicist s perspective include : s. h. strogatz , nature * 410 * , 268 ( 2001 ) ; r. albert and a .- barabsi , rev . phys . * 74 * , 47 ( 2002 ) ; s. n. dorogovtsev and j. f. f. mendes , adv . xx * , xxx ( 2002 ) . s. r. kumar , p. raphavan , s. rajagopalan , and a. tomkins , in : _ proc . 8th www conf . _ ( 1999 ) ; s. r. kumar , p. raphavan , s. rajagopalan , and a. tomkins , in : _ proc . 25th vldb conf . _ ( 1999 ) ; j. kleinberg , r. kumar , p. raghavan , s. rajagopalan , and a. tomkins , in : _ proceedings of the international conference on combinatorics and computing _ , lecture notes in computer science , vol . 1627 ( springer - verlag , berlin , 1999 ) .

complex networks have been the focus of the study of dynamical properties of complex systems in nature and society in the last decade @xcite . usually all nodes are assumed to belong to one population or class , and the interactions between two distinct populations have been reported just recently @xcite . the dynamical properties of two interacting populations , extroverts and introverts were recently studied using dynamical network evolution model @xcite . buldyrev et al @xcite developed a framework for understanding the robustness of interacting networks . using the generating functions method they present the exact analytical solutions for the critical fraction of nodes , which upon removal , will lead to a complete fragmentation into interdependent networks . the focus of the current approach in understanding the formation and evolution of terrorist networks is on middle - range perspective as opposed to the micro - level approach that considers individual terrorist and macro - level analysis of the root causes of terrorism @xcite . in particular , the interest is in placing the relationships between individuals in the context of ( i ) their interactions with each other , ( ii ) how they are influenced by ideas originating from their environment , ( iii ) their interactions with people and organizations outside of their group @xcite . motivated by this description which conceptually refers to a complex system and because networks provide a fruitful framework to model complex systems @xcite , we introduce a network model that aims to describe the interactions between potentially terrorist and non - terrorist populations . our goal is to present a model of a social network that contains two types of agents which demonstrate different affinities in establishing connections within their own population versus connections with the other population . we also aim that the model is simple enough to allow the derivation of approximate analytical expressions of the basic characteristics of the network . the latter can be achieved in the general framework of rate equation theory in the mean field approximation which has been introduced to study fundamental characteristics of growing network models @xcite . within this framework we introduce the rate equations that are specific to our model , solve them and obtain analytical expressions that predict the growth dynamics of the degree of individual vertices . , the initial contact @xmath2 and the secondary contact @xmath3 . ( a)-(d ) show the connections when the new node is an @xmath1 node . ( e)-(h ) show the connections when the new node is a @xmath0 node . empty symbols ( @xmath4 ) mark an @xmath1 node while full symbols ( @xmath5 ) mark a @xmath0 node . full lines represent links between nodes of the same type , e.g. friends , such as @xmath6 or @xmath7 . dashed lines represent links between nodes of different types such as @xmath8 , e.g. enemies . ] while the probability to create initial connections is easily defined , the probability for the secondary contacts between nodes is difficult to derive because of the interactions between the two types of nodes . as a first approximation , we assume that the initial and secondary contacts form edges with the same probabilities . to include a more precise contribution of both the initial and the secondary contacts in the rate equations we empirically obtain their functional dependences . this leads to an improved agreement between analytical and numerical simulation results . from the functional dependence of the degree of a node as a function of time using the mean - field arguments , we derive the degree distribution for each of the two types of nodes . we also derive analytical expressions of the structural , three - point correlations between nodes to study the clustering properties of the networks . the network models are broadly classified into two categories : the network evolution models in which addition of new edges depends on the local structure of the network , and nodal attribute models in which the existence of edges is determined solely by the attributes of the nodes ( for review see @xcite ) . the network evolution models can be further categorized into growing network evolution models and dynamical network evolution models . in the former the network growth starts with a small seed network and nodes and links are added according to specific rules until the network reaches a predetermined size . dynamical network evolution models start with an empty network and edges are added and deleted according to specific rules until statistical properties of the network stabilize . in this paper , we propose a model that incorporates a growing network evolution process with nodal attributes which could be thought of as a new class of model . the model has five free parameters , three describing the growing network evolution process and two describing nodal attributes . the parameters of the growing network evolution process are the number of nodes @xmath9 , the average number of nodes selected at random as initial contacts @xmath10 , and the average number of nodes selected as secondary contacts @xmath11 among the neighbors of each initial contact @xcite . two parameters quantify the type and amount of nodes , non - violent @xmath1 or potentially violent @xmath0 nodes and the type of interactions between them which are as follows : ( i ) nodes are randomly marked as non - violent @xmath1 with probability @xmath12 and potentially violent @xmath0 with probability @xmath13 ; ( ii ) when establishing initial contacts , nodes connect with probability @xmath14 if the nodes are of the same type , such as @xmath0 with @xmath0 or @xmath1 with @xmath1 , or with probability @xmath15 if the nodes are of different types such as @xmath1 with @xmath0 nodes . the secondary contacts are established with nodes among the neighbors of the initial contacts . the model combines the random attachment of initial contacts with the implicit preferential attachment of the secondary contacts . in that , the model represents a generalization with two types of node attributes of growing models such as @xcite . the definition of establishing edges can be thought of as creating links between friends ( solid lines in fig . [ fig1 ] ) or between enemies ( dashed lines ) . by varying the value of @xmath14 , we can generate different strengths of friendliness or animosity . the possible configurations of triads that arise in social networks in such a context are : ( a ) three friendly interactions ; ( b ) one friendly and two unfriendly connections ; ( c ) two friendly interactions and one unfriendly ; ( d ) three unfriendly interactions @xcite . according to the strong formulation of structural balance theory in social sciences , configurations ( a ) and ( b ) are considered stable while ( c ) and ( d ) are unstable and likely to break apart @xcite . in their empirical large - scale verification of the long - standing structural balance theory , the authors of @xcite find that the unstable triads , especially formation ( c ) are extremely underrepresented in an online social system in comparison to a null model . our model produces correctly the stable configurations ( a ) ( fig . [ fig1]a , e ) and ( b ) ( fig . [ fig1]b - d , f - h ) but can not produce the unstable configurations ( c ) and ( d ) in accordance with the strong formulation of the structural balance theory @xcite . the model algorithm consists of the following steps : ( 1 ) start with a seed network of @xmath16 connected nodes among which some are @xmath1 and some are @xmath0 , depending on @xmath12 ; ( 2 ) at each time step add a new node , which has probability @xmath12 to be a non - violent and @xmath13 to be a violent node ; ( 3 ) select on average @xmath17 random nodes as initial contacts . the probability to connect the same type of nodes is @xmath14 while @xmath15 is the probability for initial contacts if they are of different types . ( 4 ) select on average @xmath18 nodes among the neighbors of each initial contact as secondary contacts . connecting the new node with the secondary contacts is done without checking if it is the same type of node or not . there are two reasons for this choice . ( i ) because the probability to establish inter - population connections ( @xmath14 ) is higher than the probability to connect nodes intra - population ( @xmath19 ) , it is more likely that the first contact and its neighbors ( potential secondary contacts ) are of the same type than of different types . therefore the secondary contacts will be more likely to be intra - population contacts even without explicitly modifying their probability to connect based on the type . ( ii ) the secondary contacts are meant to mimic the ` friend - of - a - friend ' type of contacts in the real world , and we think that the implicit preferences given by the existing network connections would more accurately describe the nature of such contacts without including an explicit separate probability . apply steps ( 2 ) to ( 4 ) until the network reaches the necessary size . we start with constructing the rate equations that describe the change of the degree of a node on average during one time step of the network growth process for each of the non - violent @xmath1 and violent @xmath0 nodes . the degree of a node grows via two processes . one is the random attachment of connecting a new node to @xmath10 nodes that are its initial contacts . the second process is when the new node is further connected to the @xmath11 nodes among the neighbors of the initial contacts . in the following we assume that the probability of this second process is linear with respect to the degree of the node which leads to implicit preferential attachment . the rate equations are : @xmath20 @xmath21 where @xmath22 is the degree of node @xmath3 and we assumed that @xmath23 and @xmath24 . all possible combinations of triads of a new node @xmath25 , the initial contact @xmath2 , and the secondary contact @xmath3 are schematically shown in fig . [ fig1](a - h ) and presented by the third through the sixth terms in eq . ( [ n ] ) and eq . ( [ v ] ) for an @xmath1 and @xmath0 node , respectively . for example , the fifth term in eq . ( [ n ] ) describes the rate of change of the degree of vertex @xmath3 ( which is an @xmath1 node ) due to establishing the configuration of contacts shown in fig . the fifth term contains four factors . the first factor is the average number of secondary contacts which is @xmath26 . @xmath27 is the probability that the new node created at time step @xmath28 is a @xmath0 node . @xmath14 is the probability that the newly created node connects to the initial contact which is a node of the same type . finally , @xmath29 is the probability that the node selected for initial contact ( node @xmath2 in fig . [ fig1]c ) shares an edge with the node @xmath3 . this is a standard expression for preferential attachment except for the complications induced by having two distinct populations . the nominator @xmath30 is the degree of the @xmath3 node if we count only the links to different types of nodes ( in this case @xmath0 nodes ) . the denominator is the sum of all possible links which the type of node selected for initial contact ( in this case @xmath0 ) could have . assuming that the initial and secondary contacts are created with the same probabilities the nominator would be equal to @xmath24 and we will approximate it in this way and the denominator would be equal to @xmath31 . we will use this expression as a first approximation and we will also empirically derive functional dependences of the denominator and compare the results . note that the number of edges , multiplied by two , that exist between @xmath1 and @xmath1 nodes is equal to @xmath32 and include both edges created as initial contacts and edges created as secondary contacts . the same is true for the number of edges between @xmath0 and @xmath0 nodes which is @xmath33 , and the number of edges between @xmath1 and @xmath0 nodes which is @xmath34 in eqs . ( [ n ] ) and ( [ v ] ) . we know that the probabilities for creating edges as initial contacts are @xmath14 or @xmath19 if between same type of nodes or different types of nodes , respectively . we do not know , however , what these probabilities are when the edges represent connections to secondary contacts . this is the reason to express the respective summations by the following relations : [ sms ] @xmath35 eqs . ( [ sms ] ) contain the term @xmath36 because there are @xmath37 vertices at time @xmath28 and @xmath38 is the average initial degree of a vertex . @xmath39 in eq . ( [ snn ] ) is the probability that the edge is between @xmath1 nodes . @xmath40 ( [ svv ] ) is the probability that the edge is between @xmath0 nodes , and @xmath41 eq . ( [ snv ] ) is the probability that the edge is between nodes of different types . functions @xmath42 , @xmath43 , and @xmath44 contain edges established due to both initial and secondary contacts , whose contributions can not be separated and derived analytically . therefore , we will obtain these functional dependences through empirical considerations . if we assume that the edges to secondary contacts are established with the same probabilities as the edges to initial contacts , then the relations would have been : [ simple ] @xmath45 after separating the variables and integrating the rate equation of the degree @xmath46 of @xmath1 node eq.([n ] ) from @xmath47 to @xmath22 and from @xmath48 to @xmath28 we obtain the following expression for the degree as a function of time @xmath49 @xmath50 where [ knt ] @xmath51 integrating the rate equation of the degree @xmath22 of @xmath0 nodes , eq.([v ] ) produces the time dependence of the degree of any @xmath0 node @xmath52 where [ kvt ] @xmath53 if we assume that the edges to secondary contacts are established with the same probabilities as the edges to initial contacts , then the solution of the rate equation eq . ( [ n ] ) will be the following for an @xmath1 node @xmath54 where [ knts ] @xmath55 and for a @xmath0 node : @xmath56 where [ kvts ] @xmath57 after making use of eqs . ( [ simple ] ) . in the mean field approximation , the degree @xmath49 of a node @xmath3 evolves with time @xmath28 strictly monotonically after the node was added to the network at time @xmath48 . therefore , the nodes added to the network more recently will have on average lower degree than those added to the network earlier . assuming that we add nodes to the network at equal intervals , the probability density of @xmath48 is @xmath58 . using the properties of cumulative probability distribution function , we can write that the probability of a node to have degree @xmath59 is equal to the probability that the node has been added to the network at time @xmath60 @xmath61 we can derive the probability density function of @xmath1 nodes , @xmath62 by obtaining an expression for @xmath63 from eq . ( [ kin ] ) , then replacing it in eq . ( [ cdf ] ) and differentiating the resultant equation with respect to @xmath64 , which is @xmath65 . the result is : @xmath66 similarly , we can derive the probability density function of @xmath0 nodes , @xmath67 by obtaining an expression for @xmath68 from eq . ( [ kiv ] ) , then replacing it in eq . ( [ cdf ] ) and differentiating the resultant equation with respect to @xmath64 to obtain : @xmath69 if we use eqs . ( [ kins ] ) and ( [ kivs ] ) to derive expressions for @xmath63 and @xmath68 , respectively , then the degree distributions are presented by @xmath70 @xmath71 it should be noted that @xmath22 and all quantities are expectation values and can be compared to simulation results which are assemble averages . the analytical results converge to those reported in ref . @xcite in the limit of one population which means @xmath72 , @xmath73 , @xmath74 , and @xmath75 . the dependence of the clustering coefficient as a function of the degree of a node can be derived using the rate equation method @xcite . the number of triangles @xmath76 ( @xmath77 ) around a node if @xmath3 is an @xmath1 ( @xmath0 ) node is changing with time following two processes . the first process is when node @xmath3 is selected as one of the initial contacts with probability @xmath78 ( @xmath79 ) and the new node links to some of its neighbors which are @xmath11 on average . the second process is when node @xmath3 is selected as a secondary contact and a triangle is formed between the new node , the initial contact , and the secondary contact . it is possible that two neighboring initial contacts and the new node form a triangle , but the contribution of this process is negligible . the rate equation for the number of connections between the nearest neighbors of a node of degree @xmath80 ( @xmath81 ) is given by @xmath82 @xmath83 respectively . after some algebra and using eq . ( [ n ] ) if @xmath3 is an @xmath1 node and eq . ( [ v ] ) if @xmath3 is a @xmath0 node we obtain @xmath84 and @xmath85 after integrating both sides of eqs . ( [ en1 ] ) and ( [ ev1 ] ) with respect to @xmath28 , using the initial condition @xmath86 , and @xmath47 given by eq . ( [ knt6 ] ) , we obtain the expressions for the change with time of the number of connections between the nearest neighbors of a node of degree @xmath80 @xmath87 if @xmath3 is an @xmath1 node and of a node of degree @xmath81 @xmath88 if @xmath3 is a @xmath0 node . we use eqs . ( [ pn ] ) and ( [ pv ] ) to obtain expressions for @xmath89 and insert them in eqs . ( [ en2 ] ) and ( [ ev2 ] ) , respectively . finally , the degree - dependent clustering coefficient @xmath90 as a function of the degree @xmath91 of the node which is also referred as the clustering spectrum , is given by @xmath92 } { k(k-1 ) } \label{ckn}\ ] ] for an @xmath1 node and by @xmath93 } { k(k-1 ) } \label{ckv}\ ] ] for a @xmath0 node , where we make use of @xmath94 $ ] , which defines the clustering coefficient of a vertex as the ratio of the total number of existing connections between all @xmath91 of its neighbors and the number @xmath95 of all possible connections between them . the degree - dependent clustering coefficient @xmath90 defines the local clustering properties of the network . the global clustering characteristics of a network include the mean clustering coefficient @xmath96 as averaged over the vertex degree , the mean clustering @xmath97 as averaged over the nodes of the network ( where @xmath98 is the clustering coefficient of node @xmath3 ) , and the so - called transitivity @xmath99 @xcite . making use of degree - dependent local clustering coefficient ( eqs . ( [ ckn ] ) and ( [ ckv ] ) ) and the degree distribution ( eqs . ( [ pns ] ) and ( [ pvs ] ) ) for @xmath1 or @xmath0 node one can define the respective mean clustering coefficient as : @xmath100 transitivity is a measure of the ratio of the total number of loops of length three in a graph to the total number of connected triples and is defined as @xcite @xmath101 the mean clustering coefficient @xmath96 and the transitivity @xmath99 assess in a different manner the clustering properties of a graph . in real networks they could have very different values for the same network @xcite . we compare three outputs : the analytical derivation of degree distribution @xmath102 ( eqs . ( [ pns ] ) and ( [ pvs ] ) ) that was obtained assuming that the links to secondary contacts are established with the same probabilities as the links to initial contacts eq . ( [ simple ] ) , the derivation of @xmath102 ( eqs . ( [ pn ] ) and ( [ pv ] ) ) obtained by using functional dependences eq . ( [ sms ] ) , and the numerical simulation results . nodes @xmath39 as a function of @xmath103 , where @xmath104 for @xmath105 ( @xmath106 ) and @xmath107 for @xmath108 ( @xmath109 ) . ( b ) probability to establish both initial and secondary contacts between @xmath0 nodes @xmath110 as a function of @xmath111 , where @xmath112 for @xmath113 ( @xmath106 ) and @xmath114 for @xmath115 ( @xmath109 ) . ( c ) probability to establish both initial and secondary contacts between different types of nodes as a function of @xmath116 for @xmath117 $ ] at fixed @xmath105 ( @xmath106 ) , and fixed @xmath108 ( @xmath109 ) , respectively . values on y - axis in ( a ) represent matrix multiplication of the vector of @xmath39 as a function of @xmath118 $ ] at fixed @xmath119 ( @xmath106 ) ( @xmath120 ( @xmath109 ) ) multiplied by the vector of normalized number of @xmath6 edges as a function of @xmath121 $ ] at fixed @xmath105 ( @xmath106 ) ( @xmath108 ( @xmath109 ) ) . values on y - axis in ( b ) represent matrix multiplication of the vector of @xmath110 as a function of @xmath122 $ ] at fixed @xmath119 ( @xmath106 ) ( @xmath120 ( @xmath109 ) ) multiplied by the vector of normalized number of @xmath7 edges as a function of @xmath121 $ ] at fixed @xmath113 ( @xmath106 ) ( @xmath115 ( @xmath109 ) ) . case i : one node as initial contact and two nodes as secondary contacts . , title="fig : " ] nodes @xmath39 as a function of @xmath103 , where @xmath104 for @xmath105 ( @xmath106 ) and @xmath107 for @xmath108 ( @xmath109 ) . ( b ) probability to establish both initial and secondary contacts between @xmath0 nodes @xmath110 as a function of @xmath111 , where @xmath112 for @xmath113 ( @xmath106 ) and @xmath114 for @xmath115 ( @xmath109 ) . ( c ) probability to establish both initial and secondary contacts between different types of nodes as a function of @xmath116 for @xmath117 $ ] at fixed @xmath105 ( @xmath106 ) , and fixed @xmath108 ( @xmath109 ) , respectively . values on y - axis in ( a ) represent matrix multiplication of the vector of @xmath39 as a function of @xmath118 $ ] at fixed @xmath119 ( @xmath106 ) ( @xmath120 ( @xmath109 ) ) multiplied by the vector of normalized number of @xmath6 edges as a function of @xmath121 $ ] at fixed @xmath105 ( @xmath106 ) ( @xmath108 ( @xmath109 ) ) . values on y - axis in ( b ) represent matrix multiplication of the vector of @xmath110 as a function of @xmath122 $ ] at fixed @xmath119 ( @xmath106 ) ( @xmath120 ( @xmath109 ) ) multiplied by the vector of normalized number of @xmath7 edges as a function of @xmath121 $ ] at fixed @xmath113 ( @xmath106 ) ( @xmath115 ( @xmath109 ) ) . case i : one node as initial contact and two nodes as secondary contacts . , title="fig : " ] nodes @xmath39 as a function of @xmath103 , where @xmath104 for @xmath105 ( @xmath106 ) and @xmath107 for @xmath108 ( @xmath109 ) . ( b ) probability to establish both initial and secondary contacts between @xmath0 nodes @xmath110 as a function of @xmath111 , where @xmath112 for @xmath113 ( @xmath106 ) and @xmath114 for @xmath115 ( @xmath109 ) . ( c ) probability to establish both initial and secondary contacts between different types of nodes as a function of @xmath116 for @xmath117 $ ] at fixed @xmath105 ( @xmath106 ) , and fixed @xmath108 ( @xmath109 ) , respectively . values on y - axis in ( a ) represent matrix multiplication of the vector of @xmath39 as a function of @xmath118 $ ] at fixed @xmath119 ( @xmath106 ) ( @xmath120 ( @xmath109 ) ) multiplied by the vector of normalized number of @xmath6 edges as a function of @xmath121 $ ] at fixed @xmath105 ( @xmath106 ) ( @xmath108 ( @xmath109 ) ) . values on y - axis in ( b ) represent matrix multiplication of the vector of @xmath110 as a function of @xmath122 $ ] at fixed @xmath119 ( @xmath106 ) ( @xmath120 ( @xmath109 ) ) multiplied by the vector of normalized number of @xmath7 edges as a function of @xmath121 $ ] at fixed @xmath113 ( @xmath106 ) ( @xmath115 ( @xmath109 ) ) . case i : one node as initial contact and two nodes as secondary contacts . , title="fig : " ] first , let us focus on the functional dependence of the probability that the edge is between @xmath1 nodes @xmath123 on @xmath124 ( eq . ( [ snn ] ) ) , which represents the combined probabilities that the edge is between same type of nodes @xmath14 and the probability that they are @xmath1 nodes . we empirically estimate @xmath39 by calculating the number of @xmath6 edges , multiplying it by two and dividing it by the average number of nodes at time @xmath28 which is @xmath125 . we obtain the functional dependence of empirically estimated @xmath123 by a matrix multiplication of two vectors ; one is the vector of @xmath123 values as a function of @xmath12 at fixed @xmath119 and the other is the vector of @xmath123 values as a function of @xmath14 at fixed @xmath105 . next , we aim to construct a function of @xmath12 and @xmath14 , @xmath126 such that the empirically estimated @xmath39 which express the probabilities for both initial and secondary contacts is a linear function of @xmath124 . we plot the result for case i ( one initial contact and two secondary contacts ) in fig . [ fig2]a for @xmath118 $ ] at fixed values of @xmath119 and for @xmath121 $ ] at fixed values of @xmath105 ( circles @xmath106 ) . squares ( @xmath109 ) fig . [ fig2]a mark results for @xmath118 $ ] at fixed values of @xmath120 and for @xmath121 $ ] at fixed values of @xmath108 . we obtain that the combined probability @xmath126 of the form @xmath127 produces the following least - squares linear fit [ parn ] @xmath128 for @xmath104 ( @xmath106 ) and @xmath107 ( @xmath109 ) , respectively . the prediction error estimate was generated for @xmath123 and found to be @xmath129 ( for @xmath106 ) and @xmath130 and ( for @xmath109 ) which allows us to obtain a range of values for @xmath123 limited by @xmath131 . we use these functional dependences within their range to express @xmath39 in the solutions of the rate equations eqs . ( [ kin ] ) and ( [ kiv ] ) and in the expression of respective degree distributions and clustering coefficients . applying the same reasoning , we obtain the combined probability @xmath110 as a function of @xmath132 which represents both the initial and secondary contacts . results are shown in fig . [ fig2]b for @xmath122 $ ] at fixed @xmath119 and @xmath121 $ ] at fixed @xmath113 ( @xmath106 ) . squares ( @xmath109 ) mark results for @xmath122 $ ] at fixed @xmath120 and @xmath121 $ ] at fixed @xmath115 . for an argument of the form @xmath133 the least - square fit produces [ parv ] @xmath134 for @xmath112 ( @xmath106 ) and @xmath114 ( @xmath109 ) , respectively . the prediction error estimate defines the range of values for @xmath135 , where @xmath136 ( for @xmath106 ) and @xmath137 ( for @xmath109 ) . we obtain the probability @xmath41 for creating a link between different type of nodes both as initial and as secondary contacts to be [ pard ] @xmath138 the dependence of @xmath41 as a function of @xmath139 is shown in fig . [ fig2]c for @xmath140 $ ] at fixed @xmath105 ( @xmath106 ) and @xmath108 ( @xmath109 ) . we applied the linear least - square fit for an argument of the form @xmath141 the prediction error estimate is obtained to be @xmath142 ( for @xmath106 ) and @xmath143 ( for @xmath109 ) . we use the above parameterization procedures of the probabilities to establish both initial and secondary contacts between @xmath1 nodes @xmath39 , between @xmath0 nodes @xmath110 , and between different types of nodes @xmath41 in obtaining the degree distribution ( eqs . ( [ pn ] ) and ( [ pv ] ) ) and clustering coefficient ( eqs . ( [ ckn ] ) and ( [ ckv ] ) ) for each of the cases considered below . nodes ( @xmath4 ) @xmath105 and 20% @xmath0 nodes @xmath113 ( @xmath5 ) . edges are formed with probability @xmath119 if they connect @xmath1 with @xmath1 or @xmath0 with @xmath0 nodes and with probability @xmath144 if they connect @xmath1 with @xmath0 nodes . symbols represent results from numerical simulations done on a network with @xmath145 agents and averaged over 100 runs . lines represent the results of degree distribution @xmath102 given by eq . ( [ pn ] ) for @xmath1 nodes ( solid ) and by eq . ( [ pv ] ) for @xmath0 nodes ( dashed ) using functional dependences shown in fig . [ fig2 ] to obtain functions @xmath146 , @xmath147 , and @xmath148 . the set of two lines correspond to using @xmath149 or @xmath150 for each of @xmath151 , where @xmath152 . the dash - dotted line represents analytical distribution @xmath102 obtained using eq . ( [ pns ] ) for @xmath1 nodes . the dotted line represents analytical distribution @xmath102 obtained using eq . ( [ pvs ] ) for @xmath0 nodes . ( a ) case i : one node as initial contact and two nodes as secondary contacts ; ( b ) case ii : one node as initial contact with probability 0.9 and two nodes as initial contacts with probability 0.1 ; the number of nodes as secondary contacts is from uniform distribution @xmath153 $ ] ; ( c ) case iii : two nodes as initial contacts ; the number of nodes as secondary contacts is from uniform distribution @xmath154$].,title="fig : " ] nodes ( @xmath4 ) @xmath105 and 20% @xmath0 nodes @xmath113 ( @xmath5 ) . edges are formed with probability @xmath119 if they connect @xmath1 with @xmath1 or @xmath0 with @xmath0 nodes and with probability @xmath144 if they connect @xmath1 with @xmath0 nodes . symbols represent results from numerical simulations done on a network with @xmath145 agents and averaged over 100 runs . lines represent the results of degree distribution @xmath102 given by eq . ( [ pn ] ) for @xmath1 nodes ( solid ) and by eq . ( [ pv ] ) for @xmath0 nodes ( dashed ) using functional dependences shown in fig . [ fig2 ] to obtain functions @xmath146 , @xmath147 , and @xmath148 . the set of two lines correspond to using @xmath149 or @xmath150 for each of @xmath151 , where @xmath152 . the dash - dotted line represents analytical distribution @xmath102 obtained using eq . ( [ pns ] ) for @xmath1 nodes . the dotted line represents analytical distribution @xmath102 obtained using eq . ( [ pvs ] ) for @xmath0 nodes . ( a ) case i : one node as initial contact and two nodes as secondary contacts ; ( b ) case ii : one node as initial contact with probability 0.9 and two nodes as initial contacts with probability 0.1 ; the number of nodes as secondary contacts is from uniform distribution @xmath153 $ ] ; ( c ) case iii : two nodes as initial contacts ; the number of nodes as secondary contacts is from uniform distribution @xmath154$].,title="fig : " ] nodes ( @xmath4 ) @xmath105 and 20% @xmath0 nodes @xmath113 ( @xmath5 ) . edges are formed with probability @xmath119 if they connect @xmath1 with @xmath1 or @xmath0 with @xmath0 nodes and with probability @xmath144 if they connect @xmath1 with @xmath0 nodes . symbols represent results from numerical simulations done on a network with @xmath145 agents and averaged over 100 runs . lines represent the results of degree distribution @xmath102 given by eq . ( [ pn ] ) for @xmath1 nodes ( solid ) and by eq . ( [ pv ] ) for @xmath0 nodes ( dashed ) using functional dependences shown in fig . [ fig2 ] to obtain functions @xmath146 , @xmath147 , and @xmath148 . the set of two lines correspond to using @xmath149 or @xmath150 for each of @xmath151 , where @xmath152 . the dash - dotted line represents analytical distribution @xmath102 obtained using eq . ( [ pns ] ) for @xmath1 nodes . the dotted line represents analytical distribution @xmath102 obtained using eq . ( [ pvs ] ) for @xmath0 nodes . ( a ) case i : one node as initial contact and two nodes as secondary contacts ; ( b ) case ii : one node as initial contact with probability 0.9 and two nodes as initial contacts with probability 0.1 ; the number of nodes as secondary contacts is from uniform distribution @xmath153 $ ] ; ( c ) case iii : two nodes as initial contacts ; the number of nodes as secondary contacts is from uniform distribution @xmath154$].,title="fig : " ] the analytical distribution @xmath102 is given by eq . ( [ pns ] ) for @xmath1 nodes and by eq . ( [ pvs ] ) for @xmath0 nodes and plotted in fig . [ fig3 ] with dash - dotted and dotted lines , respectively . the degree distribution @xmath102 obtained using functional dependences is given by eq . ( [ pn ] ) for @xmath1 nodes and by eq . ( [ pv ] ) for @xmath0 nodes and plotted in fig . [ fig3 ] with solid and dashed lines , respectively . simulations are conducted on a network with @xmath145 nodes starting with a seed network of 8 nodes and are averaged over a 100 runs . all three cases considered are for value of the probability to create an @xmath1 node @xmath105 and for the value of the probability to establish a link between the same type of nodes @xmath119 . to touch upon the versatility of the model we consider three different cases . they are case i : one node as initial contact @xmath155 and two nodes as secondary contacts @xmath156 ( fig . [ fig3]a ) ; case ii : one node as initial contact with probability 0.9 and two nodes as initial contacts with probability 0.1 , which gives @xmath157 ; the number of nodes as secondary contacts is from uniform distribution @xmath153 $ ] and therefore , @xmath158 ( fig . [ fig3]b ) ; case iii : two nodes as initial contacts @xmath159 ; the number of nodes as secondary contacts is from uniform distribution @xmath154 $ ] , @xmath160 ( fig . [ fig3]c ) . results demonstrate that for all cases considered the simulations compare relatively well with the analytical derivation of the degree distribution even though using functional dependences in @xmath102 derivation improves the agreement within the limits of the simulations . nodes ( @xmath4 ) and @xmath0 nodes ( @xmath5 ) for @xmath105 and @xmath119 . symbols represent results from numerical simulations . lines ( indistinguishable ) represent the results of theoretical derivation of clustering coefficient as a function of the degree of the node using eq . ( [ ckn ] ) for @xmath1 nodes and eq . ( [ ckv ] ) for @xmath0 nodes . case iii : two nodes as initial contacts ; the number of nodes as secondary contacts is from uniform distribution @xmath154 $ ] . ] the simulation results for the clustering coefficient as a function of the degree of the node @xmath90 for @xmath1 and @xmath0 nodes for case iii are plotted in fig . [ fig4 ] with empty and full circles , respectively . the analytical solution for clustering coefficient @xmath90 using eq . ( [ ckn ] ) for @xmath1 nodes and eq . ( [ ckv ] ) for @xmath0 nodes are plotted with lines which coincide with each other . a clear @xmath161 trend is observed which indicates the hierarchy in the system . the global clustering properties of the network are assessed by the mean clustering coefficient @xmath96 ( eq . ( [ bc ] ) ) which is averaged over vertex degree , and the transitivity @xmath99 ( eq . ( [ t ] ) ) . we study how @xmath96 and @xmath99 change as a function of @xmath12 for a fixed value of @xmath14 ( plotted in fig . [ fig5]a , c ) and as a function of @xmath14 for a fixed value of @xmath12 ( plotted in fig . [ fig5]b , d ) . the mean clustering coefficient @xmath96 for case i ( circles in fig . [ fig5]a ) has values in the range between 0.41 and 0.43 as a function of @xmath12 for both @xmath1 and @xmath0 nodes for fixed value of @xmath119 . in both case ii ( squares ) and case iii ( diamonds ) , where the number of secondary contacts is drawn from a uniform distribution , the @xmath162 for @xmath1 and @xmath0 nodes is symmetrical with respect to its value at @xmath108 . values of @xmath96 for @xmath1 and @xmath0 nodes as a function of @xmath14 ( fig . [ fig5]b ) demonstrate a tendency to converge for @xmath14 approaching one . the transitivity @xmath99 ( fig . [ fig5]c ) as a function of @xmath12 shows a symmetrical pattern for @xmath1 and @xmath0 nodes with respect to its value at @xmath108 similar to @xmath96 behavior but with a wider difference between the results for @xmath1 and @xmath0 nodes and different values . as the probability to establish a link between same type of nodes @xmath14 increases the values of transitivity @xmath99 ( fig . [ fig5]d ) for @xmath1 and @xmath0 nodes converge . higher values of both @xmath96 and @xmath99 among the three cases are obtained for case ii when there is an option to create either one or two initial contacts and the number of secondary contacts vary as well , e.g. in @xmath163 $ ] . ( eq . ( [ bc ] ) ( a ) as a function of the probability to create an @xmath1 node @xmath12 for a fixed value of the probability to create a link between same type of nodes @xmath119 and ( b ) as a function of @xmath14 for a fixed value of @xmath105 for @xmath1 nodes ( empty symbols ) and @xmath0 nodes ( full symbols ) . ( c , d ) same as ( a , b ) but for the values of transitivity @xmath99 ( eq . ( [ t ] ) . circles ( @xmath1@xmath4/@xmath0@xmath5 ) represent case i. squares ( @xmath1@xmath109/@xmath0@xmath164 ) represent case ii and diamonds ( @xmath1@xmath165/@xmath0@xmath166 ) mark case iii . cases i , ii , and iii are as defined in fig . 3.,title="fig : " ] ( eq . ( [ bc ] ) ( a ) as a function of the probability to create an @xmath1 node @xmath12 for a fixed value of the probability to create a link between same type of nodes @xmath119 and ( b ) as a function of @xmath14 for a fixed value of @xmath105 for @xmath1 nodes ( empty symbols ) and @xmath0 nodes ( full symbols ) . ( c , d ) same as ( a , b ) but for the values of transitivity @xmath99 ( eq . ( [ t ] ) . circles ( @xmath1@xmath4/@xmath0@xmath5 ) represent case i. squares ( @xmath1@xmath109/@xmath0@xmath164 ) represent case ii and diamonds ( @xmath1@xmath165/@xmath0@xmath166 ) mark case iii . cases i , ii , and iii are as defined in fig . 3.,title="fig : " ] ( eq . ( [ bc ] ) ( a ) as a function of the probability to create an @xmath1 node @xmath12 for a fixed value of the probability to create a link between same type of nodes @xmath119 and ( b ) as a function of @xmath14 for a fixed value of @xmath105 for @xmath1 nodes ( empty symbols ) and @xmath0 nodes ( full symbols ) . ( c , d ) same as ( a , b ) but for the values of transitivity @xmath99 ( eq . ( [ t ] ) . circles ( @xmath1@xmath4/@xmath0@xmath5 ) represent case i. squares ( @xmath1@xmath109/@xmath0@xmath164 ) represent case ii and diamonds ( @xmath1@xmath165/@xmath0@xmath166 ) mark case iii . cases i , ii , and iii are as defined in fig . 3.,title="fig : " ] ( eq . ( [ bc ] ) ( a ) as a function of the probability to create an @xmath1 node @xmath12 for a fixed value of the probability to create a link between same type of nodes @xmath119 and ( b ) as a function of @xmath14 for a fixed value of @xmath105 for @xmath1 nodes ( empty symbols ) and @xmath0 nodes ( full symbols ) . ( c , d ) same as ( a , b ) but for the values of transitivity @xmath99 ( eq . ( [ t ] ) . circles ( @xmath1@xmath4/@xmath0@xmath5 ) represent case i. squares ( @xmath1@xmath109/@xmath0@xmath164 ) represent case ii and diamonds ( @xmath1@xmath165/@xmath0@xmath166 ) mark case iii . cases i , ii , and iii are as defined in fig . 3.,title="fig : " ] as a function of the probability to create a link between same type of nodes @xmath14 for a fixed value of @xmath105 of ( a ) @xmath6 network ; ( b ) @xmath7 network ; ( c ) @xmath8 network . circles ( @xmath5 ) represent case i. squares ( @xmath164 ) represent case ii and diamonds ( @xmath166 ) mark case iii . cases i , ii , and iii are as defined in fig . 3.,title="fig : " ] as a function of the probability to create a link between same type of nodes @xmath14 for a fixed value of @xmath105 of ( a ) @xmath6 network ; ( b ) @xmath7 network ; ( c ) @xmath8 network . circles ( @xmath5 ) represent case i. squares ( @xmath164 ) represent case ii and diamonds ( @xmath166 ) mark case iii . cases i , ii , and iii are as defined in fig . 3.,title="fig : " ] as a function of the probability to create a link between same type of nodes @xmath14 for a fixed value of @xmath105 of ( a ) @xmath6 network ; ( b ) @xmath7 network ; ( c ) @xmath8 network . circles ( @xmath5 ) represent case i. squares ( @xmath164 ) represent case ii and diamonds ( @xmath166 ) mark case iii . cases i , ii , and iii are as defined in fig . 3.,title="fig : " ] the assortative properties of the network describe the degree correlation of the nodes at the ends of an edge and are quantified by the pearson correlation coefficient @xmath167 . the network is said to be assortative when high degree nodes tend to connect to other high - degree nodes and @xmath168 $ ] @xcite . the network is characterized by disassortative mixing when high degree nodes tend to connect to low degree nodes and @xmath169 . we calculate @xmath167 of each of @xmath6 , @xmath7 , and @xmath8 networks obtained for a fixed value of @xmath105 and plot the @xmath167-dependence as a function of @xmath14 in fig . [ fig6]a - c for the three cases considered . @xmath6 and @xmath8 networks of case i ( @xmath5 ) show slight disassortative mixing owing to the two initial contacts and one secondary contact . increasing the probability @xmath14 to create an edge between the same type of nodes for case i ( @xmath5 in fig . [ fig6]b ) results in changing from disassortative for @xmath170 to slightly assortative for @xmath171 . varying the number of initial and secondary contacts in case ii ( @xmath164 ) and iii ( @xmath166 ) produces assortatively mixed @xmath6 and @xmath7 networks ( fig . [ fig6]a , b ) . the assortativity coefficient @xmath172 of the @xmath7 networks ( @xmath164 and @xmath166 in fig . [ fig6]b ) increases linearly with increasing the probability to create a link between the same type of nodes reaching a plateau around @xmath173 and then decreasing for @xmath174 probably because all available nodes are already connected . the values of @xmath172 are larger versus @xmath175 because of the smaller number ( @xmath176 ) of @xmath0 nodes available for contacts . a recent visualization of the connections in a terrorist network such as the global salafi jihad depicts them to form an assortative network @xcite . for all three cases the @xmath8 networks ( fig . [ fig6]c ) show disassortative mixing of their degree only except for very high values of @xmath177 and @xmath178 for case iii ( @xmath166 ) . a recent empirical study of an online social system reports that relationships driven by aggression lead to markedly different systemic characteristics than relations of a non - aggressive nature @xcite . assortativity is a characteristic of global properties of the system . in agreement with the empirical findings the assortativity of @xmath8 network ( fig . [ fig6]c ) produced by our model which represents relationships driven by aggression is clearly different from the assortativity of @xmath6 and @xmath7 networks ( fig . [ fig6]a , b ) which are driven by non - aggressive relationships . we introduced a model intended to characterize the interactions between two distinct populations , which form links more easily within their group than between groups . we aim to describe the interactions of potentially violent terrorist groups within the context of a largely non - violent population , although the same model could , in principle , be applied to other non - mainstream social groups . the model is kept simple enough so that analytical solutions could be derived and compared with empirical parameterizations and numerical simulation results . the model produces networks with relatively high mean clustering coefficient @xmath96 and transitivity @xmath99 . their values vary with the balance between the initial and secondary contacts . this is expected because of the interplay between the random and preferential attachments of the initial and secondary connections , respectively . the assortativity pattern of modeled networks show that the potentially violent @xmath7 network qualitatively resembles the connectivity pattern in terrorist networks reported in @xcite . the assortativity behavior of @xmath8 network which is driven by aggression is clearly different than the assortativity pattern of @xmath6 and @xmath7 networks which are non - aggressive relationships ; a finding which is in agreement with the results of recent empirical study of an online social system @xcite .

molecular clouds are observed through the emission of a number of molecular transitions that provide a wealth of information about their chemical composition , gas temperature and density , magnetic field strength , fractional ionization , structure and kinematics . this information is essential to our understanding of the process of star formation . the interpretation of molecular emission line maps is not always unique . the main source of uncertainty is the absence of the third spatial dimension ( along the line of sight ) in the observational data . statistical properties of the velocity and density distributions along the line of sight are difficult to disentangle . furthermore , the components of the gas velocity on the plane of the sky are unknown . two dimensional images of molecular clouds are usually converted into three dimensional `` objects '' using the radial velocity instead of the third spatial dimension . this method can be useful to separate individual mass condensations from each other , since it is conceivable that their relative velocity is larger than their internal one . however , velocity blending or the lack of well defined condensations along the line of sight may cause significant uncertainties ( issa , maclaren & wolfendale 1990 ; adler & roberts 1992 ; ballesteros paredes , vazquez semadeni & scalo 1999 ; pichardo et al . 2000 ; ostriker , stone & gammie 2001 ; lazarian et al . 2001 ; ballesteros paredes & mac low 2002 ) . due to the difficulty of a direct interpretation of the observational data , a `` forward approach '' that starts from a rather general theoretical model and synthesizes its observational properties can be more instructive . different models may sometimes satisfy the same set of observational constraints , but they should also provide guidance for further observational studies that could help select the correct model . ideally , numerical models to be compared with observed spectral line data cubes should be based on the numerical solutions of the mhd equations , in the regime of highly super sonic turbulence , and on radiative transfer calculations . in some works , stochastic fields are used instead of the solution of the mhd equations and in most studies the radiative transfer calculation is omitted , in favor of density weighted velocity profiles . the first large synthetic spectral maps of molecular transitions computed by solving the non lte radiative transfer through the density and velocity data cubes obtained as the numerical solution of the mhd equations were presented by padoan et al . ( 1998 ) , based on juvela s radiative transfer code ( juvela 1997 ) , and were used in a number of works ( e.g. padoan et al . 1999 , 2000 , 2001 ) . another new radiative transfer code has also been used more recently to generate synthetic spectral maps from mhd simulations ( ossenkopf 2002 ) . a number of statistical methods have been proposed to compare numerical models of turbulence with large spectral maps of molecular clouds ( see for example scalo 1984 ; kleiner & dickman 1985 , 1987 ; stutzki & gusten 1990 ; gill & henriksen 1990 ; houlahan & scalo 1992 ; hobson 1992 ; langer , wilson , & anderson 1993 ; williams , de geus & blitz 1994 ; miesch & bally 1994 ; miesch & scalo 1995 ; lis et al . 1996 ; blitz & williams 1997 ; heyer & schloerb 1997 ; stutzki et al . 1998 ; miesch , scalo & bally 1999 ; falgarone et al . 1994 ; padoan et al . 1999 ; mac low & ossenkopf 2000 ; bensch , stutzki & ossenkopf 2001 ) . in this work we apply the spectral correlation function ( scf ) method , proposed by rosolowsky et al . ( 1999 ) and further developed in padoan , rosolowsky & goodman ( 2001 ) , to a number of observational and synthetic spectral maps . we show that the slope and normalization of the scf of observational maps correlate with the spectral line width . theoretical models of molecular clouds should therefore yield synthetic spectral maps reproducing such correlations , but not all of them can . in the next section we briefly define the scf , and in 3 we present the observational data used in this work . the computation of the theoretical models and synthetic spectral maps is presented in 4 . results from numerical models are compared with the observational data in 5 and are discussed in 6 . conclusions are drawn in 7 . the spectral correlation function ( scf ) measures the spatial correlation of spectral line profiles within a spectral map . it is sensitive to the properties of both the gas mass distribution and the gas velocity field ( rosolowsky et al . 1999 ; padoan , rosolowsky & goodman 2001 ; padoan et al . 2001 ; ballesteros paredes , vazquez semadeni & goodman 2002 ) . let @xmath2 be the antenna temperature as a function of velocity channel @xmath3 at map position @xmath4 . the scf for spectra with spatial separation @xmath5 is : @xmath6 where the average is computed over all map positions @xmath4 . @xmath7 is the scf uncorrected for the effects of noise , @xmath8 ^ 2 } { \sigma_vt(\vecr , v)^2+\sigma_vt(\vecr+\vecdr , v)^2 } } \right\rangle _ { \vecdr } , \label{2}\ ] ] where the average is limited to separation vectors @xmath9 with @xmath10 , and @xmath11 is the scf due to noise alone , @xmath12 and @xmath13 is the `` spectrum quality '' ( see discussion in padoan , rosolowsky & goodman 2001 ) . @xmath13 is defined as the ratio of the rms signal within a velocity window @xmath14 and the rms noise , @xmath15 ( over all velocity channels ) , @xmath16 where @xmath17 is the width of the velocity channels . in the present work we compute the scf of both observational and synthetic spectral maps , obtained by computing the radiative transfer through the three dimensional density and velocity fields of numerical simulations of super sonic mhd turbulence . the result is typically a power law for @xmath18 that extends up to a separation @xmath5 comparable to the map size , reflecting the self similarity of super sonic turbulence ( padoan , rosolowsky & goodman 2001 ) . the power law behavior is sometimes interrupted at an intermediate scale , possibly suggesting the presence of a physical mechanism limiting the inertial range of turbulence . an example of a scf that defines an intermediate scale is the scf of the hi survey of the large magellanic cloud ( lmc ) by kim et al . ( 1998 , 1999 ) . padoan et al . ( 2001 ) have recently been able to map the gas disk thickness of the lmc , assuming it is related to the intermediate scale defined by the break in the scf power law . the absolute value of @xmath18 at any @xmath19 and the slope of the @xmath18 power law for any given region depends on which molecular tracer is used ( padoan , rosolowsky & goodman 2001 ) . transitions probing higher gas density produce more fragmented integrated intensity maps than transitions probing lower gas density , and their scf is therefore steeper . in order to compare the scf of observational and synthetic maps it is therefore important to solve the radiative transfer through the model density and velocity fields accurately for the same molecular transition that is observed . in this work our aim is to compute the scf of observational data in order to provide constraints for theoretical models . the best constraints come from computing the scf of spectral maps of a specific molecular transition over a large range of line width and linear size . observationally , small scale and narrow line width objects are usually mapped out with high density tracers , while larger objects are instead usually probed with lower density tracers . @xmath0co provides a good compromise , since it is the only molecule for which very large maps containing thousands of spectra have been obtained with a significant range of resolution . in this work we have therefore chosen to use observational and synthetic maps of the j=10 line of @xmath0co . we have used 11 @xmath0co maps . for each map , we have listed in table 1 the approximate size , the distance , the rms velocity computed as the standard deviation of the line profile averaged over the whole map , the telescope beam size , the spatial sampling , the width of the velocity channels and the spectral quality defined in the previous section . smaller maps have been obtained from portions of the maps of the taurus , perseus and rosette molecular cloud complexes and the scf has been computed for each of them . the position of these smaller maps within the molecular cloud complexes is shown in figures 1 and 2 . they have been called t1 to t7 in taurus , p1 to p5 in perseus , r1b in the rosette molecular cloud map by blitz & stark ( 1986 ) and r1 and r2 in the rosette molecular cloud map by heyer et al . ( 2001 ) . the scf of each map has been approximated with a power law , over the range of spatial separations where a power law fit is relevant . for each power law fit we compute its slope , , and its absolute value at 1 pc , : @xmath20 the values of , , @xmath1 ( the line of sight rms velocity ) and the galactic coordinates of the center of each map are given in table 2 . the scf of maps of molecular cloud complexes and some smaller regions are shown in figure [ fig5 ] . we solve the compressible mhd equations in a staggered mesh of @xmath21 computational cells , with volume centered mass density and thermal energy , face centered velocity and magnetic field components , edge centered electric currents and electric fields and with periodic boundary conditions . the code uses shock and current sheet capturing techniques to ensure that magnetic and viscous dissipation at the smallest resolved scales provide the necessary dissipation paths for magnetic and kinetic energy . a more detailed presentation of the numerical method can be found elsewhere . ( padoan & nordlund 1999 ) . for the purpose of the present work we have computed numerical solutions of the mhd equations using an isothermal equation of state , and a random driving force . in all experiments , the initial density is uniform , and the initial velocity is random . we generate the velocity field in fourier space , and we give power , with a normal distribution , only to the fourier components in the shell of wave - numbers @xmath22 . we perform a helmholtz decomposition , and use only the solenoidal component of the initial velocity . however , a compressional component of the velocity field develops almost immediately due to the flow compressibility . the external driving force is generated on large scales in the same way as the velocity field . the initial magnetic field is uniform , and is oriented parallel to the @xmath23 axis : @xmath24 . because of the limited numerical resolution we have chosen not to model the collapse of turbulent density fluctuations . gravity has therefore been neglected . we have recently started to compute turbulent self gravitating flows with a numerical mesh of 500@xmath25 cells . results of the analysis of these larger simulations including self gravity will be presented in future works . we have run a number of mhd simulations in a @xmath21 computational mesh , with periodic boundary conditions . the simulations are intended to describe the turbulent dynamics in the interior of molecular clouds . the two most important numerical parameters in the models are the rms sonic and alfvnic mach numbers , @xmath26 and @xmath27 . the rms sonic mach number is here defined as the ratio of the rms flow velocity and the speed of sound . the alfvnic mach number is defined as the ratio of the rms flow velocity and the alfvn velocity , @xmath28 , where b is the volume averaged magnetic field strength . all the models used in this work have @xmath29 , except for model e that has @xmath30 , according to the suggestion that the dynamics of molecular clouds is essentially super alfvnic ( padoan & nordlund 1999 ) . our numerical simulations conserve magnetic flux , and so the volume averaged magnetic field is constant in time . as a consequence , also the value of @xmath27 as defined above remains constant . however , the value of @xmath31 grows with time ( until equilibrium is reached ) due to compression and stretching of magnetic field lines ( see padoan & nordlund 1999 ) . if we define the alfvn velocity using the rms value of the magnetic field strength , instead of its volume average , then the typical alfvnic mach number in our super alfvnic runs is @xmath32 , because of the formation of regions with large value of magnetic field strength ( mainly dense regions , as found in observations ) . the sonic mach number of observed turbulent motions in molecular clouds is @xmath33 on the scale of several parsecs , and decreases toward smaller scale . the turbulent velocity becomes comparable to the speed of sound only on very small scale , @xmath34 pc . in order to study the effect of the sonic mach number on the scf , we have computed mhd models with different values of @xmath26 , @xmath35 , 5 , 2.5 , 1.25 and 0.625 . each model has been run for approximately six dynamical times ( the dynamical time is here defined as the ratio of half the size of the computational box and the rms flow velocity ) , in order to achieve a statistically relaxed state , independent of the initial conditions . the velocity and density fields from the final snapshot of each model have been used to compute @xmath0co @xmath36 spectra , solving the radiative transfer with a non lte monte carlo code ( 4.2 ) . while the mhd calculations are independent of the physical value of the average gas density , the size of the computational mesh ( or the column density ) and the kinetic temperature , these physical parameters are necessary inputs for the radiative transfer calculations . the models are scaled to physical units assuming a value for i ) the kinetic temperature , @xmath37 , that determines the physical unit of velocity ( the numerical unit of velocity is the speed of sound ) ; ii ) the average gas density , @xmath38 ; iii ) the size of the computational box , @xmath39 . for all models we have assumed @xmath40 k , typical of molecular clouds . the dependence of observed average gas density and cloud size on the observed rms turbulent velocity ( or sonic mach number , assuming a constant value of @xmath37 ) is well approximated by empirical larson type relations ( larson 1981 ) . however , the size velocity relation has a large intrinsic scatter ( falgarone , puget & perault 1992 ) , and both the size velocity and density size relations have been criticized by several authors ( loren 1989 ; kegel 1989 ; scalo 1990 ; issa , maclaren & wolfendale 1990 ; adler & roberts 1992 ; vazquez - semadeni , ballesteros - paredes & rodriguez 1997 ; ostriker , stone & gammie 2001 ; ballesteros paredes & mac low 2002 ) . for these reasons , we scale the mhd models in four different ways . these four sets of models are all based on the same five mhd turbulence models and differ from each other only in the way they are rescaled to physical units when computing the radiative transfer . models a1 to a5 and b1 to b5 have all the same value of the average density , @xmath41 @xmath42 . models a1 to a5 have all the same size @xmath43 pc and column density @xmath44 @xmath45 ; models b1 to b5 have @xmath46 pc and @xmath47 @xmath45 . models a1r to a5r and b1r to b5r are rescaled using the larson type relations : @xmath48 where a temperature @xmath40 k is assumed , and @xmath49 that is equivalent to a constant mean surface density . models a1r to a5r have the same column density as models a1 to a5 , that is @xmath50 @xmath42 in equation ( [ larson2 ] ) ; they also have sizes @xmath5110 , 2.5 , 0.625 , 0.156 and 0.039 pc respectively , which implies @xmath52 in equation ( [ larson1 ] ) . models b1r to b2r have the same column density as models b1 to b5 , that is @xmath53 @xmath42 in equation ( [ larson2 ] ) ; they have sizes @xmath5120 , 5 , 1.25 , 0.31 and 0.078 pc respectively , which implies @xmath54 in equation ( [ larson1 ] ) . finally , the equipartition model ( model e ) has been computed only for one value of the rms sonic mach number , @xmath35 . it is rescaled to the larson type relations only once , for a size of 10 pc and a column density of @xmath44 @xmath45 . for this model we have computed spectral maps along 5 different directions , three orthogonal to the faces of the numerical mesh , as in the other experiments , and two along diagonal directions , at an angle of 54.7@xmath55 with the average magnetic field ( @xmath23 axis ) . maps from diagonal directions sample lines of sight of different length at different map positions ( longer at the central position than near the corners ) . however , the number of computational cells along each line of sight is on the average even larger than in maps from orthogonal directions , since a diagonal line of sight often cuts through the computational cells away from their center ( close to their corners ) . furthermore , the maps are computed only for a region of size equivalent to that of maps from orthogonal directions ( @xmath56 cells ) , eliminating the corners of the computational mesh . as a result , only a few percent of the spectra are generated from lines of sight sampling less than 50 computational cells . a fraction of the the lines of sight close to the map edges are nevertheless shorter than the energy injection scale ( approximately half the size of the computational mesh ) . this may introduce a bias toward smaller line width , since velocity differences are expected to grow with increasing distances . this bias or its effect on the scf should be small , since our results seem to vary smoothly as a function of the angle between the line of sight and the direction of the average magnetic field . the models a4 , a5 , b3 , b4 and b5 have velocity dispersion significantly smaller than found observationally at the scale of 5 pc . they are not used here to test the validity of models with such low velocity dispersion , but rather to test the ability of the scf method to rule them out as poor description of molecular cloud turbulence . in order to test the ability of the scf to rule out unphysical models , we have also computed two stochastic models , s2 and s4 . in both models the density field is a random field with a log normal probability distribution function , and a power law power spectrum with power law exponent equal to -1 ( the approximate value found in our mhd models ) . the velocity field is generated as a gaussian field , also with power law power spectrum . the power law exponent of the velocity field power spectrum is -2 ( close to the actual value in the mhd models ) in model s2 and -4 in model s4 . for the purpose of computing the radiative transfer and the synthetic spectral maps , both models have been scaled to a physical size @xmath46 pc and a column density @xmath57 @xmath45 . these two stochastic models are unphysical in the sense that they are not solutions of the fluid equations . statistical properties such as the power spectrum and the probability density function of density and velocity may be similar to those of flows obtained by solving the fluid equations , but their phase correlations are unphysical . this is in part illustrated by the fact that these stochastic models look clumpy , rather than filamentary as real clouds and mhd models . furthermore , their velocity and density fields can not be self consistent because they are computed independently of each other . it is shown below that the scf method can indeed rule out these unphysical models . the sonic rms mach number , @xmath26 , the average gas density , @xmath38 , and the physical size , @xmath39 , of the different models used for the radiative transfer computations are given in table 3 . the radiative transfer calculations were carried out with a monte carlo program which is a generalization of the one - dimensional monte carlo method ( bernes 1979 ) into three dimensions . the model cloud is divided into small , cubic cells in which physical properties are assumed to be constant . the discretization allows the inclusion of arbitrary kinetic temperature and molecular abundance variations . however , in the present calculations , the temperature and relative abundances are kept constant . the 2.73 k cosmic background is used as the external radiation field . there are important differences between our program and the normal monte carlo method , and some principles of the implementation are given below . a detailed description is given elsewhere ( juvela 1997 ) . in the basic monte carlo method radiation field is simulated with photon packages , each representing a number of real photons . the packages are created at random velocities at random locations and sent toward random directions . each package is followed through the cloud and interactions between photons and molecules are counted . later this information is used to solve new estimates for the level populations of the molecules . in our method the radiative transfer is simulated along random lines going through the cloud . initially , as a photon package enters the cloud it contains only background photons . as the package goes through a cell in the cloud some photons emitted by this cell are added to the package and , in particular , the number of photons absorbed within the emitting cell is calculated explicitly . this becomes important when cells are optically thick and , compared with normal monte carlo simulation , ensures more accurate estimation of the energy transfer between cells . in our program each simulated photon package represents intensity of all simulated transitions and doppler shifts at the same time . the number of individual photon packages is correspondingly smaller , and in the present case we use 240000 photon packages per iteration . the lines are divided into 70 fixed velocity channels . there is no noise associated with random sampling of doppler shifts . the simulated velocity range was adjusted according to the velocity range found in the model clouds . the channels are narrow compared with the total line widths and smaller than or equal to the smallest intrinsic line widths in the cells . the velocity discretization is therefore not expected to affect the results of the calculations . the density and velocity fields from the mhd simulations are sampled on a numerical mesh of @xmath21 cells . to speed up the radiative transfer calculations the density and velocity fields were rebinned into a mesh of @xmath58 cells by linear interpolation . the velocity dispersion between neighboring cells in the original @xmath21 data cube was used to approximate the turbulent line width within each cell of the new @xmath58 data cube . this velocity dispersion should apply to a scale slightly larger than the size of the cells in the @xmath58 mesh . however , this is approximately compensated by the fact that numerical dissipation in the mhd simulations decreases significantly the velocity dispersion on very small scale , below the actual turbulent inertial range value at that scale . on each iteration new level populations are solved from the equilibrium equations and iterations are stopped when the relative change is below @xmath592.0@xmath6010@xmath61 in all cells . only the six lowest levels were tested for convergence . the relative changes tend to be largest on the upper levels where the level populations become very small and , on the average , the convergence of the relevant first energy levels is much better than the quoted limit . the total number of energy levels included in the calculations was nine , a number clearly sufficient in case of excitation temperatures below 10k . the collisional coefficients were taken from flower & launay ( 1985 ) and green & thaddeus ( 1976 ) . the final level populations were used to calculate maps of 90@xmath6290 spectra toward three directions perpendicular to the faces of the mhd data cube . for the equipartition model e spectra were calculated also along two diagonal directions . in these cases the maps of 90@xmath6290 spectra do not extend over the whole projected cloud area . each spectrum corresponds to the intensity calculated along one line of sight ( spectra are not convolved with a larger beam ) . the spectra contain 60 velocity channels as in the monte carlo simulation . the results were compared with spectra calculated assuming lte conditions . the comparison showed that for typical physical conditions found in molecular clouds the lte assumption would be unsuitable ( padoan et al . most observational and theoretical spectral maps yield a scf that can be approximated by a single power law within a range of spatial separations , often spanning over an order of magnitude . from each power law fit we compute its slope , , and its value at 1 pc , , defined as in ( [ alphadef ] ) . we also compute the value of the velocity dispersion , @xmath1 , from each map , measured as the standard deviation of the @xmath0co @xmath36 spectrum averaged over the entire map . the values of , and @xmath1 computed from the observational maps are given in table 2 , while the values of the same quantities from the theoretical models are given in table 3 . every model provides three sets of values , because spectral maps have been computed using three orthogonal directions for the line of sight . each group of three sets of values can be interpreted as the same model cloud being `` observed '' from different directions , or as three different model clouds with comparable rms velocity . in the equipartition model , e , the rms velocities inferred from different directions are very different from each other , the largest rms velocity being found in the direction parallel to the mean magnetic field ( along the @xmath23 axis ) , and the lowest in the directions perpendicular to the magnetic field . for this model we have also computed spectral maps from two more lines of sight , corresponding to diagonal directions across the computational box . figure [ fig6 ] shows the scf of the model a1r , with line of sight parallel to the direction of the mean magnetic field . the scf of the equipartition model e is also shown for four lines of sight , two diagonal , one parallel to the direction of the mean magnetic field and one perpendicular to it to it . the figure shows that the scf of model e is very sensitive to the line of sight , due to the large variations of the rms velocity in different directions relative to the mean magnetic field . the velocity dispersion relative to the speed of sound , or the value of the sonic mach number , is the most important physical parameter characterizing the nature of the turbulence . we therefore study the dependence of the scf on the turbulent velocity dispersion ( or the rms sonic mach number , ) and propose to use this dependence to test theoretical models against the observational data . in figure [ fig7 ] the slope of the scf is plotted against the line of sight velocity dispersion . the top panels show the models of constant size and constant average gas density ( models a1 to a5 -left , and b1 to b5 -right ) ; the bottom panels show the models rescaled according to larson type relations ( models a1r to a5r -left and b1r to b5r -right ) . the observations indicate a strong correlation between and @xmath1 , over an order of magnitude in @xmath1 . a least square fit to the observational data gives : @xmath63 and for the super sonic and super alfvnic models rescaled with larson type relations ( a1r to a5r ) : @xmath64 consistent with the observational result ( the uncertainty in the exponent is the standard deviation from the least square fit ) . the corresponding models not scaled with the larson type relations ( a1 to a5 ) are also indistinguishable from the observational result ( see figure [ fig7 ] top left panel ) . models of type b ( right panels of figure [ fig7 ] ) have instead values of significantly smaller than the average ones from the observational data . the stochastic models s2 and s4 are indistinguishable from each other ; they are also totally inconsistent with the empirical @xmath1 relation , which allows them to be ruled out as invalid by the scf method . finally , the equipartition model e provides values that are consistent with the observations , and comparable to the super alfvnic models , apart from a larger scatter of values between different lines of sight . we interpret the increase of with @xmath1 as a consequence of the increasing compressibility of the turbulent flow ( @xmath1 is roughly proportional to the rms sonic mach number of the flow because the temperature in all the models is @xmath40 k , and approximately the same in the observed regions ) . the value of is in general found to grow with increasing density contrast , probably due to the increasing concentration of the mass along the line of sight around one or few dense cores , which helps decorrelating the spectra from each other . the value of is plotted against @xmath1 in figure [ fig8 ] . the top panels show the models of constant size and the bottom panels the models scaled with the larson type relations , as in figure [ fig7 ] . the values of and @xmath1 from the observational maps are weakly correlated , with slightly increasing with increasing @xmath1 : @xmath65 a tight correlation is instead found in the models of constant size ( top panels of figure [ fig8 ] ) , with decreasing with increasing @xmath1 . this inconsistency between the models and the observations is most likely due to the fact that molecular clouds of 5 to 20 pc of size ( as assumed by these models ) are never found with velocity dispersion as low as assumed in models a4 , a5 and b3 , b4 and b5 . the bottom panels of figure [ fig8 ] show that the inconsistency is in fact mostly resolved as soon as the model sizes are scaled according to the larson type relation . for the models a1r to a5r we obtain : @xmath66 if models with @xmath67 km / s were not included ( justified by the absence of such low velocity dispersions in the observational sample ) , the slope of the least square fit would be @xmath68 , fully consistent with the observations . the equipartition model yields values of and @xmath1 consistent with the observations in all directions , but the one parallel to the mean magnetic field . it could be concluded that either none of the observed objects has a significant component of the magnetic field along the line of sight , or that all of them have a magnetic field weaker than predicted by the equipartition model , consistent with the super alfvnic models . in figure [ fig9 ] we have plotted observations and models on the plane . the constant size models are again inconsistent with the observations , as is expected since the observational maps span a large range of scales . when the models are scaled according with the larson type relations and the realistic average column density of @xmath69 @xmath45 ( myers & goodman 1988 ) , the observed scatter in the plane is reproduced . the trend of the absolute value of to increase with for large values of both of them is also reproduced , between models with rms mach 5 and 10 ( a2r and a1r respectively ) ; however , models with rms mach of 20 or 30 would be necessary to fit the and values measured for the rosette molecular cloud , which can be appropriately resolved only with a numerical resolution in excess of @xmath70 computational cells . while the stochastic models s2 and s4 are only marginally inconsistent with the observations in this plot , the line of sight parallel to the direction of the mean magnetic field and one of the two diagonal lines of sight in the equipartition model e are again inconsistent with the observational data . the scf has been proposed as a statistical tool to test the validity of theoretical models describing the structure and dynamics of star forming clouds ( rosolowsky et al . 1999 ) . in a previous work we improved the scf method by studying its dependence on spatial and velocity resolution and on instrumental noise ( padoan , rosolowsky & goodman 2001 ) . here we have applied that improved scf to a number of large @xmath0co maps of molecular cloud complexes and obtained empirical correlations that can be used to test theoretical models . of the theoretical models we have computed some compare well with the empirical correlations and some do not , which shows that the scf can be used as an effective tool to rule out inappropriate or unphysical models . the empirical correlations we have obtained relate the values of , and @xmath1 with each other . the @xmath1 correlation rules out the unphysical stochastic models ( s2 and s4 ) . such models were found to produce spectral line profiles similar to observational ones by dubinski , narayan & phillips ( 1995 ) . they have also been used as models of the density field in molecular clouds by stutzki et al . ( 1998 ) and to calibrate their principal component analysis by brunt & heyer ( 2002 ) . the scf @xmath1 correlation shows that stochastic models are inappropriate to describe the structure of molecular cloud complexes . models not scaled with larson type relations ( a1a5 , b1b5 ) and models with larger than average column density ( b1b5 , b1r b5r ) have also been compared with the empirical scf correlations to show that incorrectly scaled models are readily ruled out by the scf method . the @xmath1 and the correlations do not favor the model with equipartition of kinetic and magnetic energies ( model e ) . such model yields too small values of or too large values of compared with the observational data , when seen in the direction parallel to the average magnetic field . of the two diagonal directions , one is consistent with the data and the other is not . a possible interpretation is that none of the observed regions has an average magnetic field oriented close to the direction of the line of sight . the equipartition model starts to be inconsistent with the observational data when seen along the diagonal directions , at an angle of 54.7@xmath55 to the average magnetic field . the line of sight should be within such an angle to the magnetic field in approximately 40% of the cases , assuming random orientation of the average magnetic field in the observed regions . an alternative interpretation is that all the observed regions have an average magnetic field strength smaller than in the equipartition model , and consistent with super alfvnic conditions . the super alfvnic models rescaled with larson type relations are in fact able to reproduce the empirical scf correlations . however , the total number of truly independent directions on the sky in the present observational sample is still small . more regions should be studied to rule out the equipartition model based on the scf results . the analysis of the mhd models could in principle give different results if self gravity was taken into account . however , the introduction of self gravity is not expected to decrease the value of and increase the value of , as necessary to make the equipartition model consistent with the observational correlation . the main effect of self gravity is the collapse of the densest regions , increasing the density contrast beyond the level due to the turbulence alone . this could slightly increase the value of because we interpret the increase of with @xmath1 in the mhd models as due to the increased compressibility of the turbulent flow . an increase in the value of is not expected because the local collapse of dense cores can not increase the correlation between spectra at large distances ( for a given value of an increase in would correspond to an increase in the scf at large spatial separation ) . nevertheless , the effect of self gravity should be tested by including it in the numerical solution of the mhd equations . the numerical resolution should also be larger than in the present work to resolve the initial phase of the gravitational collapse of dense cores . we have only recently started to compute self gravitating flows in a numerical mesh of @xmath71 cells , and their analysis will be presented in future works . padoan & nordlund ( 1997 , 1999 ) have proposed that the dynamics of molecular clouds on large scales is consistent with super alfvnic turbulence and inconsistent with the equipartition model . in numerical simulations of super - alfvnic turbulence the average magnetic energy grows with time , even if flux is conserved ( the average magnetic field is constant ) . the magnetic field strength is increased locally mainly in regions of compression in super sonic turbulence , and in part by stretching of field lines . even if initial conditions are such that the turbulence is highly super - alfvnic , magnetic pressure is often larger than thermal pressure in the postshock gas , due to the amplification of the magnetic field components perpendicular to the shock direction . equipartition of dynamic pressure , @xmath72 , and magnetic pressure , @xmath73 , is therefore achieved locally , but not necessarily over the whole flow . for example , in super sonic and super - alfvnic runs at a resolution of @xmath74 , the ratio of volume average magnetic and dynamic pressures relaxes at a value @xmath75 , starting from initial conditions with @xmath76 ( padoan et al . comparable values are found in the numerical experiments used in this work . the amplification of the magnetic field by the turbulence therefore does not alter the super alfvnic character of the flow . the correlation between local magnetic field strength and gas density in super sonic and super alfvnic turbulence has a very large scatter , and a well defined upper envelope with @xmath77 , both consistent with the observational data ( padoan & nordlund 1997 , 1999 ; ostriker , stone & gammie 2001 ; passot & vazquez semadeni 2002 ) . the largest values of the magnetic field strength are generally found in dense cores , but some dense cores may have a relatively weak magnetic field . however , dense cores assembled by turbulent shocks are not expected to have internal super alfvnic turbulence , because of the dissipation of kinetic energy in the shocks and of the amplification of the magnetic field in the compressed gas . observational evidence for an approximate equipartition of turbulent and magnetic energy in dense cores would therefore not be inconsistent with the super alfvnic character of the large scale flow that assembles them . the comparison between our theoretical models and the observational data could be improved if more regions with sub sonic turbulence were available in the observational sample . small velocity dispersion is found in small objects , according to larson s velocity size relation , or to the power spectrum of turbulence . the spatial resolution in single dish surveys is typically too low to sample a small object ( fraction of a parsec ) with a very large spectral maps ( several thousands of spectra ) . the only exceptions in the observational sample used in this work are l1512 and l134a . these two large maps of nearby clouds with very low velocity dispersion were obtained by falgarone et al . ( 1998 ) as part of their iram key project , focused on regions of relatively low column density at the edges of molecular cloud complexes . maps of large regions with very large velocity dispersion are instead more easily obtained from observations than in numerical simulations . assuming a gas kinetic temperature of the order of 10 k , a line of sight ( one dimensional ) velocity dispersion in excess of 2 km / s corresponds to a sonic rms mach number of the flow @xmath78 . in the present work we have not computed numerical flows with @xmath79 , since that would require a larger numerical resolution ( the density contrast grows linearly with the alfvnic mach number and therefore with the value of @xmath26 if the average magnetic field strength is not varied ) . for this reason the models do not reach the largest values of @xmath1 , and obtained from the observations ( from the maps of the rosette molecular cloud complex ) . the progression of models toward increasing values of @xmath26 suggests that a model with @xmath78 would likely fit the observed values found in the rosette molecular cloud complex , where the observed velocity dispersion is in excess of 2 km / s . this is illustrated in figure [ fig10 ] . the top panel of figure [ fig10 ] shows the for the observational data . the shaded area shows the range of values covered by the theoretical models a1r to a5r . the bottom panel shows the same plot for the model a1r to a5r . each diagonal segment connects the values for the three directions of each model . the values of and @xmath39 of the models are also given in the plot . the arrow marks the direction of increasing suggesting that models with @xmath80 may fit the observations with the largest velocity dispersion . in the present work we have computed the spectral correlation function ( scf ) of spectral maps of molecular cloud complexes and regions within them , observed in the j=10 transition of @xmath0co . we have found that the scf is a power law over approximately an order of magnitude in spatial separation . the power law slope of the scf , , its normalization , , and the spectral line width averaged over the whole map , @xmath1 , have been computed for all the observational maps . we have obtained empirical correlations between these quantities and have proposed to use them to test the validity of theoretical models of molecular clouds . theoretical models of spectral line maps have been generated by computing the radiative transfer through the numerical solutions ( density and velocity fields ) of the magneto - hydrodynamic ( mhd ) equations , for turbulent flows with different values of the rms sonic and alfvnic mach numbers , and also through stochastic density and velocity fields with different power spectra . super - alfvnic mhd models rescaled according to larson type relations are in the best agreement with the empirical correlations . unphysical stochastic models are instead ruled out . mhd models with equipartition of magnetic and kinetic energy of turbulence do not reproduce the observational data when their average magnetic field is oriented approximately parallel to the line of sight . finally , mhd models not rescaled according to larson type relations are also inconsistent with the observational data . we can not exclude the possibility that different physical models for the dynamics of molecular clouds , or even unphysical models , that we have not tested here , would satisfy the empirical correlations found in this work . however , we have shown that the scf method is able to rule out certain unphysical or incorrectly scaled models . reproducing these scf results should be considered as a necessary ( but not sufficient ) condition for the validity of theoretical models describing the structure and the dynamics of molecular clouds . models for which the scf or similar statistical tests can not be computed to allow a quantitative comparison with observed spectral maps can not be legitimately evaluated . the comparison between theory and observations presented in this work requires significant computational resources . numerical simulations of three dimensional turbulent flows must be run at large resolution and the radiative transfer has to be computed in three dimensions in order to generate synthetic spectral maps of the observed molecular transitions . the type of models and the physical parameters investigated in this work are therefore limited to a few significant cases . future work should investigate the scf of a larger variety of models , including different magnetic field intensities and flows with gravitationally collapsing cores . we are grateful to eve ostriker and jim stone for helpful comments on our model data comparison . the referee report by enrique vazquez semadeni has also contributed to improve this work . this work was supported by an nsf galactic astronomy grant to ag . the work of pp was partially performed while pp held a national research council associateship award at the jet propulsion laboratory , california institute of technology . mj acknowledges the support of the academy of finland grant no . 1011055 . * figure captions : * + * table [ t1 ] * main parameters of the observed spectral maps : approximate size , distance , rms velocity over the whole map , telescope beam , spatial sampling , velocity channel width , average spectrum quality and bibliographic reference . + * table [ t2 ] * spectral line width averaged over the whole map , @xmath1 , power law slope of the scf , and scf normalization , , galactic longitude , @xmath81 , and galactic latitude , @xmath82 , of the center of all the observed maps and selected regions within them . + * table [ t3 ] * first three columns from the left : model name , rms sonic mach number of the flow and physical size of the computational mesh . following columns : line of sight velocity dispersion , scf slope and scf normalization , repeated for the three orthogonal directions for which synthetic spectral maps have been computed in each model . values for the diagonal directions of model e are not given ( they are within the ranges of values covered by the other three directions parallel and perpendicular to the mean magnetic field ) . + * figure [ fig1 ] : * top panel : velocity integrated intensity map of the perseus molecular cloud complex in the j=1 - 0 transition of @xmath0co ( padoan et al . bottom panel : same as top panel , but for the taurus molecular cloud complex ( mizuno et al . smaller regions within the maps where the scf has also been computed are highlighted . + * figure [ fig2 ] : * same as in figure [ fig1 ] , but for the rosette molecular cloud complex . top panel from heyer et al . ( 2001 ) ; bottom panel from blitz & stark ( 1986 ) . + * figure [ fig5 ] : * top left panel : the scf averaged over the entire map of the perseus , rosette and taurus molecular cloud complexes . solid lines are least square fits to the power law sections of the scf . the exponents @xmath83 of the power law fits are also given in the figure . top right panel : scf of the whole map of the perseus molecular cloud complex and of smaller regions within the same map . bottom left panel : scf of the taurus molecular cloud complex and of smaller regions within the same map . bottom right panel : scf of pvceph and hh300 . + * figure [ fig6 ] : * the scf computed from mhd models . asterisk symbols are for the super - alfvnic model a1r in the @xmath84 direction ( parallel to the mean magnetic field ) . diamond symbols are for the equipartition model in the @xmath85 and @xmath84 direction ( perpendicular and parallel to the average magnetic field direction respectively ) and along two diagonal directions ( @xmath86 and @xmath87 ) . the slope of the scf increases with increasing rms velocity . the scf is therefore weakly dependent on the direction of the line of sight for the super - alfvnic model , while it is much steeper in the direction parallel to the magnetic field ( larger rms velocity ) than in the perpendicular direction in the equipartition model . + * figure [ fig7 ] : * scf slope versus velocity dispersion . the top panels show the models of constant size and constant average gas density as asterisks ( models a1 to a5 left , b1 to b5 right ) ; the bottom panels show the models rescaled according to larson type relations as asterisks ( models a1r to a5r left , b1r to b5r right ) . observational values are shown as squares , the equipartition model as triangles and the stochastic models as diamonds . + * figure [ fig8 ] : * scf value at 1 pc versus velocity dispersion . different panels show different models as in figure [ fig7 ] . symbols are also as in figure [ fig7 ] . + * figure [ fig9 ] : * scf slope versus scf value at 1 pc . symbols and panels as in figure [ fig7 ] . + * figure [ fig10 ] : * top panel : values of and from the observations . some of the symbols are labeled with the region name . the shaded area shows the range of values covered by the models a1r to a5r . bottom panel : same shaded area as in the top panel . diagonal segments shows the range of values of and for the three directions of each model . the rms sonic mach number of the corresponding model is given on the right hand side of each segment , while the value of the linear size is given on the left hand side . the arrow indicates the progression of models toward larger values of sonic mach number , . lcc|ccc|ccc|ccc & & & & x & & & y & & & z & + model & & 0 & & & & & & & & & + + a1 & 10.0 & 5 & 1.13 & 0.30 & 0.37 & 1.36 & 0.32 & 0.34 & 1.24 & 0.35 & 0.31 + a2 & 5.0 & 5 & 0.56 & 0.25 & 0.46 & 0.71 & 0.30 & 0.39 & 0.67 & 0.28 & 0.42 + a3 & 2.5 & 5 & 0.30 & 0.19 & 0.55 & 0.38 & 0.24 & 0.48 & 0.36 & 0.21 & 0.52 + a4 & 1.2 & 5 & 0.17 & 0.13 & 0.66 & 0.17 & 0.15 & 0.64 & 0.18 & 0.16 & 0.61 + a5 & 0.6 & 5 & 0.12 & 0.13 & 0.68 & 0.11 & 0.10 & 0.71 & 0.12 & 0.14 & 0.66 + b1 & 10.0 & 20 & 1.21 & 0.26 & 0.64 & 1.42 & 0.27 & 0.61 & 1.33 & 0.27 & 0.59 + b2 & 5.0 & 20 & 0.61 & 0.22 & 0.70 & 0.77 & 0.26 & 0.65 & 0.72 & 0.24 & 0.67 + b3 & 2.5 & 20 & 0.33 & 0.18 & 0.73 & 0.41 & 0.23 & 0.68 & 0.39 & 0.20 & 0.71 + b4 & 1.2 & 20 & 0.18 & 0.14 & 0.77 & 0.19 & 0.15 & 0.77 & 0.21 & 0.17 & 0.75 + b5 & 0.6 & 20 & 0.13 & 0.13 & 0.79 & 0.13 & 0.11 & 0.81 & 0.14 & 0.14 & 0.78 + + a1r & 10.0 & 10 & 1.13 & 0.30 & 0.44 & 1.36 & 0.32 & 0.42 & 1.23 & 0.35 & 0.39 + a2r & 5.0 & 2.5 & 0.56 & 0.23 & 0.41 & 0.72 & 0.30 & 0.32 & 0.66 & 0.26 & 0.37 + a3r & 2.5 & 0.62 & 0.30 & 0.18 & 0.41 & 0.37 & 0.21 & 0.34 & 0.35 & 0.20 & 0.37 + a4r & 1.2 & 0.16 & 0.16 & 0.12 & 0.45 & 0.15 & 0.13 & 0.42 & 0.17 & 0.15 & 0.38 + a5r & 0.6 & 0.04 & 0.10 & 0.11 & 0.41 & 0.10 & 0.08 & 0.49 & 0.11 & 0.12 & 0.39 + b1r & 10.0 & 20 & 1.21 & 0.26 & 0.64 & 1.42 & 0.27 & 0.61 & 1.33 & 0.27 & 0.59 + b2r & 5.0 & 5 & 0.60 & 0.22 & 0.55 & 0.76 & 0.23 & 0.49 & 0.69 & 0.20 & 0.53 + b3r & 2.5 & 1.25 & 0.33 & 0.15 & 0.52 & 0.40 & 0.18 & 0.45 & 0.37 & 0.16 & 0.49 + b4r & 1.2 & 0.31 & 0.18 & 0.12 & 0.50 & 0.18 & 0.13 & 0.47 & 0.19 & 0.14 & 0.44 + b5r & 0.6 & 0.08 & 0.12 & 0.10 & 0.46 & 0.12 & 0.09 & 0.52 & 0.13 & 0.12 & 0.42 + + e & 10.0 & 5 & 0.73 & 0.21 & 0.46 & 0.81 & 0.25 & 0.43 & 1.90 & 0.49 & 0.22 + + s2 & 10.0 & 20 & 1.37 & 0.15 & 0.52 & 1.29 & 0.15 & 0.53 & 1.31 & 0.14 & 0.53 + s4 & 10.0 & 20 & 1.23 & 0.16 & 0.47 & 1.17 & 0.15 & 0.49 & 1.17 & 0.17 & 0.46 +

in recent years , mapreduce has emerged as a computational paradigm for processing large - scale data sets in a series of rounds executed on conglomerates of commodity servers @xcite , and has been widely adopted by a number of large web companies ( e.g. , google , yahoo ! , amazon ) and in several other applications ( e.g. , gpu and multicore processing ) . ( see @xcite and references therein . ) informally , a mapreduce computation transforms an input set of key - value pairs into an output set of key - value pairs in a number of _ rounds _ , where in each round each pair is first individually transformed into a ( possibly empty ) set of new pairs ( _ map step _ ) and then all values associated with the same key are processed , separately for each key , by an instance of the same reduce function ( simply called _ reducer _ in the rest of the paper ) thus producing the next new set of key - value pairs ( _ reduce step _ ) . in fact , as already noticed in @xcite , a reduce step can clearly embed the subsequent map step so that a mapreduce computation can be simply seen as a sequence of rounds of ( augmented ) reduce steps . the mapreduce paradigm has a functional flavor , in that it merely requires that the algorithm designer decomposes the computation into rounds and , within each round , into independent tasks through the use of keys . this enables parallelism without forcing an algorithm to cater for the explicit allocation of processing resources . nevertheless , the paradigm implicitly posits the existence of an underlying unstructured and possibly heterogeneous parallel infrastructure , where the computation is eventually run . while mostly ignoring the details of such an underlying infrastructure , existing formalizations of the mapreduce paradigm constrain the computations to abide with some local and aggregate memory limitations . in this paper , we look at both modeling and algorithmic issues related to the mapreduce paradigm . we first provide a formal specification of the model , aimed at overcoming some limitations of the previous modeling efforts , and then derive interesting tradeoffs between memory constraints and round complexity for the fundamental problem of matrix multiplication and some of its applications . the mapreduce paradigm has been introduced in @xcite without a fully - specified formal computational model for algorithm design and analysis . triggered by the popularity quickly gained by the paradigm , a number of subsequent works have dealt more rigorously with modeling and algorithmic issues @xcite . in @xcite , a mapreduce algorithm specifies a sequence of rounds as described in the previous section . somewhat arbitrarily , the authors impose that in each round the memory needed by any reducer to store and transform its input pairs has size @xmath0 , and that the aggregate memory used by all reducers has size @xmath1 , where @xmath2 denotes the input size and @xmath3 is a fixed constant in @xmath4 . the cost of local computation , that is , the work performed by the individual reducers , is not explicitly accounted for , but it is required to be polynomial in @xmath2 . the authors also postulate , again somewhat arbitrarily , that the underlying parallel infrastructure consists of @xmath5 processing elements with @xmath5 local memory each , and hint at a possible way of supporting the computational model on such infrastructure , where the reduce instances are scheduled among the available machines so to distribute the aggregate memory in a balanced fashion . it has to be remarked that such a distribution may hide non negligible costs for very fine - grained computations ( due to the need of allocating multiple reducer with different memory requirements to a fixed number of machines ) when , in fact , the algorithmic techniques of @xcite do not fully explore the larger power of the mapreduce model with respect to a model with fixed parallelism . in @xcite the same model of @xcite is adopted but when evaluating an algorithm the authors also consider the total work and introduce the notion of work - efficiency typical of the literature on parallel algorithms . an alternative computational model for mapreduce is proposed in @xcite , featuring two parameters which describe bandwidth and latency characteristics of the underlying communication infrastructure , and an additional parameter that limits the amount of i / o performed by each reducer . also , a bsp - like cost function is provided which combines the internal work of the reducers with the communication costs incurred by the shuffling of the data needed at each round . unlike the model of @xcite , no limits are posed to the aggregate memory size . this implies that in principle there is no limit to the allowable parallelism while , however , the bandwidth / latency parameters must somewhat reflect the topology and , ultimately , the number of processing elements . thus , the model mixes the functional flavor of mapreduce with the more descriptive nature of bandwidth - latency models such as bsp @xcite . a model which tries to merge the spirit of mapreduce with the features of data - streaming is the mud model of @xcite , where the reducers receive their input key - value pairs as a stream to be processed in one pass using small working memory , namely polylogarithmic in the input size . a similar model has been adopted in @xcite . mapreduce algorithms for a variety of problems have been developed on the aforementioned mapreduce variants including , among others , primitives such as prefix sums , sorting , random indexing @xcite , and graph problems such as triangle counting @xcite minimum spanning tree , @xmath6-@xmath7 connectivity , @xcite , maximal and approximate maximum matching , edge cover , minimum cut @xcite , and max cover @xcite . moreover simulations of the pram and bsp in mapreduce have been presented in @xcite . in particular , it is shown that a @xmath8-step erew pram algorithm can be simulated by an @xmath9-round mapreduce algorithm , where each reducer uses constant - size memory and the aggregate memory is proportional to the amount of shared memory required by the pram algorithm @xcite . the simulation of crew or crcw pram algorithms incurs a further @xmath10 slowdown , where @xmath11 denotes the local memory size available for each reducer and @xmath12 the aggregate memory size @xcite . all of the aforementioned algorithmic efforts have been aimed at achieving the minimum number of rounds , possibly constant , provided that enough local memory for the reducer ( typically , sublinear yet polynomial in the input size ) and enough aggregate memory is available . however , so far , to the best of our knowledge , there has been no attempt to fully explore the tradeoffs that can be exhibited for specific computational problems between the local and aggregate memory sizes , on one side , and the number of rounds , on the other , under reasonable constraints of the amount of total work performed by the algorithm . our results contribute to filling this gap . matrix multiplication is a building block for many problems , including matching @xcite , matrix inversion @xcite , all - pairs shortest path @xcite , graph contraction @xcite , cycle detection @xcite , and parsing context free languages @xcite . parallel algorithms for matrix multiplication of dense matrices have been widely studied : among others , we remind @xcite which provide upper and lower bounds exposing a tradeoff between communication complexity and processor memory . for sparse matrices , interesting results are given in @xcite for some network topologies like hypercubes , in @xcite for pram , and in @xcite for a bsp - like model . in particular , techniques in @xcite are used in the following sections for deriving efficient mapreduce algorithms . in the sequential settings , some interesting works providing upper and lower bounds are @xcite for dense matrix multiplication , and @xcite for sparse matrix multiplication . the contribution of this paper is twofold , since it targets both modeling and algorithmic issues . we first formally specify a computational model for mapreduce which captures the functional flavor of the paradigm by allowing a flexible use of parallelism . more specifically , our model generalizes the one proposed in @xcite by letting the local and aggregate memory sizes be two independent parameters , @xmath11 and @xmath12 , respectively . moreover our model makes no assumption on the underlying execution infrastructure , for instance it does not impose a bound on the number of available machines , thus fully decoupling the degree of parallelism exposed by a computation from the one of the machine where the computation will be eventually executed . this decoupling greatly simplifies algorithm design , which has been one of the original objectives of the mapreduce paradigm . ( in section [ sec : preliminary ] , we quantify the cost of implementing a round of our model on a system with fixed parallelism . ) our algorithmic contributions concern the study of attainable tradeoffs in mapreduce for several variants of the fundamental primitive of matrix multiplication . in particular , building on the well - established three - dimensional algorithmic strategy for matrix multiplication @xcite , we develop upper and lower bounds for dense - dense matrix multiplication and provide similar bounds for deterministic and/or randomized algorithms for sparse - sparse and sparse - dense matrix multiplication . the algorithms are parametric in the local and aggregate memory constraints and achieve optimal or quasi - optimal round complexity in the entire range of variability of such parameters . finally , building on the matrix multiplication results , we derive similar space - round tradeoffs for matrix inversion and matching , which are important by - products of matrix multiplication . the rest of the paper is structured as follows . in section [ sec : preliminary ] we introduce our computational model for mapreduce and describe important algorithmic primitives ( sorting and prefix sums ) that we use in our algorithms . section [ sec : intromatrix ] deals with matrix multiplication in our model , presenting theoretical bounds to the complexity of algorithms to solve this problem . we apply these results in section [ sec : applications ] to derive algorithms for matrix inversion and for matching in graphs . our model is defined in terms of two integral parameters @xmath12 and @xmath11 , whose meaning will be explained below , and is named . algorithms specified in this model will be referred to as _ mr - algorithms_. an mr - algorithm specifies a sequence of _ rounds _ : the @xmath13-th round , with @xmath14 transforms a multiset @xmath15 of key - value pairs into two multisets @xmath16 and @xmath17 of key - value pairs , where @xmath16 is the input of the next round ( empty , if @xmath13 is the last round ) , and @xmath17 is a ( possibly empty ) subset of the final output . the input of the algorithm is represented by @xmath18 while the output is represented by @xmath19 , with @xmath20 denoting the union of multisets . the universes of keys and values may vary at each round , and we let @xmath21 denote the universe of keys of @xmath15 . the computation performed by round @xmath13 is defined by a _ reducer _ function @xmath22 which is applied independently to each multiset @xmath23 consisting of all entries in @xmath15 with key @xmath24 . let @xmath2 be the input size . the two parameters @xmath12 and @xmath11 specify the memory requirements that each round of an mr - algorithm must satisfy . in particular , let @xmath25 denote the space needed to compute @xmath26 on a ram with @xmath27-bit words , including the space taken by the input ( i.e. , latexmath:[$m_{r , k } \geq output , which contributes either to @xmath17 ( i.e. , the final output ) or to @xmath16 . the model imposes that @xmath29 , for every @xmath14 and @xmath24 , that @xmath30 , for every @xmath14 , and that @xmath31 . the complexity of an mr - algorithm is the number of rounds that it executes in the worst case , and it is expressed as a function of the input size @xmath2 and of parameters @xmath11 and @xmath12 . the dependency on the parameters @xmath11 and @xmath12 allows for a finer analysis of the cost of an mr - algorithm . as in @xcite , we require that each reducer function runs in time polynomial in @xmath2 . in fact , it can be easily seen that the model defined in @xcite is equivalent to the model with @xmath32 and @xmath33 , for some fixed constant @xmath34 , except that we eliminate the additional restrictions that the number of rounds of an algorithm be polylogarithmic in @xmath2 and that the number of physical machines on which algorithms are executed are @xmath35 , which in our opinion should not be posed at the model level . compared to the model in @xcite , our model introduces the parameter @xmath12 which limits the size of the aggregate memory required at each round , whereas in @xcite this size is virtually unbounded . moreover , the complexity analysis in focuses on the tradeoffs between @xmath11 and @xmath12 , on one side , and the number of rounds on the other side , while in @xcite a more complex cost function is defined which accounts for the overall message complexity of each round , the time complexity of each reducer computation , and the latency and bandwidth characteristics of the executing platform . sorting and prefix sum primitives are used in the algorithms presented in this paper . the input to both primitives consists of a set of @xmath2 key - value pairs @xmath36 with @xmath37 and @xmath38 , where @xmath39 denotes a suitable set . for sorting , a total order is defined over @xmath39 and the output is a set of @xmath2 key - value pairs @xmath40 , where the @xmath41 s form a permutation of the @xmath42 s and @xmath43 for each @xmath44 . for prefix sums , a binary associative operation @xmath45 is defined over @xmath39 and the output consists of a collection of @xmath2 pairs @xmath40 where @xmath46 , for @xmath47 . by straightforwardly adapting the results in @xcite to our model we have : [ prefixsorting ] the sorting and prefix sum primitives for inputs of size @xmath2 can be implemented in with round complexity @xmath48 for @xmath49 . we remark that the each reducer in the implementation of the sorting and prefix primitives makes use of @xmath50 memory words . hence , the same round complexity can be achieved in a more restrictive scenario with fixed parallelism . in fact , our model can be simulated on a platform with @xmath51 processing elements , each with internal memory of size @xmath50 , at the additional cost of one prefix computation per round . therefore , @xmath52 can be regarded as an upper bound on the relative power of our model with respect to one with fixed parallelism . goodrich @xcite claims that the round complexities stated in theorem [ prefixsorting ] are optimal for any @xmath53 as a consequence of the lower bound for computing the or of @xmath2 bits on the bsp model @xcite . it can be shown that the optimality carries through to our model where the output of a reducer is not bounded by @xmath11 . let @xmath54 and @xmath55 be two @xmath56 matrices and let @xmath57 . we use @xmath58 and @xmath59 , with @xmath60 , to denote the entries of @xmath61 and @xmath62 , respectively . in this section we present upper and lower bounds for computing the product @xmath62 in . the algorithms we present envision the matrices as conceptually divided into submatrices of size @xmath63 , and we denote these matrices with @xmath64 , @xmath65 and @xmath66 , respectively , for @xmath67 . clearly , @xmath68 . all our algorithms exploit the following partition of the @xmath69 products between submatrices ( e.g. , @xmath70 ) into @xmath71 _ groups _ : group @xmath72 , with @xmath73 , consists of products @xmath74 , for every @xmath67 and for @xmath75 . observe that each submatrix of @xmath54 and @xmath55 occurs exactly once in each group @xmath72 . we focus our attention on matrices whose entries belong to a semiring @xmath76 such that for any @xmath77 we have @xmath78 , where @xmath79 is the identity for @xmath45 . in this setting , efficient matrix multiplication techniques such as strassen s can not be employed . moreover , we assume that the inner products of any row of @xmath54 and of any column of @xmath55 with overlapping nonzero entries never cancel to zero , which is a reasonable assumption when computing over natural numbers or over real numbers with a finite numerical precision . in our algorithms , any input matrix @xmath80 ( @xmath81 ) is provided as a set of key - value pairs @xmath82 for all elements @xmath83 . key @xmath84 represents a progressive index , e.g. , the number of nonzero entries preceding @xmath85 in the row - major scan of @xmath80 . we call a @xmath86 matrix _ dense _ if the number of its nonzero entries is @xmath87 , and we call it _ sparse _ otherwise . we suppose that @xmath12 is sufficiently large to contain the input and output matrices . in what follows , we present different algorithms tailored for the multiplication of dense - dense ( section [ sec : ddmult ] ) , sparse - sparse ( section [ sec : ssmult ] ) , and sparse - dense matrices ( section [ sec : sdmult ] ) . we also derive lower bounds which demonstrate that our algorithms are either optimal or close to optimal ( section [ sec : lb ] ) , and an algorithm for estimating the number of nonzero entries in the product of two sparse matrices ( section [ sec : evaluation ] ) . in this section we provide a simple , deterministic algorithm for multiplying two dense matrices , which will be proved optimal in subsection [ sec : lb ] . the algorithm is a straightforward adaptation of the well - established three - dimensional algorithmic strategy for matrix multiplication of @xcite , however we describe a few details of its implementation in since the strategy is also at the base of algorithms for sparse matrices . we may assume that @xmath88 , since otherwise matrix multiplication can be executed by a trivial sequential algorithm . we consider matrices @xmath54 and @xmath55 as decomposed into @xmath89 submatrices and subdivide the products between submatrices into groups as described above . in each round , the algorithm computes all products within @xmath90 consecutive groups , namely , at round @xmath91 , all multiplications in @xmath72 are computed , with @xmath92 . the idea is that in a round all submatrices of @xmath54 and @xmath55 can be replicated @xmath93 times and paired in such a way that each reducer performs a distinct multiplication in @xmath94 . then , each reducer sums the newly computed product to a partial sum which accumulates all of the products contributing to the same submatrix of @xmath62 belonging to groups with the same index modulo @xmath93 dealt with in previous rounds . at the end of the @xmath95-th round , all submatrix products have been computed . the final matrix @xmath62 is then obtained by adding together the @xmath93 partial sums contributing to each entry of @xmath62 through a prefix computation . we have the following result . [ th : upddmult ] the above -algorithm multiplies two @xmath96 dense matrices in @xmath97 rounds . the algorithm clearly complies with the memory constraints of since each reducer multiplies two @xmath98 submatrices and the degree of replication is such that the algorithm never exceeds the aggregate memory bound of @xmath12 . also , the @xmath69 products are computed in @xmath99 rounds , while the final prefix computation requires @xmath100 rounds we remark that the multiplication of two @xmath96 dense matrices can be performed in a constant number of rounds whenever @xmath101 , for constant @xmath102 , and @xmath103 . consider two @xmath86 sparse matrices @xmath54 and @xmath55 and denote with @xmath104 the maximum number of nonzero entries in any of the two matrices , and with @xmath105 the number of nonzero entries in the product @xmath106 . below , we present two deterministic mr - algorithms ( d1 and d2 ) and a randomized one ( r1 ) , each of which turns out to be more efficient than the others for suitable ranges of parameters . we consider only the case @xmath107 , since otherwise matrix multiplication can be executed by a trivial one - round mr - algorithm using only one reducer . we also assume that the value @xmath108 is provided in input . ( if this were not the case , such a value could be computed with a simple prefix computation in @xmath52 rounds , which does not affect the asymptotic complexity of our algorithms . ) however , we do not assume that @xmath105 is known in advance since , unlike @xmath108 , this value can not be easily computed . in fact , the only source of randomization in algorithm r1 stems from the need to estimate @xmath105 . this algorithm is based on the following strategy adapted from @xcite . for @xmath109 , let @xmath42 ( resp . , @xmath41 ) be the number of nonzero entries in the @xmath110th column of @xmath54 ( resp . , @xmath110th row of @xmath55 ) , and let @xmath111 be the set containing all nonzero entries in the @xmath110th column of @xmath54 and in the @xmath110th row of @xmath55 . it is easily seen that all of the @xmath112 products between entries in @xmath111 ( one from @xmath54 and one from @xmath55 ) must be computed . the algorithm performs a sequence of _ phases _ as follows . suppose that at the beginning of phase @xmath7 , with @xmath113 , all products between entries in @xmath111 , for each @xmath114 and for a suitable value @xmath13 ( initially , @xmath115 ) , have been computed and added to the appropriate entries of @xmath62 . through a prefix computation , phase @xmath7 computes the largest @xmath116 such that @xmath117 . then , all products between entries in @xmath118 , for every @xmath119 , are computed using one reducer with constant memory for each such product . the products are then added to the appropriate entries of @xmath62 using again a prefix computation . algorithm d1 multiplies two sparse @xmath96 matrices with at most @xmath108 nonzero entries each in @xmath120 rounds , on an . the correctness is trivial and the memory constraints imposed by the model are satisfied since in each phase at most @xmath12 elementary products are performed . the theorem follows by observing that the maximum number of elementary products is @xmath121 and that two consecutive phases compute at least @xmath12 elementary products in @xmath122 rounds . the algorithm exploits the same three - dimensional algorithmic strategy used in the dense - dense case and consists of a sequence of phases . in phase @xmath7 , @xmath113 , all @xmath63-size products within @xmath116 consecutive groups are performed in parallel , where @xmath116 is a phase - specific value . observe that the computation of all products within a group @xmath72 requires space @xmath123 $ ] , since each submatrix of @xmath54 and @xmath55 occurs only once in @xmath72 and each submatrix product contributes to a distinct submatrix of @xmath62 . however , the value @xmath124 can be determined in @xmath125 space and @xmath52 rounds by `` simulating '' the execution of the products in @xmath72 ( without producing the output values ) and adding up the numbers of nonzero entries contributed by each product to the output matrix . the value @xmath116 is determined as follows . suppose that , at the beginning of phase @xmath7 , groups @xmath72 have been processed , for each @xmath126 and for a suitable value @xmath13 ( initially , @xmath115 ) . the algorithm replicates the input matrices @xmath127 times . subsequently , through sorting and prefix computations the algorithm computes @xmath128 for each @xmath129 and determines the largest @xmath130 such that @xmath131 . then , the actual products in @xmath132 , for each @xmath133 are executed and accumulated ( again using a prefix computation ) in the output matrix @xmath62 . we have the following theorem . algorithm @xmath134 multiplies two sparse @xmath96 matrices with at most @xmath108 nonzero entries each in @xmath135 rounds on an , where @xmath105 denotes the maximum number of nonzero entries in the output matrix . the correctness of the algorithm is trivial . phase @xmath7 requires a constant number of sorting and prefix computations to determine @xmath116 and to add the partial contributions to the output matrix @xmath62 . since each value @xmath124 is @xmath136 and the groups are @xmath71 , clearly , @xmath137 , and the theorem follows . we remark that the value @xmath105 appearing in the stated round complexity needs not be explicitly provided in input to the algorithm . we also observe that with respect to algorithm d1 , algorithm d2 features a better exploitation of the local memories available to the individual reducers , which compute @xmath89-size products rather than working at the granularity of the single entries . by suitably combining algorithms d1 and d2 , we can get the following result . [ d12round ] there is a deterministic algorithm which multiplies two sparse @xmath96 matrices with at most @xmath108 nonzero entries each in @xmath138 rounds on an , where @xmath105 denotes the maximum number of nonzero entries in the output matrix . algorithm d2 requires @xmath122 rounds in each phase @xmath7 for computing the number @xmath116 of groups to be processed . however , if @xmath105 were known , we could avoid the computation of @xmath116 and resort to the fixed-@xmath93 strategy adopted in the dense - dense case , by processing @xmath139 consecutive groups per round . this would yield an overall @xmath140 round complexity , where the @xmath141 additive term accounts for the complexity of summing up , at the end , the @xmath93 contributions to each entry of @xmath62 . however , @xmath105 may not be known a priori . in this case , using the strategy described in section [ sec : evaluation ] we can compute a value @xmath142 which is a 1/2-approximation to @xmath105 with probability at least @xmath143 . ( we say that @xmath142 @xmath144-approximates @xmath105 if @xmath145 . ) hence , in the algorithm we can plug in @xmath146 as an upper bound to @xmath105 . by using the result of theorem [ otilde ] with @xmath147 and @xmath148 , we have : [ r1round ] let @xmath149 . algorithm r1 multiplies two sparse @xmath96 matrices with at most @xmath108 nonzero entries in @xmath150 rounds on an , with probability at least @xmath143 . by comparing the rounds complexities stated in corollary [ d12round ] and theorem [ r1round ] , it is easily seen that the randomized algorithm r1 outperforms the deterministic strategies when @xmath151 , for any constant @xmath3 , @xmath152 , and @xmath153 . for a concrete example , r1 exhibits better performance when @xmath154 , @xmath155 , and @xmath11 is polylogarithmic in @xmath12 . moreover , both the deterministic and randomized strategies can achieve a constant round complexity for suitable values of the memory parameters . observe that a @xmath156-approximation to @xmath105 derives from the following simple argument . let @xmath42 and @xmath41 be the number of nonzero entries in the @xmath110th column of @xmath54 and in the @xmath110th row of @xmath55 respectively , for each @xmath157 . then , @xmath158 . evaluating the sum requires @xmath159 sorting and prefix computations , hence a @xmath156-approximation of @xmath105 can be computed in @xmath160 rounds . however , such an approximation is too weak for our purposes and we show below how to achieve a tighter approximation by adapting a strategy born in the realm of streaming algorithms . let @xmath161 and @xmath162 be two arbitrary values . an @xmath144-approximation to @xmath105 can be derived by adapting the algorithm of @xcite for counting distinct elements in a stream @xmath163 , whose entries are in the domain @xmath164=\{0,\ldots , n-1\}$ ] . the algorithm of @xcite makes use of a very compact data structure , customarily called _ sketch _ in the literature , which consists of @xmath165 lists , @xmath166 . for @xmath167 , @xmath168 contains the @xmath169 distinct smallest values of the set @xmath170 , where @xmath171\rightarrow [ n^3]$ ] is a hash function picked from a pairwise independent family . it is shown in @xcite that the median of the values @xmath172 , where @xmath173 denotes the @xmath7th smallest value in @xmath168 , is an @xmath144-approximation to the number of distinct elements in the stream , with probability at least @xmath174 . in order to compute an @xmath144-approximation of @xmath105 for a product @xmath175 of @xmath176 matrices , we can modify the algorithm as follows . consider the stream of values in @xmath164 $ ] where each element of the stream corresponds to a distinct product @xmath177 and consists of the value @xmath178 . clearly , the number of distinct elements in this stream is exactly @xmath105 . ( a similar approach has been used in @xcite in the realm of sparse boolean matrix products . ) we now show how to implement this idea on an . the mr - algorithm is based on the crucial observation that if the stream of values defined above is partitioned into segments , the sketch for the entire stream can be obtained by combining the sketches computed for the individual segments . specifically , two sketches are combined by merging each pair of lists with the same index and selecting the @xmath7 smallest values in the merged list . the -algorithm consists of a number of phases , where each phase , except for the last one , produces set of @xmath179 sketches , while the last phase combines the last batch of @xmath179 sketches into the final sketch , and outputs the approximation to @xmath105 . we refer to the partition of the matrices into @xmath180 submatrices and group the products of submatrices as done before . in phase @xmath7 , with @xmath181 , the algorithm processes the products in @xmath182 consecutive groups , assigning each pair of submatrices in one of the @xmath93 groups to a distinct reducer . a reducer receiving @xmath183 and @xmath184 , each with at least a nonzero entry , either computes a sketch for the stream segment of the nonzero products between entries of @xmath183 and @xmath184 , if the total number of nonzero entries of @xmath183 and @xmath184 exceeds the size of the sketch , namely @xmath185 words , or otherwise leaves the two submatrices untouched ( observe that in neither case the actual product of the two submatrices is computed ) . in this latter case , we refer to the pair of ( very sparse ) submatrices as a _ pseudosketch_. at this point , the sketches produced by the previous phase ( if @xmath186 ) , together with the sketches and pseudosketches produced in the current phase are randomly assigned to @xmath179 reducers . each of these reducers can now produce a single sketch from its assigned pseudosketches ( if any ) and merge it with all other sketches that were assigned to it . in the last phase ( @xmath187 ) the @xmath179 sketches are combined into the final one through a prefix computation , and the approximation to @xmath105 is computed . [ otilde ] let @xmath188 and let @xmath189 and @xmath190 be arbitrary values . then , with probability at least @xmath191 , the above algorithm computes an @xmath144-approximation to @xmath105 in @xmath192 rounds , on an the correctness of the algorithm follows from the results of @xcite and the above discussion . recall that the value computed by the algorithm is an @xmath144-approximation to @xmath105 with probability @xmath174 . as for the rounds complexity we observe that each phase , except for the last one , requires a constant number of rounds , while the last one involves a prefix computation thus requiring @xmath122 rounds . we only have to make sure that in each phase the memory constraints are satisfied ( with high probability ) . note also that a sketch of size @xmath193 is generated either in the presence of a pair of submatrices @xmath183 , @xmath184 containing at least @xmath194 entries , or within one of the @xmath179 reducers . by the choice of @xmath93 , it is easy to see that in any case , the overall memory occupied by the sketches is @xmath195 . as for the constraint on local memories , a simple modification of the standard balls - into - bins argument @xcite and the union bound suffices to show that with probability @xmath174 , in every phase when sketches and pseudosketches are assigned to @xmath179 reducers , each reducer receives in @xmath196 words . the theorem follows . ( more details will be provided in the full version of the paper . ) let @xmath54 be a sparse @xmath56 matrix with at most @xmath108 nonzero entries and let @xmath55 be a dense @xmath56 matrix ( the symmetric case , where @xmath54 is dense and @xmath55 sparse , is equivalent ) . the algorithm for dense - dense matrix multiplication does not exploit the sparsity of @xmath54 and requires @xmath197 rounds . also , if we simply plug @xmath198 in the complexities of the three algorithms for the sparse - sparse case ( where @xmath108 represented the maximum number of nonzero entries of @xmath54 or @xmath55 ) we do not achieve a better round complexity . however , a careful analysis of algorithm d1 in the sparse - dense case reveals that its round complexity is @xmath199 . therefore , by interleaving algorithm d1 and the dense - dense algorithm we have the following corollary . the multiplication on of a sparse @xmath56 matrix with at most @xmath108 nonzero entries and of a dense @xmath56 matrix requires a number of rounds which is the minimum between @xmath200 and @xmath201 . observe that the above sparse - dense strategy outperforms all previous algorithms for instance when @xmath202 . in this section we provide lower bounds for dense - dense and sparse - sparse matrix multiplication . we restrict our attention to algorithms which perform all nonzero elementary products , that is , on _ conventional _ matrix multiplication @xcite . although this assumption limits the class of algorithms , ruling out strassen - like techniques , an elaboration of a result in @xcite shows that computing all nonzero elementary products is necessary when entries of the input matrices are from the semirings @xmath203 and @xmath204 . semiring , where @xmath205 is the identity of the @xmath206 operation , is usually adopted while computing the shortest path matrix of a graph given its connection matrix . ] indeed , we have the following lemma which provides a lower bound on the number of products required by an algorithm multiplying any two matrices of size @xmath56 , containing @xmath207 and @xmath208 nonzero entries and where zero entries have fixed positions ( a similar lemma holds for @xmath209 ) . as a consequence of the lemma , an algorithm that multiplies any two arbitrary matrices in the semiring @xmath203 must perform all nonzero products . consider an algorithm @xmath210 which multiples two @xmath56 matrices @xmath54 and @xmath55 with @xmath207 and @xmath208 nonzero entries , respectively , from the semiring @xmath211 and where the positions of zero entries are fixed . then , algorithm @xmath210 must perform all the nonzero elementary products . @xcite shows that each @xmath59 can be computed only by summing all terms @xmath212 , with @xmath213 , if the algorithm uses only semiring operations . the proof relies on the analysis of the output for some suitable input matrices , and makes some assumptions that force the algorithm to compute even zero products . however , the result still holds if we allow all the zero products to be ignored , but some adjustments are required . in particular , the input matrices used in @xcite do not work in our scenario because may contain less than @xmath207 and @xmath208 nonzero entries , however it is easy to find inputs with the same properties working in our case . more details will be provided in the full version . the following theorem exhibits a tradeoff in the lower bound between the amount of local and aggregate memory and the round complexity of an algorithm performing conventional matrix multiplication . the proof is similar to the one proposed in @xcite for lower bounding the communication complexity of dense - dense matrix multiplication in a bsp - like model : however , differences arise since we focus on round complexity and our model does not assume the outdegree of a reducer to be bounded . in the proof of the theorem we use the following lemma which was proved using the red - blue pebbling game in @xcite and then restated in @xcite as follows . [ lem : nummult ] consider the conventional matrix multiplication @xmath214 , where @xmath54 and @xmath55 are two arbitrary matrices . a processor that uses @xmath215 elements of @xmath54 , @xmath216 elements of @xmath55 , and contributes to @xmath217 elements of @xmath62 can compute at most @xmath218 multiplication terms . [ th : lbddmult ] consider an -algorithm @xmath210 for multiplying two @xmath56 matrices containing at most @xmath207 and @xmath208 nonzero entries , using conventional matrix multiplications . let @xmath219 and @xmath105 denote the number of nonzero elementary products and the number of nonzero entries in the output matrix , respectively . then , the round complexity of @xmath210 is @xmath220 let @xmath210 be an @xmath221-round -algorithm computing @xmath214 . we prove that @xmath222 . consider the @xmath13-th round , with @xmath223 , and let @xmath224 be an arbitrary key in @xmath21 and @xmath225 . we denote with @xmath226 the space taken by the output of @xmath26 which contributes either to @xmath17 or to @xmath16 , and with @xmath25 the space needed to compute @xmath26 including the input and working space but excluding the output . clearly , @xmath227 , @xmath228 , and @xmath229 . suppose @xmath230 . by lemma [ lem : nummult ] , the reducer @xmath22 with input @xmath231 can compute at most @xmath232 elementary products since @xmath233 and @xmath234 , where @xmath215 and @xmath216 denote the entries of @xmath54 and @xmath55 used in @xmath26 and @xmath217 the entries of @xmath62 for which contributions are computed by @xmath26 . then , the number of terms computed in the @xmath13-th round is at most @xmath235 since @xmath236 and the summation is maximized when @xmath237 for each @xmath238 . suppose now that @xmath239 . partition the keys in @xmath21 into @xmath240 sets @xmath241 such that @xmath242 for each @xmath243 ( the lower bound may be not satisfied for @xmath244 ) . clearly , @xmath245 . by lemma [ lem : nummult ] , the number of elementary products computed by all the reducers @xmath26 with keys in a set @xmath246 is at most @xmath247 . since @xmath248 for each non negative assignment of the @xmath249 variables and since @xmath250 , it follows that at most @xmath251 elementary products can be computed using keys in @xmath246 , where @xmath252 . therefore , the number of elementary products computed in the @xmath13-th round is at most @xmath253 since @xmath254 and the sum is maximized when @xmath255 for each @xmath243 . therefore , in each round @xmath256 nonzero elementary products can be computed , and then @xmath257 . the second term of the lower bound follows since there is at least one entry of @xmath62 given by the sum of @xmath258 nonzero elementary products . we now specialize the above lower bound for algorithms for generic dense - dense and sparse - sparse matrix multiplication . an -algorithm for multiplying any two dense @xmath86 matrices , using conventional matrix multiplication , requires @xmath259 rounds . on the other hand , an -algorithm for multiplying any two sparse matrices with at most @xmath108 nonzero entries requires @xmath260 rounds . in the dense - dense case the lower bound follows by the above theorem [ th : lbddmult ] since we have @xmath261 and @xmath262 when @xmath263 . in the sparse - sparse case , we set @xmath264 and we observe that there exist assignments of the input matrices for which @xmath265 , and others where @xmath266 the deterministic algorithms for matrix multiplication provided in this section perform conventional matrix multiplication , and hence the above corollary applies . thus , the algorithm for dense - dense matrix multiplication described in section [ sec : ddmult ] is optimal for any value of the parameters . on the other hand , the deterministic algorithm d2 for sparse - sparse matrix multiplication given in section [ sec : dssmult ] is optimal as soon as @xmath267 , @xmath268 and @xmath11 is polynomial in @xmath12 . our matrix multiplications results can be used to derive efficient algorithms for inverting a square matrix and for solving several variants of the matching problem in a graph . the algorithms in this section make use of division and exponentiation . to avoid the intricacies of dealing with limited precision , we assume each memory word is able to store any value that occurs in the computation . a similar assumption is made in the presentation of algorithms for the same problems on other parallel models ( see e.g. @xcite ) . in this section we study the problem of inverting a lower triangular matrix @xmath54 of size @xmath269 . we adopt the simple recursive algorithm which leverages on the easy formula for inverting a @xmath270 lower triangular matrix ( * ? ? ? @xmath271^{-1 } = \left [ \begin{array}{cc } a^{-1 } & 0 \\ -c^{-1}ba^{-1 } & c^{-1 } \end{array } \right].\ ] ] for @xmath272 and @xmath273 , let @xmath274 be the @xmath275 submatrix resulting from the splitting of @xmath54 into submatrices of size @xmath276 . since equation holds even when @xmath277 are matrices , we have that @xmath278 can be expressed as in equation in figure [ fig : prodmatrinv ] . note that @xmath279 . @xmath280 , 0\leq i\leq 2^k-1.\ ] ] the -algorithm for computing the inverse of @xmath54 works in @xmath281 phases . let @xmath282 for @xmath283 . in the first part of phase @xmath79 , the inverses of all the lower triangular submatrices @xmath284 , with @xmath285 , are computed in parallel . since each submatrix has size @xmath98 , each inverse can be computed sequentially within a single reducer . in the second part of phase @xmath79 , each product @xmath286 for @xmath287 , is computed within a reducer . in phase @xmath13 , with @xmath288 , each term @xmath289 for @xmath290 , is computed in parallel by performing two matrix multiplications using @xmath291 aggregate memory and local size @xmath11 . therefore , at the end of phase @xmath292 we have all the components of @xmath293 , i.e. , of @xmath294 . [ thm : trianmatrinv ] the above algorithm computes the inverse of a nonsingular lower triangular @xmath269 matrix @xmath54 in @xmath295 rounds on an . the correctness of the algorithm follows from the correctness of which in turns easily follows from the correctness of the formula to invert a lower triangular @xmath296 matrix . from the above discussion it easy to see that the memory requirements are all satisfied . we now analyze the round complexity of the algorithm . at phase @xmath13 we have to compute @xmath297 products between matrices of size @xmath298 each product is computed in parallel by using @xmath299 aggregate memory and thus each phase @xmath13 requires @xmath300 rounds by using the algorithm described in section [ sec : ddmult ] . the cost of the lower triangular matrix inversion algorithm is then @xmath301 which gives the bound stated in the theorem . if @xmath302 is @xmath303 and @xmath101 for some constant @xmath3 , the complexity reduces to @xmath27 rounds , which is a logarithmic factor better than what could be obtained by simulating the pram algorithm . it is also possible to compute @xmath294 using the closed formula derived by unrolling a blocked forward substitution . in general , the closed formula contains an exponential number of terms . there are nonetheless special cases of matrices for which a large number of terms in the sum are zero and only a polynomial number of terms is left . this is , for instance , the case for triangular band matrices . ( note that the inverse of a triangular band matrix is triangular but not necessarily a triangular band matrix . ) if the width of the band is @xmath304 , then we have a polynomial number of terms in the formula . in this case we can do matrix inversion in constant rounds for sufficiently large values of @xmath11 and @xmath12 . a complete discussion of this method will be presented in the full version of the paper . building on the inversion algorithm for triangular matrices presented in the previous subsection , and on the dense - dense matrix multiplication algorithm , in this section we develop an -algorithm to invert a general @xmath269 matrix @xmath54 . let the trace @xmath305 of @xmath54 be defined as @xmath306 , where @xmath307 denotes the entry of @xmath54 on the @xmath110-th row and @xmath110-th column . the algorithm is based on the following known strategy ( see e.g. , ( * ? ? ? 8.8 ) ) . 1 . compute the powers @xmath308.[step : inv1 ] 2 . compute the traces @xmath309 , for @xmath310.[step : inv2 ] 3 . compute the coefficients @xmath311 of the characteristic polynomial of @xmath54 by solving a lower triangular system of @xmath156 linear equations involving the traces @xmath312 ( the system is shown below).[step : inv3 ] 4 . compute @xmath313[step : inv4 ] we now provide more details on the mr implementation of above strategy . the algorithm requires @xmath314 , which ensures that enough aggregate memory is available to store all the @xmath156 powers of @xmath54 . in step [ step : inv1 ] , the algorithm computes naively the powers in the form @xmath315 , @xmath316 , by performing a sequence of @xmath317 matrix multiplications using the algorithm in section [ sec : ddmult ] . then , each one of the remaining powers is computed using @xmath318 aggregate memory and by performing a sequence of at most @xmath319 multiplications of the matrices @xmath315 obtained earlier . in step [ step : inv2 ] , the @xmath156 traces @xmath312 are computed in parallel using a prefix like computation , while the coefficients @xmath311 of the characteristic polynomial are computed in step [ step : inv3 ] by solving the following lower triangular system : @xmath320 \left [ \begin{array}{c } c_{n-1 } \\ c_{n-1 } \\ c_{n-3 } \\ \vdots \\ c_0 \end{array } \right ] = - \left [ \begin{array}{c } s_1 \\ s_2 \\ s_3 \\ \vdots \\ s_n \end{array } \right].\ ] ] if we denote with @xmath321 the matrix on the left hand side , with @xmath62 the vector of unknowns , and with @xmath39 the vector of the traces on the right hand side , we have @xmath322 . in order to compute the coefficients in @xmath62 the algorithm inverts the @xmath269 lower triangular matrix @xmath321 as described in section [ sec : lowertrianinv ] , and computes the product between @xmath323 and @xmath39 , to obtain @xmath62 . finally , step [ step : inv4 ] requires a prefix like computation . we have the following theorem . [ thm : genmatrinv ] the above algorithm computes the inverse of any nonsingular @xmath269 matrix @xmath54 in @xmath324 rounds on , with @xmath314 . for the correctness of the algorithm see ( * ? ? ? it is easy to check that the memory requirements of the model are satisfied . we focus here on analyzing the round complexity . computing the powers in the form @xmath315 , @xmath316 requires @xmath325 rounds , since the algorithm performs a sequence of @xmath317 products . the remaining powers are computed in @xmath326 rounds since each power is computed by performing at most @xmath317 product using @xmath327 aggregate memory . the prefix like computation for finding the @xmath156 traces @xmath312 requires @xmath52 rounds , while the linear system takes @xmath328 rounds . the final step takes @xmath52 rounds using a prefix like computation . the round complexity in the statement follows . if @xmath302 is @xmath329 and @xmath101 for some constant @xmath3 , the complexity reduces to @xmath27 rounds , which is a quadratic logarithmic factor better than what could be obtained by simulating the pram algorithm . the above algorithm for computing the inverse of any nonegative matrix requires @xmath314 . in this section we provide an -algorithm providing a strong approximation of @xmath294 assuming @xmath53 . a matrix @xmath55 is a _ strong approximation _ of the inverse of an @xmath269 matrix @xmath54 if @xmath330 , for some constant @xmath331 . the norm @xmath332 of a matrix @xmath54 is defined as @xmath333 where @xmath334 denotes the euclidean norm of a vector . the condition number @xmath335 of a matrix @xmath54 is defined as @xmath336 . an iterative method to compute a strong approximation of the inverse of a @xmath269 matrix @xmath54 is proposed in ( * ? ? ? the method works as follows . let @xmath337 be a @xmath269 matrix satisfying the condition @xmath338 for some @xmath339 and where @xmath340 is the @xmath269 identity matrix . for a @xmath269 matrix @xmath62 let @xmath341 . we define @xmath342 , for @xmath343 . we have @xmath344 by setting @xmath345 where @xmath346 , we have @xmath347 @xcite . then , if @xmath348 for some constant @xmath349 , @xmath350 provides a strong approximation when @xmath351 . from the above discussion , it is easy to derive an efficient -algorithm to compute a strong approximation of the inverse of a matrix using the algorithm for dense matrix multiplication in section [ sec : ddmult ] . the above algorithm provides a strong approximation of the inverse of any nonegative @xmath352 matrix @xmath54 in @xmath353 rounds on an when @xmath348 for some constant @xmath349 . the correctness of the algorithm derives from @xcite . once again we only focus on the round complexity of the algorithm . computing @xmath354 requires a a constant number of prefix like computations , and hence takes @xmath52 rounds . to compute @xmath350 , @xmath343 from @xmath355 , we need the value @xmath356 which involves a multiplication between two @xmath269 matrices and a subtraction between two matrices . hence , each phase requires @xmath357 rounds . since the algorithm terminates when @xmath358 , the theorem follows . a strategy for computing , with probability at least 1/2 , a perfect matching of a general graph using matrix inversion is presented in @xcite . the strategy is the following : 1 . let the input of the algorithm be the adjacency matrix @xmath54 of a graph @xmath359 with @xmath156 vertices and @xmath224 edges . [ step : match1 ] 2 . let @xmath55 be the matrix obtained from @xmath54 by substituting the entries @xmath360 corresponding to edges in the graph with the integers @xmath361 and @xmath362 respectively , for @xmath363 , where @xmath364 is an integer chosen independently and uniformly at random from @xmath365 $ ] . we denote the entry on the @xmath110th row and @xmath366th column of @xmath55 as @xmath367.[step : match2 ] 3 . compute the determinant @xmath368 of @xmath55 and the greatest integer @xmath369 such that @xmath370 divides @xmath368.[step : match3 ] 4 . compute @xmath371 , the adjugate matrix of @xmath55 , and denote the entry on the @xmath110th row and @xmath366th column as @xmath372.[step : match4 ] 5 . for each edge @xmath373 , compute @xmath374 if @xmath375 is odd , then add the edge @xmath376 to the matching.[step : match5 ] an -algorithm for perfect matching easily follows by the above strategy . we now provide more details on the mr implementation which assumes @xmath314 . in step [ step : match2 ] , @xmath55 is obtained as follows . the algorithm partitions @xmath54 into square @xmath98 submatrices @xmath377 , @xmath378 , and then assigns each pair of submatrices @xmath379 to a different reducer . this assignment ensures that each pair of entries @xmath380 of @xmath54 is sent to the same reducer . consider now the reducer receiving the pair of submatrices @xmath381 and consider the set of pairs @xmath380 of @xmath54 such that @xmath360 , where @xmath382 , @xmath383 , and @xmath384 . for each of these pairs the reducer chooses a @xmath364 independently and uniformly at random from @xmath365 $ ] , and sets @xmath367 to @xmath361 and @xmath385 to @xmath362 . for all the other entries @xmath386 , the reducer sets @xmath387 . let @xmath388 , @xmath389 be the coefficients of the characteristic polynomial of @xmath55 , which can be computed as described in section [ sec : genmatrinv ] . steps [ step : match3 ] and [ step : match4 ] can be easily implemented since the determinant of @xmath55 is @xmath390 and @xmath391 finally , in step [ step : match5 ] , matrices @xmath55 and @xmath371 are partitioned in square submatrices of size @xmath98 , and corresponding submatrices assigned to the same reducer , which computes the values @xmath375 for the entries in its submatrices and outputs the edges belonging to the matching . the above algorithm computes , with probability at least 1/2 , a perfect matching of the vertices of a graph @xmath392 , in @xmath324 rounds on , where @xmath314 . the correctness of the algorithm follows from the correctness of @xcite and it is easy to see that the memory requirements of the model are satisfied . we focus here on the round complexity . from the above description , it is easy to see that the computation of @xmath55 and the @xmath364 s in step [ step : match2 ] only takes one round . steps [ step : match3 ] and [ step : match4 ] require the computation of the coefficients of the characteristic polynomial of @xmath55 , and so takes a number of rounds equal to the algorithm for matrix inversion described in section [ sec : genmatrinv ] , i.e. , @xmath393 . step [ step : match5 ] takes one round . since the round complexity is dominated by the number of rounds needed to compute the coefficients of the characteristic polynomial of @xmath55 , the theorem follows . we note that matching is as easy as matrix inversion in the model . the above result can be extend to minimum weight perfect matching , to maximum matching , and to other variants of matching in the same way as in ( * ? ? ? * sect . 5 ) . in this paper , we provided a formal computational model for the mapreduce paradigm which is parametric in the local and aggregate memory sizes and retains the functional flavor originally intended for the paradigm , since it does not require algorithms to explicitly specify a processor allocation for the reduce instances . performance in the model is represented by the round complexity , which is consistent with the idea that when processing large data sets the dominant cost is the reshuffling of the data . the two memory parameters featured by the model allow the algorithm designer to explore a wide spectrum of tradeoffs between round complexity and memory availability . in the paper , we covered interesting such tradeoffs for the fundamental problem of matrix multiplication and some of its applications . the study of similar tradeoffs for other important applications ( e.g. , graph problems ) constitutes an interesting open problem . the work of pietracaprina , pucci and silvestri was supported , in part , by miur of italy under project algodeep , and by the university of padova under the strategic project stpd08ja32 and project cpda099949/09 . the work of riondato and upfal was supported , in part , by nsf award iis-0905553 and by the university of padova through the visiting scientist 2010/2011 grant .

@xmath3 after totally geodesic @xmath4-spheres , clifford tori represent the next simplest minimal embeddings of closed surfaces in the round unit @xmath1-sphere @xmath5 . in fact marques and neves @xcite , in their proof of the willmore conjecture , have identified clifford tori as the unique area minimizers among all embedded closed minimal surfaces of genus at least one , and brendle @xcite has shown they are the only embedded minimal tori , affirming a conjecture of lawson . the first examples of higher - genus minimal surfaces in @xmath5 were produced by lawson himself @xcite , and further examples were found later by karcher , pinkall , and sterling @xcite . both constructions proceed by solving plateau s problem for suitably chosen boundary and extending the solution to a closed surface by reflection . in this article we carry out certain constructions by using gluing techniques by singular perturbation methods . one begins with a collection of known embedded minimal surfaces . these ingredients are then glued together to produce a new embedded surface , called the initial surface , having small but nonvanishing mean curvature introduced in gluing . the construction is successful when the initial surfaces are close to a singular limit . the construction is then completed by perturbing the surface to minimality without sacrificing embeddedness . of course the size of the mean curvature and the feasibility of perturbing the surface so as to eliminate it both depend crucially on the design of the initial surface . gluing methods have been applied extensively and with great success in gauge theories by donaldson , taubes , and others . in many geometric problems similar to the one studied in this article obstructions appear to solving the linearized equation . an extensive methodology has been developed to deal with this difficulty in a large class of geometric problems , starting with @xcite , @xcite and with further refinements in @xcite . we refer to @xcite for a general discussion of this gluing methodology and @xcite for a detailed general discussion of doubling and desingularization constructions for minimal surfaces . in this article , however , we limit ourselves to constructions of unusually high symmetry ( except in section [ further ] ) and this way we avoid these difficulties entirely . the first desingularization construction by gluing methods for minimal surfaces with intersection curves which are not straight lines was carried out in @xcite and serves as a prototype ( see for example @xcite ) for desingularizations of rotationally invariant surfaces with transverse intersections without triple or higher points . ( an independent construction by traizet @xcite has straight lines as intersections . ) for one earlier application of the gluing methodology in the context of minimal surfaces in @xmath5 see @xcite , where a `` doubling '' construction of the clifford torus is carried out ; this work has been extended in @xcite for `` stackings '' of the clifford torus and in @xcite for doublings of the equatorial two - sphere . the present construction also glues tori , but by desingularization rather than doubling . the idea of a desingularization construction for intersecting minimal surfaces in a riemannian three - manifold is to start with a collection of minimal surfaces intersecting along some curves and to produce a single embedded minimal surface by desingularizing the curves of intersection . assuming transverse intersection , this is accomplished , at the level of the initial surface , through the replacement of a tubular neighborhood of each component curve of the intersection set by a surface which on small scales approximates a minimal surface in euclidean space desingularizing the intersection of a collection of planes along a single line . in prior desingularization constructions the appropriate models for these desingularizing surfaces were furnished by the classical scherk towers of @xcite , which desingularize the intersection of two planes . one novelty of the present article is our use of the more general karcher - scherk towers , introduced in @xcite , which come in families desingularizing any number of intersecting planes and so accommodate curves of intersection whose complements , in small neighborhoods of the curves , contain more than four components . note that although having more than two minimal surfaces intersect along a curve is not a generic situation , it can happen in rotationally invariant cases as for example in the case of coaxial catenoids . extending the results of @xcite to such situations for example is an interesting but difficult problem because one would have to use the full family of karcher - scherk towers as studied in @xcite . a motivation for our construction is the observation ( * ? ? ? * section 2.7 ) that lawson s surfaces may be regarded as desingularizations of a collection of great @xmath4-spheres intersecting with maximal symmetry along a common equator . in this article we pursue analogous constructions with tori instead of spheres as proposed in ( * ? ? ? * section 4 , page 300 ) . pitts and rubinstein have described one class of surfaces ( item 10 on table 1 of @xcite ) , similar to some of our surfaces , to be obtained by min - max methods . recently choe and soret @xcite have produced examples by solving plateau s problem for a suitably selected boundary , in the spirit of @xcite . their examples resemble the simpler examples we construct . ( to prove they are the same one would have to prove that the solution of the plateau problem is unique ; see remark [ r : unique ] ) . our construction has been developed independently and is more general in ways we describe below , and our strategy is quite different , based as we already mentioned on gluing techniques by singular perturbation methods . on the other hand our methods work only for high - genus surfaces . to outline , we construct two infinite families of embedded minimal surfaces in @xmath5 . the first family consists of desingularizations of a configuration @xmath6 ( see [ e : wk ] ) of @xmath7 clifford tori intersecting symmetrically along a pair of disjoint great circles @xmath8 and @xmath9 which lie on two orthogonal two - planes in @xmath10 . the second family consists of desingularizations of a configuration @xmath11 ( see [ e : wk ] ) which is the previous one augmented by the clifford torus which is equidistant from @xmath8 and @xmath9 . in both cases the construction is based on choosing `` scaffoldings '' , that is unions of great circles contained in the given configuration , and which we demand to be contained in the minimal surfaces we construct . moreover , reflections with respect to the great circles contained in the scaffoldings are required to be symmetries of our constructions . we denote the scaffoldings we choose by @xmath12 or @xmath13 ( see [ scaff ] ) . to construct the initial surfaces we replace tubular neighborhoods of the intersection circles with surfaces modeled on appropriately scaled and truncated maximally symmetric karcher - scherk towers . towers with @xmath14 ends are used along @xmath8 and @xmath9 , while classical scherk towers with @xmath15 ends are used along other circles of intersection ( present only in @xmath11 ) . the replacements are made so that each initial surface is closed and embedded , contains the applicable scaffolding , and is invariant under reflection through every scaffold circle ( see definition [ initsdef ] ) . these initial surfaces are perturbed then to minimality in a way which respects the reflections , so the surfaces produced are closed embedded minimal and still contain the scaffolding ( see the main theorem [ mainthm ] ) . note that lawson s approach also makes use of a scaffolding . our approach , however , gives much more freedom in the number of handles we include in the fundamental domain , while in lawson s method the fundamental domain is a disc so that plateau s problem can be solved . this makes no difference when considering the original construction of lawson in @xcite : we expect that the construction with more handles in the fundamental domain will still produce a lawson surface even though it does not a priori impose all the symmetries of the surface . when there are more than two circles of intersection involved , however , we can choose different numbers of handles on each of them and this gives a plethora of new surfaces as in the present constructions ( see the main theorem [ mainthm ] ) . it seems also rather daunting to try to construct even the simplest of our surfaces desingularizing @xmath11 by lawson s method . the present constructions motivate two important new directions for further study . first , what other similar desingularization constructions can be carried out in cases where there are obstructions due to less symmetry ? one has to deal then with the obstructions in the usual way by introducing smooth families of initial surfaces with the parameters corresponding to the obstructions as in earlier work ( see @xcite ) . we discuss this question in section [ further ] and we provide some partial answers . second , as remarked in the end of ( * ? ? ? * section 4.2 ) , there are various natural questions about rigidity and uniqueness for the surfaces presently constructed which are similar to those asked ( * ? ? ? * questions 4.3 , 4.4 and 4.5 ) about the lawson surfaces . in particular we are currently working to prove that the surfaces desingularizing @xmath6 can not be smoothly deformed to surfaces desingularizing @xmath2 clifford tori still intersecting along @xmath8 and @xmath9 but with different angles ( that is they can not `` flap their wings '' ) . more precisely we hope to prove that even with reduced symmetries imposed so `` flapping the wings '' is allowed , there are no new jacobi fields on our surfaces . @xmath3 as we have already mentioned the main difficulty of this construction as compared to earlier results is proving that under the symmetries imposed there is no kernel on the karcher - scherk towers . as for the classical singly periodic scherk surfaces ( @xmath16 ) @xcite , this is achieved by subdividing the surface suitably and applying the montiel - ros approach @xcite . our approach is also somewhat different than usual in some other aspects and we employ the high symmetry we have available . @xmath3 in section 2 we study in sufficient detail the maximally symmetric karcher - scherk towers using the enneper - weirstrass representation and following @xcite . in section 3 we study the geometry of the clifford tori and the initial configurations we will be using later , their symmetries , and the symmetries and scaffoldings we will impose in our constructions later . in section 4 we discuss in detail the construction of the initial surfaces and we study their geometry . in section 5 we provide estimates for the geometric quantities on the initial surfaces . in section 6 we study the linearized equation and estimate its solutions on the initial surfaces . we finally combine these results to establish the main theorem in section 7 . finally in section 8 we discuss further results using more technology . given an open set @xmath17 of a submanifold immersed in an ambient manifold endowed with metric @xmath18 , an exponent @xmath19 , and a tensor field @xmath20 on @xmath17 , possibly taking values in the normal bundle , we define the pointwise hlder seminorm @xmath21_\alpha(x ) = \sup_{y \in b_x } \frac{{\left\lvertt(x)-\tau_{yx}t(y)\right\rvert}_g}{d(x , y)^\alpha},\ ] ] where @xmath22 denotes the open geodesic ball , with respect to @xmath18 , with center @xmath23 and radius the minimum of @xmath24 and the injectivity radius at @xmath23 ; @xmath25 denotes the pointwise norm induced by @xmath18 ; @xmath26 denotes parallel transport , also induced by @xmath18 , from @xmath27 to @xmath23 along the unique geodesic in @xmath22 joining @xmath27 and @xmath23 ; and @xmath28 denotes the distance between @xmath23 and @xmath27 . given further a continuous positive function @xmath29 and a nonnegative integer @xmath30 , assuming that @xmath20 is a section of the bundle @xmath31 over @xmath17 all of whose order-@xmath30 partial derivatives ( with respect to any coordinate system ) exist and are continuous , we set @xmath32_\alpha(x)}{f(x)},\ ] ] where @xmath33 denotes the levi - civita connection determined by @xmath18 . in case @xmath31 is the trivial bundle @xmath34 , instead of @xmath35 we write @xmath36 . when @xmath37 , we write just @xmath38 , and when @xmath39 , we write just @xmath40 . additionally , if @xmath41 is a group acting a on a set @xmath42 and if @xmath43 is a subset of @xmath42 , then we refer to the subgroup @xmath44 as the _ stabilizer _ of @xmath43 in @xmath41 . when @xmath43 is a subset of euclidean @xmath1-space ( or the round @xmath1-sphere ) , we will set @xmath45 where @xmath46 ( or @xmath47 ) . for a subset @xmath48 of euclidean space ( or the round @xmath1-sphere ) we define @xmath49 to be the group generated by reflections with respect to the lines ( or great circles ) contained in @xmath48 . here reflection through a great circle can be defined as the restriction to the @xmath1-sphere of reflection in @xmath10 through the @xmath4-plane containing the circle . if @xmath41 is a group of isometries of a riemannian manifold with a two - sided immersed submanifold @xmath50 , then we call @xmath51 _ even _ if @xmath52 preserves the sides of @xmath50 and _ odd _ if it exchanges them . in this two - sided case , sections of the normal bundle of @xmath50 may be represented by functions , on which @xmath53 then acts according to @xmath54 , where @xmath55 is defined to be @xmath24 for @xmath52 even and @xmath56 for @xmath52 odd . we append a subscript @xmath41 to the spaces of functions defined above to designate the subspace which is @xmath57-equivariant in this sense so that for instance @xmath58 finally , we make extensive use of cutoff functions , and for this reason we fix a smooth function @xmath59 $ ] with the following properties : a. @xmath60 is nondecreasing , b. @xmath61 on @xmath62 $ ] and @xmath63 on @xmath64 $ ] , and c. @xmath65 is an odd function . given then @xmath66 with @xmath67 , we define a smooth function @xmath68:{\mathbb{r}}\to[0,1]$ ] by @xmath69=\psi\circ l_{a , b},\ ] ] where @xmath70 is the linear function defined by the requirements @xmath71 and @xmath72 . clearly then @xmath68 $ ] has the following properties : a. @xmath68 $ ] is weakly monotone , b. @xmath68=1 $ ] on a neighborhood of @xmath73 and @xmath68=0 $ ] on a neighborhood of @xmath74 , and c. @xmath68+\psi[b , a]=1 $ ] on @xmath75 . the authors would like to thank richard schoen for his continuous support and interest in the results of this article . was partially supported by nsf grants dms-1105371 and dms-1405537 . in @xcite karcher introduced generalizations of the classical singly periodic scherk surfaces , including the maximally symmetric karcher - scherk towers of order @xmath0 , which we will denote by @xmath76 . @xmath76 is a singly periodic , complete minimal surface embedded in euclidean @xmath1-space , asymptotic to @xmath2 planes intersecting at equal angles along a line . this line , which we call the axis of @xmath76 , is parallel to the direction of periodicity . the classical singly periodic scherk tower @xcite asymptotic to two orthogonal planes is recovered by taking @xmath16 . although in this article we will only use @xmath76 in our constructions , it is worth mentioning that there is a continuous family of singly periodic minimal surfaces with scherk - type ends which has been studied by prez and traizet in @xcite and in which family @xmath76 is the most symmetric member . we proceed to outline the construction of karcher @xcite . note that @xmath76 , which we will define in detail later , differs by a scaling and rotation from the surface described now . karcher considered the enneper - weierstrass data of @xmath77 on the closed unit disc in @xmath78 punctured at the @xmath14^th^ roots of @xmath56 . the embedding defined by the data maps the origin to a saddle point , the @xmath2 line segments joining opposite roots of @xmath56 to horizontal lines of symmetry intersecting at equal angles , the @xmath14 radii terminating at roots of unity to alternately ascending and descending curves of finite length lying in @xmath2 vertical planes of symmetry , and the @xmath14 circumferential arcs between consecutive roots of @xmath56 to curves of infinite length lying alternately in one of two horizontal planes of symmetry . the union of this region with its reflection through either horizontal plane of symmetry is a fundamental domain for the tower under periodic vertical translation . the images of small neighborhoods of the roots of @xmath56 are asymptotic to vertical half - planes , which bisect the vertical planes of symmetry . for future reference the following proposition fills in the details of the above outline and summarizes the basic geometric properties of @xmath76 , including its symmetries and asymptotic behavior . to state the lemma we make a few preliminary definitions . first we define the sets @xmath79 , a union of horizontal planes , and @xmath80 , a union of vertical planes , by @xmath81 whose intersection is the union of horizontal lines @xmath82 we enumerate the connected components of the complement of @xmath83 by @xmath84 for each @xmath85 , and we also define a partition of @xmath86 into disjoint sets @xmath87 and @xmath88 given by @xmath89 to describe the symmetries of @xmath76 we write @xmath90 for reflection in @xmath91 through the plane @xmath92 , @xmath93 for translation in @xmath91 by @xmath74 units along the directed line @xmath30 , and @xmath94 for rotation in @xmath91 through angle @xmath95 about the directed line @xmath30 ( according to the usual orientation conventions ) . a horizontal bar over a subset of @xmath91 denotes its topological closure in @xmath91 , and angled brackets enclosing a list of elements ( or sets of elements ) of @xmath96 indicate the subgroup generated by all the elements listed or included in the sets mentioned . [ tower ] for every integer @xmath0 there is a complete embedded minimal surface @xmath97 such that a. @xmath98 , @xmath99 , and every straight line on @xmath76 is contained in @xmath100 ; b. for any connected component @xmath42 of @xmath87 the intersection @xmath101 is an open disc with @xmath102 ( the union of four horizontal rays ) ; c. for each non negative integer @xmath103 the quotient surface @xmath104 has @xmath14 ends and genus @xmath105 ; d. @xmath106 ( recall [ gsym ] and [ grefl ] ) and @xmath107 acts transitively on the set of connected components of @xmath108 , the set of connected components of @xmath87 , and the set of connected components of @xmath88 ; e. @xmath109 ; f. @xmath110 ; g. @xmath111 ; and h. there exists @xmath112 so that @xmath113 has @xmath14 connected components , all isometric by the symmetries , exactly one of which lies in the region @xmath114 and the intersection of this component with @xmath115 is the graph @xmath116 over the @xmath117-plane of a function @xmath118 , which decays exponentially in the sense that @xmath119 we have @xmath120 the usual inner - weierstrass recipe ( see @xcite for example or any standard reference for the classical theory of minimal surfaces ) defines from the data [ weidata ] a minimal surface in @xmath91 parametrized by the closed unit disc in @xmath78 punctured at the @xmath121 roots of @xmath56 . requiring the parametrization to take the origin in @xmath78 to the origin in @xmath91 , it takes the form @xmath122 where @xmath123 is the @xmath124^th^ @xmath14^th^ root of @xmath56 and @xmath125 is the imaginary part of the branch of the logarithmic function cut along the ray from @xmath126 to @xmath127 and taking the value @xmath126 at @xmath24 . the symmetries can be read from the data by the following standard argument . looking at the expression for the metric @xmath128 in terms of the enneper - weierstrass data [ weidata ] , one identifies as intrinsic geodesics both circumferential arcs on the unit circle and diametric segments joining opposite @xmath129 roots of @xmath130 . looking next at the expression for the second fundamental form @xmath131\ ] ] in terms of the enneper - weierstrass data , it becomes apparent that the diametric segments joining opposite @xmath14^th^ roots of @xmath56 are extrinsic geodesics as well , which are therefore mapped to euclidean lines , while the diametric segments joining opposite @xmath14^th^ roots of unity along with the circumferential arcs are lines of curvature , which , being also intrinsic geodesics , are necessarily mapped to planar curves . consultation of [ weimap ] confirms that ( a ) the straight lines lie along the intersection of the single horizontal plane @xmath132 with the vertical planes of the form @xmath133 for @xmath134 , ( b ) the images of the circumferential arcs lie , alternatingly , in the two horizontal planes @xmath135 , and ( c ) the images of the remaining lines of curvature lie , consecutively , in the vertical planes of the form @xmath136 for @xmath134 . below we will check that the parametrization [ weimap ] is in fact an embedding of the punctured unit disc . in fact we will show that the image of the punctured sector @xmath137 is embedded and intersects the planes @xmath132 , @xmath138 , @xmath139 , and @xmath140 only along the curves just mentioned . the reflection principle for harmonic functions then implies that this image may be extended to a complete embedded minimal surface @xmath141 , invariant under reflection through any line in @xmath142 and through any plane of the form @xmath143 or @xmath144 for @xmath145 . we define @xmath146 . items ( ii ) and ( iii ) and the first two claims of item ( i ) follow immediately , as do the containments @xmath147 , considering that item ( v ) and the equalities @xmath148 are clear from the definitions ( [ ck ] and [ ak ] ) of @xmath100 , @xmath87 , and @xmath149 alone . to see the containment @xmath150 note that any symmetry of the tower must permute the asymptotic planes , so must preserve their intersection , so must take any line in @xmath100 to a line orthogonally intersecting the @xmath151-axis ; we are assuming ( and confirm with the maximum - principle argument below ) that @xmath76 intersects the @xmath151-axis only where @xmath100 does , and from the expression ( [ tow2ff ] ) for the second fundamental form we know we have already accounted for all lines on @xmath76 through such points . the equalities @xmath152 are also immediate consequences of the definitions . in particular reflection through the @xmath132 plane preserves @xmath100 , but it does not preserve @xmath76 ( because , for example , the normal to @xmath76 is not constantly vertical along the lines @xmath153 ) , so @xmath154 . now , since @xmath155 has index @xmath4 in @xmath156 and @xmath157 , in fact @xmath158 . thus we have checked items ( ii)-(vii ) , as well as the first two claims of ( i ) , and the transitivity claim in ( iv ) is now obvious . to verify the remaining claim in ( i ) , note that any straight line in @xmath76 , by virtue of the latter s minimality , is a line of reflectional symmetry . on the other hand reflection through a straight line in @xmath91 preserves @xmath87 only if the line is the @xmath151-axis ( and then only for @xmath2 even ) or if it is contained in @xmath100 or @xmath159 . it is easy to see , however , that of these lines only those contained in @xmath100 lie on the surface . ( for example [ weidata ] and [ weimap ] reveal that at easily identified points where any of the other lines do intersect the surface the normal to the surface there is parallel to the line . ) in addition to checking ( viii ) , it remains to show that the image of the region @xmath33 is embedded and intersects the planes @xmath132 , @xmath138 , @xmath139 , and @xmath140 as described above . that the tower is embedded may be established by recognizing the conjugate surface of the region between two consecutive horizontal planes of symmetry as a graph specifically the solution to the jenkins - serrin problem @xcite on a regular @xmath14-gon and then appealing to a theorem of krust . see @xcite for details on this approach . alternatively we show embeddedness more directly as follows and in the process identify the intersection of the image of @xmath33 with these four planes . recall that @xmath33 is the punctured sector @xmath160 . from [ weimap ] it is clear that @xmath161 is positive along the radial segment from the origin to @xmath24 and vanishes along both the radial segment from the origin to @xmath162 and the circular arc from @xmath24 to @xmath162 . a sufficiently small circular arc , centered at @xmath163 , which originates on the segment from @xmath126 to @xmath162 and terminates on the circumferential arc from @xmath162 to @xmath24 , can be seen from [ weimap ] to have height monotonically increasing from @xmath126 to @xmath164 . the maximum principle then implies that the image of @xmath33 is contained in the slab @xmath165 and intersects @xmath132 only along the ( straight , horizontal ) image of the radial segment to @xmath162 and @xmath138 only along the ( horizontal ) image of the circumferential arc . similarly , from [ weimap ] one may readily check monotonicity of @xmath166 , @xmath167 , and @xmath168 on the boundary curves of @xmath33 in order to establish that the boundary has image contained in the wedge @xmath169 . moreover , [ weimap ] reveals that in @xmath33 @xmath170 = 0,\end{aligned}\ ] ] so that another application of the maximum principle ( to the harmonic coordinate functions @xmath171 , @xmath172 , and @xmath173 , the last extended to the closure of @xmath33 ) establishes that the image of @xmath33 is contained in @xmath174 and intersects @xmath139 only along the image of the radial segment to @xmath24 and @xmath140 only along the image of the radial segment to @xmath162 . because of the symmetries it therefore suffices to show that the enneper - weierstrass parametrization restricts to @xmath33 as an embedding . to this end observe first that each level curve of @xmath172 in @xmath33 is connected . indeed one can verify that @xmath167 vanishes nowhere on @xmath33 and has norm ( relative to @xmath175 ) tending to infinity at @xmath162 , so @xmath176 defines a smooth vector field on the closure of @xmath33 . the corresponding flow for time @xmath177 , when it exists , maps points ( other than the fixed point @xmath162 ) with @xmath178 to points with @xmath179 . examining the field at the boundary , one can check that the backward flow of a point in @xmath33 exits @xmath33 only after it reaches the segment joining @xmath126 and @xmath162 , while the forward flow leaves @xmath33 only through the real boundary segment . thus , if the flow for time @xmath180 of a point @xmath23 on the real segment from @xmath126 to @xmath24 lies in @xmath33 , then the flow for time @xmath177 of any real point to the right of @xmath23 will also lie in @xmath33 . now given @xmath181 with @xmath182 , the flow for time @xmath183 takes each point to a point on the real segment of @xmath33 ( since by the maximum principle and earlier monotonicity arguments this segment is the entirety of the @xmath184 curve in @xmath33 ) . by the previous considerations the flow for time @xmath185 exists for every point on the real segment joining these points , so , the flow for time @xmath185 being a continuous function of the initial point , we get a level @xmath178 path joining @xmath186 and @xmath187 . now suppose there exist points @xmath181 such that @xmath188 and @xmath189 . then there is a path contained in a single level curve of @xmath172 joining @xmath186 and @xmath187 , and , if @xmath186 and @xmath187 are distinct , by the mean value theorem there exists a third point @xmath190 between @xmath186 and @xmath187 on this path at which @xmath166 vanishes along the path . thus at @xmath190 the gradients @xmath191 and @xmath192 must be parallel . one finds from [ weimap ] that these gradients are parallel only on the circular boundary of @xmath33 , where they are tangential to the boundary , and therefore they are endpoints of level curves , so that we may assume @xmath190 is not such a point . thus we conclude that @xmath193 , showing not only that the enneper - weierstrass parametrization is an embedding but also that its image of the unit disc is actually a graph over a region in the @xmath132 plane . now we prove ( viii ) . it is clear from [ weimap ] that for @xmath194 sufficiently large the set of @xmath195 in the unit disc with @xmath196 has @xmath14 components , each containing exactly one of the @xmath14^th^ roots of @xmath56 . we define @xmath197 so that @xmath198 is the signed distance of the image of @xmath195 from the plane @xmath140 and @xmath199 the three terms on the right being pairwise orthogonal . we will show that for @xmath200 sufficiently large there is a correspondingly small neighborhood @xmath201 of @xmath162 in the closed unit disc such that the map @xmath202 defined by @xmath203 restricts to a diffeomorphism from @xmath201 onto the half - strip @xmath204 $ ] . working from [ weimap ] we find @xmath205 @xmath206 and @xmath207 where @xmath208 and @xmath209 and @xmath210 are defined by the equalities where they are introduced . then @xmath211 , and for each nonnegative integer @xmath30 there exists a constant @xmath212 such that @xmath213 defining the map @xmath214 by @xmath215 we see that for @xmath194 sufficiently large the composite @xmath216 is well - defined ( identifying @xmath78 with @xmath217 as usual ) and @xmath218 so , since @xmath219 and @xmath220 for each integer @xmath221 , by taking @xmath194 large enough we can ensure that @xmath216 is a small perturbation of the identity and so a diffeomorphism with image containing the half - strip @xmath204 $ ] for some @xmath222 . thus @xmath223 itself restricts to a diffeomorphism from some region @xmath201 onto this half - strip , as asserted above , which shows that the image of @xmath201 under the enneper - weierstrass parametrization is the graph of @xmath224 over the half - strip @xmath225 . since @xmath226 is smooth on @xmath201 , @xmath227 , and @xmath228 , we finally obtain the estimates @xmath229 the union @xmath100 of all horizontal lines on @xmath76 can be regarded as a scaffolding for the tower , but we emphasize that a tower is not uniquely determined by a choice of scaffold : [ towerremark ] ( i ) the surface @xmath230 satisfies all conditions in the lemma provided the roles of @xmath87 and @xmath88 are reversed in the statement , so in particular its intersection with @xmath79 is @xmath100 . \(ii ) for @xmath103 a positive integer the surface @xmath231 also has @xmath100 as its intersection with @xmath79 , but the quotient surface @xmath232 has genus @xmath233 instead of @xmath126 ( recall [ tower].iii ) . in this subsection we discuss the clifford tori and their geometry . we first introduce some helpful notation . given a great circle @xmath234 in @xmath5 we will write @xmath235 for the furthest great circle from it . ( note that the points of @xmath235 are at distance @xmath236 in @xmath5 from @xmath234 and any point of @xmath237 is at distance @xmath238 from @xmath234 ) . as viewed from @xmath10 , the planes containing @xmath234 and @xmath235 are orthogonal complements . on the other hand , @xmath234 and @xmath235 may be regarded as parallel in that the function on @xmath5 measuring distance from one of the circles is constant on the other . ( this relation of parallelism between two great circles in @xmath5 is not transitive . ) another useful characterization of @xmath235 identifies it as the set of poles of great two - spheres with equator @xmath234 . [ d : tori ] if @xmath234 and @xmath239 are as above , then we call them _ totally orthogonal . _ we define the clifford torus @xmath240 $ ] with `` axis - circles '' @xmath234 and @xmath235 to be the set of points in @xmath5 equidistant from @xmath234 and @xmath235 . @xmath240 $ ] can be alternatively defined as the set of points which are at a distance @xmath241 from @xmath234 , or equivalently at a distance @xmath241 from @xmath235 . clearly @xmath240={\mathbb{t}}[c^\perp]$ ] . the set of points at distance @xmath242 from @xmath240 $ ] is @xmath243 and the set of points at distance @xmath244 from @xmath240 $ ] is @xmath245 . @xmath240 $ ] is a flat , square , embedded torus , foliated by the circles , of radius @xmath246 in @xmath10 , where great two - spheres having @xmath234 as equator and poles on @xmath235 intersect @xmath247 , and also by the circles , orthogonally intersecting these , where great two - spheres having equator @xmath235 and poles on @xmath234 intersect @xmath247 in pairs on opposite sides of the equator . evidently any element of @xmath47 that preserves @xmath248 as a set is a symmetry of @xmath240 $ ] . the group of these symmetries includes arbitrary rotation or reflection in the two circles as well as orthogonal transformations exchanging the circles . to proceed further we have the following . [ d : rot ] given @xmath234 as above , @xmath249 , and assuming an orientation chosen on the totally orthogonal circle @xmath235 , we define @xmath250 , rotation about @xmath234 by angle @xmath95 , to be the element of @xmath251 preserving @xmath234 pointwise and rotating the totally orthogonal circle @xmath235 through an angle @xmath95 , according to the chosen orientation on @xmath235 . just as well @xmath250 may be called rotation in @xmath235 by angle @xmath95 . we assume now that orientations have been chosen for both @xmath234 and @xmath235 . ( of course , after orienting @xmath5 , an orientation on a circle @xmath234 determines an orientation on @xmath235 . ) we define then two @xmath252 subgroups of @xmath47 by @xmath253 each of whose elements rotates @xmath234 and @xmath235 simultaneously by a common angle , the two subgroups being distinguished by the relative sense of rotation in the circles . it is easy to see that any such one - parameter subgroup of @xmath47 , acting by common rotation in a pair of totally orthogonal circles , has only great circles as orbits . moreover by an easy calculation the orbits of @xmath254 intersect the orbits of @xmath255 at an angle equal to twice the distance from @xmath234 . consequently , @xmath240 $ ] itself is foliated by two such families of great circles , with the circles of one family intersecting the circles of the other orthogonally . more explicitly let @xmath256 $ ] be great circles through some given point @xmath257 $ ] with @xmath33 an orbit of @xmath254 and @xmath258 an orbit of @xmath255 . then one family consists of the images of @xmath33 under the action of @xmath255 and the other of the images of @xmath258 under the action of @xmath259 . reflection through @xmath33 ( that is @xmath260 ) preserves the great circles in @xmath240 $ ] which are orthogonal to @xmath33 . since these great circles foliate @xmath240 $ ] we conclude that the reflection @xmath260 is a symmetry of @xmath240 $ ] . it follows that any point of a clifford torus lies on a circle of reflection ( two in fact ) and this immediately implies the minimality of the torus ( since the symmetry allows the mean curvature nowhere to point ) . of course the two great circles through a point are asymptotic lines for the second fundamental form ; the short circles mentioned above latitudes @xmath242 from the poles of great two - spheres with equator one of the axis - circles bisect these and as such are circles of principal curvature . these circles have curvature @xmath261 in @xmath10 and @xmath24 in @xmath5 , showing that the second fundamental form of @xmath240 $ ] has constant norm @xmath261 . this also implies that there are no great circles on @xmath240 $ ] other than the orbits of @xmath262 as above , and therefore reflection through any great circle in @xmath240 $ ] is a symmetry of @xmath240 $ ] . given now @xmath33 and @xmath258 as above , @xmath263 rotates the points of @xmath33 to a parallel great circle along orthogonal geodesic segments parallel to @xmath258 . taking @xmath264 we conclude that @xmath265 , and similarly @xmath266 , are great circles which are parallel to @xmath33 and @xmath258 respectively on @xmath240 $ ] at distance @xmath236 . this implies that @xmath267 moreover , the elements of @xmath254 which were originally expressed as common rotations in @xmath234 and @xmath235 , are just as well common rotations in @xmath33 and @xmath268 ( since @xmath269 and @xmath270 obviously have the same action on @xmath33 and @xmath268 , which are orbits of @xmath271 and are the intersections with @xmath5 of two @xmath4-planes spanning @xmath10 ) and hence @xmath272 if we choose orientations on @xmath33 and @xmath268 appropriately ( or if we choose an orientation on @xmath5 ) . similarly @xmath273 ( since @xmath258 and @xmath274 are orbits of @xmath275 ) . by the above @xmath240 $ ] is ruled by two families of great circles and any great circle on @xmath240 $ ] belongs to one of the two families . any two circles in a single family are thereby not only parallel in @xmath240 $ ] but also parallel in @xmath5 in the sense of distance as defined above . moreover @xmath240 $ ] is generated by twisting about any one of its great circles @xmath33 an orthogonally intersecting great circle @xmath258 , where the twisting is common rotation in @xmath33 and @xmath268 : traversing @xmath33 , the conormal co - rotates with the position vector , tracing out @xmath33 and @xmath268 at equal rates . in this sense the clifford torus resembles the helicoid , but the helicoid is just singly ruled , while all of the clifford torus great circles are on equal footing . additionally , while each helicoid is either right or left - handed , every clifford torus is ambidextrous , being right - handed along the circles of one foliation and left - handed along the others . more precisely , given an orientation on @xmath5 , a clifford torus @xmath240 $ ] , and a great circle @xmath276 $ ] , we call @xmath33 right - handed if @xmath277={\mathbb{t}}[c]$ ] and left - handed if @xmath278={\mathbb{t}}[c]$ ] . furthermore , an element @xmath279 of @xmath47 fixing a great circle @xmath33 on @xmath240 $ ] pointwise but rotating @xmath268 along itself will of course preserve @xmath33 and @xmath268 as sets but rotate @xmath240 $ ] . such considerations reveal the existence of a one - parameter family @xmath280 \ , : \ , \phi \in { \mathbb{r}}\}$ ] of clifford tori , each intersecting @xmath240 $ ] transversely along @xmath33 and @xmath268 at a constant angle @xmath95 . since two great circles intersecting @xmath33 orthogonally can meet only on @xmath33 or @xmath268 ( unless they coincide everywhere ) , the ruling forbids distinct members of this family from intersecting anywhere else . more precisely for @xmath281 , @xmath282 = { \mathsf{r}}_d^{\phi_2}{\mathbb{t}}[c]$ ] when @xmath283 and @xmath282 \cap { \mathsf{r}}_d^{\phi_2}{\mathbb{t}}[c]= d\cup d^\perp$ ] otherwise . there is a second one - parameter family of clifford tori through @xmath33 and @xmath268 , having the opposite chirality along both , compared to the first family . this family can be obtained from the first by , for example , reflection through a great two - sphere containing @xmath33 ( or @xmath268 ) . a member of one family then intersects each member of the other family along two pairs of circles : @xmath33 and @xmath268 as well as two more circles , @xmath258 and @xmath274 , where @xmath33 and @xmath268 intersect @xmath258 and @xmath274 orthogonally . the angle of intersection varies ( from @xmath126 to @xmath284 ) along these circles , with tangency of the surfaces occurring at the eight points where the four circles intersect in pairs . note that by the above if @xmath33 is any great circle , then any clifford torus @xmath247 which contains @xmath33 contains also @xmath268 . moreover @xmath247 belongs to one of two families of clifford tori containing @xmath285 as described above . also for each point @xmath286 there is a unique torus in each family which contains it , and on that torus there is a unique great circle through @xmath287 parallel to @xmath33 and @xmath268 in the torus . additionally note that if @xmath288 $ ] , then , fixing a point @xmath289 , there is a point @xmath290 at distance @xmath242 from @xmath291 , and in fact @xmath287 must lie a distance @xmath242 from @xmath33 , since otherwise there would be a point on @xmath33 less than @xmath242 from @xmath234 , violating the definition of @xmath240 $ ] . on the other hand , @xmath33 is preserved by @xmath271 or @xmath275 while both subgroups act transitively on @xmath234 , so actually the distance from every point on @xmath234 to @xmath33 is @xmath242 . we conclude that whenever @xmath33 is a great circle on @xmath240 $ ] , @xmath234 is also a great circle on @xmath292 $ ] . now suppose two clifford tori @xmath240 $ ] and @xmath293 $ ] have intersection @xmath294 , where @xmath33 is a great circle . since @xmath276 $ ] and @xmath295 $ ] , it follows from the last observation in the previous paragraph that @xmath296 $ ] . moreover @xmath240 $ ] and @xmath293 $ ] must have the same chirality along @xmath33 ( since otherwise their intersection would be larger as above ) , so one torus can be obtained from the other by a rotation about @xmath33 , and therefore @xmath234 and @xmath297 ( and @xmath235 and @xmath298 ) must be parallel great circles on @xmath292 $ ] . conversely , if @xmath234 and @xmath297 are parallel great circles on @xmath292 $ ] , then @xmath297 can be obtained from @xmath234 by a rotation about @xmath33 , so @xmath293 $ ] is obtained from @xmath240 $ ] by the same rotation , and therefore @xmath240 $ ] and @xmath293 $ ] intersect along just @xmath33 and @xmath268 as described above . thus two clifford tori intersect along a single pair of totally orthogonal great circles precisely when all four of their axis - circles are parallel on the clifford torus equidistant from the intersection circles . moreover if @xmath240 $ ] and @xmath293 $ ] orthogonally intersect along great circles @xmath33 and @xmath268 , then each contains the axis - circles of the other : by definition of @xmath240 $ ] , geodesics emanating from @xmath247 orthogonally hit @xmath248 at distance @xmath242 from @xmath247 , but geodesics intersecting @xmath240 $ ] orthogonally at @xmath33 by assumption lie on @xmath293 $ ] . on the other hand two clifford tori @xmath240 $ ] and @xmath293 $ ] have intersection @xmath299 precisely when @xmath234 and @xmath297 intersect orthogonally , in which case @xmath235 and @xmath298 also intersect orthogonally and moreover @xmath300\cap{\mathbb{t}}[d']$ ] . indeed , suppose first that @xmath240 \cap { \mathbb{t}}[c']=d\cup d^\perp\cup d'\cup{d'}^\perp$ ] . since @xmath33 lies on @xmath301 $ ] and @xmath302 $ ] , @xmath234 and @xmath297 must be great circles on @xmath292 $ ] ; if @xmath234 and @xmath297 were parallel , then @xmath301 $ ] and @xmath302 $ ] would intersect transversely along @xmath33 and @xmath268 only , as in the preceding paragraph , so they must intersect orthogonally . by identical reasoning @xmath235 and @xmath298 must also intersect orthogonally and in fact all four axis - circles lie on @xmath303 $ ] as well as on @xmath292 $ ] , whose intersection is therefore precisely the union of these circles . conversely , if we assume @xmath234 and @xmath297 are two orthogonally intersecting great circles , then we can construct the two clifford tori @xmath247 and @xmath304 containing @xmath234 and @xmath297 ( so that @xmath305 ) and their corresponding axes @xmath294 and @xmath306 ; it follows from the previous three sentences ( exchanging the roles of all @xmath234 and @xmath33 circles ) that @xmath33 and @xmath258 as well as @xmath268 and @xmath274 intersect orthogonally and their union is all of @xmath301 \cap t[c']$ ] . starting with orthogonally intersecting axis - circles @xmath234 and @xmath297 as above , by varying the angle between @xmath234 and @xmath297 while fixing their two intersection points one obtains pairs of tori intersecting along ( noncircular ) curves with four points of tangency . the full space of intersections is two - dimensional , corresponding to the relative configuration of representative axis - circles . it can be parametrized by , for instance , the minimum and maximum distances from one of the axis - circles to points on the other . the case of tangency corresponds to the minimum vanishing , while the case of constant - angle intersection along a pair of orthogonal great circles corresponds to equality of the minimum and maximum . we remark finally that any two clifford tori necessarily intersect . this is a consequence of a general result of frankel @xcite , but for an elementary argument note that every single clifford torus divides @xmath5 into two connected components , which are open tubular neighborhoods of each axis - circle , so if two clifford tori failed to intersect , then one side of one of them would have to be properly contained in one side of the other , but the two sides have equal volume , precluding this situation . although in the previous subsection we discussed clifford tori in some generality , in this article we will only be concerned with finite maximally symmetric subcollections of a one - parameter family of tori intersecting along a pair of totally orthogonal great circles and the clifford torus equidistant from the circles of intersection . to fix the notation we start by taking @xmath5 to be the unit sphere in @xmath10 , to which we give its standard orientation and which we will routinely identify with @xmath307 via the map @xmath308 , so that @xmath309 . we write @xmath310 for the unit circles in the coordinate planes , oriented by increasing @xmath177 . as described in the previous subsection there is a unique clifford torus @xmath311={\mathbb{t}}[c_2 ] = \left\ { \frac{1}{\sqrt{2}}\left(e^{i{\ensuremath{\mathrm{x } } } } , e^{i{\ensuremath{\mathrm{y}}}}\right ) : { \ensuremath{\mathrm{x } } } , { \ensuremath{\mathrm{y}}}\in { \mathbb{r}}\right\}\ ] ] equidistant from @xmath8 and @xmath9 . there is also a one - parameter family of clifford tori containing @xmath8 ( so also @xmath9 ) and right - handed along it ( so also along @xmath9 ) ; we distinguish the one @xmath312 that contains the great circle @xmath313 in the real plane @xmath314 in @xmath307 ( see [ e : cphph ] for the notation ) . for each integer @xmath0 we intend to desingularize the configurations @xmath315 of @xmath2 and @xmath316 clifford tori respectively , where for each integer @xmath124 @xmath317 so that @xmath318 and @xmath319 for all @xmath145 . thus @xmath320 intersects @xmath321 at constant angle @xmath322 along @xmath8 and @xmath9 , while @xmath320 intersects @xmath304 orthogonally along the two totally orthogonal geodesics on @xmath320 equidistant to @xmath8 and @xmath9 . reflections through certain great circles will play an important role in the construction , so we name some of these circles now . first we define for each @xmath323 @xmath324 oriented by increasing @xmath151 . clearly for each @xmath325 @xmath326 so in total there are @xmath327 such great circles , pairwise disjoint ; @xmath328 consists of the @xmath14 great circles where @xmath304 intersects the other clifford tori in @xmath329 : @xmath330 and @xmath331 and @xmath332 are parallel on @xmath320 to @xmath8 and @xmath9 at a distance @xmath242 ; and for @xmath333 @xmath334 is the closer of two totally orthogonal great circles on @xmath304 equidistant from @xmath331 and @xmath335 . we also mention that @xmath336 = { \mathbb{t}}[c'_{j+3k/2}].\ ] ] next for @xmath337 we label the great circle orthogonally intersecting @xmath8 at @xmath338 and @xmath9 at @xmath339 by @xmath340 thus @xmath341 and @xmath342 are disjoint unless @xmath343 or @xmath344 is an integral multiple of @xmath345 , in which case they intersect only at two antipodal points on @xmath8 ( when @xmath346 ) , only at two antipodal points on @xmath9 ( when @xmath347 ) , or they coincide ; in particular @xmath348 for all @xmath349 . note also that @xmath350 where the intersections at the four points are orthogonal , and that @xmath320 is foliated by the disjoint great circles @xmath341 satisfying @xmath351 for @xmath352 . last we define for each @xmath353 the great circle @xmath354 oriented by increasing @xmath151 . note that the clifford torus @xmath304 is foliated by the disjoint great circles @xmath355 with @xmath356 , @xmath357 and the parameter @xmath358 measures the angle at the point @xmath359 between @xmath247 and @xmath360 , which is a clifford torus through @xmath8 and @xmath9 but left - handed along both . note also that the circles @xmath355 and @xmath331 intersect orthogonally at two points given by @xmath361 for future reference we first note the identity @xmath362 for any two great circles @xmath234 and @xmath297 and angles @xmath363 , as well as the particular products @xmath364 we now describe the symmetry groups of the above configurations . [ configsym ] ( i ) for @xmath0 @xmath365 in particular this group contains reflection through @xmath341 whenever @xmath366 , through @xmath331 for all @xmath367 , and through @xmath355 for all @xmath353 . \(ii ) @xmath368 for @xmath369 . \(iii ) @xmath370 . first suppose @xmath0 . it is clear that @xmath371 contains the group on the right - hand side of equation [ gsymwk ] . a symmetry of @xmath6 will either exchange the intersection circles @xmath8 and @xmath9 or will preserve each as a set . in particular it will preserve @xmath372={\mathbb{t}}[c_2]$ ] , so @xmath373 . moreover , each element of @xmath47 is completely determined by its action on @xmath8 and @xmath9 . if a symmetry of @xmath6 preserves @xmath8 , then it also preserves @xmath9 and it must act by rotation or reflection in each circle . an orthogonal transformation reversing the orientation of one circle but preserving the orientation of the other can not be a symmetry of @xmath6 ( since @xmath8 and @xmath9 are right - handed in all of the tori of the configuration and a reflection in just one of them reverses chirality ) . a symmetry acting by a reflection in one circle must therefore act by reflection in both circles ; in other words such a symmetry must be a reflection through a geodesic of @xmath5 orthogonally intersecting @xmath8 and @xmath9 . reflection through a geodesic orthogonal to both @xmath8 and @xmath9 will preserve the configuration precisely when the geodesic lies in @xmath374 , meaning on one of the tori or halfway between two tori . reflections through these geodesics clearly belong to the group on the right - hand side of [ gsymwk ] . any rotation in each circle preserving @xmath6 can be composed with a symmetry of the form @xmath375 to produce a symmetry fixing @xmath8 pointwise and rotating @xmath9 . clearly any such symmetry likewise belongs to the group on the right - hand side of [ gsymwk ] , so we have now accounted for all symmetries of @xmath6 preserving @xmath8 and @xmath9 separately . any transformation exchanging the two circles is the product of a transformation preserving them with the reflection through a geodesic equidistant from @xmath8 and @xmath9 . these are the great circles on the torus @xmath304 , and reflection through such a circle is a symmetry of the configuration precisely when the geodesic either orthogonally intersects all the other tori or else lies on either one of the other tori or halfway between a consecutive pair of these . all these reflections belong to the right - hand side of [ gsymwk ] too , so we have finished checking ( i ) . if @xmath369 , then any element of @xmath329 must permute the intersection circles @xmath8 and @xmath9 ( and can not exchange them with any intersection circles on @xmath304 ) and therefore must preserve @xmath6 , confirming ( ii ) . in the case @xmath16 , however , the three tori @xmath376 , @xmath377 , and @xmath304 are all equivalent under the symmetries , as are all six intersection circles @xmath8 , @xmath9 , @xmath378 , @xmath379 , @xmath380 , and @xmath381 . in particular @xmath382 belongs to @xmath383 and exchanges @xmath376 and @xmath304 . any symmetry of @xmath384 exchanging @xmath304 with either of the other clifford tori can be composed with @xmath382 and ( possibly ) an element of @xmath385 to obtain an element of @xmath385 , completing the proof of ( iii ) . the choices we are about to make to desingularize the initial configurations will break many of the symmetries just described , including in particular the continuous symmetries . nevertheless , to simplify the construction we will insist on retaining reflections through a collection of great circles which will be included in their entirety on the surfaces we construct and which serve as a sort of scaffolding for the construction . more precisely for any integers @xmath0 and @xmath386 we introduce the scaffoldings @xmath387 and corresponding groups @xmath388 motivation for the choice of scaffolds and corresponding symmetries can be found in the next section . for any integers @xmath0 and @xmath386 a. @xmath389 is the union of @xmath390 great circles , of which the @xmath391 circles @xmath392 with @xmath393 are parallel on @xmath321 ; b. @xmath394 is the union of @xmath395 great circles , of which the @xmath396 circles @xmath397 with @xmath398 are parallel on @xmath321 and the @xmath396 circles @xmath399 with @xmath398 are parallel on @xmath304 ; c. @xmath389 intersects each of @xmath8 and @xmath9 at the @xmath400 roots of unity , at each of which points exactly @xmath2 great circles in @xmath389 intersect , one on each @xmath321 ; d. @xmath394 intersects each of @xmath8 and @xmath9 at the @xmath401 roots of unity , at each of which points exactly @xmath2 great circles in @xmath394 intersect , one on each @xmath321 ; e. @xmath394 intersects each @xmath402 with @xmath403 at the @xmath404 points @xmath405 , with @xmath406 , at each of which exactly two great circles in @xmath394 intersect , namely @xmath407 on @xmath321 and @xmath408 on @xmath304 ; f. @xmath409 ; and g. @xmath410 . items ( i)-(v ) follow immediately from the definitions , as do the containments in the last two items . referring to [ symcomp ] , we observe @xmath411 completing the proof . we define @xmath412 by @xmath413 observe that @xmath414 takes planes of constant @xmath415 to clifford tori through @xmath8 and @xmath9 , cylinders of constant @xmath416 to constant - mean - curvature tori ( degenerating to @xmath8 or @xmath9 when @xmath416 is respectively an even or odd multiple of @xmath236 ) with axis - circles @xmath8 and @xmath9 , horizontal planes to great two - spheres with equator @xmath9 , and radial and vertical lines to great circles . in particular @xmath417 , @xmath418 , @xmath419 , @xmath420 , and @xmath421 . on the other hand the clifford torus left - handed along @xmath8 and containing the circle @xmath422 is the image under @xmath414 of the helicoid @xmath423 , and @xmath355 and @xmath424 are the images of the two helices where this helicoid intersects the cylinder @xmath425 . in particular @xmath426 . though great spheres do not play an important role in the construction , to aid understanding of the geometry of @xmath414 we also mention that the great sphere with equator @xmath8 and poles @xmath427 is the image of the helicoid @xmath428 . moreover , the open solid torus @xmath429 is mapped diffeomorphically by @xmath414 onto the open solid torus of points within @xmath430 of @xmath8the @xmath8 side of @xmath304and @xmath414 is an approximate isometry for small @xmath416 . more precisely , writing @xmath431 for the round metric on @xmath5 and @xmath432 for the euclidean metric on @xmath91 , we have @xmath433 we also note that @xmath414 intertwines some symmetries of interest : for every @xmath434 @xmath435 since @xmath414 maps the asymptotic planes of the karcher - scherk towers to clifford tori , we can smoothly glue the towers imported by @xmath414 to the tori of our configurations by straightening the wings to exact half - planes before applying @xmath414 . we accomplish this transition using the cut - off functions defined in [ epsiab ] . specifically , given integers @xmath0 and @xmath386 , we set @xmath436 and we define the map @xmath437 by modifying the inclusion map @xmath438 ( recall [ tower ] ) as follows . we will specify @xmath439 on the wedge @xmath440 and complete its global definition by requiring it to commute with all elements of @xmath107 . we assume that @xmath103 is large enough so that @xmath441 . the new map @xmath439 agrees with @xmath442 on @xmath443 . we recall from [ tower ] that the complement @xmath444 is the graph @xmath445 of the function @xmath446 over the half - plane @xmath447 and further that this graph misses the boundary of the wedge . we decree that @xmath448}({\ensuremath{\mathrm{x}}})w_k({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z}}})\ ] ] for each point @xmath449 in this region . imposing the symmetries as just described completes the definition of @xmath450 , and we also define its truncated image @xmath451 for any @xmath452 controlling the truncation . each initial surface @xmath453 desingularizing @xmath6 is specified by a quintuple of data @xmath454 , and each initial surface @xmath455 desingularizing @xmath11 is specified by a septuple @xmath456 , where @xmath0 and @xmath457 are integers and @xmath458 . the role of @xmath2 in the definition of the initial configurations is already clear : it is the number of clifford tori intersecting along @xmath8 and @xmath9 . along with @xmath103 it determines the scaffolding ( @xmath389 or @xmath394 ) according to [ scaff ] . we have also used @xmath103 in [ adef ] to set the distance from a tower s axis at which its wings are straightened . together with @xmath2 and @xmath103 the remaining positive integers ( either @xmath459 , @xmath460 , and @xmath461 or @xmath462 and @xmath463 ) specify the number of fundamental periods , or equivalently the scale , of the towers along each circle of intersection . finally , the relative alignment of towers along the various circles is prescribed by either @xmath464 or @xmath465 and @xmath466 . we will now explain the roles of the data in slightly more detail and simultaneously offer some brief motivation for the definitions of the scaffoldings ( [ scaff ] above ) and the initial surfaces ( [ initsdef ] below ) . the initial surfaces are to be constructed from the initial configurations they are intended to desingularize by replacing a tubular neighborhood of each intersection circle @xmath234 with a truncated karcher - scherk tower with straightened wings , scaled in @xmath91 by some factor @xmath467 , mapped into @xmath5 by @xmath414 , and then positioned as desired along @xmath234 by a rotation @xmath468 \in o(4)$ ] : @xmath469\phi \left ( { m_{_c}}^{-1 } \widetilde{\mathcal{s}}_{{k_{_c}},m}\left({m_{_c}}s_{_c}\right ) \right),\ ] ] where @xmath470 is the number ( either @xmath2 or @xmath4 ) of clifford tori intersecting along @xmath234 in the corresponding initial configuration and where @xmath471 is picked , somewhat arbitrarily , to ensure each tower is truncated well away from towers on neighboring circles . since @xmath76 has fundamental period @xmath472 and @xmath414 is periodic in @xmath151 with period @xmath472 ( each circle of intersection having length @xmath472 ) , obviously @xmath473 must be an integer in order for the initial surface to be embedded . further constraints on each @xmath473 are placed by the following symmetry requirements which we make of the initial surface to simplify the analysis of the linearized operator on the towers in section [ linear ] . [ sa ] let @xmath50 be an initial surface . we require @xmath474 and @xmath475 , and for each intersection circle @xmath234 in the corresponding initial configuration we require @xmath476 . the assumption ensures triviality of the kernel of each tower s jacobi operator restricted to the space of deformations respecting @xmath477 ; we do not claim that these conditions are necessary , but they are natural and sufficient . it is not hard to check ( using [ tower ] and [ symcomp ] for instance ) that imposing [ sa ] is equivalent to demanding @xmath478 and @xmath479 ( when @xmath50 desingularizes @xmath6 ) or @xmath480 and @xmath481 ( when @xmath50 desingularizes @xmath11 ) for some integer @xmath386 . in particular each tower @xmath482 must itself contain the appropriate scaffolding , forcing @xmath473 to divide @xmath391 ( when @xmath478 ) or @xmath396 ( when @xmath483 . since @xmath484 includes no symmetries exchanging @xmath8 and @xmath9 , the quotients @xmath485 and @xmath486 are independent and are given by @xmath460 and @xmath461 . on the other hand modulo @xmath487 @xmath8 and @xmath9 are equivalent to one another but to no other intersection circles , while for every @xmath145 the circles @xmath331 and @xmath488 are equivalent but @xmath331 and @xmath335 are inequivalent . thus only @xmath489 , @xmath490 , and @xmath491 can be independently prescribed as @xmath459 , @xmath462 , and @xmath463 respectively . the data so far described completely determine the periods and sizes of the towers replacing the circles of intersection , but this information and the particular scaffolding do not quite fix the initial surface @xmath50 , even up to congruence . ( we call two initial surfaces @xmath492 and @xmath493 _ congruent _ in @xmath5 if there exists @xmath494 such that @xmath495 . ) specifically there is some not entirely inconsequential freedom in the choice of @xmath468 $ ] in [ sigmac ] : it may be replaced by @xmath496 $ ] ; equivalently we could replace the tower @xmath76 defining @xmath497 by its `` dual '' tower mentioned in remark [ towerremark ] and having the same scaffolding and symmetry group as @xmath76 but occupying @xmath88 instead of @xmath87 . accordingly we allow the values of @xmath464 to make a choice of one model tower or the other along @xmath9 for initial surfaces desingularizing @xmath6 and we use @xmath465 and @xmath466 for identical purposes along the towers @xmath331 with @xmath124 odd and even respectively for initial surfaces desingularizing @xmath11 . thus these data control the _ alignment _ of the towers . it is not necessary to allow for the two possibilities on every ( inequivalent ) intersection circle , since up to congruence it is only the relative alignment that matters . in fact , even allowing realignment on just the circles mentioned , we still sometimes produce congruent initial surfaces with different values of @xmath464 ( or @xmath465 and @xmath466 ) , depending on the parities of the other data . for example , assuming @xmath460 and @xmath461 relatively prime ( since we may absorb a common divisor into @xmath103 ) , the surfaces @xmath498 and @xmath499 ( formally defined in [ initsdef ] below ) desingularizing @xmath6 are congruent precisely when either @xmath103 or exactly one @xmath500 is odd . indeed , if @xmath501 is odd , then @xmath502 if @xmath460 is even but @xmath461 odd , then @xmath503 and if @xmath460 is odd but @xmath103 and @xmath461 even , then @xmath504 to see that @xmath505 and @xmath506 are not congruent when @xmath103 is even and both @xmath460 and @xmath461 are odd , recall that each initial surface will be defined to include the scaffold circles whose union is @xmath507 , which orthogonally intersect @xmath8 and @xmath9 at the @xmath400 roots of unity on each . choosing the global normal which at @xmath359 points in the positive direction of @xmath8 then determines the direction of that normal at all @xmath400 roots of unity on @xmath8 and @xmath9 . we call the unit normal at such a point positive if it points in the positive direction of the circle on which it lies and we call it negative otherwise . then , given two such points joined by an arc of a great circle in @xmath389 , we say that the two normals there are aligned if they are either both positive or both negative and otherwise say that they are antialigned . the parities assumed for the data imply that , for a given initial surface with that data , this alignment does not depend on the pair of chosen points and therefore defines a property of the initial surface which is invariant under congruences that preserve @xmath389 , but it is reversed by altering the value of @xmath464 . this shows that @xmath505 is not congruent to @xmath506 . similar considerations apply to surfaces desingularizing @xmath11 , but to avoid complicating the definition of the initial surfaces we do not make a systematic effort to eliminate completely duplication of congruence classes within the collection of initial surfaces . with the foregoing in mind we define the initial surfaces as follows , recalling [ adef ] and [ stildekm ] . [ initsdef ] given integers @xmath0 , @xmath386 , and relatively prime @xmath508 , as well as @xmath509 , set @xmath510 given instead integers @xmath0 , @xmath511 , and relatively prime @xmath512 , as well as @xmath513 , set @xmath514 \right ) \right ) \\ & \;\;\ ; \cup \bigcup_{j=0}^{2k-1 } { \mathsf{r}}_{c_1}^{j\pi / k } \left({\mathsf{r}}_{c_1}^{\pi / kmn'_{(-1)^j } } { \mathsf{r}}_{c_2}^{\pi / kmn'_{(-1)^j}}\right ) ^{\sigma'_{(-1)^j } } { \mathsf{r}}_{c_{0,0}^{\pi/4}{\mathsf{r}}_{c_{\frac{\pi}{2},\frac{\pi}{2}}}^{\pi/4}}\phi \left(\frac{1}{2kmn'_{(-1)^j } } \widetilde{\mathcal{s}}_{2,m } \left(2kmn'_{(-1)^j}\frac{\pi}{4k}\right ) \right ) . \end{aligned}\ ] ] we will abbreviate the initial surfaces @xmath515 and @xmath516 by @xmath453 and @xmath455 respectively or sometimes indiscriminately by @xmath50 , when context permits . the divisibility assumptions are made to avoid listing a single initial surface multiple times under different labels , but as already acknowledged some redundancy persists in the list in that certain items are congruent to others . in such cases the resulting minimal surfaces ultimately produced will also be congruent . we next collect some basic properties of the initial surfaces . [ initsprop ] for every choice of data , assuming @xmath517 , the initial surfaces @xmath515 and @xmath518 are closed , smooth surfaces embedded in @xmath5 . moreover a. @xmath453 has genus @xmath519 ; b. @xmath455 has genus @xmath520 ; c. @xmath521 and @xmath522 ; and d. @xmath523 and @xmath524 . the closedness , smoothness , and embeddedness of the initial surfaces are clear from the definition and preceding discussion . for ( i ) , the components of @xmath525 may be grouped into pairs of consecutive ( in the sense of rotations about either circle ) components , and the two members of each pair may then be glued to each other along @xmath8 and @xmath9 to form @xmath2 new ( topological ) tori . we get the connected sum of these tori , a surface of genus @xmath2 , at the cost of one fundamental period of a tower along @xmath8 ( or @xmath9 ) . each additional fundamental period , of the towers along @xmath8 and @xmath9 , then contributes @xmath526 handles to the resulting surface . the genus of @xmath455 is similarly calculated . each portion of @xmath304 between two consecutive circles of intersection is glued to two portions of tori , both from alternately the @xmath8 or the @xmath9 side , orthogonally intersecting it , where these latter two are themselves glued along the circle where they intersect ( so either @xmath8 or @xmath9 ) . in this way we obtain @xmath14 topological tori , whose connected sum we take at the cost of one fundamental period for all but one tower on @xmath304 . each additional period of each of these towers contributes one handle to the resulting surface , while each period of each of the remaining two towers contributes @xmath526 handles . the great circles in the scaffoldings pass uninterrupted through the desingularized circles of intersection by virtue of the positioning and scaling in [ initsdef ] and the fact that @xmath414 maps the horizontal lines on the euclidean towers to great circles . that reflections through these geodesics belong to the stabilizers of the initial surfaces follows from the intertwining [ phintertwine ] by @xmath414 of symmetries of @xmath76 ( [ tower ] ) with symmetries of the configurations ( [ configsym ] ) to be desingularized . [ unitnormal ] since each initial surface is embedded in @xmath5 , it is also orientable and therefore possesses a unique global unit normal , henceforth denoted @xmath527 , which points in the positive direction ( meaning toward @xmath528 ) along @xmath8 at @xmath529 . we will write @xmath18 for the metric on each initial surface induced by its defining embedding in @xmath530 , @xmath43 for its @xmath527-directed scalar - valued second fundamental form , and @xmath531 for its @xmath527-directed scalar - valued mean curvature . before proceeding with the construction , we pause to elaborate briefly on the symmetry groups , that is the full stabilizers in @xmath47 , of the initial surfaces , in one particular class of highly symmetric cases . the groups presented in the above proposition are the minimum symmetry groups enforced throughout the construction , but in general each such group will be properly contained in the symmetry group of a given initial surface ( consistent with that group ) as well as of the corresponding final minimal surface . to illustrate , consider the initial surfaces @xmath532 of type @xmath453 with @xmath533 . whatever the values of @xmath2 , @xmath103 , and @xmath464 , the full symmetry group here will always contain reflection through not just @xmath534 for all @xmath535 ( already represented in @xmath484 ) but also @xmath536 for @xmath537 odd . when @xmath103 is odd , @xmath538 and @xmath539 are equivalent under ambient isometries ( as discussed in the alignment subsection immediately preceding [ initsdef ] ) , so we may assume @xmath540 . the full symmetry group then admits @xmath541 , excluded from @xmath484 , and therefore also @xmath542 for every odd @xmath124 , because ( see [ symcomp ] ) for every integer @xmath124 the product @xmath543 . ( when @xmath544 , one has instead reflection through @xmath331 for each even @xmath124 , because @xmath545 and by assumption @xmath546 is an odd multiple of @xmath547 ) . in this case there are no other circles of reflection on @xmath304 that are parallel to @xmath8 and @xmath9 through tori right - handed along them . indeed the only such circles through which reflection preserves @xmath6 are the @xmath548 for @xmath549 . that even values of @xmath124 are inadmissible follows from the last parenthetical remark . that half - integer values are inadmissible follows also from [ symcomp ] since @xmath550 takes for example @xmath551 to @xmath552 , since @xmath553 is a nonintegral multiple of the half - period @xmath554 when @xmath103 is odd there are , however , also circles of reflection orthogonal to the @xmath331 : since @xmath313 and @xmath555 are circles of reflection of @xmath538 ( for @xmath103 odd still ) , by [ symcomp ] @xmath556 is also a circle reflection , so by [ symcomp ] again @xmath557 is too for every @xmath145 . when @xmath103 is even , @xmath538 and @xmath539 are inequivalent . the first ( @xmath540 ) surface has reflectional symmetry through @xmath331 for every integer @xmath124 ( since @xmath558 , @xmath103 is even , and @xmath559 ) . again one also has @xmath557 as a circle of reflection for every @xmath145 . because @xmath550 preserves the initial surface if and only if @xmath553 is an odd number of half - periods , one has as well reflectional symmetry through @xmath331 for every half - integer @xmath124 precisely when @xmath103 is not divisible by @xmath15 . the second ( @xmath544 ) surface has symmetry group including @xmath560 and excluding @xmath542 for every integer @xmath124 . thus @xmath556 is not a circle of reflection , but instead @xmath561 is for every @xmath562 . when @xmath103 is divisible by @xmath15 the symmetry group also includes reflection through @xmath331 for every half - integer @xmath124 ( but for no integer @xmath124 ) , but when @xmath103 is not divisible by @xmath15 there are no circles of reflection on @xmath304 parallel to @xmath8 and @xmath9 through tori right - handed along them . note that this description of the initial surfaces @xmath532 with @xmath103 even is consistent with the properties of the surfaces constructed by choe and soret in @xcite ; specifically the genus and symmetries of the @xmath540 and @xmath544 surfaces match those of , respectively , the _ odd _ and _ even _ surfaces in @xcite . [ r : unique ] note that to prove that the surfaces we construct in [ mainthm ] of type @xmath453 with @xmath103 even and @xmath533 are the same surfaces found in @xcite , it is enough to prove the uniqueness of the solutions to the plateau problems in @xcite . although this seems very likely to be true , we do not have a proof at the moment . note also that in @xcite lawson claims uniqueness for the solution to his plateau problem . every initial surface is covered by certain open sets , which we call _ extended standard regions _ , of two types . regions of the first type are indexed by the circles of intersection . given a particular initial surface @xmath50 , for any circle @xmath234 of intersection in the corresponding initial configuration we define @xmath473 ( suppressing dependence on the given initial surface ) to be the number of fundamental periods of the tower in @xmath50 wrapped around @xmath234 , so that ( recall [ initsdef ] ) @xmath473 is the product of @xmath103 with the appropriate factor involving @xmath2 or @xmath4 as well as @xmath459 , @xmath460 , or @xmath461 , depending on @xmath50 and @xmath234 : @xmath563 we similarly define @xmath564 then , recalling [ adef ] , we let @xmath565:=\left\{p \in \sigma : d_{{g_{_s}}}(p , c ) < \frac{a}{{m_{_c}}}\right\},\ ] ] where @xmath566 denotes the distance in @xmath5 between @xmath287 and @xmath234 . this region is naturally identified with a truncated karcher - scherk tower @xmath567 via the map @xmath568 \phi \left(\frac{({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{y}}},{\ensuremath{\mathrm{z}}})}{{m_{_c}}}\right),\end{aligned}\ ] ] where @xmath468 $ ] is an element of @xmath251 chosen so that @xmath569 = \varphi_{c , m } \left ( { \mathcal{s}_{{k_{_c}}}}(a ) \right)$ ] . sometimes we may suppress the dependence on @xmath103 , writing simply @xmath570 . in turn we define @xmath571 $ ] by @xmath572 a new constant @xmath573 , to be determined later , dictates the extent of the second type of region . it will be chosen independently of @xmath103 but large enough so that each such region closely approximates a clifford torus . regions of the second type in an initial surface @xmath50 are indexed by the connected components of the complement of the circles of intersection in the initial configuration that @xmath50 desingularizes . given such a component @xmath20 , with boundary @xmath574 , we may assume if necessary by redefining @xmath575 by precomposition with a symmetry of @xmath576 and @xmath577 by precomposition with a symmetry of @xmath578that @xmath20 has inward unit conormals @xmath579 and @xmath580 along @xmath234 and @xmath33 . then , recalling [ tower ] and for any integer @xmath581 setting @xmath582}w_j({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) \in { \mathbb{r}}^3 : { \ensuremath{\mathrm{x}}}\in ( b , a+1 ] , \ , { \ensuremath{\mathrm{z}}}\in { \mathbb{r}}\},\ ] ] we define @xmath583 : = { \phi_{_c}}\left(h_{{k_{_c}}}\right ) \cup \phi_{_d}\left(h_{k_{_d}}\right ) \cup ( \sigma \cap t \cap \{d_{{g_{_s}}}(\cdot , c)>b/{m_{_c}}\ } \cap d_{{g_{_s}}}(\cdot , d)>b / m_{_d}\}).\ ] ] this region is naturally identified with @xmath584 via the map @xmath585 \to t_b , \text { defined by } \\ & \varpi_{_{t , m}}(p ) = \begin{cases } { \phi_{_c}}\circ \pi_{{\ensuremath{\mathrm{x}}}{\ensuremath{\mathrm{z } } } } \circ { \phi_{_c}}^{-1}(p ) \text { if } p \in { \phi_{_c}}\left(h_{{k_{_c}}}\right ) \\ \phi_{_d } \circ \pi_{{\ensuremath{\mathrm{x}}}{\ensuremath{\mathrm{z } } } } \circ \phi_{_d}^{-1}(p ) \text { if } p \in \phi_{_d } \left(h_{k_{_d}}\right ) \\ p \text { otherwise } , \end{cases}\end{aligned}\ ] ] where @xmath586 is euclidean orthogonal projection onto the @xmath117-plane . it is immediate that @xmath587 is well - defined and smooth . given an initial surface @xmath50 , we write @xmath588 for the collection of circles of intersection in the corresponding initial configuration and @xmath589 for the collection of components of the complement of @xmath590 in the initial configuration . then @xmath591 \cup \bigcup_{t \in { \mathcal{t}}(\sigma ) } s[t]$ ] , the members of @xmath592 : c \in { \mathcal{c}}(x)\}$ ] are pairwise disjoint , the members of @xmath593 : t \in { \mathcal{t}}(\sigma)\}$ ] are pairwise disjoint , and @xmath569 \cap s[t ] = \emptyset$ ] unless @xmath594 . using the diffeomorphisms just defined , the next two propositions compare the extended standard regions , as embeddings in @xmath595 , to standard karcher - scherk towers and planes in euclidean space . [ stdtor ] let @xmath50 be an initial surface and @xmath596 a toral component . write @xmath597 for the flat metric on @xmath20 , and for each point @xmath598 let @xmath599 denote the distance from @xmath287 to the boundary circles of @xmath600 . then for every nonnegative integer @xmath30 there exists a constant @xmath601independent of @xmath103such that a. @xmath602^{\otimes 2},\ ; m^2\varpi_t^*g_{_t},\ ; e^{-m\varpi_t^*d_{{g_{_s}}}(\partial t_b,\cdot ) } \right)\right\| } \leq c(\ell)$ ] ; b. @xmath603,\ ; m^2\varpi_t^*g_{_t},\ ; e^{-m\varpi_t^*d_{{g_{_s}}}(\partial t_b,\cdot ) } \right)\right\| } \leq c(\ell)m^{-1}$ ] ; and c. @xmath604,\ ; m^2\varpi_t^*g_{_t},\ ; e^{-m\varpi_t^*d_{{g_{_s}}}(\partial t_b,\cdot ) } \right)\right\| } \leq c(\ell)m^{-1}$ ] . we select a boundary circle @xmath234 of @xmath20 and a rotation @xmath468 \in so(4)$ ] so that @xmath605 \phi(\{{\ensuremath{\mathrm{y}}}=0,\;x > 0\})$ ] . it suffices to establish the estimates within @xmath606 . using [ phullback ] we find @xmath607^*{g_{_s}}= & d{\ensuremath{\mathrm{x}}}^2 + d{\ensuremath{\mathrm{y}}}^2 + d{\ensuremath{\mathrm{z}}}^2 + \frac{\sin^2 { \ensuremath{\mathrm{r}}}- { \ensuremath{\mathrm{r}}}^2}{{\ensuremath{\mathrm{r}}}^4 } \left ( { \ensuremath{\mathrm{y}}}^2\ , d{\ensuremath{\mathrm{x}}}^2 + { \ensuremath{\mathrm{x}}}^2\,d{\ensuremath{\mathrm{y}}}^2 - 2{\ensuremath{\mathrm{x}}}{\ensuremath{\mathrm{y}}}\,d{\ensuremath{\mathrm{x}}}\,d{\ensuremath{\mathrm{y}}}\right ) \\ & + 2\frac{\sin^2 { \ensuremath{\mathrm{r}}}}{{\ensuremath{\mathrm{r}}}^2 } \left ( { \ensuremath{\mathrm{x}}}\,d{\ensuremath{\mathrm{y}}}\,d{\ensuremath{\mathrm{z}}}- { \ensuremath{\mathrm{y}}}\,d{\ensuremath{\mathrm{x}}}\,d{\ensuremath{\mathrm{z}}}\right ) , \end{aligned}\ ] ] whose components and whose inverse s components have ( coordinate ) derivatives of all orders bounded on @xmath608^{-1}\mathcal{u } \subset \{\sqrt{{\ensuremath{\mathrm{x}}}^2+{\ensuremath{\mathrm{y}}}^2}<\pi/4\ } \subset { \mathbb{r}}^3 $ ] . moreover @xmath609 identifies @xmath610 $ ] as a graph over @xmath20 so that @xmath611 \cap \mathcal{u } = { \mathsf{r}}[c]\phi\{({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) \ ; : \ ; { \mathsf{r}}[c]\phi({\ensuremath{\mathrm{x}}},0,{\ensuremath{\mathrm{z } } } ) \in \varpi_t^{-1}\left(s[t ] \cap \mathcal{u}\right)\ } , \mbox { where } \\ & f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z } } } ) = { m_{_c}}^{-1}w_{{k_{_c}}}({m_{_c}}{\ensuremath{\mathrm{x}}},{m_{_c}}{\ensuremath{\mathrm{z } } } ) { \psi\left [ a+1,a \right]}({m_{_c}}{\ensuremath{\mathrm{x } } } ) , \end{aligned}\ ] ] recalling @xmath612 from [ tower ] . from [ towerdecay ] we have for any nonnegative integers @xmath124 and @xmath30 the existence of a constant @xmath613 ensuring the estimate @xmath614 for any @xmath615 $ ] . now , via @xmath468\phi$ ] , the coordinates @xmath616 on @xmath91 transfer to @xmath617 and the functions @xmath618 restrict to coordinates on @xmath20 , so that @xmath619 and @xmath620_{ij}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z } } } ) = & \left[{g_{_s}}\right]_{ij}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) + f_{,i}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\left[{g_{_s}}\right]_{i{\ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) \\ & + f_{,j}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\left[{g_{_s}}\right]_{j{\ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) + f_{,i}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})f_{,j}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\left[{g_{_s}}\right]_{{\ensuremath{\mathrm{y}}}{\ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) , \end{aligned}\ ] ] whence follows the estimate ( i ) for the metric , in light of [ fest ] and the boundedness of all ( coordinate ) derivatives of all components of @xmath431 and its inverse with respect to the @xmath616 coordinate system as established in [ gsph ] . assuming the normal @xmath527 on @xmath610 $ ] has positive inner product with @xmath621 , we calculate also @xmath622^{{\ensuremath{\mathrm{y}}}{\ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) - f_{,k}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\left[{g_{_s}}\right]^{k { \ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) + f_{,k}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})f_{,\ell}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\left[{g_{_s}}\right]^{k \ell}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) \right)^{-\frac{1}{2 } } \cdot \\ & [ \gamma_{ij}^{{\ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) + f_{,ij}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z } } } ) + f_{,j}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\gamma_{i{\ensuremath{\mathrm{y}}}}^{{\ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) \\ & + f_{,i}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\gamma_{j{\ensuremath{\mathrm{y}}}}^{{\ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) + f_{,i}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})f_{,j}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\gamma_{{\ensuremath{\mathrm{y}}}{\ensuremath{\mathrm{y}}}}^{{\ensuremath{\mathrm{y}}}}({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) \\ & - f_{,k}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\gamma_{ij}^k({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) - f_{,k}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})f_{,j}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\gamma_{i{\ensuremath{\mathrm{y}}}}^k({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) \\ & - f_{,k}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})f_{,i}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})\gamma_{j{\ensuremath{\mathrm{y}}}}^k({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) - f_{,k}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})f_{,i}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}})f_{,j}({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z } } } ) \gamma_{{\ensuremath{\mathrm{y}}}{\ensuremath{\mathrm{y}}}}^k({\ensuremath{\mathrm{x}}},f({\ensuremath{\mathrm{x}}},{\ensuremath{\mathrm{z}}}),{\ensuremath{\mathrm{z } } } ) ] , \end{aligned}\ ] ] where @xmath623 and each instance of @xmath624 is a christoffel symbol of @xmath431 in the @xmath616 coordinate system . noting that the squared norm of the second fundamental form of @xmath20 is simply @xmath4 and that @xmath20 is minimal , we obtain ( ii ) and ( iii ) , using again the estimates [ fest ] and the boundedness exhibited by [ gsph ] . [ stdtow ] let @xmath50 be an initial surface and @xmath625 an intersection circle . write @xmath626 for the second fundamental form of the inclusion @xmath627 relative to the rescaled euclidean metric @xmath628 and the unit normal whose pushforward by @xmath629 has positive @xmath431 inner product with @xmath527 , and set @xmath630 . then for each nonnegative integer @xmath30 there exists a constant @xmath601independent of @xmath103such that a. @xmath631^{\otimes 2},\ ; m^2g\right)\right\| } \leq c(\ell)m^{-1/2}$ ] ; b. @xmath632 , \ ; m^2g)\right\| } \leq c(\ell)m^{-1/2}$ ] ; and c. @xmath633 . from [ gsph ] @xmath634 since the second fundamental form of @xmath76 is bounded , as is each of its covariant derivatives , we obtain @xmath635 , \ , m^2 g , \ , m^{-1}+d_{{g_{_s}}}(c,\cdot ) ) \right\| } \leq c(\ell ) , \\ & { \left\| m^{-2}{\left\lverta\right\rvert}^2 - { { x_{_c}}^{-1}}^*{\left\lvert{{\widehat{a}}}_{_c}\right\rvert}^2 : c^\ell(s[c ] , \ , m^2 g , \ , m^{-1}+d_{{g_{_s}}}(c,\cdot ) ) \right\| } \leq c(\ell ) , \mbox { and } \\ & { \left\| m^{-1}h : c^\ell(s[c ] , \ , m^2 g , \ , m^{-1}+d_{{g_{_s}}}(c,\cdot ) ) \right\| } \leq c(\ell ) , \end{aligned}\ ] ] establishing the estimates of the proposition on the subset of @xmath569 $ ] within the tubular neighborhood of center @xmath234 and @xmath431-radius @xmath636 . by assuming @xmath637 , we ensure that the complement of this subset falls under the regime of the preceding proposition , which in conjunction with the asymptotic geometry of @xmath576 itself completes the proof , under the further assumption that @xmath638 . given an initial surface @xmath50 , embedded in @xmath5 by @xmath639 , along with a function @xmath640 with sufficiently small @xmath641 norm , and recalling ( [ unitnormal ] ) the choice @xmath642 of global unit normal , we define the immersion @xmath643 where @xmath644 is the exponential map on @xmath530 . write @xmath645 for the global unit normal on @xmath646 which has nonnegative inner product with the velocity field for the geodesics generated by @xmath527 and write @xmath647 $ ] for the scalar mean curvature of @xmath646 relative to @xmath645 . write @xmath41 for ( recalling [ symm ] ) @xmath484 when @xmath50 is type @xmath453 and for @xmath487 when @xmath50 is type @xmath455 . the main theorem will be proven by selecting a solution @xmath648 ( the space of smooth @xmath41-odd functions on @xmath50recall [ invfun ] ) to @xmath647=0 $ ] , small enough that @xmath646 is an embedding . to that end we next study the linearization @xmath649 at @xmath126 of @xmath650 , given by @xmath651 = \left(\delta + { \left\lverta\right\rvert}^2 + 2\right)u,\ ] ] where the constant term @xmath4 arises as the ricci curvature of @xmath431 contracted twice with @xmath527 . actually , to secure bounds uniform in @xmath103 and to facilitate the comparison of this operator on the extended standard regions to certain limit operators , we focus on @xmath652 in view of the estimates of the second fundamental form contained in [ stdtor ] and [ stdtow ] , this operator is bounded for any @xmath653 . the present section is devoted to obtaining a bounded inverse by first analyzing the operator `` semilocally''meaning when restricted to spaces of functions defined on each of the various extended standard regions and by afterward applying an iteration scheme to piece together a global solution . we first solve the jacobi equation for the inclusion map @xmath438 of the exact karcher - scherk towers of standard size , given data with sufficiently small support . to avoid the introduction of substitute kernel needed in more complicated gluing constructions , we impose the symmetries @xmath41 induces on the limit tower . specifically , set @xmath654 , write @xmath655 for the second fundamental form of @xmath438 , and for each positive integer @xmath459 let @xmath656 then the jacobi operator @xmath657 defines a bounded linear map @xmath658 for any @xmath653 and integers @xmath0 and @xmath659 . given @xmath660 we recall that @xmath661 and we set @xmath662 [ towsol ] fix @xmath653 , @xmath660 , and integers @xmath0 and @xmath659 . then there exists a linear map @xmath663 and there exists a constant @xmath234depending on just @xmath664 , @xmath73 , @xmath2 , and @xmath459such that for any @xmath223 in the domain of @xmath665 we have @xmath666 and @xmath667 the proof will be completed by ( i ) introducing a conformal metric @xmath668 , ( ii ) establishing that the schrdinger operator @xmath669 acting between sobolev spaces defined with respect to this metric has discrete spectrum omitting @xmath126 , ( iii ) extracting a @xmath641 bound for solutions , and finally ( iv ) applying schauder estimates . for @xmath369 the tower @xmath76 has umbilic points where the gauss map takes vertical values , so the pullback @xmath670 of the round spherical metric @xmath431 by the gauss map , as applied in @xcite and other constructions , degenerates there . instead we will pull back the spherical metric by a different map . recall that above we studied the enneper - weierstrass representation [ weimap ] on the unit disc , which , after a similarity transformation , parametrized a half - period of @xmath76 . in fact we observe that [ weimap ] extends to a diffeomorphism @xmath671 from the extended complex plane @xmath672 punctured at the @xmath14 roots of @xmath56 to the corresponding tower modulo vertical translation by @xmath673 . moreover , the inverse extends to a covering map @xmath674 of the punctured extended plane by the full tower . by composing with the appropriate rotation , scaling , and the inverse of stereographic projection @xmath675 , we obtain a smooth covering @xmath676 ( the gauss map @xmath677 is then just @xmath678 . ) referring further to the enneper - weierstrass data [ weidata ] we deduce ( see for example @xcite or any standard reference for the classical theory of minimal surfaces ) @xmath679 so since @xmath680 we find @xmath681 so that the conformal factor @xmath682 in front of @xmath683 in [ eta ] and its reciprocal @xmath684 are bounded on the inverse image under @xmath685 of every compact subset of @xmath686 . also from [ weidata ] we find @xmath687 whence @xmath688 so that the potential term of @xmath669 is a smooth function with absolute value bounded on all @xmath76 . note that pullback @xmath689 of the conformal factor in [ eta ] is even under reflection through every line through opposite @xmath14^th^ roots of @xmath56 ; therefore ( recall the discussion of the symmetries in the proof of [ tower ] ) @xmath682 itself is in particular @xmath690-even , and so @xmath691 , like @xmath692 , takes @xmath690-odd functions to @xmath690-odd functions . for each nonnegative integer @xmath30 write @xmath693 for the sobolev space consisting of all @xmath690-odd ( in the distributional sense ) measurable functions whose weak covariant derivatives up to order @xmath30 , with respect to @xmath694 , have squared norms with finite integrals on the quotient @xmath695 ; define the @xmath693 norm of such a function to be the square root of the sum , from order @xmath126 to order @xmath30 , of these integrals . although @xmath76 is not compact ( nor is the quotient @xmath696 ) , @xmath697 is nevertheless a union of closed round hemispheres punctured on their equators : @xmath698 where the overline indicates topological closure in @xmath76 and @xmath699 is the open region on @xmath76 between two consecutive horizontal planes of symmetry ; each @xmath700 is isometric to an open round hemisphere of radius @xmath24 . thus , from a sequence bounded in @xmath701 , by applying the rellich - kondrashov lemma successively to @xmath702 contiguous such hemispheres and bearing in mind the @xmath690 equivariance , we can extract a subsequence converging in @xmath703 . using also the boundedness of @xmath704 and a standard application of the riesz representation theorem for hilbert spaces , we conclude that @xmath691 has discrete spectrum . in proposition [ nokernel ] below we show moreover that @xmath691 has trivial kernel . now , given @xmath705 , we have @xmath706 with @xmath707 , so by proposition [ nokernel ] there exists @xmath708 weakly solving @xmath709 and satisfying the estimate @xmath710 . by standard elliptic regularity theory and the bounded geometry of @xmath711 in fact @xmath712 with @xmath713 for each @xmath714 , where @xmath715 is the open ball with center @xmath287 and @xmath683 radius @xmath716 . next , from the bochner formula together with the divergence theorem , the recognition that the compactly supported smooth functions are dense in @xmath701 , and the equality @xmath717 , one secures the further estimate @xmath718 now , given @xmath719 and @xmath714 , there is a sector @xmath720 of a spherical cap ( relative to @xmath694 ) , with center @xmath287 , vertex angle @xmath242 , and radius @xmath721 , entirely contained in @xmath76 ( so missing the roots of unity at the equator ) . then , writing @xmath722 for the @xmath694 geodesic through @xmath287 , parametrized by arc length @xmath723 from @xmath287 , and with initial angle @xmath415 measured from one edge of @xmath724 to the other , we have @xmath725}(\gamma_\theta(s))v(\gamma_\theta(s ) ) ) \ , ds = \int_0^{1/4 } s\frac{d^2}{ds^2}({\psi\left [ 1/4,1/8 \right]}(\gamma_\theta(s))v(\gamma_\theta(s ) ) ) \ , ds , $ ] using the fundamental theorem of calculus and integrating by parts . integrating in @xmath415 from @xmath126 to @xmath242 and applying the cauchy - schwarz inequality yields the simple morrey - sobolev inequality @xmath726 this estimate , in conjunction with [ h2est ] and [ schauder ] , completes the proof . [ nokernel ] the operator @xmath691 acting on @xmath727 , as defined in the proof of proposition [ towsol ] , has trivial kernel . to show that @xmath691 has trivial @xmath727 kernel we will first count its nullity on the somewhat larger domain @xmath728 , where @xmath729 is the subgroup of @xmath690 having the same generators save the reflections through lines , which are excluded this count is performed by adapting the variational proofs given by montiel and ros for lemma 12 and lemma 13 in @xcite . there they calculate the multiplicity of the eigenvalues of the laplacian on the round sphere as eigenvalues of the laplacian on certain coverings of the sphere . here we are interested in the multiplicity of @xmath126 only , for the more complicated operator @xmath691 . recalling [ omegadef ] we see that @xmath730 . write @xmath731 for the space of restrictions to @xmath732 of elements of @xmath733 and write @xmath734 for the closure in @xmath733 of the space of smooth functions compactly supported in @xmath732 and invariant under the ( rotational ) symmetries of @xmath735 that preserve @xmath732 . we define the bilinear form @xmath736 on @xmath737 @xmath738 : = -(du , dv)_{l^2(\eta)}+\left(e^{-2\phi}{\left\lvert{{\widehat{a}}}\right\rvert}^2u , v\right)_{l^2(\eta)}.\ ] ] then mimicking @xcite , we define the subspace @xmath739 of @xmath733 by @xmath740 = \lambda ( u , v)_{l^2(\eta ) } \right\rangle \\ & \,\,\ , \oplus \ , \bigoplus_{j=2}^{2n } \left\langle \left . h^1_{0 , { { \widehat{\mathscr{h}}}}}(\omega_j,\eta ) \ ; \right| \ ; \exists \lambda \geq 0 \;\ ; \forall v \in h^1_{0,{{\widehat{\mathscr{h}}}}}(\omega_j,\eta ) \;\ ; b_{{{\widehat{l}}}_\eta,\omega_j}[u , v ] = \lambda ( u , v)_{l^2(\eta ) } \right\rangle , \end{aligned}\ ] ] and the vector space @xmath741 by @xmath742 = \lambda ( u , v)_{l^2(\eta ) } \right\rangle \\ & \,\,\ , \oplus \ , \bigoplus_{j=2}^{2n } \left\langle \left . h^1_{{{\widehat{\mathscr{h}}}}}(\omega_j,\eta ) \ ; \right| \ ; \exists \lambda>0 \;\ ; \forall v \in h^1_{{{\widehat{\mathscr{h}}}}}(\omega_j,\eta ) \;\ ; b_{{{\widehat{l}}}_\eta,\omega_j}[u , v ] = \lambda ( u , v)_{l^2(\eta ) } \right\rangle , \end{aligned}\ ] ] where in the first equation we take the direct sum within @xmath733 , while in the second we use the abstract direct sum , and in both equations angled brackets indicate the linear span in @xmath733 . using the variational characterization of eigenvalues and the unique - continuation principle we find as in @xcite that ( i ) orthogonal projection in @xmath733 onto the subspace spanned by the @xmath743 eigenfunctions with strictly negative eigenvalues has injective restriction to @xmath739 and ( ii ) orthogonal projection in @xmath744 onto @xmath741 has injective restriction ( after precomposition with the obvious inclusion ) to the subspace of @xmath733 spanned by @xmath743 eigenfunctions with nonpositive eigenvalues according to lemma [ hemisphere ] below @xmath743 on @xmath745 ( with dirichlet condition ) has one - dimensional kernel and no strictly negative eigenvalues , so from ( i ) in the preceding paragraph we deduce that @xmath743 on @xmath733 has at least @xmath746 strictly negative eigenvalues , counted with multiplicity . on the other hand lemma [ hemisphere ] also states that @xmath743 on @xmath731 with neumann condition has trivial kernel and precisely one simple negative eigenvalue , so from ( ii ) in the preceding paragraph we deduce that @xmath743 on @xmath733 has no more than @xmath702 nonnegative eigenvalues , counted with multiplicity . thus @xmath691 has nullity at most one on @xmath733 . since the vertical component of the gauss map is a jacobi field for @xmath76 and is @xmath735 but not @xmath690 equivariant , we see that the @xmath701 kernel of @xmath691 is indeed trivial . [ hemisphere ] for each @xmath145 , with notation as in [ omegadef ] and the proof of proposition [ nokernel ] , a. @xmath743 on @xmath734 ( with dirichlet boundary condition ) has @xmath24-dimensional kernel and no strictly negative eignevalues , and b. @xmath743 on @xmath731 with neumann boundary condition has trivial kernel and exactly one simple negative eigenvalue . the hemisphere @xmath700 with its pole deleted is conformal , via stereographic projection ( from the antipodal pole ) and a logarithm , to the standard half - cylinder with flat metric . concretely , using polar coordinates on the unit disc pulled back by the same stereographic projection to @xmath732 , we have @xmath747 we find ( recall [ potential ] ) that for the corresponding operator @xmath748 we have @xmath749 evidently @xmath750 , so whenever @xmath195 is an eigenfunction of the flat laplacian @xmath751 , @xmath752 is an eigenfunction ( when nonzero ) of @xmath753 with the same eigenvalue . moreover @xmath754 and @xmath755 are linearly independent for every real @xmath756 except @xmath757 and obviously @xmath758 . corresponding to the latter exception we have for @xmath759 the linearly independent eigenfunctions @xmath760 and @xmath761 with eigenvalue @xmath126 . as for the former exception , noting that we have already accounted for the eigenfunction @xmath762 of @xmath759 with eigenvalue @xmath763 and that @xmath764 , we find that nonzero @xmath195 solving @xmath765 is another , independent such eigenfunction . thus we deduce that an eigenfunction for @xmath766 of the form @xmath767 with eigenvalue @xmath768 has radial factor @xmath769 a linear combination of @xmath770 and @xmath771 given by @xmath772 or a linear combination of @xmath773 and @xmath774 in case @xmath775 or a linear combination of @xmath776 and @xmath777 in case @xmath778 . separating variables , we need only consider eigenfunctions of the above form , and , because of the rotational symmetries imposed , @xmath30 must take values in @xmath779 . now suppose @xmath780 on @xmath732 . then @xmath781 as well , on @xmath732 punctured at its pole @xmath782 , so @xmath783 and @xmath784 for some constants @xmath785 and @xmath786 , unless @xmath787 , in which case @xmath788 . ( the other exceptional case of @xmath757 is excluded by the symmetries . ) if we impose dirichlet conditions , we find @xmath789 when @xmath790 , but then @xmath791 is singular at @xmath792 unless @xmath793 , which must therefore hold , since @xmath767 is an eigenfunction on all @xmath732 . on the other hand @xmath794 while @xmath795 , so @xmath796 when @xmath758 . thus dirichlet @xmath691 has kernel spanned by @xmath797 , confirming the nullity asserted in ( i ) . if we instead impose neumann conditions , since @xmath798 ( again @xmath757 is excluded by the symmetries ) , we find @xmath799 when @xmath790 , so again we need @xmath793 to avoid a singularity at @xmath792 . on the other hand @xmath800 while @xmath801 , so in this case @xmath802 , but @xmath803 is also singular at @xmath792 . thus neumann @xmath691 indeed has trivial kernel , as claimed in ( ii ) . for @xmath804 one can not expect agreement of solutions to @xmath805 and to @xmath806 , but the variational characterization of eigenvalues reveals that the number ( counting multiplicity ) of strictly negative ( dirichlet or neumann ) eigenvalues will agree for the two operators , at least on compact subsets of @xmath807 , where the conformal factor is bounded with bounded inverse . furthermore , again using the variational characterization of eigenvalues one sees that for @xmath808 sufficiently small , the number ( counted with multiplicity ) of strictly negative eigenvalues for @xmath691 on @xmath732 with dirichlet condition on the equator is the same as the number of strictly negative eigenvalues of @xmath691 on @xmath732 less a spherical cap @xmath809 with radius @xmath810 and center the pole @xmath126 , imposing dirichlet conditions on both the equator and boundary of the cap . ( to show the former number is at least the latter extend test functions vanishing on @xmath811 to test functions vanishing on @xmath809 ; for the reverse inequality use a logarithmic cut - off , identically @xmath126 on @xmath809 and identically @xmath24 on @xmath812 . ) likewise , assuming @xmath810 small enough , the number of strictly negative eigenvalues of @xmath691 on @xmath732 with neumann condition on the equator is the same as the number of strictly negative eigenvalues of @xmath691 on @xmath813 with neumann condition on the equator and dirichlet condition on @xmath811 . now suppose @xmath814 is an eigenfunction of @xmath691 on @xmath813 with strictly negative eigenvalue , imposing either of the above boundary conditions . then @xmath815 , so in particular @xmath804 . if @xmath816 , then @xmath817 . if we impose the dirichlet condition on the equatorial circle @xmath818 , then , since @xmath819 and @xmath820 , we must have @xmath802 , but @xmath821 , so the dirichlet condition on @xmath822 means @xmath796 as well . thus there are no such dirichlet eigenfunctions . if instead we impose the neumann condition on @xmath818 , then , since @xmath823 and @xmath824 , we need @xmath796 , but @xmath825 vanishes only at @xmath792 , so the dirichlet condition on @xmath822 means @xmath802 too . thus there are no such neumann eigenfunctions either . now assume @xmath826 . then @xmath827 . as when studying the kernel above , imposition of the dirichlet condition @xmath818 forces @xmath789 , but a quick calculation shows that imposition of the dirichlet condition on @xmath822 then requires @xmath828 = ( k-1 ) \tanh [ ( k-1 ) \ln \epsilon]$ ] . since for every real @xmath829 the function @xmath830 is even and on @xmath831 strictly monotonic , this last condition implies @xmath757 , contradicting the initial assumption of the paragraph and completing the proof of ( i ) . imposing instead the neumann condition on @xmath818 forces @xmath799 , so the dirichlet condition on @xmath822 now demands @xmath832 = ( k-1 ) \tanh [ ( k-1 ) \ln \epsilon]$ ] . for any @xmath808 the function on the left is even in @xmath833 , strictly monotonic in @xmath833 on @xmath831 , and has limit @xmath834 as @xmath833 goes to @xmath126 ; moreover , as @xmath810 tends to @xmath126 , the right - hand side goes to @xmath835 . thus this equation has exactly one solution , completing the proof of ( ii ) . [ r : nk ] a simpler proof of [ hemisphere ] is also possible using the @xmath836 metric instead of the @xmath694 metric and without reference to the @xmath837 operators . [ towsol ] can also be proved without using the @xmath694 metric . now we state some estimates for solutions to the poisson equation @xmath838 on the euclidean strip @xmath839 of given width @xmath840 , with @xmath841 subject to dirichlet data and @xmath223 odd under reflection through the horizontal line @xmath842 for given @xmath843 and every @xmath145 . we set @xmath844 in the applications to follow , @xmath845 will tend to infinity with @xmath103 , while @xmath846 will be bounded independently of @xmath103 , so it is important that the estimates here do not depend on @xmath845 . the additional decay estimate included in the proposition will be necessary to guarantee convergence of the iterative scheme used to construct global solutions on the initial surfaces . [ torsol ] with notation as in the preceding paragraph , given @xmath847 and @xmath653 , there exists a linear map @xmath848 and there exists a constant @xmath849depending on @xmath664 and @xmath846 but not on @xmath845such that if @xmath850 and @xmath851 , then @xmath852 , @xmath841 vanishes on @xmath853 , and @xmath854 moreover , if @xmath223 vanishes outside @xmath855 \times { \mathbb{r}}$ ] for @xmath856 , then @xmath857 where @xmath23 is the coordinate on the @xmath858 $ ] factor of @xmath859 . define @xmath860 to be the dirichlet solution @xmath841 to the poisson equation @xmath861 . then @xmath862 and for each @xmath863 @xmath864 where @xmath715 is the intersection with @xmath859 of the euclidean disc with center @xmath287 and radius @xmath716 . defining @xmath865 for each positive integer @xmath459 we have @xmath866 for which equation one finds dirichlet green s function @xmath867 since @xmath868 we have @xmath869 ) } \leq \frac{y^2}{n^2}{\left\|f_n\right\|}_{c^0([0,x\pi ] ) } \leq \frac{\sqrt{2y^5\pi}}{n^2}{\left\|f\right\|}_{c^0(t_x}),\ ] ] and so @xmath870 which upgrades the local schauder estimates above to the first inequality asserted in the proposition . if moreover @xmath223 vanishes outside @xmath855 \times { \mathbb{r}}$ ] , then @xmath871}(x ) = \frac{u_n(a ) \sinh \frac{nx}{y}}{\sinh \frac{na}{y } } \mbox { and } \\ & u_n|_{[b , x\pi]}(x ) = \frac{u_n(b ) \sinh \frac{n(x\pi - x)}{y}}{\sinh \frac{n(x\pi - b)}{y } } , \end{aligned}\ ] ] establishing in conjunction with [ modest ] the decay estimates . the final task of this section is to apply propositions [ towsol ] and [ torsol ] iteratively on the extended standard regions to prove existence and obtain estimates of global solutions to the equation @xmath872 on each initial surface . [ rcal ] fix @xmath653 and data @xmath873 or @xmath874 for an initial surface . there is a positive integer @xmath875 such that for every @xmath876 and for every initial surface @xmath50 defined by the corresponding data there exists a linear map @xmath877 and there exists a constant @xmath849independent of @xmath103such that if @xmath878 , then @xmath879 and @xmath880 for @xmath596 with boundary circles @xmath881 , set @xmath882 and define the diffeomorphism ( recalling definitions [ stdef ] and [ txdef ] ) @xmath883,\ ] ] the cutoff function @xmath884)$ ] , and the linear map @xmath885,m^2 g ) \to c_{{\mathscr{g}}}^{2,\beta}(s[t],m^2g)\ ] ] by @xmath886 } \circ x + { \psi\left [ a_t , a_t-1 \right ] } \circ x \right ) , \mbox { and } \\ & x_{_t } : = \varpi_t^{-1 } \circ \kappa , \end{aligned}\ ] ] where @xmath887 is any isometry mapping @xmath888 to a scaffold circle on the torus containing @xmath600 . for @xmath625 recall the diffeomorphism @xmath889\ ] ] and define the cutoff function @xmath890)$ ] and the linear map @xmath891,m^2g)\ ] ] by @xmath892 } \circ { \ensuremath{\mathrm{r}}}\right ) . \end{aligned}\ ] ] next , given @xmath893 , let @xmath894{\mathcal{r}}_tf|_{s[t ] } , \\ & f_2 : = \sum_{c \in { \mathcal{c}}(\sigma ) } [ \psi_{_c},m^{-2}{\mathcal{l } } ] { \mathcal{r}}_cf_1|_{s[c ] } , \mbox { and } \\ & \widetilde{{\mathcal{r } } } f : = \sum_{t \in { \mathcal{t}}(\sigma ) } \psi_t{\mathcal{r}}_t ( f+f_2)|_{s[t ] } + \sum_{c \in { \mathcal{c}}(\sigma ) } \psi_c{\mathcal{r}}_c f_1|_{s[c]}. \end{aligned}\ ] ] the idea behind the definition of @xmath895 is as follows . first we construct approximate solutions on each toral region ( the @xmath896}$ ] terms ) and cut them off smoothly . these solutions are only approximate since we have obtained them by applying the solution operator for the model problem on the euclidean strip , and the resulting error is controlled by the deviation of the initial surface s geometry from the model geometry . additional error , supported in the tower regions , is created by cutting off the approximate solution with @xmath897 . we know only that its size is controlled by that of the original @xmath223 , and we account for it in @xmath898 along with the restriction of the original @xmath223 to the tower regions , where we next construct and cut off approximate solutions in a similar fashion . again there is error controlled by the geometry and also cutoff error , for which we have no better bound than the norm of @xmath223 but which is supported inside the toral regions far from their boundary , so we can construct an approximate solution to correct for them and apply the decay estimate in [ towsol ] . more precisely we now check that @xmath899 } \\ & + \sum_{t \in { \mathcal{t}}(\sigma ) } \left ( { x_{_t}^*}^{-1}\delta_{{g_{_e}}}x_t^*-m^{-2}{\mathcal{l}}\right ) { \mathcal{r}}_t \left(f+f_2\right)|_{s[t ] } \\ & + \sum_{t \in { \mathcal{t}}(\sigma ) } \left [ \psi_{_t } , m^{-2}{\mathcal{l}}\right ] { \mathcal{r}}_t f_2|_{s[t ] } , \end{aligned}\ ] ] so , noting that @xmath900}$ ] is supported far away from @xmath901 $ ] , we find from [ stdtor ] , [ stdtow ] , [ towsol ] , and [ torsol ] @xmath902 where @xmath903 is a constant depending on @xmath73 but not on @xmath103 and where @xmath234 is a constant depending on neither @xmath73 nor @xmath103 . thus we may at this stage fix @xmath73 ( finally determining the extent of the toral regions ) sufficiently large in terms of @xmath234 and then take @xmath103 sufficiently large in terms of @xmath903 so as to ensure that @xmath904 is invertible . the proof is then concluded by taking @xmath905 . recall that given an initial surface @xmath50 with defining embedding @xmath639 and a function @xmath906 , we have defined the map @xmath907 by @xmath908 , @xmath909 being the exponential map on @xmath5 and @xmath527 a global unit normal for the initial surface . for @xmath910 sufficiently small @xmath646 is an immersion with well - defined mean curvature @xmath647 $ ] relative to the global unit normal @xmath645 having positive inner product with the parallel translates of @xmath527 along the geodesics it generates . we now prove the main theorem by solving @xmath647=0 $ ] . [ mainthm ] given data ( a ) @xmath873 or ( b ) @xmath874 for an initial surface ( recalling [ initsdef ] ) , there exist @xmath911 and @xmath849 such that whenever @xmath912 , the initial embedding @xmath639 corresponding to the data can be perturbed to a minimal embedding @xmath907 by a function @xmath648 ( depending on @xmath103 ) that satisfies the estimate @xmath913 . here @xmath41 is either ( a ) @xmath484 or ( b ) @xmath487 ( recalling [ symm ] ) . in particular @xmath914 has the same genus as @xmath50 ( see [ initsprop ] ) , is invariant under @xmath41 , and contains the scaffolding ( a ) @xmath389 or ( b ) @xmath394 ( recalling [ scaff ] ) . moreover , in the complement in @xmath5 of any tubular neighborhood of the circles of intersection of the initial configuration ( a ) @xmath6 or ( b ) @xmath11 , for @xmath103 sufficiently large @xmath914 is the graph over some subset of the initial configuration of a smooth function converging smoothly to @xmath126 as @xmath915 . fix @xmath916 . by [ stdtor ] and [ stdtow ] the initial mean curvature satisfies @xmath917 : c^{2,2\beta}(\sigma , m^2g)\right\| } \leq cm^{-3/2}\ ] ] for a constant @xmath234 independent of @xmath103 . setting @xmath918,\ ] ] then [ rcal ] implies that @xmath919 for a ( possibly different ) constant @xmath234 independent of @xmath103 . the function @xmath920 represents the first - order correction to the initial surface . to complete the perturbation we need to estimate the nonlinear part of @xmath650 near @xmath126 , defined by @xmath921 : = { \mathcal{h}}[u ] - { \mathcal{h}}[0 ] - { \mathcal{l}}u.\ ] ] to proceed efficiently we consider the blown - up metric @xmath922 on @xmath5 . given @xmath906 we can define @xmath923 by @xmath924 , where @xmath925 is the exponential map on @xmath595 and @xmath926 is the @xmath922 unit normal for @xmath50 parallel to @xmath527 ; of course @xmath927 , @xmath928 , and @xmath929 . for @xmath930 sufficiently small we can define also @xmath931 $ ] to be the mean curvature of @xmath932 relative to @xmath922 ( and @xmath933 ) . obviously @xmath934 = m{\mathcal{h}}_{m^2{g_{_s}}}[mu],\ ] ] so @xmath935 = m^{-2}\int_0 ^ 1 \int_0^t \frac{d^2}{ds^2}{\mathcal{h}}[su ] \ , ds \ , dt = m^{-1 } \int_0 ^ 1 \int_0^t \frac{d^2}{ds^2}{\mathcal{h}}_{m^2g_s}[smu ] \ , ds \ , dt.\ ] ] now , if @xmath936 is sufficiently small in terms of the riemannian curvature of @xmath595 and the second fundamental form of @xmath845 relative to @xmath922 , then @xmath646 will be an immersion , @xmath647 $ ] will be well - defined , and moreover @xmath937 } { \left\|\frac{d^2}{ds^2}{\mathcal{h}}_{m^2g_s}[smu ] : c^{0,2\beta}(\sigma , m^2g)\right\| } \leq c{\left\|mu : c^{2,2\beta}(\sigma , m^2g)\right\|}^2,\ ] ] where @xmath234 is a constant controlled by finitely many covariant derivatives of the riemannian curvature of the ambient space @xmath595 and finitely many covariant derivatives of the second fundamental form of @xmath845 relative to @xmath922 . of course the riemannian curvature of @xmath595 is bounded uniformly in @xmath103 ( tending to @xmath126 in fact ) and all of its derivatives vanish ; while @xmath845 itself depends on @xmath103 , each derivative of its second fundamental form , relative to @xmath922 , is bounded independently of @xmath103 . consequently , if @xmath42 is the closed ball of radius @xmath938 in @xmath939 and @xmath940 , we have @xmath941 : c^{2,2\beta}(\sigma , m^2g)\right\| } \leq cm^{-2}.\ ] ] evidently then , taking @xmath103 large enough , @xmath942 $ ] defines a map @xmath943 which is continuous with respect to the @xmath944 norm on @xmath42 as well as the @xmath939 norm , so by the schauder fixed point theorem admits a fixed point @xmath945 . accordingly @xmath946 $ ] and @xmath947 = { \mathcal{h}}[0 ] + { \mathcal{l}}u_0 + { \mathcal{l}}v_0 + { \mathcal{q}}[u_0+v_0 ] = 0.\ ] ] the higher regularity of @xmath948 then follows immediately , and the @xmath641 decay estimate of @xmath949 ensures embeddedness . in this subsection we briefly outline a highly symmetric construction where the symmetry imposed is not so great that there are no obstructions . the obstruction space is nontrivial but of finite dimension independent of the symmetries and the genus of the surfaces constructed . the construction can easily be explained in terms of the earlier presentation : the initial configuration used is @xmath11 and the symmetry group imposed is @xmath484 ( and not @xmath487 ) . this corresponds to using the scaffolding @xmath950 . the towers desingularizing @xmath8 and @xmath9 are then symmetric enough that they carry no kernel . the construction in this respect can proceed as the earlier one in [ mainthm ] . on the other hand the towers desingularizing the circles @xmath951 are classical scherk singly periodic surfaces and the symmetries imposed fix @xmath320 but not @xmath304 . this situation is similar to many recent constructions @xcite where there is enough symmetry to simplify the obstruction space in comparison to the more general situation in @xcite , but not enough to render it trivial as in [ mainthm ] . more precisely we have a two - dimensional kernel , one dimension for each circle of intersection @xmath555 and @xmath952 . ( note that modulo the symmetries these are the only circles of intersection besides @xmath8 and @xmath9 ) . there are no circles in the scaffolding contained in @xmath304 and therefore @xmath304 is not held fixed by the construction . we introduce then two continuous parameters in the construction , @xmath953 and @xmath954 . @xmath555 is replaced by a parallel copy on @xmath376 at ( signed ) distance @xmath953 and similarly @xmath952 is replaced by a parallel copy on @xmath377 at ( signed ) distance @xmath954 . by the symmetries then all @xmath331 are appropriately replaced also . @xmath304 is a union of annuli with boundaries the @xmath331 . these are replaced then by minimal graphs so that the new annuli span the @xmath331 s . this way @xmath304 is replaced by a new torus with derivative discontinuities along its circles of intersection with the @xmath320 s . the construction of the initial surfaces then proceeds as usual by using towers appropriately . note that modulo the symmetries there are four circles which get desingularized : @xmath955 and the ( perturbed to new positions ) @xmath956 . following the same conventions as in section [ sinit ] we denote by @xmath957 the number of half periods the desingularizing towers will have between successive circles of reflection in @xmath389 along @xmath958 respectively . this together with three alignment parameters @xmath959 and the continuous parameters @xmath953 and @xmath954 determine the initial surfaces . we have the following . [ thm-2 ] given data @xmath960 for an initial surface as outlined above there exists @xmath911 such that whenever @xmath912 , one of the initial surfaces ( for some appropriate values of @xmath961 ) described above can be perturbed to a minimal surface which contains @xmath389 , is symmetric under the action of @xmath484 , and has genus @xmath962 . moreover as @xmath963 the minimal surfaces converge as varifolds to @xmath11 . in this subsection we discuss corollaries in our setting of a general desingularization theorem announced in ( * ? ? ? * theorem f ) and ( * ? ? ? * theorem 3.1 ) . the statement of this theorem is motivated in @xcite , and its proof is outlined in detail in @xcite and will be presented in detail in @xcite . we will refer to this theorem in the rest of the discussion as the `` general theorem '' . the general theorem applies to situations where the intersection curves are transverse and have double points only , because the corresponding general construction is understood only when classical scherk surfaces are used to model the desingularizing regions in the vicinity of the intersection curves . therefore we can only consider the cases where the initial configurations in our setting are @xmath964 or @xmath965 ( recall [ e : wk ] ) excluding the possibility @xmath966 . recall that in the first case we have two clifford tori @xmath376 and @xmath377 intersecting orthogonally along two totally orthogonal circles @xmath8 and @xmath9 . in the second case we have three pairwise orthogonal clifford tori @xmath967 with six intersection circles @xmath8 , @xmath9 , @xmath555 , @xmath952 , @xmath968 , and @xmath969 , where we also have @xmath970={\mathbb{t}}[c'_4]$ ] , @xmath971={\mathbb{t}}[c'_1]$ ] , @xmath372={\mathbb{t}}[c_2]$ ] , @xmath972 , @xmath973 , and @xmath974 , as follows from [ e : cp ] , [ e : cpperp ] , and [ e : cpinter ] with @xmath16 . following the general theorem we define @xmath975 and @xmath976 or @xmath977 ( recall [ e : wk ] ) the abstract surfaces with connected components the closures of the connected components of @xmath978 or @xmath979 . recall now that by the discussion of the clifford tori in section [ initial ] , any clifford torus @xmath247 is covered isometrically by @xmath78 with deck transformations generated by @xmath980 and @xmath981 . the linearized operator for the mean curvature is @xmath982 , which clearly has a four dimensional kernel with basis @xmath983 where @xmath984 are the standard coordinates on @xmath78 . an alternative basis is given by @xmath985 the existence of kernel means that the general theorem can not be applied unless we impose enough symmetry to ensure that the kernel modulo the symmetries becomes trivial . to impose these symmetries we consider the scaffolding @xmath986 defined by @xmath987 ( recall [ e : cphph ] ) and the corresponding group @xmath988 . it is easy to calculate then that @xmath989 note that for @xmath990 we have @xmath991 , @xmath992 , @xmath993 , and @xmath994 . if we impose @xmath995 as the group of symmetries of the construction , then @xmath996 has to be contained in the nodal lines of any eigenfunction allowed by the symmetries on @xmath376 . @xmath376 this way is subdivided into two flat squares of side length @xmath345 . the eigenvalues for the laplacian on each square with dirichlet boundary data are of the form @xmath997 with @xmath998 . @xmath15 is not included then . working similarly on @xmath377 we conclude that there is no kernel modulo the symmetries on @xmath964 . we have also to check that there is no kernel on @xmath976 . in this case we have to impose an extra dirichlet condition on @xmath9 , and then @xmath376 ( or @xmath377 ) is subdivided into four flat rectangles of sides @xmath345 by @xmath236 and the eigenvalues allowed are @xmath999 with @xmath1000 , and so @xmath15 is again not included . we study now the case of @xmath384 . first we check that there is no kernel on @xmath304 . because of the symmetry @xmath1001 we can assume that we are working on a rectangular ( instead of a square ) flat torus with sides of length @xmath1002 and @xmath1003 . the eigenvalues of the laplacian then are @xmath1004 with @xmath1005 , which do not include @xmath15 . to check that there is no kernel on @xmath1006 note first that on @xmath376 dirichlet conditions are imposed on @xmath8 , @xmath9 , @xmath313 , @xmath555 , and @xmath1007 . this subdivides @xmath376 into eight flat rectangles of sides @xmath242 by @xmath345 where the laplacian with dirichlet conditions on the boundary have eigenvalues @xmath1008 with @xmath998 . similarly for @xmath377 so it remains only to check @xmath304 . this has dirichlet conditions imposed on @xmath555 , @xmath952 , @xmath1007 , and @xmath1009 . @xmath304 is then subdivided into four flat cylindrical annuli of width @xmath242 and so without even using the symmetries we have that the smallest eigenvalue is @xmath1010 so that @xmath15 is again not included . applying then the general desingularization theorem announced in ( * ? ? ? * theorem f ) and ( * ? ? ? * theorem 3.1 ) we have the following as a corollary . [ thm-3 ] @xmath964 can be desingularized to produce embedded closed minimal surfaces in @xmath5 symmetric under @xmath995 of genus @xmath1011 , where the towers desingularizing @xmath8 and @xmath9 include @xmath460 and @xmath461 periods respectively , provided @xmath460 and @xmath461 are large enough in absolute terms . as @xmath1012 the minimal surfaces tend to @xmath964 . similarly @xmath384 can be desingularized to produce embedded closed minimal surfaces in @xmath5 symmetric under @xmath995 of genus @xmath1013 , where the towers desingularizing @xmath8 and @xmath9 include @xmath460 and @xmath461 periods respectively , the towers desingularizing @xmath555 and @xmath1007 include @xmath462 periods , and the towers desingularizing @xmath952 and @xmath1009 include @xmath1014 periods , provided @xmath957 are large enough in absolute terms . as @xmath1015 the minimal surfaces tend to @xmath384 . note that the main difference of this result compared with the earlier ones is the small symmetry imposed and and that the ( still large ) number of periods along each circle can be prescribed independently on each circle ( except for the identifications by the symmetries of @xmath555 with @xmath1007 and @xmath952 with @xmath1009 ) , as opposed to requiring that all numbers have a large common divisor @xmath103 .

topological insulators ( tis ) are amongst the most actively investigated systems in condensed matter physics @xcite . in reality , there is evidence for their existence in two @xcite and three @xcite spatial dimensions . due to bulk - boundary correspondence , non - trivial topological states of matter have edge states at their boundaries with peculiar transport and optical properties . for instance , the two - dimensional ( 2d ) , time - reversal symmetric quantum spin hall state that is realized in hg(cd)te quantum wells ( qws ) is known to come along with helical edge states that are protected against elastic backscattering of non - magnetic impurities @xcite . however , not only the edge state physics of these systems is interesting but also the 2d bulk physics bears exciting novelties . the low - energy excitations of hg(cd)te qws are described by a model the bernevig - hughes - zhang ( bhz ) model @xcite that interpolates between the limiting cases of schrdinger and dirac fermions . this interplay between schrdinger and dirac physics constitutes an opportunity for new phenomena to emerge . we have , for instance , recently discovered collective charge excitations at zero doping , i.e. intrinsic plasmons , in this system which are absent in both separate limits @xcite . in this article , we complement our study of the screening properties and the collective charge excitations of hg(cd)te qws on the basis of random phase approximation ( rpa ) , and hence present a comprehensive analysis of its polarization function in the static and full dynamic limit , at zero and finite doping . continuously tuning the parameters of the bhz model , we reproduce the limits of pure dirac and pure schrdinger fermions and explore intermediate regimes , in order to understand how analogies and differences emerge . we support our numerical calculations of the polarization functions with analytical expressions derived by f - sum rules . in the static limit , we calculate the screening properties due to the intrinsic system and at finite doping , analyzing the induced charge density ( with friedel oscillations ) in response to a charged impurity . different to the dirac fermion system graphene , where static screening in the intrinsic limit is momentum independent and can therefore be absorbed into an effective dielectric constant @xcite , the bhz model shows a significant momentum dependence that translates into a finite extent of the induced charge density . in the dynamic limit , we are particularly interested in a better understanding of the plasmon excitations of this system away from zero doping where we previously found a new plasmon due to the interplay between schrdinger and dirac fermion physics @xcite . at finite doping , under certain conditions specified below that are e.g. applicable to hg(cd)te qws , we find a coexistence between this novel ( interband ) plasmon and an ordinary ( intraband ) plasmon . both plasmons can be rather weakly damped by single - particle excitations and should therefore be observable . interestingly , the two plasmons respond to the topology of the bandstructure with a distinctive behavior . they seem to merge one into the other in a normal insulating phase , while they remain clearly resolved when the system realizes a topological insulator . generally , rpa is known to provide reliable predictions at large densities and in systems with a large number of fermionic degrees of freedom . while its validity was indeed questioned for the intrinsic dirac limit , where the system is unable to screen the coulomb interaction and strong renormalization effects are expected @xcite , rpa has been shown to yield a quantitative description of many - body effects in graphene @xcite . it has been widely used for the study of plasmons in the dirac model , including various forms of ( multilayer ) graphene and ti surface states , see ref . for a comprehensive review . closely related to our work , the intraband plasmons of black phosphorous have been studied on the basis of rpa and an extended version of the bhz model including anisotropy @xcite . a similar study has been done for mos@xmath0 @xcite . our article is organized as follows . in sec . [ sec_model ] , we introduce the bhz model and present the general formalism we employ to calculate the static and dynamical dielectric function and the induced charge density . the nature of the nontrivial pseudospin , the origin of possible interband plasmons , experimentally relevant parameters and the different contributions to the f - sum rule are also discussed here . subsequently , in sec . [ sec_undoped ] , we present the static screening properties , the dynamical excitation spectrum ( new interband plasmon ) and the f - sum rule in the undoped regime . here we revisit and go beyond the results from ref . . [ sec_doped ] , this analysis is extended to the case of finite doping where inter- and intraband excitations equally matter . we begin by discussing the ability of the bhz model to interpolate between dirac and schrdinger physics . afterwards , we have a closer look at parameters which are experimentally relevant for hg(cd)te qws , see sec . [ sec : hg(cd)te quantum wells ] . in this limit , we find a coexistence of inter- and intraband plasmons occuring for energies and momenta which are suitable for raman or electron loss spectroscopy . we close this chapter by investigating the influence of a non - trivial topology on the plasmonic excitation spectrum . finally , in sec . [ sec_con ] , a conclusion and a brief outlook are given . the bhz hamiltonian @xcite for a two - dimensional electron gas ( 2deg ) near the @xmath1-point has the form @xmath2 here @xmath3 are the pauli matrices associated with the band - pseudospin degree of freedom ( band @xmath4 and @xmath5 in hg(cd)te quantum wells ( qws ) ) , @xmath6 , @xmath7 with @xmath8 . the system possesses time - reversal symmetry and @xmath9 is block diagonal in the kramer s partner or spin degree of freedom . restricting ourselves to the block @xmath10 , the results can be extended to the other one by applying the time reversal operator . @xmath10 describes fermions with intermediate properties between a dirac and a conventional 2deg system . the off - diagonal term ( @xmath11 parameter ) is typical for a dirac system ( @xmath12 in graphene ) , with @xmath13 the dirac mass ( corresponding to a gap of @xmath14 ) . we consider positive and negative masses , where the latter one corresponds to an inversion of the bandstructure and the system is topologically non - trivial @xcite . for simplicity , we restrict ourselves to a bandstructure with a minimum at the @xmath1-point , which limits the mass to @xmath15 . in analogy to a 2deg , the diagonal elements bear kinetic energy elements which preserve ( @xmath16 parameter ) and break ( @xmath17 parameter ) particle - hole ( p - h ) symmetry ( @xmath18 for schrdinger fermions with @xmath19 the quasi - particle mass ) . the eigenstates of eq . ( [ eq : hamiltonian ] ) are described by the following dispersion and pseudospin @xmath20 with @xmath21 for valence and conduction band . note that we consider electrons to be perfectly localized on the 2d x - y plane and therefore we neglect the real shapes of the envelope functions due to the quantum confinement along @xmath22 direction @xcite . the bhz model is characterized by intrinsic scales for momentum , @xmath23 , and energy , @xmath24 , which reflect the interpolating character of the model between dirac ( @xmath11 parameter ) and schrdinger ( @xmath16 parameter ) system . fermi momentum @xmath25 and chemical potential @xmath26 provide externally tunable momentum and energy scales , which we call fermi scales in the following . we expect the ratio between fermi and intrinsic scales to govern the physics of this system . we therefore define the dimensionless quantities @xmath27 where we set @xmath28 in the following . @xmath29 is defined to be the energy to the wave vector @xmath30 , such that @xmath31 if @xmath32 . for @xmath33 , we therefore expect intermediate physics , while in the limit @xmath34 ( @xmath35 ) the dirac ( 2deg ) physics should be recovered . the linear response of an homogeneous system to an external applied potential is described by the density - density generalized susceptibility or retarded polarization function @xmath36 . this response comprises two main phenomena : screening , described by the real part @xmath37 $ ] , and dissipation by single - particle excitations ( spes ) , given by the imaginary part @xmath38 $ ] . the polarization function in rpa yields the expression @xmath39 with @xmath40 , @xmath41 a positive infinitesimal , @xmath42 for spin degeneracy , @xmath43 and @xmath44 the fermi - dirac function with @xmath45 and @xmath46 the boltzmann constant . in the following we will assume zero temperature , @xmath47 . the overlap factor is given by @xmath48.\label{eq : f factor}\ ] ] eq . ( @xmath49 ) implies that @xmath50 is only a function of the reduced dimensionless variables @xmath51 and @xmath52 and parametrically depends on @xmath53 , @xmath54 and @xmath30 . in the massless dirac limit ( @xmath55 ) , eigenspinors are characterized by their helicity and consequently the overlap factor @xmath56 only depends on the angle @xmath57 between @xmath58 and @xmath59 . it is strictly one ( zero ) for states with the same ( opposite ) helicity . in the bhz model , the quadratic terms have the effect of turning the pseudospin of the eigenstates out of plane in opposite directions for conduction and valence bands at large @xmath51 , see fig . [ fig : overlap ] . , and a ti phase ( b ) , @xmath60 . the bands are separated by an additional @xmath61 for better illustration of the pseudospin . [ fig : overlap],title="fig:",width=158 ] , and a ti phase ( b ) , @xmath60 . the bands are separated by an additional @xmath61 for better illustration of the pseudospin . [ fig : overlap],title="fig:",width=158 ] this results in a decay of the overlap factor down to @xmath62 in the limit of a conventional 2deg system ( @xmath63 or @xmath64 ) . a finite mass @xmath65 has a similar effect , but in the limit of @xmath66 . the pseudospin turns in the same ( opposite ) direction as for the quadratic term for positive ( negative ) mass , see fig . [ fig : overlap ] . this has the direct consequence that for a normal insulator ( ni ) phase the interband overlap factor is reduced , while it is increased for a ti phase . on the contrary , a positive ( negative ) mass enhances ( diminishes ) the intraband overlap factor . this picture is also confirmed in section [ f - sum rule introduction ] by calculating the f - sum rule . the bare coulomb interaction @xmath67 in an electron gas is modified by screening into the effective interaction @xmath68 . there , screening is described by the dynamical dielectric function . employing dimensionless units , it acquires the form @xmath69 where we have introduced the interaction strength parameter @xmath70 ( effective dirac fine structure constant @xcite ) and the dimensionless function @xmath71 @xmath72 in graphene one finds @xcite @xmath73 , while in hg(cd)te qws it is of the order @xmath74 @xcite . here , @xmath75 is the background dielectric constant , accounting for screening of internal electronic shells , while @xmath76 gives the dynamic screening due to electrons in the low energy bands . zeros of @xmath77 describe a density - density ( longitudinal ) perturbation of the system that it is able to sustain itself , which forms a collective mode called plasmon . it is defined by @xmath78 with the plasma frequency @xmath79 , and the finite imaginary part @xmath80 accounts for the possible damping due to single - particle excitations @xcite . the dissipation of the interacting system , including both single - particle excitation and the plasmon mode , is then described by the imaginary part of the interacting polarization function @xmath81 . in order to compare to the non - interacting one , we will plot the normalized functions @xmath82 , \ \pi^{im}\equiv \im\left[\pi^{r}\right ] , \ \pi^{re}\equiv \re\left[\pi^{r}\right ] \nonumber\end{aligned}\ ] ] in the following , with @xmath83 . in rpa , eq . ( [ eq : epsi ] ) characterizes the screening of the interaction between two electrons exchanging momentum @xmath51 and energy @xmath52 , by the creation of electron - hole pairs in the electron gas with the same momentum @xmath51 . if these pairs are resonant in energy @xmath84 , they correspond to a physical process leading to dissipation and a lowering of the coulomb interaction - described by the imaginary part of the polarization function , eq . ( [ eq : pi_par ] ) . when @xmath85 , we have only virtual electron - hole pairs , which either still screen the interaction , if @xmath86 , or even enhance it ( antiscreening effect ) , if @xmath87 . these effects depend on the energy of the created pair , for @xmath88 one finds antiscreening , while @xmath89 leads to a screening of the bare coulomb interaction . this can be directly seen from the definition of the polarization function , eq . ( [ eq : pi_par ] ) . for every allowed excitation , the real part of the integrand in eq . ( [ eq : pi_par ] ) becomes @xmath90}{\omega^2-\omega_{eh } \left[\boldsymbol{\tilde{x}},\boldsymbol{\tilde{x}'}\right]^{2 } } , \label{eq : integrand}\ ] ] with @xmath91 ( @xmath92 ) for intraband ( interband ) excitations . therefore every process with energy less than @xmath52 increases @xmath93 , lowering @xmath94 and thus increasing the interaction . in the intrinsic dirac system within rpa one finds @xmath95 for all energies @xmath52 where electron - hole excitations are allowed @xcite . thus the screening effect of virtual excitations with @xmath89 cancels exactly with the one from excitations with @xmath88 , such that the only screening comes from the resonant process @xmath84 . in the bhz model , the high energy excitations become less likely as the electron and the hole band get decoupled for large @xmath52 . additionally their excitation energy is higher as in the dirac case for the same momentum @xmath51 , leading to an additional reduction of their influence on @xmath93 due to the lorentzian in eq . ( [ eq : integrand ] ) . further , low energy excitations become more important , as processes are allowed that where forbidden in the dirac system by helicity ( see sec . [ sec : bhz model no mass no d ] for details ) . combining these effects , one finds the virtual excitations which increase the coulomb interaction , @xmath88 , dominating for larger frequency @xmath52 , leading to an increased effective interaction and the possibility of intrinsic plasmons in the bhz model @xcite . more mathematically speaking , the described effects alter the high energy behaviour of @xmath96 from a decay like @xmath97 in the dirac case to a @xmath98 decay in the bhz model , as is shown in sec . [ sec : expansion of pi ] . taking the kramers - kronig relation @xmath99 one finds directly that the real part of the polarization changes sign for @xmath100 , but not for @xmath101 . in more general terms , one can expect intrinsic interband plasmons to appear in all models for which @xmath96 decays faster as @xmath97 for high energies . the static limit of the polarization function is obtained by sending @xmath102 at finite momentum @xmath51 in eq . ( [ eq : pi_par ] ) . in this limit we can easily analyze the response of the system to the application of a static ( or sufficiently slowly varying ) external potential . an important physical problem of this kind is the screening of a charged impurity by the electronic system . the static polarization is a strictly real function , that we define as @xmath103 in a multiband system , like the bhz model , it is useful to separate the contributions to the static polarization coming from the intrinsic neutral system , @xmath104 ( obtained for @xmath105 ) , and the contribution due to a finite charge density , @xmath106 ( finite @xmath26 ) . consistently with the notation of eq . ( [ eq : pi_static ] ) , the dielectric function , eq . ( [ eq : epsi ] ) , can therefore be rearranged into @xmath107.\end{aligned}\ ] ] from the static dielectric constant we can find the induced charge density in response to a test charge @xmath108 placed at the origin . the variation of the electronic charge density in momentum space corresponds to @xmath109 , where @xmath110 is given by @xcite @xmath111}-1= \\ & = & n_r(x ) + n_0(x ) + n_\mu(x ) . \nonumber\end{aligned}\ ] ] here the induced charge density can be seen as a sum of three contributions of different physical nature . the first is due to the background polarization @xmath112 ( high energy polarization of the system ) , the second to the intrinsic polarization @xmath113 ( polarization of the natural system ) and the third to the polarization of the finite charge density in the system @xmath114 , with @xmath115 in real space , the density fluctuation ( using physical dimensional units ) is given by @xmath116 with @xmath117 the zero - th order bessel function . including coulomb interaction , we now have a 4-dimensional parameter space consisting of @xmath53 , @xmath54 , @xmath30 and @xmath70 . this parameter space will be explored systematically in the following . while the exploration of the different physical behaviors featured by the bhz model in different regions of this parameter space has a clear theoretical significance , we want to stress that our discussion is also relevant for experiments . in particular , realistic parameters for hg(cd)te qw structures @xcite are roughly @xmath118 , @xmath119 , @xmath120 and masses @xmath13 with absolute values up to several mev . the interaction strength is around @xmath121 with an average @xmath122 from the cdte substrate ( @xmath123 ) and hgte ( @xmath124 ) . considering the experimental acceptable damping rate for plasmons , we refer to experiments on the surface states of a 3d ti @xcite . there , plasmons with a ratio of @xmath125 are perfectly resolvable . the f - sum rule for the polarization function provides the total spectral weight of all excitations in the system . it is identical for the interacting and noninteracting system , as the interaction conserves the number of particles . thus the sum rule is a powerful tool to check our numerics . additionally , it offers a deeper insight concerning the shift of spectral weight between the inter and intra spes as well as the different plasmons in the system . the f - sum rule is defined by @xcite @xmath126=g_s\left\langle 0\left| \left[\left[n_{\mathbf{q}},h^0\right],n_{\mathbf{q}}^{\dagger}\right]\right|0\right\rangle\ \label{eq : f - sum def}\ ] ] with the density operator @xmath127 and the hamiltonian @xmath128 with @xmath10 as defined in eq . ( [ eq : hamiltonian ] ) . @xmath129 is a spinor associated with the band - pseudospin degree of freedom ( band @xmath4 and @xmath5 in hg(cd)te qws ) . the spin degree of freedom enters via the degeneracy factor @xmath130 . for the calculation we follow the steps outlined in the appendix of ref . , where the f - sum rule for the dirac model is obtained . for the bhz model the computational steps are the same , therefore we only present important intermediate results and differences to the dirac limit . the commutator in eq . ( [ eq : f - sum def ] ) is given by @xmath131,n_{\mathbf{q}}^{\dagger}\right]&= & \sum_\mathbf{k}\left(\psi^\dag_{\mathbf{k}}h^0_{\mathbf{k},\mathbf{q}}\psi_{\mathbf{k}}-\psi^\dag_{\mathbf{k}+\mathbf{q}}h^0_{\mathbf{k}+\mathbf{q},\mathbf{q}}\psi_{\mathbf{k}+\mathbf{q}}\right)\nonumber\\ & -&2q^2\sum_\mathbf{k}\psi^\dag_{\mathbf{k}+\mathbf{q}}\left(d\sigma_0+b\sigma_z\right)\psi_{\mathbf{k}+\mathbf{q}}\label{eq : f - sum commutator}\end{aligned}\ ] ] with @xmath132 . a simple shift of the momentum sums in eq . ( [ eq : f - sum commutator ] ) would put the first line to zero , but this is not allowed . in the same way as in the dirac system , the operators are unbounded and one has to work with a large momentum cutoff @xmath133 . while in the dirac limit one finds simply @xmath134 and the second line of eq . ( [ eq : f - sum commutator ] ) would be zero , now the latter gives rise to a contribution depending on the chemical potential , as one would expect for a 2deg . the sums in eq . ( [ eq : f - sum commutator ] ) are then converted into integrals and solved in the limit of large @xmath133 . care has to been taken when converting the momentum cutoff @xmath133 into the frequency cutoff @xmath135 , such that both integrals cover the same phase space . for a pure dirac system one find the f - sum rule @xcite @xmath136=-\frac{g_s q^{2}\lambda}{16 } \label{eq : dirac f - sum rule}\ ] ] where the cutoff @xmath135 is needed as the dirac spectrum is unbounded . in a 2deg system one finds @xmath137=\frac{g_s}{4 } \left(b\pm d\right ) k_{f}^{2}q^{2}=-\frac{\pi nq^{2}}{2 m } \label{eq:2deg f - sum rule}\ ] ] with @xmath138 the electron density and @xmath139\neq0 $ ] only over a finite range of @xmath140 . similar to a dirac system , the bhz spectrum is unbounded which complicates the evaluation of the sum rule and makes it necessary to introduce a high - energy cutoff @xmath141 . we find approximately for @xmath142 @xmath143 \label{eq : bhz f - sum rule}\\ = & \frac{g_{s}}{8}x^{2}\biggl[\ln\left(\frac{2\lambda e^{-1 - 2\xi_{m}+2\left|\omega_{f}\right|}}{1 + 2x_{f}^{2}\left(1+\gamma\xi_{d}\right)+2\xi_{m}+2\left|\omega_{f}\right|}\right ) \nonumber \\ + & \frac{1-x^{2}+4\xi_{m}}{\lambda}-\frac{2x^{4}+\left(1 + 4\xi_{m}\right)^{2}-4x^{2}\left(2 + 7\xi_{m}\right)}{4\lambda^{2}}\biggl ] \nonumber \\ + & \mathcal{o}\left(\frac{\xi_{d}}{\lambda^{2}}\right)+\mathcal{o}\left(\frac{1}{\lambda^{3}}\right ) \nonumber \end{aligned}\ ] ] with @xmath144 $ ] and euler s number @xmath145 , so the leading order term diverges logarithmically with @xmath146 . this is due to the fact that @xmath147 $ ] decays like @xmath98 for @xmath148 , and not as @xmath97 as for a dirac system . the sum rule is exact up to order @xmath149 ( @xmath150 ) for finite ( zero ) @xmath151 . the f - sum rules for bhz , dirac and 2deg models are always proportional to @xmath152 in the leading order , but otherwise distinct from one another . taking the limit @xmath63 in the bhz result , eq . ( [ eq : bhz f - sum rule ] ) , gives the 2deg case , eq . ( [ eq:2deg f - sum rule ] ) ) , the same is not possible for the limit @xmath153 , as there the defined cutoff @xmath154 would go to zero . ] . we begin our discussion of eq . ( [ eq : bhz f - sum rule ] ) by comparing the contributions from the different orders @xmath155 , @xmath156 and @xmath157 . in the limit of @xmath158 we find @xmath159 and @xmath160 , thus the ratio @xmath161 determines the importance of higher order corrections for @xmath162 . we take @xmath163 for the cutoff in the following . already for @xmath164 and a maximal momentum @xmath165 , the corrections of order @xmath156 are 2% of order @xmath155 , while contributions of order @xmath157 are smaller than 0.1% . a modest cutoff @xmath166 works best for comparing eq . ( [ eq : bhz f - sum rule ] ) to numerical data , as the latter one is only given over a finite range of @xmath52 . a larger @xmath146 makes it necessary to extrapolate the data , providing a source for errors . next , we investigate changes to the f - sum rule and therefore to the total spectral weight by varying the mass . the influence of a finite mass is studied in fig . [ fig : f - sum rule ] of the lowest order f - sum rule including mass over the one without mass . @xmath169 and @xmath32 . ( b ) ratio @xmath170 of the lowest order f - sum rule including finite doping over the one without doping , @xmath171 ( @xmath172 ) as a black , solid ( red , dashed ) line . @xmath173 and @xmath174 . @xmath164 and @xmath175 in both plots . [ fig : f - sum rule],title="fig:",width=158 ] of the lowest order f - sum rule including mass over the one without mass . @xmath169 and @xmath32 . ( b ) ratio @xmath170 of the lowest order f - sum rule including finite doping over the one without doping , @xmath171 ( @xmath172 ) as a black , solid ( red , dashed ) line . @xmath173 and @xmath174 . @xmath164 and @xmath175 in both plots . [ fig : f - sum rule],title="fig:",width=158 ] \(a ) for @xmath169 and @xmath32 . a positive mass lowers the f - sum rule , while a negative mass increases it linearly . this is a direct consequence from the change of the overlap factor : a negative mass enhances the coupling between the two bands , while a positive mass diminishes it , as in the latter case the pseudospins do not match . it is also consistent with the increase in the optical conductivity observed in the undoped limit with negative mass @xcite . last , we consider the effects of finite doping . it blocks interband transitions close to the dirac point , but due to the small density of states , these transitions carry only a small spectral weight . on the other hand , doping enables intraband transitions , which carry a large spectral weight due the combined effects of larger overlap factor , density of states and smaller excitation energies compared to interband transitions . therefore , a finite doping usually increases the f - sum rule , as seen in fig . [ fig : f - sum rule ] ( b ) , where we plot @xmath170 for positive ( black , solid line ) and negative ( red , dashed line ) doping with @xmath174 and @xmath176 . a finite @xmath54 adds a term @xmath177 to the leading order of the f - sum rule , @xmath178 ( @xmath179 ) for positive ( negative ) doping . it can be seen as an increased ( decreased ) contribution from the 2deg part of the spectrum , eq . ( [ eq:2deg f - sum rule ] ) , and leads to the slight decrease of the f - sum rule for negative doping in panel ( b ) . in order to compare the importance of different excitations in the system , one should compare their spectral weight and thus their contribution to the f - sum rule . the latter has the benefit of being independent of the coulomb interaction strength and the position of the excitation peaks , in contrast to the polarization function @xmath180 . as an example , we assume that the excitation spectrum , @xmath180 , is governed by a single plasmonic peak following a lorentzian shape with width @xmath1 and peak height @xmath181 . then the f - sum rule is proportional to @xmath182 . the value of this integral should be independent of @xmath70 and thus of @xmath183 . therefore we find @xmath184 , such that the peak height of a resonance in @xmath180 naturally has to scale with @xmath185 to fulfill the f - sum rule . we conclude that the importance of a resonance in @xmath180 should be judged by its spectral weight , which can be estimated by multiplying the peak height with its position @xmath183 . the relevant width of the peak is given by @xmath186 , with @xmath1 being the width of the resonance in @xmath180 . in this section , we focus on an intrinsic ( undoped limit @xmath105 ) bhz model system . first , we analyze the static polarization function and the static screening properties . then we consider the long wavelength limit of the dynamical polarization function , providing an analytical expansion . finally , we add some complementary arguments elucidating the origin of the new interband plasmon ( absent both in the dirac and 2deg cases ) , whose appearence for the intrinsic bhz model has been proposed in ref . . in order to set a reference with a closely related and analytically solvable model , we discuss the static intrinsic polarization for a massive dirac limit , given by @xcite @xmath187 \overset{m\rightarrow0}{\longrightarrow } \frac{-g q}{16 a } , \label{eq : pi_dirac_static}\end{aligned}\ ] ] where the index @xmath62 stands for intrinsic limit @xmath105 , @xmath188 account for possible spin and band degeneracy , and @xmath189 . when the dirac system is massless ( @xmath190 ) , @xmath191 is a linear function of the momentum @xmath192 . a finite dirac mass suppresses the polarization for @xmath193 , where @xmath191 shows a super - linear behavior . for @xmath194 , the mass is negligible instead and the result of the massless limit is reproduced . the static polarization function of the bhz model is simply obtained by direct numerical evaluation of eq . ( [ eq : pi_par ] ) at zero frequency . in fig . [ fig : pi_0_static ] , we show @xmath104 calculated for a particle - hole symmetric bhz system ( @xmath195 ) . note that we obtain the massless dirac case in the limit @xmath196 ( and therefore @xmath197 ) , where @xmath198 . a finite @xmath16 parameter determines a fundamental qualitative change with respect to a dirac system . indeed , @xmath104 reaches a maximum at @xmath199 and then decays as @xmath200 for @xmath201 as shown in the inset of fig . [ fig : pi_0_static ] . and @xmath195.,width=302 ] a finite and positive dirac mass @xmath13 leads to a general suppression of the polarization function with respect to the massless case . in the region @xmath202 ( where quadratic terms are less important ) , @xmath104 resembles the massive dirac case , with a super - linear increase in the region @xmath203 , due to the suppression of the interband overlap factor determining a reduction of the polarization at small momentum . for intermediate values @xmath204 , analogously to the massive dirac limit , @xmath104 is approximatively linear in @xmath51 . considering larger momenta @xmath205 , the behavior is dominated by the quadratic terms and the polarization eventually vanish for @xmath206 . in general , the interplay of quadratic terms and a finite dirac mass shifts the maximum of @xmath104 . when the dirac mass @xmath13 is negative ( topological insulator phase ) , we observe a less pronounced suppression of the polarization for @xmath207 , with respect to a massive dirac system ( normal insulator ) with equal modulus of @xmath13 . moreover , on the contrary to the @xmath208 case , @xmath104 is enhanced at large @xmath51 with respect to the massless , particle - hole symmetric limit . this behavior is due to the enhanced overlap factor between electron and hole bands in the topological insulator phase . in fig . [ fig : d](a - c ) , value.,width=302 ] we analyze the effects of a finite value of the parameter @xmath151 in the bhz model , for @xmath209 , @xmath62 and @xmath210 . a finite @xmath151 breaks particle - hole symmetry by changing the effective masses of conduction and valence bands . we only found quantitative changes to @xmath104 , which is progressively reduced for increasing @xmath151 . in a massless dirac system , where the static polarization is linear in @xmath192 [ eq . ( [ eq : pi_dirac_static ] ) ] , the dielectric function is a constant @xmath211 therefore the intrinsic polarization contribution can be absorbed into an effective background dielectric constant @xmath212 . as a consequence , a test charge @xmath213 , placed at the origin , induces a screening electronic density @xmath214= z e \left ( \frac{1-\varepsilon}{\varepsilon}\right),\ ] ] which in real space corresponds to a screening image charge [ a fraction @xmath215 of the external one ] placed exactly at the same position @xmath216 note that the screening charge only due to the electronic system ( without background contribution ) is a fraction @xmath217 of the external one . in a massive dirac system , the large @xmath192 behavior of @xmath191 reproduces the massless limit and therefore a screening charge given by eq . ( [ eq : charge_dirac ] ) is also developed at vanishing distances @xmath218 in response to an external test charge . however , in the long wave length limit ( @xmath219 ) @xmath191 has a superlinear behavior and thus @xmath220 . thus an induced charge density of the same sign as the external charge is developed at finite distances @xcite [ summing up to @xmath221 , so that the test charge feels only the background screening over long distances , as expected in an insulator . for the bhz model , we find similar to eq . ( [ constant screening dirac ] ) @xmath222 in the long wavelength limit , but @xmath223 . in order to understand this , we discuss next the induced charge density in real space for the bhz model . it is given by @xmath224 with @xmath225 a natural charge density constant of the model . we note that @xmath226 is proportional to @xmath227 and @xmath70 , but @xmath226 has an additional dependence on @xmath70 ( and thus on @xmath11 ) through its integrand . it also parametrically depends on @xmath167 and @xmath151 through @xmath104 and @xmath228 . in fig . [ fig : n_int ] , due to a test charge in the intrinsic limit of the bhz model for @xmath229 . the plot is invariant under a change of @xmath16 parameter and only depend on the effective fine structure constant @xmath70 . in the inset , @xmath230 calculated for @xmath231 and finite dirac mass.,width=302 ] we plot the induced charge density @xmath226 in real space for @xmath229 with different values of @xmath70 . opposite to a dirac system , the induced charge density has a finite extent over a distance of the order of @xmath232 , which is clearly related to the decay of @xmath233 at large wavevector due to the presence of quadratic @xmath16 terms . @xmath226 decays at large distances as @xmath234 . an electron far away from this induced charge , @xmath235 , does not see the finite extent of it and is therefore screened in the same way as in the dirac system , leading to the similarity of eqs . ( [ constant screening dirac ] ) and ( [ constant screening bhz ] ) . in the opposite limit where the electron sits on top of the induced charge , @xmath236 , it does not feel it at all , resulting in no screening besides @xmath237 . in the inset of fig . [ fig : n_int ] , we study the effect of a finite dirac mass term . with a finite @xmath167 , the induced density ( as in the case of pure dirac systems ) shows a qualitatively different behavior . @xmath226 changes sign for sufficiently large @xmath218 , ensuring a vanishing total induced charge . from a quantitative point of view , a finite negative ( positive ) @xmath13 enhances ( suppresses ) the features of @xmath226 , due to its effect on the interband overlap factor . an analytic discussion of the polarization function is only possible in the limit @xmath197 . here , we focus on the limit of vanishing mass @xmath238 to extract an analytic formula of the plasmon dispersion . an expansion of @xmath239 in @xmath51 gives , for @xmath240 @xmath241 where one finds an @xmath98 behaviour with an additional logarithmic correction for the real part in the high frequency limit . calculating the plasmon dispersion by performing an expansion of eq . ( [ eq : plasmon equation ] ) up to second order in @xmath242 , one finds the linear dispersion @xmath243 which is only valid for sufficiently large @xmath70 , such that the conditions @xmath244>0 $ ] and @xmath240 are fulfilled . the linearity of the dispersion follows from eq . ( [ eq : pi undoped expanded ] ) only by inclusion of the damping via @xmath1 . without the substitution @xmath245 , @xmath246=0 $ ] has no sensible solution for @xmath183 . the damping ratio is given by @xmath247 underlining the importance of damping in this limit . the plasmon is only well defined for a finite @xmath248 , with @xmath249 where @xmath250 sets the limit for the detectability of the plasmons , for example in the recent experiment @xcite @xmath251 was shown to be of the order @xmath252 . ( [ eq : plasmonfrequency linear ] ) translates this into a finite momentum scale @xmath253 with the intrinsic plasmon length scale @xmath254 , given by the coulomb interaction strength times the charge decay length @xmath255 , see sec . [ sec : screening_int ] . we interpret @xmath256 as the length scale up to which charge separation due to coulomb interaction can occur and give rise to the interband plasmons , in an undoped and therefore overall neutral system . in the opposite limit of high frequencies , the term @xmath257 spoils a simple @xmath258 behaviour of the plasmon dispersion . in this limit , we can extract the analytic form of the damping rate @xmath259 with euler s number @xmath145 , yet the plasmon dispersion can only be calculated numerically . in the following discussion of the different excitation spectra , we will use these analytic results to check our numerics in the limits of small momenta and low and high frequencies . the non - interacting single - particle excitation spectrum is given by @xmath96 , which we plot in fig . [ fig : intrinsic spectra ] ( a ) ( a ) and @xmath180 for @xmath260 ( b ) and @xmath261 ( c ) . ( d ) shows linecuts for fixed @xmath262 with @xmath263 in black solid , red dot - dashed , blue long dashed and green short dashed lines , respectively . @xmath173 and @xmath174 . [ fig : intrinsic spectra],title="fig:",width=158 ] ( a ) and @xmath180 for @xmath260 ( b ) and @xmath261 ( c ) . ( d ) shows linecuts for fixed @xmath262 with @xmath263 in black solid , red dot - dashed , blue long dashed and green short dashed lines , respectively . @xmath173 and @xmath174 . [ fig : intrinsic spectra],title="fig:",width=158 ] ( a ) and @xmath180 for @xmath260 ( b ) and @xmath261 ( c ) . ( d ) shows linecuts for fixed @xmath262 with @xmath263 in black solid , red dot - dashed , blue long dashed and green short dashed lines , respectively . @xmath173 and @xmath174 . [ fig : intrinsic spectra],title="fig:",width=158 ] ( a ) and @xmath180 for @xmath260 ( b ) and @xmath261 ( c ) . ( d ) shows linecuts for fixed @xmath262 with @xmath263 in black solid , red dot - dashed , blue long dashed and green short dashed lines , respectively . @xmath173 and @xmath174 . [ fig : intrinsic spectra],title="fig:",width=158 ] for @xmath173 and @xmath174 . due to energy conservation , there are no excitations beneath a frequency @xmath264 . in contrast to graphene , where one observes a diverging behaviour of the polarization at @xmath264 , here @xmath96 increases continously from @xmath62 . this is due to the broken particle - hole symmetry ( @xmath265 ) which ensures that the lowest energy excitations correspond to processes exciting particles from the valence band to the proximity of the dirac point , where , however , the density of states is zero . the excitation spectrum shows a maximum for small momenta @xmath202 which lies beneath the plasmon dispersion given by the black line , perturbatively calculated from eq . ( [ eq : plasmon equation ] ) up to order @xmath266 for @xmath260 . considering a finite coulomb interaction , the excitation spectrum is given by @xmath180 plotted in fig . [ fig : intrinsic spectra ] ( b ) for @xmath260 and ( c ) for @xmath261 . the maximum of the spectrum shifts to higher energies compared to the non - interacting one , indicating the formation of a collective excitation in the system , i.e. a plasmon . this is proven by solving the plasmon equation ( eq . ( [ eq : plasmon equation ] ) ) perturbatively up to order @xmath267 , with the dispersion plotted as a black line on top of the spectrum . additionally , the dispersion based on the expansion of @xmath239 in the limit @xmath197 for @xmath268 and @xmath148 are plotted as gray lines in fig . [ fig : intrinsic spectra](c ) . the plasmon dispersion relation starts linearly for small @xmath192 , as one would expect for a neutral system without doping . at high energies on the other hand , a free - particle behaviour could be expected , leading to the usual @xmath269 dispersion known from doped systems . although eq . ( [ eq : pi undoped expanded ] ) shows that this picture is only partly true due to the logarithmic correction of @xmath93 , fig . [ fig : intrinsic spectra ] ( c ) indicates a qualitative agreement . [ fig : intrinsic spectra ] ( d ) shows linecuts of @xmath180 for fixed @xmath262 with @xmath263 . additionally , the black vertical lines indicate the plasmon frequency for @xmath270 ( left line ) and @xmath261 ( right line ) . for @xmath271 the maximum of the interacting spectrum lies between the maximum of the non - interacting spectrum and the plasmon frequency , indicating that single - particle and collective excitations are equally strong . increasing the interaction to @xmath260 , the maximum of the interacting spectrum and the plasmon frequency almost coincide , therefore the plasmon dominates over the single - particle excitation . at very large interactions @xmath261 , the plasmon is the only relevant excitation in the system . increasing the coulomb interaction broadens the plasmon peak and reduces its height as shown in fig . [ fig : intrinsic spectra ] ( d ) . this seems contrary to the picture of a plasmon as a sharp interaction - induced charge resonance , suggesting that these interband plasmons may not be well - defined for high energies . yet this is a false conclusion . in sec . [ f - sum rule interpretation exc spectrum ] we discussed that the contribution of the resonance to the f - sum rule is the actual measure of importance of a resonance . it can be estimated by multiplying the peak height in @xmath180 by @xmath183 , while the relevant peak width is given by @xmath186 . the latter is decreasing with @xmath183 according to eq . ( [ eq : damping expansion ] ) . from this normalization of the peak we conclude that the discussed interband plasmons fulfill the interpretation as sharp interaction - induced charge resonances , with the width @xmath186 decreasing with increasing plasmon frequency , above the critical frequency @xmath272 as defined in sec . [ sec : expansion of pi ] . the f - sum rule provides a check for our numerics . in fig . [ fig : f - sum numerical undoped ] , with @xmath273 being the difference between the numercial and analytical f - sum rule . black dots are for the non - interacting spectrum , while blue stars stand for @xmath260 and green triangles for @xmath261 . @xmath274 , @xmath175 , @xmath173 and @xmath174 . the deviations around @xmath275 stem from numerical instabilities , which are however negligibly small . [ fig : f - sum numerical undoped],width=226 ] we plot the ratio @xmath276 , with @xmath277 where @xmath278 is the numerical calculated f - sum rule and @xmath279 the analytic one . the deviation are of the order @xmath280 , comparable to the analytical uncertainty , see sec . [ f - sum rule introduction ] , and thus negligible . the f - sum has to be the same for interacting and not interacting systems . we find a slight dependence on the interaction strength @xmath70 , which could be a numerical artifact , depending on @xmath146 , or a real @xmath70 dependence like in graphene , where spectral weight is missing for small frequencies , cf . ( 14 ) in ref . ( @xmath281 for the undoped dirac model ) . as the effect declines with increasing cutoff @xmath146 , we conclude that the rpa approximation in the bhz model misses no spectral weight compared to the full coulomb interaction , even in the undoped limit . in this section , we extend our analysis to finite doping @xmath282 , where a net charge density is present in the system . doping the system has two effects : one is the fermi blocking of interband excitations ( red arrow in fig . [ fig : ab bandstructure ] ( a ) ) for small @xmath51 and @xmath52 . the other is the appearance of intraband excitations ( green arrow in fig . [ fig : ab bandstructure ] ( a ) ) , which are absent in the intrinsic limit . again , first we study the polarization and screening properties of the system in the static limit , where we also study friedel oscillations due to the scattering on a charged impurity . then , we study the dynamical polarization function in the long - wavelength limit , where we obtain an analytical expression for the collective plasmonic modes of the system . finally we numerically compute the dynamical polarization function in the full range of momenta and frequencies in the full parameter space of the bhz model , analyzing the effect of each of the model parameters . particular emphasis is put on the coexistence of interband and intraband plasmons and on how the bhz model interpolates between the dirac and 2deg behavior . in fig . [ fig : normalized ] , of the bhz model for @xmath283 at finite doping for different value of @xmath284 , normalized by the dos @xmath285 . in the inset , details on the value of @xmath286 as a function of @xmath284 are given . , width=302 ] we present the static polarization function @xmath287 at finite doping , conveniently normalized by the density of states at the fermi level @xmath285 . this normalization stands out naturally from the long wavelength property of the polarization function @xmath288 for the bhz model at finite doping , @xmath289 has a pronounced dependence on the extrinsic parameter @xmath290 . for @xmath291 ( @xmath292 ) the fermi level falls in a region where locally the dispersion curve has predominant dirac ( 2deg ) character . in a 2deg system , the static polarization assumes the following analytic form @xcite @xmath293 while in the dirac limit we have @xcite @xmath294.\end{aligned}\ ] ] our calculations for the bhz model with @xmath283 correctly reproduce the dirac and 2deg limits for @xmath291 and @xmath292 , respectively . we note that with a finite @xmath16 term and nonzero @xmath284 the polarization will always have a decay behavior for @xmath295 . in the 2deg and dirac limit one finds @xmath296 for @xmath297 , coincidence due to the balancing effect of dispersion curve and overlap factor . interestingly , in the bhz model we observe instead a deviation from unity , shown in details in the inset of fig . [ fig : normalized ] , which has a maximum for @xmath298 . in the 2deg limit , @xmath299 , @xmath300 has a strong discontinuity in its first derivative at @xmath301 , while for decreasing @xmath284 this discontinuity decreases and finally vanishes in the dirac limit , where the discontinuity affects only the second derivative . we already analyzed in section [ sec : screening_int ] the intrinsic response of a bhz system to a test charge , when no net charge density is present in the system . while the intrinsic response is realized on intrinsic scales of the model @xmath232 , the metallic response ( at finite electronic density ) is characterized by the fermi wave length @xmath302 . therefore it is convenient to express @xmath114 as a function of dimensionless units @xmath303 , due to the presence of a discontinuity at @xmath304 . the induced charge density @xmath305 is given by @xmath306 where , using the property eq . ( [ eq : lwl_pi ] ) , we have emphasized the dependence of the induced density on the dos at the fermi level , which now appears in the scaling factor @xmath307 . we note that the integral also depends on the parameters @xmath70 and @xmath284 ( and naturally on @xmath167 and @xmath151 , when finite ) . and @xmath308 , @xmath252 and @xmath309 . data in different panels belong to systems with @xmath310 , @xmath252 , @xmath309 , @xmath311 , @xmath312 and @xmath313 . all calculation are obtained by keeping @xmath314 and @xmath315 nm@xmath316 , and varying @xmath317 , @xmath318 and @xmath319 ev nm , for @xmath308 , @xmath252 and @xmath309 , respectively , while varying the parameter @xmath16 accordingly to @xmath284 . in the panel @xmath320 ( 2deg limit ) , the three curves with @xmath308 , @xmath252 and @xmath309 are quite close and correspond to similar @xmath321 parameter ( @xmath322 , @xmath323 and @xmath324 , respectively ) . , width=321 ] in fig . [ fig : n_dop ] , we present the induced screening electronic radial density for the bhz model for @xmath283 due to a point - like test charge . each panel corresponds to a different value of the ratio @xmath290 , and within each panel curves differing by the dirac fine structure constant @xmath70 are presented . friedel oscillations appear of period @xmath302 , which become more defined for larger @xmath70 . we also note that density oscillations are more prominent for @xmath325 than in the dirac ( @xmath326 ) and 2deg limits ( @xmath327 ) . in the 2deg limit the @xmath70 parameter is ill defined and should be replaced by the more general parameter @xmath328 , characterizing the dielectric response of the system . the presence of friedel oscillations and their asymptotic behavior are related through the lighthill theorem @xcite to discontinuities in the static polarization function and its derivatives ( see for example ref . for a detailed discussion ) . a discontinuity like @xmath329 in @xmath330 , with @xmath331 the heaviside step function and @xmath332 , translates into a decay of the oscillations in @xmath333 with leading order @xmath334 . one finds @xmath335 ( @xmath336 ) for the leading order discontinuity of a 2deg ( dirac ) system , such that the first ( second ) and all higher derivatives of the static polarization function are discontinuous at @xmath337 . analyzing the friedel oscillations for the bhz model , one finds a composition of two different contributions with an asymptotic decay at large distances as @xmath234 ( 2deg contribution ) and @xmath338 ( dirac contribution ) , respectively . as a consequence , the discontinuity in the rpa polarization function of the bhz model at @xmath339 can be very well approximated by a combination of 2deg ( @xmath335 ) and dirac ( @xmath336 ) contribution . in the dirac ( 2deg ) limit , the effect of the discontinuity in the second ( first ) derivative becomes predominant and oscillations purely decay in leading order as @xmath338 ( @xmath234 ) . at finite doping , for small momenta @xmath51 , the polarization function is governed by intraband excitations , as the interband excitations are fermi - blocked . we perform an expansion in this limit , for @xmath240 , to gain an analytical insight into the physics at finite doping and derive an analytical formula for the plasmon dispersion . in particular , intraband plasmons are expected to be the dominant excitation for small momenta , similarly to the 2deg and dirac case . we expand the polarization function up to order @xmath340 @xmath341= \pi_{44}\frac{x^{4}}{\omega^{4}}+\pi_{42}\frac{x^{4}}{\omega^{2}}+\pi_{40}x^{4}\\ + \pi_{22}\frac{x^{2 } } { \omega^{2 } } + \pi_{20}x^{2}+\mathcal{o}\left(\omega^{2}\right)\end{aligned}\ ] ] and use it to solve eq . ( [ eq : plasmon equation ] ) . we obtain a plasmon dispersion of @xmath342 with the leading coefficient @xmath343 with @xmath144 $ ] and @xmath344 . in the limit of zero mass , @xmath345 interpolates smoothly between @xmath62 for @xmath346 and @xmath347 for @xmath348 . the former case corresponds to the dirac limit , where one finds the plasmon frequency @xmath349 in the literature @xcite , being identical to eqs . ( [ eq : plas_freq_bhz],[eq : pi doped expanded dirac limit ] ) . the latter case is the 2deg limit , where one finds the plasmon dispersion @xmath350 in the literature @xcite , with @xmath351 the carrier density and @xmath352 . this in in agreement with eqs . ( [ eq : plas_freq_bhz],[eq : pi doped expanded 2deg limit ] ) . thus the bhz model as a function of its parameters reproduces the plasmon dispersion in the dirac and 2deg limits and interpolates between them . we note that for @xmath353 the term @xmath354 is zero and the intraband plasmon disappears . in this limit , the leading order contribution @xmath355 of the intrinsic polarization , eq . [ eq : pi undoped expanded ] , takes the place of @xmath354 . the crucial difference between the extrinsic and the intrinsic polarization is that the latter has a finite imaginary part of order @xmath355 , leading to the linear dispersion of the interband plasmons . yet for finite @xmath356 , these interband plasmons are supressed due to the fermi blockade of the interband excitations and only exist if their plasmon frequency exceeds both the chemical potential @xmath357 and the critical frequency @xmath272 as defined in sec . [ sec : expansion of pi ] , see for example fig . [ fig : im epsi bhz xf01 ] . besides the different scaling with momenta in the limit @xmath358 , also the scaling with @xmath70 is different for the inter- and intraband plasmons , eqs . ( [ eq : plasmonfrequency linear ] ) and ( [ eq : plas_freq_bhz ] ) : linear vs. square root . this will have important consequences in the following when we will discuss how to separate the two different collective excitations . we begin the discussion of the doped spectrum by looking at the limiting results of 2deg and dirac system . from this , we then find that we can interpolate between them by changing the fermi momentum . interestingly , by considering the cases of broken particle - hole symmetry and large masses , we also find regimes which are distinct from the dirac and 2deg limit . as an example , these regimes support both inter- and intraband plasmons at parameters which are realistic for hgte qws . in all the following plots , the boundaries of the single - particle spectrum will be indicated by faint black lines , the isolines @xmath95 by red lines . the plasmon dispersions are plotted as black curves ( full result from perturbation theory ) and gray curves ( expanded result in limit @xmath197 ) . in the 2deg limit , only intraband excitations are possible . the polarization function has a well - known analytical form @xcite , therefore we can easily plot the non - interacting spectrum in fig . [ fig : pi 2deg ] ( a ) . with @xmath359 and @xmath360 the degeneracy factor . ( b ) @xmath180 for @xmath361 , with @xmath362 the coulomb interaction . we add an artificial damping in the region of @xmath363 to make the plasmons visible . [ fig : pi 2deg],title="fig:",width=158 ] with @xmath359 and @xmath360 the degeneracy factor . ( b ) @xmath180 for @xmath361 , with @xmath362 the coulomb interaction . we add an artificial damping in the region of @xmath363 to make the plasmons visible . [ fig : pi 2deg],title="fig:",width=158 ] @xmath96 is peaked for @xmath364 closely to the upper boundary of the spectrum . it decays to zero instead for large momenta and frequencies like @xmath365 , if one considers a fixed ratio @xmath366 within the spe region . the interacting spectrum is shown in fig . [ fig : pi 2deg ] ( b ) . an intraband plasmon appears with the usual @xmath269 dispersion for @xmath358 . it absorbes all of the spectral weight in this limit , thus @xmath180 is suppressed in the spe region . for intermediate momenta , the plasmon dispersion lies in the spe region and the plasmon decays and broadens . for larger momenta and frequencies , the interacting and non - interacting spectra agree qualitatively . the dirac spectrum comprises both inter- and intraband excitations . the polarization function still has a well - known analytical expression @xcite , of which we plot the non - interacting spectrum @xmath96 in fig . [ fig : pi graphene ] ( a ) . with @xmath367 and @xmath360 the degeneracy factor . in the gray area , the colorscale is exceeded due to the divergency of @xmath96 . ( b ) @xmath180 for @xmath368 , with @xmath362 the coulomb interaction . we add an artificial damping in the regions of @xmath363 to make the plasmons visible . [ fig : pi graphene],title="fig:",width=158 ] with @xmath367 and @xmath360 the degeneracy factor . in the gray area , the colorscale is exceeded due to the divergency of @xmath96 . ( b ) @xmath180 for @xmath368 , with @xmath362 the coulomb interaction . we add an artificial damping in the regions of @xmath363 to make the plasmons visible . [ fig : pi graphene],title="fig:",width=158 ] the intraband excitations occur for higher energies @xmath369 , while for intraband excitations less energy is needed , @xmath370 . both excitation spectra touch at @xmath371 , where they diverge . only the fermi - blockade suppresses the interband transitions in @xmath96 for @xmath372 and cures the divergency , see fig . [ fig : pi graphene ] ( a ) for @xmath369 . one finds a @xmath373 decay for high frequencies . the interacting spectrum @xmath180 is plotted in fig . [ fig : pi graphene ] ( b ) for @xmath374 . similar to the 2deg , all of the intraband spectral weight is absorbed by a plasmon in the limit @xmath358 and the divergence at @xmath371 is cured . interestingly , for sufficient large interaction strength @xmath70 the plasmon decays in the interband spectrum . for larger momenta and frequencies , we note that the intraband polarization does not recover the non - interacting value , as it does for the 2deg , but remains much smaller . therefore single - particle intraband excitations are blocked altogether for all momenta and frequencies in this limit . the missing spectral weight goes into a charge resonance at higher frequencies in the interband spectrum @xcite . yet , this resonance is not a solution of the plasmon equation and therefore not a plasmon @xcite . the bandstructure of the bhz model without mass and particle - hole symmetry breaking is shown in fig . [ fig : ab bandstructure ] ( a ) . . interband spectrum in red , intraband spectrum in blue and mixed area in purple . [ fig : ab bandstructure],title="fig:",width=158 ] . interband spectrum in red , intraband spectrum in blue and mixed area in purple . [ fig : ab bandstructure],title="fig:",width=158 ] . interband spectrum in red , intraband spectrum in blue and mixed area in purple . [ fig : ab bandstructure],title="fig:",width=158 ] . interband spectrum in red , intraband spectrum in blue and mixed area in purple . [ fig : ab bandstructure],title="fig:",width=158 ] the interband single - particle excitations lying lowest in energy are symmetric in momentum as shown by the red arrow in fig . [ fig : ab bandstructure ] ( a ) , going from @xmath376 to @xmath377 . due to particle - hole symmetry , this leads to nesting and thus one expects these excitations to dominate the interband spectrum . interband excitations as indicated by the dashed , black arrow on the other hand , going from momentum @xmath378 to @xmath379 with @xmath380 , are suppressed due to imperfect nesting of the different sized electron and hole cones , as well as by a small overlap factor . the latter can be cured by introducing a large negative mass , as will be shown in sec . [ sec_bhz_mass ] . then these excitations have a considerable influence onto the polarization for small energies , helping with the formation of interband plasmons , following the ideas presented in sec . [ ( anti-)screening and intrinsic plasmons ] . in the pure dirac system , these processes are forbidden by helicity . by varying the doping level we can modify the excitation spectrum of the system [ see figs . [ fig : ab bandstructure ] ( b)-(d ) ] to resemble that of a dirac system ( @xmath381 ) or of a 2deg ( @xmath382 ) , or to obtain an intermediate behavior ( @xmath383 ) . in the pictures we highlight the boundaries of the excitation spectra , with the red area corresponding to the interband spectrum and the blue area to the intraband spectrum . the overlap between the two is indicated by the purple area . the boundaries of the spectra vary from the linear graphene behavior to the @xmath384 dependence of the 2deg . in general , the mixing of linear and quadratic dispersion leads to an overlap of the inter- and intraband spectrum . this affects the visibility of the interband plasmons , which can be hidden due to strong single - particle damping . [ [ weak - doping - of - x_f0.1 ] ] weak doping of @xmath385 : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + the extrinsic ( @xmath25 ) and intrinsic ( @xmath386 ) scales of the system are separated by one order of magnitude . as the fermi surface lies in the ( almost ) linear part of the spectrum , we expect that on the @xmath25 scale we resemble graphene . the physics on the @xmath386 scale on the other hand should be more or less untouched by the doping , and the system should behave as in the intrinsic limit . we plot @xmath239 in fig . [ fig : imp bhz xf01 ] . . the red line indicates @xmath95 . [ fig : imp bhz xf01],title="fig:",width=158 ] . the red line indicates @xmath95 . [ fig : imp bhz xf01],title="fig:",width=158 ] comparing panel ( a ) to fig . [ fig : pi graphene ] , one finds good agreement with the dirac case . the biggest deviation is found in the peak of @xmath96 at @xmath387 , which is not symmetric as for a dirac system due to the overlap of inter- and intraband spectrum [ fig . [ fig : ab bandstructure ] ( b ) ] . the finite quadratic part in the spectrum cures the divergency formerly occuring in the dirac limit . the real part of @xmath388 is strongly negative only at the upper boundary of the intraband spectrum . this indicates that for small interactions , only one plasmon will dominate the excitation spectrum on the fermi scale . as we are interested in the regime where both inter- and intraband plasmons are visible , we look at the interacting spectrum , given in fig . [ fig : im epsi bhz xf01 ] by plotting @xmath180 , for a strong interaction @xmath261 . for @xmath385 with @xmath261 on the @xmath389 scale ( a ) and the @xmath386 scale ( b ) . we add an artificial damping in the regions of @xmath363 to make the plasmons visible . [ fig : im epsi bhz xf01],title="fig:",width=158 ] for @xmath385 with @xmath261 on the @xmath389 scale ( a ) and the @xmath386 scale ( b ) . we add an artificial damping in the regions of @xmath363 to make the plasmons visible . [ fig : im epsi bhz xf01],title="fig:",width=158 ] on the fermi scale [ panel ( a ) ] , the intraband plasmon absorbs all spectral weight from the intraband spectrum . the dispersion agrees with the perturbative dispersion from the expansion in eq . ( [ eq : plas_freq_bhz ] ) in the limit @xmath197 , plotted as a gray curve . the green , dashed line shows the linear dispersion of the interband plasmon in the undoped limit , based on eq . ( [ eq : plasmonfrequency linear ] ) . on the fermi scale , it is not obvious that there is an interband plasmon , although the interacting polarization function develops a smeared resonance around the perturbative interband plasmon dispersion for high momenta . switching to the intrinsic scale , fig . [ fig : im epsi bhz xf01 ] ( b ) , one finds the interband plasmon , corresponding to the single peak in @xmath390 , unperturbed by doping for momenta much larger than @xmath389 . the dispersion is the same as for a plasmon in the undoped limit @xcite . the two black lines near the peak are just the boundaries of the intraband excitation spectrum , which does not play a role here . as in the limit of @xmath197 the interband plasmon dispersion scales linearly with @xmath70 , @xmath391 see eq . ( [ eq : plasmonfrequency linear ] ) , while the intraband plasmon frequency is proportional to @xmath392 , lowering the interaction strength will lead to an overlap of the two resonances below some critical @xmath70 . [ [ strong - doping - of - x_f3 ] ] strong doping of @xmath393 : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + strong doping of the system significantly increases the spectral weight , as shown in fig . [ fig : f - sum rule ] ( b ) , with the increase of intraband excitations , while most of the interband excitations are fermi - blocked , leading therefore to an effective decoupling of the two bands . we expect the overall spectrum to be governed by intraband excitations and to resemble the spectrum of a 2deg , as the fermi surface lies in the ( almost ) quadratic part of the spectrum . the corresponding @xmath239 is plotted in fig . [ fig : imp bhz xf3 ] . . the red line indicates @xmath95 . [ fig : imp bhz xf3],title="fig:",width=158 ] . the red line indicates @xmath95 . [ fig : imp bhz xf3],title="fig:",width=158 ] the single - particle spectrum in panel ( a ) is peaked at small momenta and at energies close to the upper bound of the intraband spectrum . the interband part of the spectrum leads only to minor deviations from the 2deg case [ compare with fig . [ fig : pi 2deg ] ( a ) ] . the real part of @xmath388 in panel ( b ) is strongly negative at the upper boundary of the intraband spectrum , indicating that only a single intraband plasmon will dominate the interacting spectrum . we additionally note that the static limit property for which the polarization is a constant @xmath394 for @xmath395 , discussed in sec . [ static limit ] , extends also to an area of finite @xmath52 . the interacting spectrum is shown in fig . [ fig : im epsi bhz xf3 ] by plotting @xmath180 for the interaction strength @xmath261 . for @xmath393 and @xmath261 . we add an artificial damping in the regions of @xmath363 to make the plasmons visible . [ fig : im epsi bhz xf3],width=158 ] even for this large coulomb interaction , we only find the intraband plasmon . this is as expected due to the combined effects of fermi blocking of interband excitations and increased spectral weight for intraband transitions . the interband plasmon lies in the large overlap of inter- and intraband spectrum , cf . [ fig : im epsi bhz xf01 ] ( b ) , and it is therefore heavily damped and not visible in the overall spectrum . [ [ intermediate - doping - of - x_f1 ] ] intermediate doping of @xmath396 : + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + for intermediate doping levels like @xmath396 , a mixture of dirac and 2deg behaviour is expected , due to the similar importance of inter- and intraband excitations . we plot the polarization function @xmath239 in fig . [ fig : imp bhz xf1 ] . . the red line indicates @xmath95 . [ fig : imp bhz xf1],title="fig:",width=158 ] . the red line indicates @xmath95 . [ fig : imp bhz xf1],title="fig:",width=158 ] indeed , the single - particle spectrum in panel ( a ) looks like a combination of figs . [ fig : imp bhz xf01 ] ( a ) and [ fig : imp bhz xf3 ] ( a ) . while the the shape of the polarization resembles the one of the 2deg , the interband spectrum is now more pronounced and even dominating for @xmath397 . therefore we could expect both kinds of excitations giving rise to a plasmon mode . the real part of @xmath388 in panel ( b ) shows again just a single minimum , following the upper boundary of the intraband spectrum . the deviations from the constant behaviour @xmath394 for @xmath395 in the case of intermediate doping , see sec . [ static limit ] , are also found for finite @xmath52 . the interacting spectrum is shown in fig . [ fig : im epsi bhz xf1 ] by plotting @xmath180 for an interaction strength @xmath261 . for @xmath396 with @xmath261 . we add an artificial damping in the regions of @xmath363 to make the plasmons visible . [ fig : im epsi bhz xf1],width=158 ] it is dominated by a single resonance , lying above the intraband part of the single - particle spectrum . for small momenta , this resonance corresponds to the intraband plasmon . yet for intermediate momenta , a comparison with the interband plasmon dispersion in fig . [ fig : im epsi bhz xf01 ] ( b ) indicates that also the interband plasmon contributes to the resonance . a clear distinction between the two is then not possible anymore . in summary , doping the system offers the possibility to change the excitations spectrum on the fermi scale from a dirac to a 2deg type . the interacting excitation spectrum is usually governed by a single intraband plasmon , while the interband plasmon is hidden in the single - particle background . only large interaction strengths offer a possibility to see both plasmons in the spectrum . in the following , we will now analyze the influence of both broken p - h symmetry and finite masses , which both offer a way to separate the two plasmons and make them visible in the total spectrum . a broken particle - hole symmetry , @xmath398 , with small or vanishing mass is the experimental relevant case for hgte qws . it also offers the possibility of blocking the interband spe spectrum close to the minimal excitation energy @xmath387 , resulting in less damped interband plasmons @xcite . here , we want to use a similar effect for the intraband excitations in order to separate the inter- and intraband spectrum as well as the two plasmon modes . the broken p - h symmetry introduces an inflection point into the spectrum , @xmath399 , with momentum @xmath400 and energy @xmath401 . for @xmath402 , it lies in the hole part [ @xmath92 ] of the spectrum . with a sufficiently small fermi momentum , @xmath404 , the highest energy intraband excitations involve the dirac point for momenta on the order of the fermi momentum , see fig . [ fig : bhz spectrum finite d ] . . both low energy interband excitations ( red arrow ) and high energy intraband excitations ( green arrow ) involve the dirac point . [ fig : bhz spectrum finite d],width=207 ] the same is true for the lowest energy interband excitations . due to the vanishing density of states at the dirac point , both kind of excitations are suppressed , and therefore inter- and intraband spe spectrum are effectively separated in energy and momentum . this situation is shown in fig . [ fig : polarization bhz abkinf12 ] ( a ) ( a ) and @xmath93 ( b ) for @xmath405 and @xmath174 . the red line indicates @xmath95 . [ fig : polarization bhz abkinf12],title="fig:",width=158 ] ( a ) and @xmath93 ( b ) for @xmath405 and @xmath174 . the red line indicates @xmath95 . [ fig : polarization bhz abkinf12],title="fig:",width=158 ] and fig . [ fig : im eps bhz abkinf12 ] ( c)-(e ) , where the imaginary part of the polarization goes to zero between inter- and intraband parts of the spectrum , fully separating them . there , we choose @xmath406 , @xmath174 and @xmath173 . this blocking effect holds for small momenta up to roughly @xmath407 , indicated by the black vertical line in fig . [ fig : polarization bhz abkinf12 ] at @xmath408 . for larger momenta the high energy intraband excitation go from deep in the valence band directly to the fermi surface - the blocking effect of the dirac point is gone , see dashed arrow in fig . [ fig : bhz spectrum finite d ] . in fig . [ fig : polarization bhz abkinf12 ] ( b ) , @xmath93 shows one major difference in comparison to the p - h symmetric case of weak doping in fig . [ fig : imp bhz xf01 ] ( b ) . at the border of intra- and interband spectrum a strong antiscreening region is formed . for sufficiently low @xmath70 a plasmon should exist there , clearly separated from the second antiscreening region at higher @xmath52 , giving rise to the possibility of observing both intra- and interband plasmons . this can be seen in fig . [ fig : im eps bhz abkinf12 ] , where we plot @xmath180 for @xmath405 , @xmath174 , @xmath173 and @xmath409 ( a ) and @xmath260 ( b ) . ( c)-(e ) show linecuts for fixed @xmath410 , @xmath411 and @xmath412 , respectively , with @xmath413 in black solid line , red dot - dashed line , blue long dashed line and green short dashed line , respectively . the black , vertical line separates the inter- and intraband spe region . [ fig : im eps bhz abkinf12],title="fig:",width=158 ] for @xmath405 , @xmath174 , @xmath173 and @xmath409 ( a ) and @xmath260 ( b ) . ( c)-(e ) show linecuts for fixed @xmath410 , @xmath411 and @xmath412 , respectively , with @xmath413 in black solid line , red dot - dashed line , blue long dashed line and green short dashed line , respectively . the black , vertical line separates the inter- and intraband spe region . [ fig : im eps bhz abkinf12],title="fig:",width=158 ] for @xmath405 , @xmath174 , @xmath173 and @xmath409 ( a ) and @xmath260 ( b ) . ( c)-(e ) show linecuts for fixed @xmath410 , @xmath411 and @xmath412 , respectively , with @xmath413 in black solid line , red dot - dashed line , blue long dashed line and green short dashed line , respectively . the black , vertical line separates the inter- and intraband spe region . [ fig : im eps bhz abkinf12],title="fig:",width=158 ] for @xmath405 , @xmath174 , @xmath173 and @xmath409 ( a ) and @xmath260 ( b ) . ( c)-(e ) show linecuts for fixed @xmath410 , @xmath411 and @xmath412 , respectively , with @xmath413 in black solid line , red dot - dashed line , blue long dashed line and green short dashed line , respectively . the black , vertical line separates the inter- and intraband spe region . [ fig : im eps bhz abkinf12],title="fig:",width=158 ] for @xmath405 , @xmath174 , @xmath173 and @xmath409 ( a ) and @xmath260 ( b ) . ( c)-(e ) show linecuts for fixed @xmath410 , @xmath411 and @xmath412 , respectively , with @xmath413 in black solid line , red dot - dashed line , blue long dashed line and green short dashed line , respectively . the black , vertical line separates the inter- and intraband spe region . [ fig : im eps bhz abkinf12],title="fig:",width=158 ] for @xmath409 ( a ) and @xmath260 ( b ) . panel ( c ) - ( e ) show line cuts for fixed momenta @xmath414 and different interaction strengths @xmath413 . for large interaction strength @xmath409 , the intraband plasmon decays into the interband spe spectrum , see panel ( a ) for @xmath415 and the green short dashed line in panel ( c ) . most of the spectral weight stays there also for larger momenta , as @xmath180 is close to 0 in the intraband spe region and the resonance between inter- and intraband spe spectrum is weak . the latter can be best seen in the insets of panel ( d ) and ( e ) , represented by the green short dashed line peaked slightly above [ ( d ) ] or below [ ( e ) ] the black vertical line separating intra- and interband spe region . yet even with the peak being small , it indicates the formation of a slighly damped plasmon , but with small spectral weight . the missing spectral weight is transferred to higher energies into the interband spe region . for intermediate momenta , a second plasmon branch forms , see panel ( a ) for @xmath416 and the second peak of the green short dashed line in the inset of panel ( d ) . for even higher momenta , @xmath417 , it overlaps with the forming interband plasmon leading to a broad charge resonance without clear peak , see green short dashed line in panel ( e ) for @xmath418 . the picture changes for smaller interaction strength . for @xmath260 , the intraband plasmon decays in the region between inter- and intraband spe spectrum , indicated by the strong peak of the blue long dashed line in panel ( c ) . as the single - particle excitations in this region are suppressed due to the dirac point , the plasmon leads to a high and narrow peak of @xmath180 . considering larger momenta @xmath419 , the resonance is split : one part forms an intraband plasmon in the intraband spe region , see blue long dashed line peaked slightly below the black vertical line in panels ( d ) and ( e ) . the second part stays in the interband spe region , where it enhances the spe peak [ black line in the inset of panel ( d ) ] for intermediate momenta @xmath416 . for momenta @xmath420 , an interband plasmon forms , as shown in panel ( e ) . there , the broad single - particle peak [ black line ] around @xmath421 gets reshaped into a clear peaked resonance [ blue long dashed line ] - the interband plasmon . taking the experimental parameters from sec . [ sec : experimental_parameters ] , @xmath119 and @xmath120 , one finds for the plots in fig . [ fig : im eps bhz abkinf12 ] the fermi momentum @xmath422 and chemical potential @xmath423 . the plot range is therefore @xmath424q_{0}=\left[0,0.3\right]\frac{1}{\mathrm{nm}}$ ] and @xmath425\mathrm{\frac{e_{0}}{\hbar}}=\left[0,210\right]\mathrm{thz}$ ] and thus of the right order of magnitude for experimental techniques like raman spectroscopy or electron loss spectroscopy . both plasmonic resonances in fig . [ fig : im eps bhz abkinf12 ] ( b ) overlap for @xmath415 , before they separate for higher momenta . therefore the question arises whether one can really speak of a clear distinction between inter- and intraband plasmons for larger momenta . here , we want to study the f - sum rule and thus the spectral weight of the different excitations . the relative deviations of numerical to analytical f - sum rule are again of the order @xmath280 and thus negligible . [ fig : f - sum numerical doped ] ( a ) of interacting over non - interacting f - sum rule of the interband excitations . black dots are for @xmath271 , blue stars stand for @xmath260 and green triangles for @xmath409 . ( b ) the same for the ratio @xmath426 of the intraband excitations . for all plots : @xmath427 and @xmath428 . [ fig : f - sum numerical doped],title="fig:",width=158 ] of interacting over non - interacting f - sum rule of the interband excitations . black dots are for @xmath271 , blue stars stand for @xmath260 and green triangles for @xmath409 . ( b ) the same for the ratio @xmath426 of the intraband excitations . for all plots : @xmath427 and @xmath428 . [ fig : f - sum numerical doped],title="fig:",width=158 ] shows the ratio of spectral weight in the interband spe region for the interacting over the non - interacting case , @xmath429 , and panel ( b ) the same for the intraband spe region . the intraband plasmon lying between these two regions for @xmath430 is excluded . as for cutoffs @xmath431 one usually has @xmath432 , transfer of spectral weight from one region to the other can lead to quantitatively different relative changes of spectral weight in panels ( a ) and ( b ) . as a key result , we find that there is always spectral weight missing in the intraband spe region . for small momenta , @xmath430 , the weight goes into the undamped intraband plasmon [ this follows directly from the conservation of the f - sum rule for interacting and non - interacting systems ] , while at larger momenta it is transferred to higher energies into the interband spe region . yet , the increase is only about @xmath433 at @xmath420 , such that we can conclude that the plasmon between the inter- and intraband spe region is a pure intraband plasmon with a reduced spectral weight . the plasmon in the interband region is the interband plasmon we know already from the undoped system , see fig . [ fig : intrinsic spectra ] , with a slight increased spectral weight from the intraband spe region . deviations in the thickness of a hg(cd)te qw lead to the opening of a small gap in the bandstructure , resulting in a topological trivial @xmath435 or non - trivial @xmath436 system . apart from the possible appearance of edge states , which is beyond the scope of this paper , a small mass works in opposition to the blocking effect of finite @xmath54 , as it generates a finite density of states for @xmath437 . in the following , we therefore show that the blocking effect of a finite @xmath54 is robust against the opening of small gaps . in fig . [ fig : polarization bhz mabkinf004 ] ( a ) and the interacting one @xmath180 ( b ) with @xmath260 . @xmath438 , @xmath439 and @xmath174 . [ fig : polarization bhz mabkinf004],title="fig:",width=158 ] ( a ) and the interacting one @xmath180 ( b ) with @xmath260 . @xmath438 , @xmath439 and @xmath174 . [ fig : polarization bhz mabkinf004],title="fig:",width=158 ] we plot the non - interacting and interacting spectrum for @xmath174 and a small mass @xmath440 . a comparison with figs . [ fig : polarization bhz abkinf12 ] and [ fig : im eps bhz abkinf12 ] shows that the small mass has just the effect of separating the inter- and intraband spe region additionally . thus we conclude that the idea of observing both plasmons in experiments is robust against slight deviations in the mass and therefore the thickness of the hg(cd)te qw . a finite dirac mass opens a gap in the bandstructure and changes the pseudospin , and therefore the overlap factor , in a non - trivial fashion . thus we can expect in general a quite different behavior for positive and negative mass . yet , for these differences to occur on the intrinsic scale and thus influence the interband plasmons , @xmath441 should be of the order of 1 . in the following , we study such large masses , both negative and positive , with p - h symmetry . while not experimentally relevant for hgte qws , it offers the possibility to study the effect of a topological bandstructure on the electronic excitations , including plasmons . we also note here that the dispersion of the bhz model becomes purely parabolic for the mass @xmath442 : @xmath443 . in this limit , the polarization function , eq . ( [ eq : pi_par ] ) , can be calculated analytically . for the parameters @xmath444 and @xmath60 we plot the polarization @xmath239 in fig . [ fig : im epsi bhz mnab xf033 ] ( a ) and ( b ) . . ( c ) and ( d ) show @xmath180 with @xmath261 and @xmath409 , respectively . @xmath60 and @xmath444 in all plots . [ fig : im epsi bhz mnab xf033],title="fig:",width=158 ] . ( c ) and ( d ) show @xmath180 with @xmath261 and @xmath409 , respectively . @xmath60 and @xmath444 in all plots . [ fig : im epsi bhz mnab xf033],title="fig:",width=158 ] . ( c ) and ( d ) show @xmath180 with @xmath261 and @xmath409 , respectively . @xmath60 and @xmath444 in all plots . [ fig : im epsi bhz mnab xf033],title="fig:",width=158 ] . ( c ) and ( d ) show @xmath180 with @xmath261 and @xmath409 , respectively . @xmath60 and @xmath444 in all plots . [ fig : im epsi bhz mnab xf033],title="fig:",width=158 ] the mass separates intra- and interband spe regions for @xmath445 . compared to the massless cases of @xmath446 , fig . [ fig : imp bhz xf01 ] , and @xmath447 , fig . [ fig : imp bhz xf1 ] , the interband spe spectrum is enhanced due to the combination of enhanced overlap factor and low doping , thus small fermi blockade . due to the flat bandstructure , even for @xmath448 the chemical potential is just barely above the gap . an interesting consequence of this strong interband transition can be seen in panel ( b ) , where we find two distinct areas where @xmath449 becomes negative . as a consequence , inter- and intraband plasmons will always be separated , with the intraband plasmon being confined to low energies . this stems from the fact that the electrons in the conduction band are pseudo - spin polarized , such that intraband excitations to much higher momenta and energies , where the pseudo - spin shows in the opposite direction , are not possible . this is confirmed in panels ( c ) and ( d ) , where we plot @xmath180 with @xmath261 and @xmath409 , respectively . all the spectral weight of the intraband spe region goes into the plasmon , which at least for @xmath409 follows very well the @xmath258 law . the interband spectrum is dominated by the interband plasmon , having of course a much broader peak due to damping ( finite @xmath450 $ ] ) . the dashed , green line in the interband spectrum in fig . [ fig : im epsi bhz mnab xf033 ] indicates the energy , at which excitation processes going from momentum @xmath378 to @xmath379 with @xmath380 are possible , see black , dashed arrow in fig . [ fig : ab bandstructure ] ( a ) . usually suppressed by the overlap factor , a large negative mass enhances the overlap of these excitations to near unity for small fermi momenta . [ fig : im epsi bhz mnab xf033 ] ( b ) and ( d ) show that the interband plasmons mainly occur above this line , indicating that the described excitation process is important for the collective excitation . as the process is forbidden by helicity in the pure dirac system , it is one reason why the bhz model supports intrinsic plasmons while the dirac model does not . for the parameters @xmath444 and @xmath451 we plot the polarization in fig . [ fig : im epsi bhz mab xf033 ] ( a ) and ( b ) . . ( c ) and ( d ) show @xmath180 with @xmath261 and @xmath409 , respectively . @xmath451 and @xmath444 in all plots . [ fig : im epsi bhz mab xf033],title="fig:",width=158 ] . ( c ) and ( d ) show @xmath180 with @xmath261 and @xmath409 , respectively . @xmath451 and @xmath444 in all plots . [ fig : im epsi bhz mab xf033],title="fig:",width=158 ] . ( c ) and ( d ) show @xmath180 with @xmath261 and @xmath409 , respectively . @xmath451 and @xmath444 in all plots . [ fig : im epsi bhz mab xf033],title="fig:",width=158 ] . ( c ) and ( d ) show @xmath180 with @xmath261 and @xmath409 , respectively . @xmath451 and @xmath444 in all plots . [ fig : im epsi bhz mab xf033],title="fig:",width=158 ] compared to the negative mass , the interband spectrum is much weaker . this is a result of the lower overlap factor and the higher chemical potential ( the bandstructure is not as flat as in the ti phase ) , leading to a stronger fermi blockade . for the real part of the polarization , this has the effect that the two former distinct areas of sign reversal now almost merge . the interband excitations are so weak that the minimum @xmath449 always lies closely above the intraband spe region - indicating that it is the main source for plasmons . in panel ( c ) for @xmath261 , one can identify both inter- and intraband plasmon . interestingly , the polarization is clearly higher in the pure interband spe region than in the mixed inter- and intraband spe spectrum , suggesting that the latter one serve as an additional damping for the interband plasmon . going to the smaller interaction strength @xmath409 in panel ( d ) , one finds just a single resonance following the upper boundary of the intraband spe spectrum . thus we conclude that the interacting spectrum for moderate interaction strength is governed by just intraband plasmons . the interband excitations are too weak to support an additional plasmon but for very high interactions - a consequence of the effective decoupling of the bands by the overlap factor . we have analyzed the dynamical and static polarization properties in random phase approximation of hg(cd)te quantum wells described by the bernevig - hughes - zhang ( bhz ) model . in the static undoped limit , due to the presence of quadratic terms in the bhz model and hence to the natural length scale @xmath452 , the induced charge density in response to a test charge has a finite spatial extent . this is in contrast to the point - like screening charge obtained with the continuous dirac model of graphene . in the doped regime , we have observed friedel oscillations with an intermediate decay behavior between the dirac ( @xmath338 ) and the 2deg ( @xmath234 ) cases . the discussion of the full dynamical polarization function has been focused on the appearance of new interband plasmons due to the interplay of dirac and schrdinger physics . in principle , we expect these plasmons to appear in multiband systems where the imaginary part of the polarization function decays faster with energy than the one in the dirac case ( @xmath373 ) , which is the case for the bhz model ( decay as @xmath453 ) . these plasmons appear already in the undoped system at experimentally relevant parameters , but it is also possible to observe them in the doped regime , where they coexists with the usual intraband plasmons . this is favored by broken particle hole symmetry in the bhz model , which allows for the presence of both a dirac point and an inflection point in the bandstructure . the behavior of these two collective modes is also influenced by the topology of the bandstructure . indeed the two plasmons tend to merge into one another in a gapped trivial insulator , while they remain distinct resonances in the topological insulator phase . we have shown that these new plasmons should appear for momenta and energies on the right order of magnitude for experimental techniques like raman spectroscopy or electron loss spectroscopy on hg(cd)te quantum wells . the wide range of parameters considered in this paper , including the regime of topological trivial and non - trivial insulators , should make our results applicable to all kinds of materials described by phenomenological models interpolating between dirac and schrdinger fermion physics . throughout this article , we have only discussed bulk excitations of this peculiar two - dimensional system . hence , we have totally ignored the influence of edge states in the topologically non - trivial regime of the model in the presence of physical boundaries . an extension of our analysis to finite size systems might yield exciting new physics , where we expect an interplay of one - dimensional and two - dimensional collective charge excitations . we acknowledge interesting discussions with e. hankiewicz , m. polini , t. stauber and financial support by the dfg ( spp1666 and the dfg - jst research unit _ topotronics _ ) , the helmholtz foundation ( viti ) , and the enb graduate school on topological insulators .

in euv images of the solar corona , coronal loops appear as long graceful arcs of bright plasma that trace magnetic field lines through the atmosphere . these loops can be preferentially illuminated because localized heating and inefficient cross - field diffusion lead to hot plasma spreading along individual field lines . despite the obvious magnetic nature of these structures , it has proven challenging to measure through spectroscopic means the magnetic - field strength within the corona . of course measuring the magnetic - field strength is equivalent to measuring the energy density , and is therefore a key constraint in the modeling of energetic and eruptive phenomena such as flares and cmes . coronal loops are sometimes observed to vacillate back and forth with a regular frequency . the identification of these oscillations as resonant mhd kink waves , trapped between the loop s footpoints ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , launched the field of coronal loop seismology . coronal seismology promises the opportunity to measure the magnetic - field strength along the loop through inversion of the kink - mode eigenfrequencies ( e.g. , * ? ? ? however , before such inversions can be performed the ability to construct stable , curved coronal loop models with realistic density and magnetic profiles is needed . the state of the art in the modeling of static coronal loops is nonlinear force - free field ( nlfff ) models . typically such models have used vector magnetograph measurements in the photosphere as an observational constraint . a substantial weakness to such an approach is that the force - free assumption is rather inappropriate within the chromosphere and the low corona @xcite ; thus , the region in which the force - free assumption is valid and the region in which the observational constraint is applied are disjoint . this leads to a substantial mismatch between the model field and the actual magnetic field @xcite . more recent work has partially overcome this difficulty by applying additional constraints higher in the corona . these constraints have taken the form of euv images of bright coronal loops from instruments such as stereo and the _ atmospheric imaging assembly _ ( aia ) . initially , multiple such euv images were used , each taken from a different vantage point either using simultaneous images from the two stereo spacecraft or using solar rotation to view the presumably static magnetic structure from different angles . solar stereoscopy was then used to reconstruct the 3-d curve traced by a loop ( see the reviews of * ? ? ? * ; * ? ? ? * ) . however , recent attempts to deduce the 3-d shape of a loop from just a single high - resolution euv image ( such as those from aia ) and a coeval photospheric magnetogram have had intriguing success @xcite . despite the achievements that the nlfff models have made in reconstructing the geometry of the magnetic field , the force - free assumption decouples the thermodynamic variables from the magnetic field . certainly , if the dynamic timescales are significantly shorter than the cooling times , hydrostatic balance along field lines is still valid . however , deducing the mass density and gas pressure requires either the specification of the temperature by fiat , or the inclusion of an energy equation that models the heating and cooling of the loop ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? here we constrain the mass density within the loop in a different manner . in essence , we find an equilibrium solution by considering deviations from a force - free state . this deviation only occurs within the loop and appears as a change in the field strength without a concomitant change in the field s direction . such a field strength perturbation produces both magnetic buoyancy and a small lorentz force , both of which depend intimately on the shape or geometry of the loop and which oppose each other in equilibrium . the establishment of this equilibrium requires that the mass along the loop redistributes itself with a timescale shorter than the cooling time . thus , accounting for the buoyancy of the loop allows one to directly connect the mass density and other thermodynamic variables ( such as the temperature ) to the shape of the loop and the magnetic - field strength within the loop . we will show that the temperature profile of the loop is not a function that can be freely specified , but instead has a functional form that is a direct consequence of the geometry of the loop . our goal here is to self - consistently include curvature and buoyancy in the equilibria of coronal loops and to develop models of the temperature , density , and field strength with the geometry of the loop as the primary input . in order to permit analytic solutions we treat coronal loops as slender magnetic fibrils and adopt the thin - flux - tube approximation when deriving the force balance . the balance of forces is characterized by a magnetic bond number which is the dimensionless ratio of the buoyancy force to the lorentz force . the shape and curvature of the loop is succinctly expressed in terms of the magnetic bond number , which may be a function of position along the loop . in 2 we will derive the equation that describes the balance of forces and from this equation we identify the magnetic bond number . then , assuming that the shape of the loop is provided by observations , we derive the temperature and mass density profiles that are consistent with this shape . in 3 we present a simple equilibrium solution for an embedded fibril which has a uniform magnetic bond number . we discuss the atmospheric properties that are consistent with a constant magnetic bond number and derive the resulting coronal magnetic field . in 4 we demonstrate another simple solution corresponding to a semi - circular fibril with a uniform radius of curvature . finally , in 5 we discuss the implications of our findings and summarize the results . we will model a coronal loop as a curved , magnetic fibril embedded in a larger coronal magnetic structure . we will further assume that the fibril is compact and thin . by _ thin _ we mean that the radius of the fibril is small compared to any other relevant length scale . even though the thin - tube approximation may not apply to all coronal loops , it is an appropriate approximation for many . recent high - spatial - resolution observations of the solar corona in the fe xiii 19.5 nm line by the hi - c instrument have enabled a resolution of about 150 km , sufficient to resolve the cross - section of most coronal loops . using these observations , @xcite examined brightness cross - sections for 91 loops in the solar corona and found a distribution of radii sharply peaked at 270 km . by examining the pixel - to - pixel brightness fluctuations across loop cross - sections , @xcite have argued that coronal loops are unlikely to be structured on a finer unresolved spatial scale . since the corona s pressure and density scale heights are generally a hundred times larger than this spatial scale and the lengths of loops a thousand times larger , the thin - flux - tube approximation appears to be quite relevant for a substantial fraction of coronal loops . we also assume that the external corona is magnetically dominated and its magnetic field is a force - free field . we adopt the notation that quantities evaluated in the external corona have a subscript ` e ' , while those within the fibril lack a subscript . thus , the external magnetic field is @xmath0 ; whereas , the internal magnetic field of the fibril is just @xmath1 . one way to envision the fibril is to select a bundle of field lines within the coronal magnetic field and uniformly increase or decrease the field strength within that bundle by a constant factor , @xmath2 . we call the constant @xmath3 the field - strength deviation and it can be positive or negative depending on whether the fibril is strongly or weakly magnetized compared to its surroundings . we consider only loops that are small deviations from the force - free state . thus , the field - strength deviation will be a small quantity , @xmath4 . the constant @xmath3 was chosen such that it represents the constant of proportionality between the exterior magnetic pressure and the magnetic - pressure contrast ( the difference in the magnetic pressure between the inside and outside of the fibril ) , @xmath5 the field - strength deviation @xmath3 must be constant along the fibril . otherwise the fibril and surrounding corona would not have a common flux surface where they join . since the internal magnetic field is proportional to the external magnetic field , the internal field is also force free . however , because of the discontinuity in the field strength at the edge of the fibril , there exists a current sheath that surrounds the fibril . we neglect the spherical geometry of the solar atmosphere and assume that the corona can be treated as a plane - parallel atmosphere with constant gravity @xmath6 . we employ a cartesian coordinate system , with the @xmath7@xmath8 plane corresponding to the photosphere and the @xmath9 coordinate increasing upwards ( i.e. , @xmath10 ) . we restrict our attention to coronal loops that are symmetric about the origin and that are confined to the @xmath7@xmath9 plane . such loops lack torsion . in addition to the cartesian coordinate system , both within the fibril and within the external corona we will employ the local frenet coordinates for a field line ( illustrated in figure 1 ) . the direction tangent to the magnetic field will be denoted with the unit vector @xmath11 ( thus @xmath12 ) , and the longitudinal coordinate @xmath13 is the pathlength along a field line measured from the photosphere ( @xmath14 corresponds to the footpoint intersecting the photosphere in the region @xmath15 ) . the curvature vector for the field line is indicated by @xmath16 , and points in the direction of the principle normal @xmath17 with a modulus equal to the reciprocal of the local radius of curvature @xmath18 of the field line . the direction of the unit vector in the binormal direction will be indicated with @xmath19 and the torsion of the field line with the variable @xmath20 . in the equilibrium considered here , the loop itself lacks torsion , @xmath21 , and its binormal uniformly points in the @xmath8-direction , @xmath22 . however , the exterior coronal magnetic field may be 3-d ( with spatial symmetries near the loop ) . therefore , for completeness and to aid follow - up work we consider the more general case for the moment and specialize only as necessary . the standard geometrical relations between these coordinate vectors , i.e. , the frenet - serret formulae , are given below for reference , assuming that the position vector of a field line is given by @xmath23 , @xmath24 notice the lack of ` e ' subscripts on the frenet vectors . for a sufficiently thin tube the frenet vectors will be nearly constant across the fibril with the same value as the surrounding corona . therefore , we avoid appending subscripted labels to the frenet vectors of the external field to simplify notation and to emphasize that the field lines have the same direction and shape inside and immediately outside the fibril . the forces acting on an isolated , thin , magnetic fibril embedded in a _ field - free _ atmosphere have been previously derived by a variety of authors @xcite through averaging of the mhd momentum equation over the cross - sectional area of the tube . here we rederive the equilibrium forces in the presence of a _ magnetized _ external atmosphere . following @xcite , we begin by averaging the mhd momentum equation over the cross - section of the fibril , with the goal of deriving an equation which describes the force per unit length along the fibril . in equilibrium , there will be a balance of these forces , and this balance will specify the shape of the fibril and its thermodynamic properties . let @xmath25 be a cross - sectional surface of the fibril at the location @xmath13 that is everywhere perpendicular to the magnetic field ( i.e. , perpendicular to @xmath11 ) and let @xmath26 be a differential area of this surface . we define the cross - sectional average of a general quantity @xmath27 in the natural way , @xmath28 we write the mhd momentum equation using a formulation of the lorentz force that directly acknowledges that the magnetic force is transverse to the field itself , @xmath29 where @xmath30 in equation ( [ eqn : mhdmomentum ] ) , the gas pressure , magnetic - field strength , and mass density within the loop are @xmath31 , @xmath32 , and @xmath33 , respectively and @xmath34 denotes the lagrangian time derivative . we average equation ( [ eqn : mhdmomentum ] ) over the cross - section @xmath35 , and seek a static solution by setting the acceleration to zero , @xmath36 \ , da + \frac { \overline{b^2 { { \mbox{{\boldmath$k$ } } } } } } { 4\pi } + { { \mbox{{\boldmath$g$ } } } } \bar{\rho } = 0 \ ; .\ ] ] the two terms involving gradients can be expressed as contour integrals around @xmath37 , the boundary of @xmath35 , by using the 2-d form of the divergence theorem appropriate for integration over an open surface , @xmath38 the unit vector @xmath39 is the outward normal to the tube s bounding surface . the differential @xmath40 is the differential pathlength around @xmath37 . in deriving this equation , we have decomposed the gradient of the gas pressure into longitudinal and transverse components , @xmath41 the total pressure must be continuous across the loop s bounding surface . therefore , on @xmath37 @xmath42 and the pressures that appear within the integrand of the contour integral in equation ( [ eqn : avemomentum ] ) may be replaced with those from the external fluid , @xmath43 buoyancy is the sum of the gravity acting on a body and the net gas - pressure force acting on the outer surface of the body . in this case , we also must consider the effects of the external magnetic pressure and magnetic tension . we can construct an extension of archimedes principle that is relevant for our problem by performing a similar cross - sectional average of the mhd momentum equation in the exterior fluid . this requires that we assume that the mathematical form of the external pressures and density can be analytically continued inside the loop . the transverse component of the resulting equation provides an expression for the net external pressure force , @xmath44 where @xmath45 is the component of @xmath6 perpendicular to @xmath11 . since we have assumed that the tube is thin , to lowest order in the radius of the tube , the tangent vector and curvature vector are constant across a cross - section thus , the previous equation can be rewritten , @xmath46 this expression can be used to eliminate the external pressure integrals in equation ( [ eqn : forcebalance ] ) . if we once again assume that the tube is thin and the frenet vectors do not vary significantly over the tube s cross - section , we can replace the cross - sectional averages of the densities and pressures with their axial values to obtain @xmath47 { { { \mbox{{\boldmath$\hat{s}$ } } } } } + \frac{b^2-b_{{\rm e}}^2}{4\pi } \ , { { \mbox{{\boldmath$k$ } } } } + \left ( \rho - \rho_{{\rm e}}\right ) { { \mbox{{\boldmath$g$}}}}_\perp = 0 \ ; .\ ] ] where @xmath48 is the tangential component of gravity . for simplicity we have labeled the axial values without accents or subscripts and the external magnetic field and density are to be evaluated at the location of the magnetic fibril . note , this equation is very general ; we did not assume that either of the internal or external fields were force free , nor did we assume that the loop is torsionless . therefore , this equation is valid for a fibril that describes a fully 3-d curve through an equilibrium atmosphere filled with a general 3-d external magnetic field . the term in square brackets is the longitudinal component and represents hydrostatic balance along field lines . the remaining terms are transverse and correspond to the magnetic and buoyancy forces . in some ways these two transverse terms are analogous ; the last term , buoyancy , is a combination of gravity and the net support provided by the external gas pressure , whereas the second term , the magnetic force , is the residual lorentz force that remains once magnetic tension and the net support provided by the external magnetic pressure have been combined . both forces are normal to the surface of the tube . our intuition may tell us that buoyancy is aligned with gravity ; but , this is only true for bodies with closed symmetric surfaces , such as a sphere . further , just as buoyancy can point upwards or downwards depending on whether the fibril is over- or underdense , the net magnetic force can point up or down depending on the relative strength of the magnetic field inside and outside the fibril . we now invoke the assumption that the fibril is torsionless ( @xmath49 ) and vertically oriented , therefore lacking forces in the binormal direction . under this assumption we can separate the mean force equation ( [ eqn : meanforces ] ) into only two components , @xmath50 where we have used @xmath51 , which holds in equilibrium when the fibril is confined to the @xmath7@xmath9 plane . equation ( [ eqn : equil_s ] ) expresses the balance of forces in the tangential or axial direction @xmath11 , whereas equation ( [ eqn : equil_k ] ) describes the balance in the transverse direction of the principle normal @xmath17 . the axial equation is simply hydrostatic balance along magnetic field lines . the transverse equation constrains the density contrast , @xmath52 , through the balance of magnetic and buoyancy forces . this balance can be characterized by a magnetic bond number @xcite , @xmath53 which is a nondimensional ratio of the buoyancy to the magnetic forces . the length scale that we have used in this definition , @xmath54 , is the footpoint separation , where @xmath55 is the location of each of the fibril s footpoints in the photosphere . the magnetic bond number appears in calculations of the rayleigh - taylor instability when one or more of the fluid layers are filled with a horizontal field , providing the critical wavenumber below which instability ensues @xcite . equation ( [ eqn : equil_k ] ) expresses a condition on the radius of curvature @xmath18 of the fibril s axis in terms of the magnetic bond number @xmath56 and the geometry of the fibril ( i.e. , the direction of the principle normal relative to gravity @xmath57 ) , @xmath58 to proceed we need expressions for the frenet unit vectors , @xmath11 and @xmath17 , the radius of curvature , @xmath18 , and the arclength , @xmath13 . if @xmath59 is the height of the fibril above the photosphere , given as a function of the horizontal coordinate @xmath7 , and the fibril s axis is traced by the position vector @xmath60 , these quantities can be easily derived from equations ( [ eqn : s_hat ] ) and ( [ eqn : k_hat ] ) , @xmath61 where primes denote differentiation with respect to the photospheric coordinate @xmath7 . if we insert equations ( [ eqn : k ] ) and ( [ eqn : r ] ) into equation ( [ eqn : shapeeqn ] ) , we obtain a nonlinear ode for the height of the fibril @xmath59 , @xmath62 a quick examination of this equation reveals that the fibril will be locally concave or convex depending on the sign of the magnetic bond number and that points of inflection correspond to vanishing magnetic bond number . the magnetic bond number is proportional to the signed curvature . a stable fibril with a single concave arch will have a negative bond number everywhere , whereas a multi - arched structure will have a magnetic bond number that changes sign . the temperature , density and gas pressure within the loop can all differ from the surrounding corona . we define the contrasts is these properties in the following manner : @xmath63 technically the external variables are all functions of two spatial coordinates e.g . , @xmath7 and @xmath9 . in all of these definitions , however , the external variable is evaluated at the location of the fibril . we indicate this by expressing these variables as a function of the pathlength alone . only the contrasts in the mass density @xmath64 , gas pressure @xmath65 , and magnetic pressure @xmath66 play an active role in the force balance . the temperature contrast @xmath67 is a dependent variable that can be derived from these active variables post facto . there are four primary equations that provide all of the inter - relations between these properties of the loop , @xmath68 the first of these is a direct consequence of equation ( [ eqn : pressurecont ] ) . the second results from transforming equation ( [ eqn : equil_s ] ) from the pathlength variable @xmath13 to the fibril height @xmath69 . the third relation , equation ( [ eqn:4e - transverse ] ) , arises from a combination of the definition of the magnetic bond number ( [ eqn : bond ] ) and the equation of transverse force balance ( [ eqn : ode ] ) , where the magnetic - pressure contrast has been replaced through use of pressure continuity , equation ( [ eqn:4e - pcont ] ) . the last equation is a restatement of equation ( [ eqn : matchedflare ] ) which ensures that loop and external magnetic field share a common bounding flux surface . the temperature contrast @xmath67 can be found by using the ideal gas law ( @xmath70 ) , @xmath71 where @xmath72 is the external corona s pressure scale height , @xmath73 , and @xmath74 is the scale height of the gas - pressure contrast , @xmath75 conveniently , these equations can be solved analytically to obtain the contrast variables as a function of the fibril s shape . if we combine equations ( [ eqn:4e - hydrostatics ] ) and ( [ eqn:4e - transverse ] ) , we obtain a differential equation that relates the pressure contrast to the derivative of the fibril s height , @xmath76 this equation can be directly integrated to obtain , @xmath77 \ ; , \ ] ] where @xmath78 is a positive integration constant that can be fixed in a variety of ways . one could apply a photospheric boundary condition or one could use spectroscopic observations to fix the temperature or density contrast at the apex of the loop . strictly speaking , the integration constant is the product @xmath79 . we have included the field - strength deviation in this product for later convenience and in order to remind the reader that the thermodynamic contrasts arise from a small deviation from the force - free equilibrium . if we subsequently use the equations of pressure continuity ( [ eqn:4e - pcont ] ) and hydrostatic balance ( [ eqn:4e - hydrostatics ] ) , we can derive equations for the mass density and magnetic - pressure contrasts , @xmath80 \ ; .\end{aligned}\ ] ] thus , if we are provided the shape of the loop , we can derive all of the loop s contrast variables . different values of the integration constant @xmath79 result in different temperature profiles along the loop . equation ( [ eqn : deltat ] ) can be rewritten to show how the temperature contrast depends on the geometry of the loop and the constant @xmath79 , @xmath81}{p_{{\rm e}}+ 2 \alpha c h_{{\rm e}}z_0^{\prime\prime } } \ ; .\ ] ] note that the integration constant @xmath78 was chosen such that it appears only in the combination of @xmath79 in the equations above . thus , one needs only a single constraint to define the thermodynamic variables . given the pressure contrast @xmath65 , we can use equations ( [ eqn:4e - pcont ] ) , ( [ eqn:4e - flarerate ] ) , and ( [ eqn : deltap ] ) to express the external field strength at the location of the fibril in terms of the geometry of the fibril , @xmath82 \ ; .\ ] ] this last equation shows that the integration constant @xmath78 is really just the value of the external magnetic pressure at the apex of the loop ( hence @xmath78 is positive ) . further , the equation demonstrates that not all external fields are self - consistent with an embedded fibril . if we select an arbitrary field line within a general magnetic field , equation ( [ eqn : bext ] ) may not be satisfied for a constant @xmath78 . this becomes readily apparent when we express the direction of the tangent vector in terms of the angle @xmath83 that it makes with the @xmath7-axis , @xmath84 . with a little basic manipulation , we find the following relation between the angle @xmath83 and the height function @xmath69 : @xmath85 subsequently , one can easily show that equation ( [ eqn : bext ] ) is equivalent to the statement that the @xmath7-component of the external magnetic field is constant along the length of the fibril , @xmath86 of course , not all force - free fields will have this property . if needed , a self - consistent external magnetic field can be derived by noting that the shape of the fibril and the strength of the magnetic field at the location of the fibril provide a boundary condition for the coronal magnetic field that occupies the bulk of the domain . for simplicity we assume that the external magnetic field is potential and 2d . therefore , since the field is solenoidal and 2d it can be described by a flux function @xmath87 , @xmath88 since the field is potential the flux function must be harmonic , @xmath89 therefore , we can find a self - consistent external field solution by solving equation ( [ eqn : harmonic ] ) under the boundary condition @xmath90 applied at the location of the fibril with @xmath91 fixed by equation ( [ eqn : bext ] ) . this ensures that the external field is both parallel to the fibril and the field strength is given by @xmath91 . there is an infinity of such solutions , all with different photospheric boundary conditions . for our first example solution we wish to consider one that is both simple and fully analytic . therefore , we consider a fibril which has constant magnetic bond number @xmath56 along its length . for constant @xmath56 , the transverse force equation ( [ eqn:4e - transverse ] ) can be integrated twice to obtain the height of the fibril @xmath59 . similarly , using the result of the first of these integrations , equation ( [ eqn : sprime ] ) can be integrated to obtain the arclength @xmath92 we choose the three constants of integration such that the footpoints occur at @xmath93 , the fibril is symmetric about @xmath94 , and the footpoint at @xmath95 corresponds to @xmath14 , @xmath96 with these conditions , we find the following solution for the geometrical properties of the fibril : @xmath97 figure [ fig : constbond ] displays solutions for several different values of @xmath56 . it is clear that fibril posesses a nonconstant radius of curvature , with strong curvature near the apex and weaker curvature in the legs . the length of the fibril @xmath98 is obtained by inserting the rightmost footpoint position @xmath99 into equation ( [ eqn : arclength ] ) , @xmath100 two interesting limits of this equation exist . as @xmath101 the length of the loop converges to the footpoint separation @xmath102 . this arises because the loop becomes straight , flat , and confined to the photosphere . as @xmath103 the loop length diverges logarithmically because the height of the loop grows without bound this can be seen by recognizing that the loop reaches its apex at its center ( @xmath94 ) , therefore achieving a maximum height of @xmath104 one can easily see that @xmath105 corresponds to a logarithmic singularity in the height . clearly the height @xmath59 should be a positive function for the range @xmath106 . with a little thought , from equation ( [ eqn : z0 ] ) we can see that this is only possible if @xmath107 , once again emphasizing that a fibril with a single concave arch has a negative magnetic bond number . the contrast variables have a common geometrical factor , @xmath108 , appearing in their functional forms . for the model with uniform magnetic bond number this factor takes on a simple form . using equations ( [ eqn : z0 ] ) and ( [ eqn : dz0dx ] ) , we find @xmath109 using this expression and the equations derived in subsection [ subsec : thermo ] we can solve for all of the contrast variables and the external magnetic pressure in terms of the scale height @xmath74 that depends on the bond number , @xmath110 , @xmath111 the density and pressure contrasts , as well as the external magnetic pressure , are revealed to be exponential functions of height , all with the same constant scale height @xmath74 . the density and temperature contrast are illustrated in figure 3 , for an isothermal exterior with a scale height of @xmath112 mm . the constant @xmath79 has been chosen such that the fractional density contrast at the apex of the loop is @xmath113 . the solid curves correspond to overdense loops and the dotted curves to underdense loops , with the different colors corresponding to loops with different magnetic bond numbers . we will see in the discussion section [ sec : discussion ] that overdense loops have been heated relative to the surrounding corona and underdense loops have been cooled . therefore , overdense loops are probably more physically relevant . for an isothermal exterior , where @xmath72 is a constant , one could choose the magnetic bond number such that @xmath114 . for such a model , the temperature contrast would be identically zero all along the length of the loop and the plasma parameter @xmath115 would be a constant both inside and outside the loop , while the density contrast remains nonzero . the condition @xmath116 marks a transition between disparate behavior . loops with @xmath117 have fractional density and temperature contrasts that decrease with height , whereas loops with @xmath118 have fractional density and temperature contrasts that increase with height . in the previous subsection we discovered that a fibril with constant magnetic bond number requires that the external magnetic pressure have a constant scale height as we slide along the fibril this suggests that we seek a solution in which the magnetic pressure is only a function of height and varies exponentially . a simple harmonic solution that satisfies this requirement is a sinusoid in the horizontal @xmath7-direction multiplied by a decaying exponential in height @xmath9 , @xmath119 in this solution , @xmath120 is the horizontal repetition length of the field ( @xmath87 changes sign every @xmath121 ) . with such a flux function , the magnetic field is as follows : @xmath122 \ , e^{-\kappa z } \ ; .\ ] ] since the magnetic field is periodic in the @xmath7-direction , we will only consider the arcade that is symmetric about the origin and exists in the range @xmath123 , where @xmath124 . the magnetic pressure for such a field depends only on height and falls off exponentially with a scale height of @xmath125 , @xmath126 in order to satisfy equation ( [ eqn : consteps_be ] ) , we must insist that the external field s horizontal wavenumber is proportional to the magnetic bond number and that the photospheric value of the field strength depends on @xmath78 , @xmath127 lines of constant @xmath87 in the @xmath7@xmath9 plane denote field lines . the height above the photosphere , @xmath128 , of a field line with a specified value of the flux function is easily obtained by inverting equation ( [ eqn : psi ] ) , @xmath129 where the footpoints intersect the photosphere at @xmath130 , @xmath131 we can clearly see from equation ( [ eqn : z0 ] ) that this potential field solution for the external corona is consistent with the loop model , equation , if the magnetic bond number is inversely proportional to the repetition length of the external magnetic field , @xmath132 , which is a condition we have already imposed to ensure that the external field strength has the appropriate value along the fibril . the field lines of this potential field are illustrated in figure [ fig : constbondfield ] . in the beginning of this section we discovered that stability requires that the magnetic bond number be bounded @xmath107 . in the context of this specific model , we can see that the magnetic bond number selects the field line within the coronal field that corresponds to the axis of the fibril . fibrils with different footpoint separations and magnetic bond numbers are shown as various colored field lines in figure [ fig : constbondfield ] . a small value of the magnetic bond number corresponds to a short , flat , low - lying loop confined very near the origin , while a magnetic bond number near the lower limit is a tall , long loop , with foot - points approaching the outer edge of the arcade . in fact , in the limit @xmath133 , the loop is infinitely tall with vertical legs located at @xmath134 . therefore , there is a direct mapping between the magnetic bond number and the value of the flux function that corresponds to the loop s axis , @xmath135 and one can choose to place the loop along any field line in the coronal model by dialing the magnetic bond number between @xmath136 and 0 . another simple model to consider is a fibril with a constant radius of curvature . this model is of course an example of a fibril with a nonuniform magnetic bond number . however , it is still a single concave arch and therefore the bond number will be negative everywhere . with a constant radius of curvature @xmath18 , equation ( [ eqn : shapeeqn ] ) simplifies to @xmath137 where @xmath138 is the height above the photosphere of the fibril s center of curvature . from this expression we can see that the magnetic bond number diverges when @xmath139 resulting in a divergence of all the contrast variables at the same location . therefore , for a physically meaningful solution we must insist that the center of curvature lies below the photosphere , i.e. , @xmath140 . the fibril is therefore a circular arc that is less than a full semi - circle and intersects the photosphere at oblique angles . using the procedure outlined in subsection [ subsec : thermo ] we derive all of the contrast variables and the external magnetic pressure from the magnetic bond number , @xmath141 figure [ fig : tandd_circ ] illustrates the density and temperature contrasts for loops with radii of curvature spanning 25125 mm . each is embedded in an isothermal corona with a scale height of 75 mm . the integration constant @xmath79 has been chosen such that the fractional density contrast is @xmath113 at the apex of the loop . for all of the loops the magnitude of the density contrast decreases with height . for the parameters chosen for the figure , the overdense loops tend to be cold compared to their surroundings , although those that are sufficiently large ( @xmath142 ) can have a warm crest . similarly , underdense loops tend to be hot with the tallest and widest loops possessing a cool crest . if a change of sign occurs , it will happen at a distance of two scale heights above the center of curvature . many external field solutions could be found that possess circular field lines ; however , most will not possess the needed functional form for the external field strength along the loop . consider the field generated by a line current located at the center of curvature . such a line current generates a field with concentric field lines that have constant field stength as one moves along a line . we can see that this field fails because equation ( [ eqn : constr_be ] ) dictates that the field strength must decrease with height along the fibril . the functional form for @xmath143 given by that equation and the fact that the fibril is a circular arc , suggests that we seek a solution in polar coordinates with the following form , @xmath144 where @xmath145 and @xmath146 . the origin of the polar coordinate system is located at the center of curvature , @xmath18 is the radius of curvature of the fibril , and @xmath147 points in the positive @xmath7-direction . once again , @xmath148 is the value of the external magnetic field strength at the footpoints . we can verify that this particular potential field has a circular field line at @xmath149 by noting that the flux function is constant there ( in fact , @xmath150 on the fibril ) . direct evaluation further verifies that the magnetic pressure when evaluated at the location of the fibril has the requisite functional form , equation ( [ eqn : constr_be ] ) , as long as @xmath151 the field lines for this potential field are illustrated in figure [ fig : constcurvfield ] . the external field lines are drawn in black and the circular fibril is marked in blue . note , unlike the constant magnetic bond number model presented in section [ sec : constantepsilon ] where one could dial the control parameter @xmath56 to change where the fibril appears in the corona , only one field line in figure [ fig : constcurvfield ] meets the prerequisite of constant curvature . thus , while self - similar , this external magnetic - field model differs for fibrils with different radii of curvature . our equilibrium model is predicated on the implicit assumption that force balance is achieved at a much faster time scale than radiative cooling and thermal conduction redistribute heat . hydrostatic equilibrium should be established on a dynamical time scale in between the free - fall time and the acoustic crossing time . thus , for a loop that reaches 50 mm above the photosphere and for a pressure scale height of 80 mm , the dynamical time scale lies between 1040 minutes . transverse force balance is achieved by mass redistribution along the loop and should therefore be established on a similar time scale . estimates of the radiative cooling and conduction times vary , but for active region loops radiative times can be on the order of 140 hr and conduction times are typically 1 hr @xcite . therefore , we expect that immediately after an impulsive heating event , a coronal loop will quickly re - establish force balance and will stay in force balance as the loop slowly cools . in the following section we discuss the implications of this balance . the shape of a coronal loop is determined by the force - free equilibrium of the surrounding corona . given this shape , the mass along the loop must redistribute itself such that the buoyancy forces exactly oppose the lorentz forces acting on the loop . the resulting balance of forces can be succinctly characterized by the magnetic bond number which is fully specified by the height of the loop above the photosphere as a function of position . in equilibrium , the magnetic bond number is proportional to the signed curvature of the loop , and therefore concave curvature results in a negative magnetic bond number and convex curvature in a positive bond number . of course , inflection points correspond to locations where the bond number vanishes . therefore , a loop comprised of a single concave arch must have negative magnetic bond number everywhere if in equilibrium . equilibrium loops that possess multiple arches will have negative bond number in the crests and positive bond number in the dips or troughs . all loops with the same geometric shape have exactly the same profile of the magnetic bond number . however , these loops can be achieved in a continuum of different ways , all varying in the distribution of mass and field strength . this continuum is represented by different values of an integration constant @xmath79 which is a direct measure of the magnetic - pressure contrast at the apex of the loop . for example , an underdense loop with enhanced magnetic - field strength can have exactly the same bond number as an overdense loop with reduced field strength . while the shape of these two loops will be the same , the thermodynamic and magnetic properties of these two loop might be very different . for a loop with enhanced field strength , the mass density is proportional to the second derivative of the height function with respect to the horizontal photospheric coordinate , equation ( [ eqn : deltarho ] ) . thus , we can immediately deduce that peaks or crests in a loop will have a mass deficit while dips or troughs have mass accumulation . furthermore , if we define the total mass deviation between two points as the integrated density contrast , where @xmath154 is the cross - sectional area of the loop at the apex . if we change variable from the pathlength to the photospheric @xmath7 coordinate and replace the density contrast using equation ( [ eqn : deltarho ] ) the integral can be evaluated trivially , thus , the net mass deviation between a crest and a neighboring trough is zero and the crest drains mass into the dips . for a loop comprised of a single concave arch , the mass deviation between the two footpoints is negative . mass must drain out of the loop through the photosphere into the solar interior . in the opposite case , a loop with a reduced field strength , the mass density is proportional to the negative of the second derivative . the peaks in such a loop must have enhanced density so that buoyancy can oppose the upward lorentz force . this obviously implies that the gas pressure must increase throughout the loop to increase the hydrostatic support . the additional mass is lifted from the photosphere into the loop . observations of the temperature along the length of a coronal loops have suggested that some loops might be nearly isothermal @xcite . for the relatively small density contrasts that have been explored here , the temperature contrast is also relatively small and the temperature is perforce roughly isothermal as long as the external corona is isothermal . however , we wish to point out that the temperature contrast itself is markedly constant over height for a signficant fraction of the models we have illustrated . three of the models with constant magnetic bond number , appearing in figure 3 , have a temperature variation of less than a 30% percent from footpoint to apex . these three correspond to the models with the squattest loops . given a specific shape for a coronal loop there exists a family of possible solutions , each characterized by a different value of @xmath79 , the integration constant see equations ( [ eqn : deltap])([eqn : deltat_alt ] ) . when illustrating our models , we chose to fix this parameter by specifying the fractional density contrast at the apex of the loop . we wish to point out , however , that the parameter @xmath79 is really a measure of the total heat that has been input into the coronal loop . the excess heat contained by the loop per unit length is given by where @xmath157 is the specific heat at constant volume and @xmath158 is the adiabatic exponent . we can obtain the total excess heat contained by the loop by integrating @xmath159 over the loop s length . if we eliminate the gas - pressure contrast by using equation ( [ eqn : deltap ] ) , we reduce the integral to a constant factor times a positive definite integral that depends only on the geometry of the loop , since the constant @xmath78 and the integral appearing on the right hand side of the previous equation are always positive , we can immediately discern two facts : ( 1 ) heated loops correspond to those with a negative field - strength deviation @xmath3 and cooled loops to those with a positive value . thus , heated loops are those that are undermagnetized and overdense and cooled loops are overmagnetized and underdense . ( 2 ) the integration constant @xmath78 is a measure of the magnitude of the heat that has been injected into or extracted from the loop . obviously , heat deposition is more likely to be physically relevant . the family of solutions defined by different values of @xmath78 form a continuum of loop models with different amounts of stored heat . for example , after a large impulsive heating event , the heat content of the loop jumps but force balance is quickly re - established . this requires a redistribution of mass ( and heat ) along the loop and the resulting balance is characterized by a nonzero value of @xmath78 . then as radiative cooling and conduction slowly dissipate the loop s excess heat , the loop responds by moving through a sequence of equilibria while simultaneously maintaining force balance . this sequence is represented by slow temporal attenuation of the parameter @xmath78 . seismology of a coronal loop is only sensitive to the distribution of wave speed along the loop ( e.g. , * ? ? ? if we assume that the observed waves are kink oscillations and that they propagate at the kink speed , then , at best , a seismic analysis can provide information about the ratio of the field strength to the density . it is impossible from only a seismic analysis to measure the field strength profile independently of the density profile . however , with additional constraints , provided either from observations or from basic physical principles , one might be able to disentangle field - strength and density variations along the loop . the force - balance model constructed here provides just such a constraint . the density contrast and magnetic - pressure contrast are related to the geometry of the loop through the magnetic bond number , if we assume that the interior magnetic field is nearly the same as the external field ( hence the field - strength deviation is small ) , we can use this previous equation to express the kink speed solely in terms of the density contrast , the exterior density and the loop s geometry , if we further assume that the exterior corona is isothermal , @xmath164 , this expression has only four free parameters : the integration constant @xmath78 , the field - strength deviation @xmath3 , the coronal scale height @xmath72 and photospheric value of the exterior density @xmath165 . thus , even the measurement of only a few mode frequencies that provide different spatial averages of the kink speed should be sufficient to either fix these parameters or to verify a choice made using other information . we have developed a model of curved coronal loops that self - consistently couples deviations from the force - free state to the thermodynamic properties . the coupling is accomplished by requiring that the loop is in equilibrium and the lorentz force arising from the deviation from a force - free field is balanced by buoyancy . this links the curvature of the loop to the loop s mass - density contrast and hence to the pressure and temperature through hydrostatic balance and the ideal gas law . the end result is that specification of the loop s geometry is sufficient to derive the temperature and density profiles to within a single integration constant . this integration constant can be selected in a variety of ways , either seismically through estimates of the mean kink - wave speed or spectroscopically through estimates of the density or temperature contrast at one position along the loop . further , this integration constant is a direct assessment of the thermal history of the loop providing an estimate of the excess heat contained by the loop relative to the surrounding corona .

analyses of the impulsive phase of solar flares detected in various electromagnetic wavelength ranges are important for understanding the acceleration and propagation of particles . the accelerated electrons reveal their presence e.g. in hard x - ray and radio emissions , and cause rapid heating of the chromosphere and subsequent emission in the hydrogen h@xmath1 line . one of the key questions is the form of the accelerated electron distributions . their determination from the hard x - ray spectra depends not only on the emission mechanism but also on the processes affecting the propagation of the particles . as suggested by brown , emslie , and kontar ( 2003 ) , it is the mean electron flux distribution @xmath3 depending only on the bremsstrahlung cross section that should be used as a reference for both observational hard x - ray spectral analyses and theoretical models of electron acceleration and propagation . the mean electron flux distribution can be determined either by inverting the photon spectra ( johns and lin , 1992 ; piana et al . , 2003 ) or by a forward fitting method usually assuming a power - law form of @xmath3 ( holman et al . , 2003 ) . observed hard x - ray spectra also contain a contribution of photons backscattered in the photosphere and this will modify the calculated @xmath3 ( bai and ramaty , 1978 ; alexander and brown , 2002 ) . besides the x - ray emission , numerous types of radio emission in the metric and decimetric range are usually observed during flares . they provide further information about the acceleration and propagation of the electron beams and plasma parameters in the emission region ( karlick , 1997 ) . according to standard solar flare models ( e.g. dennis and schwartz , 1989 ) , the accelerated electrons provide one of the mechanisms for the flare energy transport from the release site to the lower atmospheric layers . as the accelerated electrons propagate along the magnetic field lines toward the photosphere , they lose their kinetic energy mainly via coulomb collisions ( brown , 1971 ) in the lower corona and chromospheric layers . the chromospheric response to the beam energy deposition determines the characteristics of optical and uv emission from the flare loops . numerical models of chromospheric response to pulsed electron beam heating ( e.g. canfield and gayley , 1987 ; heinzel , 1991 ) predict the time correlation of hard x - ray and h@xmath1 emission recently analysed e.g. by trottet et al . ( 2000 ) and wang et al . their results show that hard x - rays and h@xmath1 intensities in some flare kernels exhibit time correlations in the time range from subseconds to @xmath210 s. some flare models ( e.g. heyvaerts , priest , and rust , 1977 ; cargill and priest , 1983 ) and observations ( czaykowska et al . , 1999 ) suggest that h@xmath1 kernels are located between the upflows of the beam heated plasma and the downflows of the cool plasma in the loops disconnected from the reconnection site . the leading edges of the h@xmath1 emission close to the newly reconnected loops are supposed to be also heated by the accelerated particles , which are detectable e.g. as the hard x - ray sources . comparison of the spatial distribution of yohkoh hard x - ray sources and h@xmath1 flare kernels was done by e.g. asai et al . they found that many h@xmath1 kernels brighten successively during the evolution of the flare ribbons , but only a few radiation sources were seen in the hard x - ray images . they suggest that this discrepancy may be the result of the low dynamic range capability of the yohkoh hard x - ray telescope . in this paper , we analyse the impulsive phase of the august 20 , 2002 solar flare , which is characterised by a very flat photon spectrum at the x - ray burst maximum ( spectral index @xmath4 ) . similar flat spectra have been reported , e.g. by nitta , dennis , and kiplinger ( 1990 ) and frnk , hudson , and watanabe ( 1997 ) . the event presented here is the first such flat spectrum spectrum observed with the high energy resolution of the reuven ramaty high energy solar spectroscopic imager ( rhessi ) ( lin et al . , 2002 ) . we show the influence of photons backscattered in the photosphere on the determination of the mean electron flux distribution and compare results obtained by inversion and forward fitting methods . we also study the spatial correlation of h@xmath1 emission and its time changes with hard x - ray sources . in section 2 we describe the global behaviour of the event with regard to hard x - ray and radio fluxes . sections 3 and 4 present the rhessi , h@xmath1 and magnetic field observations , methods , and results . finally , the results are discussed in section 5 and the conclusions are given in section 6 . the analysed flare is well suited for a multi - wavelength study of the electron propagation because its observations provide a set of simultaneous data of comparable time and spatial resolution in the wavelengths directly related to the accelerated particles : rhessi hard x - ray spectra and images , h@xmath1 images ( kanzelhhe observatory ) , and radio and microwave fluxes ( bern , ondejov , and zrich facilities ) . the flare was detected by goes satellites on august 20 , 2002 . it started at 08:25 ut and reached its maximum at 08:26 ut as an m3.4 flare . the 1b h@xmath1 flare was reported in noaa ar 0069 at s10w38 , starting at 08:25 ut , peaking at 08:26 ut , and ending at 08:37 ut . from the fluxes at 4299 mhz ( ondejov ) and 988 mhz ( zrich ) . attenuators states a0 , a1 , and a3 and the time interval in which the derivative of h@xmath1 intensity was analysed are indicated by the lines at the top of the rhessi data plot . the abrupt changes in rhessi flux in the 12 - 25 kev energy band at @xmath2 08:25:17 ut and @xmath2 08:25:41 ut are caused by the attenuator changes ( a0 @xmath5 a1 and a1 @xmath5 a3 , respectively ) at these times.,scaledwidth=94.0% ] a global overview of the flare evolution in dm / m radio waves and rhessi x - rays is shown in figure [ fig_hsi_radio ] . the rhessi flare was first detected at 08:24:32 ut with the increase in flux in the 12 - 25 kev energy band . then , with some time delays ( at @xmath2 08:24:45 ut ) , bursts in higher and higher energy bands followed . during the starting phase of the x - ray emission at 08:24:30 - 08:25:08 ut , no radio bursts were observed in the dm / m range . the first weak and narrowband emission ( nbe ) in the dm - range was registered at 08:25:08 ut in the 2000 - 2200 mhz frequency range , followed by the type iii burst below 500 mhz at 08:25:16 ut . the most energetic part of this flare occurred during the time interval 08:25:15 - 08:25:50 ut , when significant hard x - ray emission was detected up to 7 mev ( see also data from the song instrument on board coronas - f ; myagkova , priv . comm . ) and microwave emission up to 89.4 ghz . the turn - over frequency between optically thick and thin part of the radio emission was shifted to values between 19.6 and 50 ghz . at that time the radio emission in the dm - range also reached the maximum and a very fast drifting ( @xmath2 -4000 mhz s@xmath6 ) and short - lasting ( @xmath2 0.1 s ) type iiid burst ( relativistic type iii burst - see poquerusse , 1994 ) was observed in the 400 - 1000 mhz range at 08:25:31.8 ut . in the decimetric range ( 2000 - 4500 mhz ) this phase was characterised by broadband pulsations . rhessi x - ray data were analysed with the rhessi data analysis software ( hurford et al . , 2002 ; schwartz et al . , 2002 ) . during the flare , different attenuators were automatically placed in front of all detectors to absorb soft x - rays and reduce pulse pile - up . the attenuator states a0 , a1 , and a3 ( no attenuator , thin attenuator , and both thick and thin attenuator , respectively ) are indicated on figure [ fig_hsi_radio ] . corrections for the effects of the different attenuator states are available for all times except when the attenuators are moving , a period of @xmath2 1 s. image reconstruction was performed with both the clean and the pixon algorithms using data from the front segments of the detectors . detector 7 was used only for the images in the energy bands above 7 kev due to its energy threshold of about 7 kev . detector 1 and 2 mainly added noise to the images indicating that there was no significant source structure with an angular scale finer than @xmath2 4 arcsecond . therefore , these detectors were excluded . the clean algorithm was used for obtaining morphology and time evolution of x - ray sources , see sections [ spatial ] and [ tder ] , figures [ fig_ha_hsi_mdi_low ] , [ fig_ha_hsi_mdi_high ] , and [ fig_haderiv_hsi ] . since there was no source component as large as or larger than fwhm of the detector 8 ( 107 arcsec ) , the imaging fields of view in all clean images were set smaller than this fwhm and consequently detectors 8 and 9 were also not used in the clean image reconstruction . the pixon algorithm ( puetter and pia 1994 ; metcalf et al . , 1996 ) is known to suppress spurious sources and to have high photometric accuracy ( metcalf et al . , 1996 ; alexander and metcalf , 1997 ) . we have used the pixon imaging technique for imaging spectroscopy at the burst maximum , see section [ flatspectrum ] . we analysed data summed over the eight front segments . detector 2 was excluded since its energy threshold is about 20 kev and its energy resolution is about 10 kev . the data were corrected for pulse pile - up and decimation . full 2d detector response matrices , each corresponding to the applied attenuator state , were used to convert input photon fluxes to count rates . the response matrix accounts for the efficiency and resolution of the detectors , the absorption by the attenuators , grids , and all other material above the detectors , and all other known instrumental effects ( smith et al . , 2002 ) . the time bin for rhessi spectra was 2 s ( @xmath7 a half rotation ) , which ensures that the rapid modulation produced by the grids does not have any distorting effect on the spectra . the rhessi spectra were analysed in the time interval 08:25:10 - 08:26:00 ut and were fitted from 8 kev up to typically 400 kev , depending on the signal - to - noise ratio . rhessi data of this flare were contaminated by an electron precipitation starting at 08:16 ut and detected up to 300 kev . therefore , a special care was taken to estimate the background due to the electron precipitation in the 12 - 300 kev energy range while standard techniques were used at other energies . in the 12 - 40 kev energy range , the background was estimated using count rates from the onboard charged particle detector whose lower threshold ( 50 kev ) is generally triggered during electron precipitation events ( smith et al . , 2002 ) . analysing several precipitation events which occured on other orbits two days before and after the flare , we have found a good correlation between the particle count rate and the front count rate in this energy range . therefore , count rates from the particle detector can serve as a proxy for the bremsstrahlung of the precipitating electrons as detected by germanium detectors in this energy band . at higher energies the germanium detector count rates during the precipitation event are generally dominated by the bremsstrahlung that originates from the electrons interacting in the earth s atmosphere and generally does not follow the time history of the particle count rate ( smith , priv . comm . ) . having taken this into account , the precipitation component in the 40 - 300 kev energy range was estimated with a cubic fit to the shoulders of the precipitation event detected before and after the flare . the uncertainty of the estimated precipitation component in the 12 - 300 kev energy range was estimated to be @xmath8 . despite this high value , any residual precipitation component in the background subtracted count spectra in the 12 - 300 kev energy range is @xmath9 at the edges of the analysed time interval and decreases to several percent with the increasing flare flux . thus , the precipitation event does not affect our results in the analysed time and energy range . in order to evaluate the power - law indices of the photon spectra and to make comparisons with previous works , we adopted a standard photon spectrum model @xmath10 consisting of a double power - law non - thermal component @xmath11 ( e.g. lin and schwartz , 1987 ) and a maxwellian isothermal component @xmath12 : @xmath13 . @xmath14 where @xmath15 , @xmath16 , @xmath17 , @xmath18 , and @xmath19 are the photon energy , the break energy in kev , the power - law index below and above @xmath16 , and @xmath19 is for normalisation , respectively . we used the forward fitting method and fitted the background subtracted x - ray spectra with the model spectrum @xmath10 . the model spectrum was convolved with the full 2d detector response matrix to compute the model count spectrum . parameters of the best fit model were found by searching for a minimum @xmath20 between the measured and the model count rate spectrum . the lowest value of the power - law index @xmath17 was found at the burst maximum as seen in @xmath21 x - rays , at 08:25:22 ut , when it dropped to 1.8 . note that this value lies outside of the range of common values of photon power - law index , @xmath22 , as reported e.g. by dennis ( 1985 ) . the flatness of the spectrum does not result from the contamination by a generally flat spectrum of the precipitation event because the residual precipitation component in the background - subtracted count spectrum was estimated to be @xmath23 in the whole energy range at 08:25:22 ut . hence , we believe that the flat spectrum is of solar origin . this conclusion is further supported by the results from the imaging spectroscopy using the pixon imaging method ( hurford et al . , 2002 ) . rhessi pixon images are constructed by finding the simplest model which is consistent with the counts observed during the time interval . the model used included a source image and a component that accounts for the counts unmodulated by the grids . this additional component prevents the unmodulated counts from being included in the source image . thus , the pixon image is not contaminated by this background and the pixon image reconstruction provides an independent method of determining the spatially integrated flare spectrum . we reconstructed the pixon images in several energy bands for 4.141 s ( @xmath2 1 spin period ) centred at the burst maximum ( 08:25:22 ut ) and determined a pixon photon spectrum from the total photon fluxes above the 10% contour level of the corresponding pixon image . since the uncertainty in the spectra from the image reconstruction is @xmath210% ( hurford , priv . comm . ) , and the pixon photon spectrum agrees within uncertainties of the same order with the flat photon spectrum up to 300 kev ( i.e. the highest energy for which the sources were detectable ) , we may conclude that the background subtraction procedure , see section [ spectroscopy ] , resulted in a solar count spectrum not significantly distorted by the precipitation component . primary photons with energies larger than 15 kev emitted downwards have a high probability of being reflected due to compton backscattering in the photosphere . these albedo photons then modify both the spectrum and the intensity of the observed hard x - rays . according to bai and ramaty ( 1978 ) , the reflectivity of a power - law photon spectrum from the isotropic source , which is assumed here , is a function of the photon energy and significantly depends on the photon spectral index @xmath24 . the reflectivity peaks at @xmath25 and increases as @xmath24 decreases . the total photon spectrum , that is the primary one plus the component composed of the reflected photons , is then characterised by a lower @xmath24 than that of the primary spectrum . for a spectral index @xmath26 , the reflected photons at @xmath27 kev can be as high as 80% of the primary flux . the amount of backscattered photons generally depends on the anisotropy of the primary hard x - ray source and the pitch - angle distribution of the beam electrons . but , the dependency is rather weak due to strong angular smoothing effect of the bremsstrahlung cross section . a detailed study of beam electron distributions is out of scope of this paper . however , the assumption of isotropy of the hard x - ray source provides a lower limit for the photospheric albedo correction . for this flare we studied the influence of the photospheric albedo correction on the mean electron flux distribution @xmath3 . we used two approaches . first , @xmath3 was approximated by a double power - law function @xmath28 where @xmath29 , @xmath30 , @xmath31 , @xmath32 , @xmath33 , @xmath34 , and @xmath35 are the electron energy , the low - energy cutoff , the break energy , the high - energy cutoff in kev , the power - law index below and above @xmath31 , and normalisation , respectively . the mean electron flux distribution represents the density weighted electron flux distribution in the source , and its determination is equivalent to the determination of the electron flux distribution assuming thin - target bremsstrahlung @xmath36 ( brown , emslie and kontar , 2003 ) @xmath37 where @xmath38 is the mean intensity in the source volume @xmath39 , @xmath40 is the distance of the emitting source , and @xmath41 is the isotropic cross section for bremsstrahlung ( haug , 1997 ) . using the forward fitting method described in section [ flatspectrum ] , the background subtracted photon spectra were fitted with a maxwellian isothermal component @xmath12 plus the thin - target bremsstrahlung @xmath42 of the electron flux model @xmath3 and the best - fit values of thermal plasma and electron flux parameters ( the fit provides @xmath43 ) were obtained . @xmath32 was kept fixed at 7 mev since setting the parameter free did not significantly improve the fits . second , the mean electron flux distribution function was derived by a regularised inversion procedure ( piana et al . , 2003 ; kontar et al . , 2004 , 2005a ) using the photon spectra ( determined with the forward fitting method ) as input . electron energies were determined in the energy range typically 10 - 400 kev , the upper energy depending on the signal - to - noise ratio . background subtracted photon spectra were corrected for photospheric albedo using the method of an angle dependent green s function for x - ray compton backscattering ( kontar et al . , 2005b ) , where the green s function represents the probability density of backscattering of a photon of initial energy @xmath44 into the observer s direction with energy @xmath15 . this approach is independent of the spectral characteristics of the primary photon spectra and allows any form of the primary photon spectra to be deduced , not just a power - law spectrum statistical plus systematic uncertainties . best fit to the primary photon flux is composed of the isothermal @xmath45 and thin - target @xmath46 component . bottom panel compares forward fitted @xmath43 derived from the total ( - - - ) and primary ( height 0.6ex depth -0.4ex width 1.5em ) photon spectrum . low - energy cutoffs and their standard errors corresponding to the total and primary photon spectrum are @xmath47 and @xmath48 , respectively . see section [ comp_ff_inv ] for details on parameter error estimation.,scaledwidth=80.0% ] as was done e.g. by bai and ramaty ( 1978 ) , and alexander and brown ( 2002 ) . green s function for a given heliocentric angle ( 40 degrees for this flare ) can be applied to the primary photon spectrum but as a computational convenience we modify the detector response matrix to account for the photospheric albedo . in this way it can be used straightforwardly in the forward fitting method ( kontar et al . ) because the result of the photon model multiplied by this new modified detector matrix is exactly the same as the result from the original detector matrix multiplied by the product of the albedo and the photon spectrum . in addition to photospheric albedo , the rhessi x - ray spectra generally contain a component from photons scattered in the earth s atmosphere . the geometry of rhessi detectors is such that most of the photons scattered in earth atmosphere must first pass through the rear segments ( see figure 1 in smith et al . , 2002 ) . , where the photons are effectively absorbed ( the stopping depth of ge detectors for a photon of 30 kev is @xmath2 1 mm ) thus , only the photons coming from a forward semisphere can get directly into the front segments . since rhessi is pointed towards the sun , the front segments are shielded by the rear ones from the earth scattered photons all the time when rhessi can see the sun apart from being at the terminator . the studied flare was observed when rhessi was far from the terminator . therefore , a correction for the earth albedo photons for actual zenith angle was neglected in the analysis . the primary photon spectrum ( derived under the assumption of its isotropy ) together with the total photon spectrum at 08:25:22 ut ( burst maximum ) is shown in the top panel of figure [ elow_albedo ] . the photon spectra differ mainly below 100 kev , the primary spectrum is slightly steeper , @xmath49 , and exhibits less flattening below 50 kev than the total photon spectrum . such an albedo correction significantly affects @xmath43 mainly at electron energies below 100 kev . the bottom panel of figure [ elow_albedo ] shows that a forward fitted @xmath43 consistent with the total photon spectrum has a high value of @xmath50 , which would represent a gap in energy between the thermal and electrons producing hard x - rays . however , removing the contribution of the photospheric albedo photons results in a forward fitted @xmath43 with @xmath51 close to the photon energy where the thermal part of the photon spectrum joins the non - thermal part . consequently , no reliable dip in @xmath43 is found . the mean electron flux distribution @xmath43 corresponding to the primary photon spectrum at 08:25:22 ut derived from both the forward fitting method and the regularised inversion procedure are shown in the top panel of figure [ albedo_ff_inv ] . the ranges of parameter values were determined as a change @xmath52 while keeping the individual parameter fixed , where the level 8.2 encloses the region of joint confidence of 7 parameters with 68% confidence ( lampton , margon and bowyer , 1976 ; press et al . , 1992 ) . uncertainties of the forward fitted @xmath43 were determined by multiple generations of the forward fit parameters lying within the same level of joint confidence . @xmath531-@xmath54 limits of the regularised solution were found from multiple inversions of the primary photon flux data perturbed within corresponding statistical and systematic uncertainties . the inversion method was studied in detail in kontar et al . ( 2004 ) , where various modelled input spectra were analysed . middle and bottom panels of figure [ albedo_ff_inv ] represent normalised and cumulative residuals of the primary @xmath55 and modelled photon spectrum @xmath56 . normalised residuals are defined as @xmath57/\sigma(\epsilon)$ ] , where @xmath58 is the uncertainty in the primary photon spectrum and includes statistical plus systematic uncertainties . deduced from forward fitting ( height 0.6ex depth -0.4ex width 1.5em ) and inversion ( * - - - * ) method for primary photon spectrum at 08:25:22 ut . shaded area represents @xmath531-@xmath54 limits on the regularised solution , thin full lines correspond to 68% confidence strip for the forward fit solution . middle and bottom panels show normalised and cumulative residuals of the primary and modelled photon spectrum for the forward fitting method . thin lines in the bottom panel represent @xmath531-@xmath54 limits for a random walk process . , scaledwidth=70.0% ] the error of the albedo correction method is @xmath59 , which is larger than @xmath58 but not included in it . cumulative residuals are defined as @xmath60 , where @xmath61 is the @xmath62 energy bin used in fitting , and can be used to assess clustering of residuals in certain energy ranges ( piana et al . , 2003 ) . normalised residuals of the best fit are limited to a 3-@xmath58 interval . cumulative residuals show minor clustering in the photon energy range 20 - 80kev but they are well below @xmath531-@xmath54 limits of random walk ( @xmath63 ) , see full lines in bottom panel of figure [ albedo_ff_inv ] . both normalised and cumulative residuals also indicate that assumed systematic uncertainties ( 5% in each energy bin ) are overestimated . adopting a lower value of 3% results in residuals distributed as expected for gaussian statistics with @xmath64 of them above the 1-@xmath54 level , but it does not modify @xmath43 significantly . , and the low - energy cutoff @xmath30 of the mean electron flux distribution @xmath43 . attenuator states a0 , a1 , and a3 are indicated by lines at the top of rhessi data plot . for display purposes the rhessi count flux at the 300 - 500 kev band was scaled by a factor of 2 . , scaledwidth=94.0% ] the best forward fit parameters for the primary spectrum are : emission measure @xmath65 , temperature @xmath66 , @xmath67 , @xmath68 , @xmath69 , @xmath70 , @xmath71 , where @xmath72 is @xmath3 integrated from @xmath30 to @xmath32 . several parameters ( @xmath73 , @xmath30 , @xmath74 ) have significantly different , by a factor of 5 , estimates of high and low @xmath54 . this behaviour is mainly due to the low value of @xmath33 because a decrease in @xmath30 and consequent increase in @xmath74 does not result in significant change in the modelled primary photon spectrum . the primary photon spectrum is therefore , within uncertainties , consistent also with a mean electron flux distribution function without a low - energy cutoff as derived from the forward fitting method . large uncertainties in @xmath73 are caused by the small number of data points in the thermal component of the spectrum ( see top panel of figure [ elow_albedo ] ) . the mean electron flux distribution deduced from the forward fitting agrees within uncertainties with the regularised solution up to @xmath2 200 kev . the discrepancy at higher energies is caused by different high - energy cutoffs in the forward fitting and inversion methods . the regularised solution was cut at 0.5 mev whereas the double power - law @xmath3 extended to @xmath75 . the large uncertainties between 20 and 40 kev shown in figure [ albedo_ff_inv ] result from the small number of electrons in this energy range and the fact that the photons are primarily from higher energy electrons . a similar argument can be used to explain the deviation between the forward fit and regularised solutions above 100 kev . the high value of @xmath32 used in the forward - fit solution introduces a contribution to the photon flux above 100 kev from the higher energy electrons not included in the regularised electron spectrum . in the case of the total photon spectrum , i.e. without the albedo correction , the regularised solution indicates a dip within 1-@xmath54 limits in the energies below @xmath76 kev where the forward fitted solution resulted in a gap above the thermal component in @xmath77 , see figure [ elow_albedo ] . figure [ fthin_time ] shows the time evolution of derived parameters @xmath33 and @xmath30 of the mean electron flux distribution corresponding to the primary photon spectra . the power - law index @xmath33 decreases to its minimum of @xmath78 at the burst maximum , 08:25:22 ut , and its time history generally shows a soft - hard - soft pattern between the photon flux and hardness of the spectra ( e.g. fletcher and hudson , 2002 ) . low values of @xmath33 indicate a flat electron flux spectrum and this is confirmed by the regularised inversion method . the low - energy cutoff @xmath30 lies within or close to the energies where the thermal component dominates @xmath77 distribution . standard errors are of the order of the @xmath30 values . therefore , no reliable dips or gaps in @xmath77 throughout the time interval could be established . our analysis shows that the low - energy cutoffs above the energies of the thermal component are likely to be the result of the contribution of photospheric albedo photons and do not exist in real mean electron spectra . to study the influence of the accelerated electrons on the h@xmath1 emission , we used full - disk h@xmath1 images covering the whole interval of the flare . the images were acquired using a filter with 0.07 nm fwhm at kanzelhhe solar observatory in austria . the time resolution of the data is 3 s and the spatial resolution is 2.2 arcsec / pixel . the position of the solar centre and the spatial resolution were determined using the routines provided by a. veronig . details on the method can be found in veronig et al . ( 2000 ) . the intensities from the flare site were scaled by the averaged quiet - sun area intensity i@xmath79 for every image separately . the brightest parts of the flare area beginning at 08:25:20 ut are distorted by the saturation of detected intensities . the saturated pixels are displayed with horizontal lines in the h@xmath1 images ( figures [ fig_ha_hsi_mdi_low ] , [ fig_ha_hsi_mdi_high ] , and [ fig_haderiv_hsi ] ) . the data set was further spatially coaligned using a technique developed for trace and eit data ( gallagher ) . the cross - correlation of the h@xmath1 data set was done in the area of the nearby sunspot with respect to the chosen h@xmath1 image . that sunspot area was not affected by the flare . therefore , we assumed no time change of the h@xmath1 intensities during the analysed time interval in that area . the positional uncertainty of the h@xmath1 images was estimated to be of the order of 1 pixel ( 2.2 arcsec ) . intensities ( grey scale ) and x - ray emission ( full line contours ) in the 7 - 12 kev ( first column ) and 12 - 25 kev ( second column ) energy ranges . contours correspond to 50 , 70 , and 90 % of maximum of each rhessi clean image . there were no detectable x - ray sources in the 12 - 25 kev energy range at 08:24:39 ut ( see the image in the top right corner ) . mdi magnetic neutral lines are plotted as dashed lines . the saturated h@xmath1 pixels are displayed with horizontal lines . , scaledwidth=80.0% ] intensities ( grey scale ) and x - ray emission ( full line contours ) in the 25 - 40 kev ( first column ) and 40 - 70 kev ( second column ) energy ranges . contours correspond to 50 , 70 , and 90 % of maximum of each rhessi clean image . there were no detectable x - ray sources in the 40 - 70 kev energy range at 08:24:48 ut ( see the image in the top right corner ) . mdi magnetic neutral lines are plotted as dashed lines . the saturated h@xmath1 pixels are displayed with horizontal lines . , scaledwidth=80.0% ] the evolution of the hard x - ray sources overlayed with the h@xmath1 emission and mdi magnetic neutral lines is displayed in figures [ fig_ha_hsi_mdi_low ] and [ fig_ha_hsi_mdi_high ] . the rhessi flux was accumulated for 1 spin period ( 4.141 s ) . the h@xmath1 images correspond to the middle of the rhessi imaging time range . due to the saturation of the h@xmath1 intensities , the spatial evolution in the flare area is shown to 08:25:20 ut only . at the very beginning of the flare ( at 08:24:39 ut , figure [ fig_ha_hsi_mdi_low ] ) , a single x - ray source is detected . then it splits into two ( at 08:24:48 ut , figures [ fig_ha_hsi_mdi_low ] and [ fig_ha_hsi_mdi_high ] ) and appears to be moving eastwards . later at 08:25:14 ut , the x - ray emission starts to shift southwards and the sources split even into three components detectable in the energy range 40 - 70 kev . there is a difference in the positions of hard x - ray sources at lower and higher energy ranges detected at the same time . the x - ray sources at higher energies tend to be located more to the south than the lower energy x - ray sources ( compare the positions of the sources in different energy bands at 08:25:14 ut and 08:25:20 ut in figures [ fig_ha_hsi_mdi_low ] and [ fig_ha_hsi_mdi_high ] ) . most of x - ray source maxima at energies above 12 kev are not spatially coincident with the brightest parts of the h@xmath1 emission . rather they tend to be located near the north and eastern edges of the bright h@xmath1 kernels . the h@xmath1 kernels form a semi - circle whose inner boundary is located near the magnetic neutral line . to understand better the magnetic field topology near the flare site , the extrapolation of magnetic field in the current - free approximation ( dmoulin et al . , 1997 ) was made using the mdi magnetogram observed at 08:12:30 ut ( see figure [ fig_mag ] ) . 10 , @xmath53 100 , and @xmath53 400 g ) are expressed by thin full and dashed lines , respectively . a cut through the region of the neutral magnetic line indicated by the arrow is shown in the sketch in the top right corner ( here sizes of loops are not to scale).,scaledwidth=65.0% ] the most interesting structure , indicated by an arrow in figure [ fig_mag ] , is located along the magnetic neutral line , which is delineated by the above mentioned semi - circular h@xmath1 ribbon . namely , this structure has a quadrupolar character in the sense of papers by uchida ( 1996 ) and hirose and uchida ( 2002 ) , i.e. low altitude loops in the close vicinity of the neutral line are between magnetic field lines with the opposite orientation and all is covered by extended overlying loops . the preflare activity in the 1550 trace band was observed directly in the centre of the quadrupolar structure ( i.e. above the magnetic neutral line ) at 08:12:20 ut . however , no 3 - 6 kev x - ray emission was detected by rhessi from this flare site at this time . comparing figures [ fig_ha_hsi_mdi_low ] , [ fig_ha_hsi_mdi_high ] and [ fig_mag ] , it can be seen that the flare started at the north end of this quadrupolar structure . then the flare kernels appeared at the north - east boundary of the structure and the flare ends at the southern extremity of the quadrupolar structure . we also note that the h@xmath1 ribbon and x - ray sources are shifted @xmath80 arcsec from the quadrupolar configuration of the extrapolated magnetic field lines . we assumed that the locations of the optical flare that are directly affected by the electron beams might be recognised as areas of abrupt changes of the h@xmath1 intensity , @xmath81 , on a time scale of the order of seconds . this assumption is supported e.g. by trottet et al . ( 2000 ) , who found correlation between hard x - ray flux and h@xmath1 intensities on time scales from seconds to tens of seconds . therefore , the time derivative of @xmath81 may provide a technique to find such areas . at the location [ 560,-280 ] arcsec revealing the largest derivative of @xmath81 ( see figure [ fig_haderiv_hsi ] ) . thin lines show observed time histories at 3x3 pixel area , @xmath82 correspond to the average over that area , and the thick line is the resulting time history smoothed by 3-point box car . intensities are scaled by the quiet sun intensity @xmath83 . ] intensities ( full line contours ) overlayed on h@xmath1 images ( grey scale ) . contours correspond to @xmath84 and @xmath85 of @xmath86 . rhessi clean images at 25 - 40 kev and 40 - 70 kev are plotted as dotted and dashed contours at 50 and 90 % of maximum of each image . the saturated h@xmath1 pixels are displayed with horizontal lines.,scaledwidth=78.0% ] in order to suppress the effects of seeing and/or incorrect positions , h@xmath1 intensities were averaged over a @xmath87 pixel area and the resulting lightcurves were smoothed by a 3-point box car ( 9 s time resolution ) . such smoothing was necessary to distinguish between the real changes and those which occurred randomly throughout the whole flare area . the effects of the averaging and smoothing are shown in figure [ fig_ha_time ] . due to the saturation of h@xmath1 data beginning at 08:25:20 ut , the time derivative of h@xmath1 intensity , @xmath86 , was studied in the time interval 08:24:33 - 08:25:20 ut ( see also the label in figure [ fig_hsi_radio ] ) . the maximum of @xmath86 was @xmath88 . a threshold , @xmath89 , was estimated by fitting @xmath86 distribution outside of the flare area with the gauss distribution . the threshold corresponds to 10 standard deviations from the mean of the fit and was adopted as the level of the reliable value of @xmath86 not affected by noise or seeing . the locations of @xmath90 are displayed on figure [ fig_haderiv_hsi ] together with the rhessi clean images in the 25 - 40 kev and 40 - 70 kev energy bands . x - ray sources in the 40 - 70 kev energy band are not shown in figure [ fig_haderiv_hsi](a ) since no source at this energy band was detectable at that time . these energy bands were chosen for comparison with @xmath86 for two reasons . first , they were above the energies where the thermal component dominates x - ray emission , i.e. @xmath91 . secondly , the electrons bombarding the chromosphere need to have an energy greater than 25 kev to penetrate into the layers of h@xmath1 formation , e.g. in the f1 flare model by machado et al . the rhessi clean images were reconstructed from data accumulated in 1 spin period . the centre of each rhessi time interval corresponds to the observational times of h@xmath1 and @xmath86 . the sites of the fast h@xmath1 intensity changes located in the northern part of the flare lie on the boundaries or inside of x - ray sources in the 25 - 40 kev and/or 40 - 70 kev energy range ( see figures [ fig_haderiv_hsi](b ) and ( c ) ) . however , we have found other areas that reveal abrupt changes in h@xmath1 intensity but are located well away from x - ray emission ( e.g. figure [ fig_haderiv_hsi](a ) ) . some of these areas even precede the position of x - ray emission as it moves to the south in time ( figures [ fig_haderiv_hsi](c ) and ( d ) ) . the difference in the locations of @xmath86 and the x - ray sources can not be explained by a spatial correlation with the x - ray sources at lower or higher energies . there is no significant shift of the x - ray sources in the energy ranges from 3 - 7 kev to 7 - 12 kev or from 40 - 70 kev to higher energy bands ; the sources at energies from 7 to 25 kev are not located closer to the @xmath86 ( see figures [ fig_ha_hsi_mdi_low ] and [ fig_ha_hsi_mdi_high ] ) either . the fact that at 08:25:20 ut there are no locations of @xmath90 along the whole flare area may be the result of the saturation of the h@xmath1 data at this time ( areas covered by horizontal lines in figure [ fig_haderiv_hsi ] ) . although the described analysis technique does not eliminate the saturated pixels , they only affect the adjacent pixels and cause a decrease in @xmath86 . the photon spectrum at the burst maximum of the august 20 , 2002 flare is , to our knowledge , the flattest ( @xmath0 ) yet detected by rhessi with its high energy resolution . to eliminate any distortion of the spectra , the background subtraction also included the estimation of the electron precipitation component , which has been done for the rhessi data for the first time . the solar origin of such an unusually flat spectrum was confirmed by a comparison with the spectrum constructed from the rhessi pixon images . using a new method of the photospheric albedo correction independent of the spectral form of the primary spectrum , we have proved the importance of the photospheric albedo correction in the case of the analysed flare . we have shown that in this case of flat x - ray spectra , the photospheric albedo significantly affects the determination of the mean electron flux distribution @xmath43 , mainly the value of the low - energy cutoff . the contamination of the flare spectra by the photospheric albedo photons may result in dips in @xmath43 indicated both by the forward fitting and inversion methods . such dips could be misinterpreted as spectral features inherent to the acceleration / propagation processes of the energetic particles in the solar flares . flares of flat hard x - ray spectra ( @xmath92 ) were previously analysed e.g. by nitta , dennis , and kiplinger ( 1990 ) and frnk , hudson , and watanabe ( 1997 ) . all of these flares have weak thermal components ( goes class m3 or lower ) but intense non - thermal components up to several hundred kev with a flat part extending down to @xmath93 kev . as was shown here , such photon spectra may correspond to a flat electron flux distribution with the low - energy cutoff close to the energy of the thermal electrons . for the flares with spectral index around 3 it is the low - energy cutoff which determines the total energy content in the flare . this happens when the largest contribution comes from the low energy particles . however , for the flares with flat spectra @xmath94 it is not the case . assuming the collisional thick - target scenario ( brown 1971 ) , one can estimate the energy flux of the non - thermal electrons injected into the plasma . in this model , the injected electron flux distribution @xmath95}$ ] is related to @xmath3 of equation ( [ ffthin ] ) as ( brown and emslie , 1988 ) @xmath96 relative change of the non - thermal energy flux @xmath97}$ ] can be expressed @xmath98 adopting the spectral index @xmath99 and the low - energy cutoffs , @xmath100 kev , @xmath101 kev , consistent with the inferred flat spectrum , the ratio is @xmath102 . thus , due to the flatness of @xmath43 , the low values of the low - energy cutoff do not cause a significant increase in the non - thermal energy flux . the flat distribution function of non - thermal electrons , especially at low energies , implies that its derivative in the energy as well as in the velocity space is close to zero or even positive . such a distribution function can be formed directly in the acceleration region ( e.g. by the acceleration in the direct electric field holman , 1985 ) and/or modified by propagation effect , particle collisions in dense plasma , and by interactions of accelerated electrons with plasma waves ( melnik and kontar , 2003 ) . if the plasma waves are generated , then the radio emission can be observed at high frequencies , as in our case . however , detailed conditions in the radio source needed to confirm this idea are not known to us . our analysis of the characteristics of the hard x - ray sources and the h@xmath1 and radio emission revealed several facts about the flare process . first , the presented flare was weak in the metric range during the whole event , except for the short - lasting relativistic type iii burst ( radio manifestation of relativistic electrons ; klassen , karlick ' y , and mann , 2003 ) . it resembles the march 6 , 1989 @xmath24-ray flare , which started as silent in radio waves below 800 mhz . similarly to rieger , treumann , and karlick ( 1999 ) , we explain the lack of metric radio emission by closed geometry of the magnetic structure and a low altitude of the flare process . additionally , the high value of the turn - over frequency , 19.6 - 50 ghz , of the radio emission indicates a strong magnetic field or a large depth of the emitting source ( benka and holman , 1992 ) , thus supporting this conclusion . furthermore , the coalignment of the h@xmath1 and hard x - ray images with the magnetic field extrapolation shows that both the h@xmath1 kernels and the hard x - ray sources are located at the footpoints of the low altitude magnetic loops which are the part of the quadrupolar structure ( figure [ fig_mag ] ) . second , our results concerning the time and spatial correlation of the hard x - ray sources and h@xmath1 emission are partly in agreement with the prediction of solar flare models . we have found that most of the x - ray emission maxima , mainly those at the energies above 12 kev ( corresponding to the x - ray emission of the non - thermal electrons ) , tend to be located on the northern and eastern edges of the h@xmath1 emission ( see figures [ fig_ha_hsi_mdi_low ] and [ fig_ha_hsi_mdi_high ] ) . this is consistent with the expectation that x - ray sources should be located on the external edges of h@xmath1 ribbons and similar to those reported by czaykowska et al . ( 1999 ) . flare models also suggest that fast changes of h@xmath1 intensities might be located at positions of x - ray sources , i.e. at the areas which are bombarded by the suprathermal electrons . in some cases , this suggestion is confirmed by our analysis . however , at some times the positions of the fast changes of h@xmath1 intensities do not spatially coincide with the x - ray sources ( see southern locations of @xmath86 in the bottom row of figure [ fig_haderiv_hsi ] ) . we propose that such h@xmath1 brightenings might be caused either by thermal conduction from a hot plasma located above , or by soft x - ray irradiation . using the lowest energy ranges of rhessi ( 3 - 7 kev ) , we tried to find such a hot plasma at these places , but without any success . however , we must point out that the time resolution of @xmath86 was 9 s. therefore to confirm the existence of these areas , more analyses and better quality data are needed . finally , the time evolution of the positions of the hard x - ray sources may indicate the movement of the reconnection site . this can be seen e.g. at 08:25:14 ut ( figures [ fig_ha_hsi_mdi_low ] and [ fig_ha_hsi_mdi_high ] ) when the hard x - ray sources of different energies are distributed along the h@xmath1 ribbon in such a way that the sources of higher energy are located further to the south - east direction . such difference in the positions could be expected in a single flare loop when it would correspond to the height distribution of the hard x - ray sources along the loop as found in aschwanden , brown , and kontar ( 2002 ) . however , this is not the case here because the extrapolation of the magnetic field shows no magnetic loop along the h@xmath1 ribbon . the displacement of the h@xmath1 ribbon and x - ray sources from the quadrupolar configuration is not due to the positional errors in the h@xmath1 , mdi , and rhessi images , which are a factor of 2 and 4 smaller , respectively . the offset could be caused by neglecting the electric currents in the current - free extrapolation . although the linear force - free extrapolations show similar results , we can not exclude the presence of localised currents which deform the magnetic field structure . we have demonstrated that the influence of the photospheric albedo on solar flare x - ray spectra must be considered in order to correctly assess the models of electron acceleration and propagation and generation of hard x - ray emission . the inferred flat mean electron flux distribution could be the result of the acceleration process and/or modified by particle - wave interactions . this explanation , however , remains speculative . the accumulated information from the multi - wavelength observations of this flare shows that the flare process took place in the low altitude magnetic loops . our analysis of h@xmath1 emission may indicate that the response of the lower atmosphere to the flare energy release is not restricted to the sites of propagation of the accelerated electrons . one of the authors , jk , would like to give special thanks to the rhessi team at gsfc for their help and support during her stay there . we are thankful to w. otruba ( kanzelhhe solar observatory , austria ) and a. veronig for providing h@xmath1 data and for valuable discussion on their analysis . we also thank t. metcalf for help with the pixon reconstruction , d. zarro for preprocessing trace data , and p. dmoulin for providing his code for the magnetic field extrapolation . furthermore , we thank p. messmer and t. lthi for the zrich and bern radio data , respectively . this work was supported by nato science fellowships programme , the contract nas5 - 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the equilibrium ( thermodynamic ) isotope effect ( eie ) is defined as the effect of isotopic substitution on the equilibrium constant . denoting an isotopolog with a lighter ( heavier ) isotope by a subscript _ ( _ h _ ) , the eie is defined as the ratio of equilibrium constants@xmath6 where @xmath7 _ _ _ _ and @xmath8 are molecular partition functions per unit volume of reactant and product . we study a specific case of eie - the equilibrium ratio of two isotopomers . in this case , the eie is equal to the equilibrium constant of the isotopomerization reaction , @xmath9 where superscripts _ r _ and _ p _ refer to reactant and product isotopomers , respectively . usually , equilibrium isotope effects are computed only approximately:@xcite in particular , effects due to indistinguishability of particles and rotational and vibrational contributions to the eie are treated separately . furthermore , the vibrational motion is approximated by a simple harmonic oscillator and the rotational motion is approximated by a rigid rotor . in general , none of the contributions , not even the indistinguishability effects can be separated from the others.@xcite however , at room temperature or above the nuclei can be accurately treated as distinguishable , and the indistinguishability effects can be almost exactly described by symmetry factors . on the other hand , the effective coupling between rotations and vibrations , anharmonicity of vibrations , and non - rigidity of rotations can in fact become more important at higher temperatures . for simplicity , from now on we denote these three effects together as anharmonicity effects and the approximation that neglects them the harmonic approximation ( ha ) . in some cases the effects of anharmonicity of the born - oppenheimer potential surface on the value of eie can be substantial.@xcite ishimoto _ _ have shown that the isotope effect on certain barrier heights can even have opposite signs , when calculated taking anharmonicity effects into account and in the ha.@xcite our goal is to describe rigorously equilibria at room temperature or above . therefore , two approximations that we make are the born - oppenheimer approximation and the distinguishable particle approximation ( we treat indistinguishability by appropriate symmetry factors ) . the error due to the born - oppenheimer approximation was studied for h / d eie by bardo and wolfsberg,@xcite and by kleinman and wolfsberg,@xcite and was shown to be of the order of 1 % in most studied cases . since we assume that nuclei are point charges , the born - oppenheimer approximation implies that the potential energy surface is the same for the two isotopomers . the differences of born - oppenheimer surfaces due to differences of nuclear volume and quadrupole of isotopes can be important for heavy elements@xcite , but these are not studied in this work . symmetry factors themselves result in an eie equal to a rational ratio , which can be computed analytically . in order to separate the symmetry contributions from the mass contributions to the eie , it is useful to introduce the reduced reaction free energy,@xmath10 where @xmath11 and @xmath12 are the symmetry factors discussed in more detail in sec . [ sec : examples ] . to include the effects of quantization of nuclear degrees of freedom beyond the ha rigorously we use the feynman path integral representation ( pi ) of the partition function . the quantum reduced reaction free energy can then be computed by thermodynamic integration with respect to the mass of the isotopes . to compute the derivative of the free energy efficiently , we use a generalized virial estimator . the advantage of this estimator is that its statistical error does not increase with the number of imaginary time slices in the discretized path integral . as a consequence , converged results can be obtained in a significantly shorter simulation than with other estimators . the ultimate goal would be to combine the path integral methodology with _ ab initio _ potentials . however , since millions of samples are required , the computational expense results in the following compromise : first , the equilibrium isotope effects are computed using _ ab initio _ potentials , but as usual , within the ha . then all anharmonicity corrections are computed using the pi methodology , but with semiempirical potentials . in other words , we take advantage of the higher accuracy of the _ ab initio _ potentials to compute the harmonic contribution to the eie and then make an assumption that the anharmonicity effects are similar for _ ab initio _ and semiempirical potentials . after describing theoretical features of the method , we apply it to hydrocarbons used in experimental measurements of isotope effects on [ 1,5 ] sigmatropic hydrogen transfer reactions . two of them were recently used by doering _ @xcite who reported equilibrium ratios of their isotopomers . this allows us to validate our calculations as well as to discuss the apparent discrepancy in measurements of doering _ et al . _ from a theoretical point of view . the outline of the paper is as follows : in sec . [ sec : method ] , we describe a rigorous quantum - mechanical methodology to compute the eie . section [ sec : examples ] presents the [ 1,5 ] sigmatropic hydrogen shift reactions on which we test the methodology , explains how _ ab initio _ methods can be combined with the pi to compute the eie , and discusses in detail symmetry effects in these reactions . section [ sec : computational - details ] explains the implementation of the method in amber 10 and describes details of calculations and error analysis of our pimd simulations . results of calculations are presented and compared with experiments in sec . [ sec : results ] . section [ sec : discussion - and - conclusions ] concludes the paper . eie can be calculated by a procedure of thermodynamic integration@xcite with respect to the mass . this method takes advantage of the relationship@xmath13,\label{eq : ti}\ ] ] where @xmath14 is the ( quantum ) free energy and @xmath15 is a parameter which provides a smooth transition between isotopomers @xmath16 and @xmath17 . this can be accomplished , e.g. , by linear interpolation of masses of all atoms in a molecule according to the equation @xmath18 in contrast to the partition function itself , the integrand of eq . @xmath19 is a thermodynamic average and therefore can be computed by either monte carlo or molecular dynamics simulations . classically , the eie is trivial and eq . can be evaluated analytically . when quantum effects are important , this simplification is not possible . to describe quantum thermodynamic effects rigorously , one can use the path integral formulation of quantum mechanics.@xcite in the path integral formalism , thermodynamic properties are calculated exploiting the correspondence between matrix elements of the boltzmann operator and the quantum propagator in imaginary time.@xcite in the last decades , path integrals proved to be very useful in many areas of quantum chemistry , most recently in calculations of heat capacities,@xcite rate constants,@xcite kinetic isotope effects,@xcite or diffusion coefficients.@xcite let @xmath20 be the number of atoms , @xmath21 the number of spatial dimensions , and @xmath22 the number of imaginary time slices in the discretized pi ( @xmath23 gives classical mechanics , @xmath24 gives quantum mechanics ) . then the pi representation of the partition function @xmath25 in the born - oppenheimer approximation is @xmath26,\label{eq : pi_qr}\\ c & \equiv\left(\frac{p}{2\pi\hbar^{2}\beta}\right)^{npd/2}\prod_{i=1}^{n}m_{i}^{pd/2}.\label{eq : pi_prefactor}\end{aligned}\ ] ] where @xmath27 is the set of cartesian coordinates associated with the @xmath28@xmath29 time slice , and @xmath30 is the effective potential @xmath31 the @xmath22 particles representing each nucleus in @xmath22 different imaginary time slices are called `` beads . '' each bead interacts via harmonic potential with the two beads representing the same nucleus in adjacent time slices and via potential @xmath32 attenuated by factor @xmath33 with beads representing other nuclei in the same imaginary time slice . by straightforward differentiation of eq . we obtain the so - called thermodynamic estimator ( te),@xcite @xmath34 a problem with expression is that its statistical error grows with the number of time slices . a similar behavior is a well known property of the thermodynamic estimator for energy,@xcite where the problem is caused by the kinetic part of energy . of imaginary time slices in the path integral . all results obtained by 1 @xmath35 long simulations ( with the time step 0.05 @xmath36 ) of compound * 1 - 5,5,5-__d__@xmath37 * ( @xmath38 ) using gaff force field , normal mode pimd , and nos - hoover chains of thermostats . [ fig : gve_te_values ] ] of imaginary time slices in the path integral . all results obtained by 1 @xmath35 long simulations ( with the time step 0.05 @xmath36 ) of compound * 1 - 5,5,5-__d__@xmath37 * ( @xmath38 ) using gaff force field , normal mode pimd , and nos - hoover chains of thermostats . note that the rmse of the gve is not only non - increasing ( as expected from theory ) , but in fact decreases slightly with increasing @xmath22 , which is due to the decrease of correlation length.[fig : gve_te_errors ] ] the growth of statistical error of the thermodynamic estimator for energy is removed by expressing the estimator only in terms of the potential and its derivatives using the virial theorem.@xcite in our case , a similar improvement can be accomplished , if a coordinate transformation , @xmath39 is introduced into eq . prior to performing the derivative . here , the `` centroid '' coordinate is defined as @xmath40 in other words , first the `` centroid '' coordinate @xmath41 is subtracted , and then the coordinates are mass - scaled . resulting generalized virial estimator ( gve)@xcite takes the form @xmath42 . \label{eq : gve}\ ] ] its primary advantage is that the root mean square error ( rmse ) of the average , @xmath43 , is approximately independent on the number of imaginary time slices @xmath22,@xmath44 in this equation @xmath45 denotes the length of the simulation and @xmath46 is the correlation length . the convergence of values and statistical errors of both estimators as a function of number @xmath22 of imaginary time slices for systems studied in this paper is discussed in section [ sec : computational - details ] . as expected , up to the statistical error they give the same values as can be seen in fig . [ fig : gve_te_values ] . nevertheless , when quantum effects are important , and a high value of @xmath22 must be used , the gve is the preferred estimator since it has much smaller statistical error and therefore converges much faster than the te ( see fig . [ fig : gve_te_errors ] ) . thermodynamic average in eq . can be evaluated efficiently using the path integral monte carlo ( pimc ) or path integral molecular dynamics ( pimd ) . in pimc , gradients of @xmath47 in eq . result in additional calculations since the usual metropolis monte carlo procedure for the random walk only requires the values of @xmath47 . this additional cost can be , however , reduced either by less frequent sampling , or by using a trick in which the total derivative with respect to @xmath15 ( not the gradients ! ) is computed by finite difference.@xcite in case of pimd , the presence of gradients of @xmath47 in eq . does not slow down the calculation since forces are already computed by a propagation algorithm . although in principle , a pimc algorithm for a specific problem can always be at least as efficient as a pimd algorithm , in practice it is much easier to write a general pimd algorithm and so pimd is usually the algorithm used in general software packages . since pimd was implemented in amber 9,@xcite we have implemented the methodology described above for computing eies into amber 10.@xcite this implementation is what was used in calculations in the following sections . in pimd , equation is augmented by fictitious classical momenta @xmath48 , @xmath49.\label{eq : pi_qp}\ ] ] the normalization prefactor @xmath50 is chosen in such a way that the original prefactor @xmath51 in eq . is reproduced when the momentum integrals in eq . are evaluated analytically . the partition function is formally equivalent to the partition function of a classical system of cyclic polyatomic molecules with harmonic bonds . each such molecule represents an individual atom in the original molecule and interacts with molecules representing other atoms via a potential derived from the potential of the original molecule.@xcite this system can be studied directly using well developed methods of the classical molecular dynamics . it is useful to see how the general path integral expressions , , and behave in the low and high temperature limits . as temperature decreases , the difference of the reduced free energies of isotopomers approaches the difference of their zero point energies ( zpes ) . therefore , still assuming that indistinguishability is correctly described by symmetry factors , the low temperature limit of the eie is equal to @xmath52,\label{eq : eie_low_temp}\ ] ] where @xmath53 denotes the zpe . at high temperature , the system approaches its classical limit . in this limit , we can set the number of imaginary time slices @xmath23 , and the eie can be computed analytically using partition function , which becomes@xmath54.\end{split}\label{eq : q_class}\ ] ] again , for isotopomers the mass dependent factors cancel out upon substitution into eq . , along with all other terms except for the integrals . although the born - oppenheimer potential surface is the same for all isotopomers , the integrals do not cancel out since the integration is not performed over the whole configuration space , but only over parts attributed to reactants or products . particles are treated as distinguishable in the classical limit and the volumes of the configuration space corresponding to reactants and products are generally different . after the reduction , eie becomes exactly equal to the ratio of the symmetry factors , @xmath55 we examine the eie for four related compounds . the parent compound ( 3__z__)-penta-1,3-diene ( compound * 1 * ) is the simplest molecule to model the [ 1,5 ] sigmatropic hydrogen shift reaction . two of its isotopologs , tri - deuterated ( 3__z__)-(5,5,5-@xmath56h@xmath37)penta-1,3-diene ( * 1 - 5,5,5-__d__@xmath37 * ) and di - deuterated ( 3__z__)-(1,1-@xmath56h@xmath5)penta-1,3-diene ( * 1 - 1,1-__d__@xmath57 * ) ( see fig . [ fig : equlibrium_pent ] ) were used by roth and knig to measure an unusually high value of the kinetic isotope effect ( kie ) on the [ 1,5 ] hydrogen shift reaction with respect to the substitution of hydrogen by deuterium.@xcite this result pointed to a significant role of tunneling in [ 1,5 ] sigmatropic hydrogen transfer reactions . subsequently , much theoretical research was devoted to the study of this reaction . here we calculate final equilibrium ratios of products of this reaction , which , to our knowledge , were not theoretically predicted so far . hydrogen shift reaction in ( 3__z__)-(5,5,5-@xmath56h@xmath37)penta-1,3-diene ( * 1 - 5,5,5-__d__@xmath37 * ) and in ( 3__z__)-(1,1-@xmath56h@xmath5)penta-1,3-diene ( * 1 - 1,1-__d__@xmath57 * ) . if all contributions except for those due to symmetry factors @xmath58 and @xmath59 were neglected , one would obtain approximate equilibrium constants @xmath60 and @xmath61 in both cases.[fig : equlibrium_pent ] ] two other compounds , 2-methyl-10-(10,10-@xmath56h@xmath5)methylenebicyclo[4.4.0]dec-1-ene ( * 2 - 1,1-__d__@xmath57 * ) and 2,4,6,7,9-pentamethyl-5-(5,5-@xmath56h@xmath5)methylene-11,11a - dihydro-12_h_-naphthacene ( * 3 - 1,1-__d__@xmath57 * ) ( see fig . [ fig : equilibria_cyclic ] ) , were recently used by doering _ _ to confirm and possibly refine the experimental value of the kie on the [ 1,5 ] hydrogen shift.@xcite in contrast to ( 3__z__)-penta-1,3-diene * * ( * * compound * 1 * ) , where the s-_trans _ conformer incompetent of the [ 1,5 ] hydrogen shift is the most stable , pentadiene moiety in compounds 2 and 3 is locked in the s-_cis _ conformation . this not only increases the reaction rate , but also rules out the ( very small ) effect of the eie on the kie due to the shift in s-_cis_/s-_trans _ equilibrium . for both molecules doering _ et al . _ reported final equilibrium ratios of isotopomers . despite the similarity of compounds * 2 * and * 3 * , these ratios are qualitatively different . indeed , one motivation for measuring eie in compound * 3 * was that doering _ et al . _ suspected that in the case of * 2 - 1,1-__d__@xmath57 * the equilibrium ratio might be modified by unwanted side reactions , mainly dimerizations . one of our goals is to elucidate this discrepancy from the theoretical point of view . as can be seen in figs . [ fig : equlibrium_pent ] and [ fig : equilibria_cyclic ] , the final equilibrium of the [ 1,5 ] hydrogen shift reaction of all examined compounds can be described as an outcome of two reactions . the second reaction ( leading from deuterio - methyl - dideuterio - methylene to dideuterio - methyl - deuterio - methylene in tri - deuterated compounds and from dideuterio - methyl - methylene to deuterio - methyl - deuterio - methylene in di - deuterated compounds ) produces a mixture of species that differ by the orientation of the mono - deuterated methylene group . due to a high barrier for rotation of this group , the two products can not be properly sampled in a single pimd simulation . therefore an additional pimd simulation is necessary , as shown on the example of * 1 - 5,5,5-__d__@xmath37 * in fig . [ fig : pimd-3steps - pent ] . the reduced free energy of the second step is then calculated as @xmath62,\label{eq : f_red_2step_pimd}\ ] ] where 1/2 is the ratio of symmetry factors and @xmath63 and @xmath64 stand for equilibrium constants obtained by the second and third pimd simulations , respectively . together they represent the second reaction step . hydrogen shift reaction in ( 3__z__)-(5,5,5-@xmath56h@xmath37)penta-1,3-diene ( * 1 - 5,5,5-__d__@xmath37 * ) . half white , half black spheres represent deuterium atoms . the methyl group , in contrast to methylene , rotates during simulations . [ fig : pimd-3steps - pent ] ] unfortunately , at present the pimd method can not be used in conjunction with higher level _ ab initio _ methods , due to a high number of potential energy evaluations needed . semiempirical methods , which can be used instead , do not achieve comparable accuracy . we therefore make the following two assumptions : first , we assume that the main contribution to the eie can be calculated in the framework of ha . second , we assume that selected semiempirical methods are accurate enough to reliably estimate the anharmonicity correction . the anharmonicity correction is calculated as @xmath65 with these two assumptions , we can take advantage of both pimd and higher level methods by adding the semiempirical anharmonicity correction to the ha result calculated by a more accurate method . the ha value of @xmath66 is obtained by boltzmann averaging over all possible distinguishable conformations , @xmath67},\label{eq : f_red_ha}\ ] ] where @xmath68 is the number of `` geometrically different isomers '' of a reactant . by geometrically different isomers we mean species differing in their geometry , not species differing only in positions of isotopically substituted atoms . @xmath69 is the electronic energy ( including nuclear repulsion ) of @xmath70 isomer , @xmath11 is the symmetry factor and @xmath71 are partition functions of the nuclear motion of @xmath11 isotopomers . @xmath72 , @xmath12 , @xmath73 denote analogous quantities for the product . although symmetry effects can be computed analytically , for the reactions studied in this paper they are nontrivial and so we discuss them here in more detail . as mentioned above , we are interested in moderate temperatures ( above @xmath74 @xmath2 ) where quantum effects might be very important but the distinguishable particle approximation remains valid . in this case , effects of particle indistinguishability and of non - distinguishing several , in principle distinguishable minima , by an experiment can be conveniently unified by the concept of symmetry factor . in our setting , we will call `` symmetry factor '' the product@xmath75 here , @xmath76 refers to the number of distinguishable minima not distinguished by the experiment and @xmath77 are the usual rotational symmetry numbers * * * * of symmetric rotors . the symmetry numbers are present only if either the whole molecule or some of its parts are treated as free or hindered classical symmetrical rotors . ( in this case , the number of minima of the hindered rotor potential is not included in @xmath76 . ) the symmetry numbers are not present if rotational barriers are so high that the corresponding degrees of freedom should be considered as vibrations . the concept can be illustrated on an example of the mono- and non - deuterated methyl group in a rotational potential with three equivalent minima 120@xmath78 apart . at low temperatures , when hindered rotation of the methyl group reduces to a vibration , the symmetry factor is determined only by @xmath76 . in the case of mono - deuterated methyl group there are three in principle distinguishable minima corresponding to three rotamers , which are , as we suppose , considered to be one species by the observer . therefore @xmath79 . in the case of non - deuterated methyl there is only one distinguishable minimum and @xmath80 . at higher temperatures , when the methyl group can be treated as a hindered rotor , its contribution to the symmetry factor is determined only by rotational symmetry number @xmath81 , where @xmath82 for a mono - deuterated methyl group and @xmath83 for a non - deuterated methyl group . from definition , it is clear that the high and low temperature pictures are consistent and give the same ratios of symmetry factors . partition functions @xmath84 and @xmath85 needed in calculation of the reduced free energy in eq . are computed as sums of partition functions of @xmath86 and @xmath87 isotopomers . in this subsection , we will discuss the accuracy of four electronic structure methods used in our calculations . ab initio mp2 and the b98 density functional method,@xcite both in combination with the 6 - 311+(2df , p ) basis set , were used for calculations within the ha . semiempirical am1@xcite and scc - dftb@xcite methods were used in both ha and pimd calculations . this allowed us to compute the error introduced by the ha . aside from symmetry factors , the eie is dominated by vibrational contributions . therefore we concentrate mainly on the accuracy of ha vibrational frequencies . according to merrick _ _ et al.,__@xcite who tested the performance of mp2 and b98 methods by means of comparison with experimental data for a set of 39 molecules , rmse of zpes is 0.46 @xmath88 at mp2/6 - 311+(2df , p ) and 0.31 @xmath88 at b98/6 - 311+(2df , p ) level of theory . appropriate zpe scaling factors are equal to 0.9777 and 0.9886 respectively . corresponding rmses of frequencies are 40 @xmath89 and 31 @xmath90 therefore , a slightly higher accuracy can be expected from the b98 functional . the accuracy of both semiempirical methods is significantly worse than the accuracy of higher level _ ab initio _ methods . according to witek and morokuma , the rmse of am1 frequencies in comparison to experimental values for 66 molecules is 95 @xmath89 with frequency scaling factor equal to 0.9566.@xcite the error of vibrational frequencies obtained using scc - dftb depends on parametrization . we tested two parameter sets : the original scc - dftb parametrization@xcite and the parameter set optimized with respect to frequencies by malolepsza _ _ et al.__@xcite ( further designated scc - dftb - mwm ) . the error of vibrational frequencies calculated with the original scc - dftb parameters was studied by krger _ _ et al.__@xcite the mean absolute deviation from the reference values calculated at blyp / cc - pvtz for a set of 22 molecules was 75 @xmath89 . the error of the reference method itself , as compared to an experiment with a slightly smaller set of molecules was 31 @xmath89 . in the above mentioned study performed by witek and morokuma , rmse of 82 @xmath89 with scaling factor 0.9933 was obtained.@xcite the scc - dftb - mwm mean absolute deviation of experimental and calculated frequencies for a set of 14 hydrocarbons is indeed better and is equal to 33 @xmath89 instead of 59 @xmath89 for the original parametrization . @xcite the suitability of am1 , scc - dftb , scc - dftb - mwm , and several other semiempirical methods for our systems was tested by comparison of eie values in the ha and of potential energy scans with the corresponding quantities computed with mp2 and b98 . results of this comparison are presented in the appendix . the am1 method is shown to reproduce _ ab initio _ eies in the ha very well , but fails to reproduce scans of potential energies of methyl and vinyl group rotations . therefore , in addition to am1 method we used the scc - dftb method , which is , among the semiempirical methods tested by us , the best in reproducing _ ab initio _ potential energy scans . on the other hand , compared to am1 , scc - dftb gives a worse eie in the ha . statistical rmses of averages of pimd simulations were calculated from the equation @xmath91 where @xmath81 is the rmse of one sample . correlation lengths @xmath46 were estimated by the method of block averages.@xcite at constant temperature , the correlation length decreases with increasing number of imaginary time slices . also , for constant number of imaginary time slices , the correlation length decreases with increasing temperature . finally , the correlation length stays approximately constant at different temperatures , if the number of imaginary time slices is chosen so that @xmath92 is converged to approximately the same precision . in our systems , correlation length is close to 3.5 @xmath93 . the eie was studied at four different temperatures : 200 , 441.05 , 478.45 , and 1000 @xmath2 using the normal mode version of pimd.@xcite to control temperature , the nos - hoover chains with four thermostats coupled to each pi degree of freedom were used.@xcite . different numbers @xmath22 of imaginary time slices had to be used at different temperatures , since at lower temperatures quantum effects become more important , and the number of imaginary time slices necessary to maintain the desired accuracy increases . to examine the required value of @xmath22 as a function of temperature we used the gaff force field.@xcite whereas the accuracy of vibrational frequencies calculated using the gaff force field is relatively low [ rms difference of gaff and b98/6 - 311+(2df , p ) frequencies of compound * 1 * is equal to 125 @xmath89 ] , the potential should be realistic enough for the assessment of the convergence with respect to @xmath22 . for example , the difference of potential energies @xmath95 between s-_cis _ and s-_trans _ conformations of compound * 1 * is 4.3 @xmath96 as compared to 2.7 @xmath96 obtained by mp2/6 - 311+(2df , p ) or 3.5 @xmath96 obtained by b98/6 - 311+(2df , p ) . to check the convergence at 478.45 @xmath2 , we calculated values of the integral in eq . for the deuterium transfer reaction in * 1 - 5,5,5-__d__@xmath37 * with 40 and 48 imaginary time slices ( using simpson s rule with 5 points ) . their difference is equal to 0.00005 @xmath97 0.00040 @xmath96 . this is less than the statistical error of the calculation on the model system , which itself is smaller than the error of production calculations , since the model calculation was 10 times longer than the longest production calculation . therefore , taking into account the accuracy of production calculations , @xmath98 can be considered the converged number of imaginary time slices . this is further supported by the observation of the convergence of the single value of the gve at @xmath38 . the relative difference of gve values obtained with 40 and 48 imaginary time slices is equal to 0.22 @xmath97 0.02 % , whereas the difference between 40 and 72 imaginary time slices equals to 0.39 @xmath97 0.04 % . the discretization error is asymptotically proportional to @xmath99 . by fitting this dependence to the calculated values , we estimated the difference between the values for @xmath98 and for the limit @xmath24 to be less than 0.6 % . this is only three times more than the difference between @xmath98 and @xmath100 . therefore , if the integral converges similarly as the gve , we can use the aforementioned difference between 40 and 48 imaginary time slices as the criterion of convergence . the convergence of the gve with the number of imaginary time slices is displayed in fig . [ fig : gve_te_values ] . since 441.05 @xmath2 is close enough to 478.45 @xmath2 we used the same value of @xmath22 at this temperature . to check the convergence at 200 and 1000 @xmath2 , we observed only the convergence of the single value of the gve . at 200 @xmath2 the relative difference of gve values at @xmath38 obtained with @xmath101 and @xmath102 is 0.03 @xmath97 0.07 % . based on the comparison with the previous result , @xmath101 is considered sufficient . at 1000 @xmath2 , the relative difference of gve values at @xmath38 for @xmath103 and @xmath104 equals to 0.1 @xmath97 0.02 % , so that 24 imaginary time slices are used further . the time step at 441.05 and 478.45 @xmath2 was 0.05 @xmath36 to satisfy the requirement of energy conservation . at 200 and 1000 @xmath2 , a shorter step of 0.025 @xmath36 was used due to the increased stiffness of the harmonic bonds between beads at 200 @xmath2 and due to the increased average kinetic energy at 1000 @xmath2 . the simulation lengths differed for different molecules . for both isotopologs of compound * 1 * simulation length of 1 @xmath35 ensured that the system properly explored both the s-_trans _ and the s-_cis _ conformations . convergence was checked by monitoring running averages and by comparing the ratio of the s-_trans _ and s-_cis _ conformers with the ratio calculated in the ha . the length of converged pimd simulations of compound * 2 * was 500 @xmath93 . convergence was checked again using running averages and by visual analysis of trajectories to ensure that the system properly explored all local minima . for compound * 3 , * the simulation length was 400 @xmath93 . the integral in eq . was calculated using the simpson s rule . using the am1 potential , the gve was evaluated for five values of @xmath15 , namely for @xmath105 . convergence was checked by comparison with values obtained using the trapezoidal rule . since the dependence of the estimator on the parameter @xmath15 is almost linear , the difference between the two results remained well under the statistical error . using the scc - dftb potential , the dependence on @xmath15 was less smooth . as a result , nine equidistant values of @xmath15 were needed to achieve similar convergence . in one case , as many as 17 values of @xmath15 had to be used . lcccc & 1@xmath106 step @xmath66 & @xmath107 & 2@xmath108 step @xmath66 & @xmath107 + + am1 ( pimd ) & 0.0395 & -0.0041@xmath970.0009 & -0.0154 & 0.0022@xmath970.0007 + scc - dftb ( pimd ) & 0.1245 & -0.0039@xmath970.0007 & -0.0616 & 0.0026@xmath970.0005 + b98 ( ha ) + @xmath109 & 0.0587 & & -0.0248 & + mp2 ( ha ) + @xmath109 & 0.0770 & & -0.0338 & + + am1 ( pimd ) & -0.0283 & 0.0063@xmath970.0009 & 0.0191 & -0.0023@xmath970.0007 + scc - dftb ( pimd ) & -0.1142 & 0.0049@xmath970.0006 & 0.0610 & -0.0017@xmath970.0005 + b98 ( ha ) + @xmath109 & -0.0466 & & 0.0282 & + mp2 ( ha ) + @xmath109 & -0.0645 & & 0.0372 & + the pimd calculations were performed using amber 10.@xcite the part of amber 10 code , which computes the derivative @xmath110 with respect to the mass was implemented by one of us and can be invoked by setting itimass variable in the input file . several posible ways to compute the derivative are obtained by combining one of two implementations of pimd in ` sander ( ` either the multisander implementation or the les implementation ) with either the nos - hoover chains of thermostats or the langevin thermostat , with the normal mode or `` primitive '' pimd , and with the te or gve . calculating the value of @xmath110 for compound * 1 - 5,5,5-__d__@xmath37 * using gaff force field , at @xmath38 , @xmath111 @xmath2 , and for several values of @xmath22 , we confirmed that all twelve possible combinations give the same result . for example , fig . [ fig : gve_te_values ] shows the agreement of the gve and te . nevertheless , the twelve methods differ by rmses of @xmath110 ( due to different statistical errors of estimators and correlation lengths ) and by computational costs . as expected , the most important at higher values of @xmath22 is the difference of statistical errors of gve and te . figure [ fig : gve_te_errors ] compares the dependence of the rmse of the gve and te on the number @xmath22 of imaginary time slices . for @xmath112 , the converged simulation with gve is approximately 100 times faster than with te . less significant differences of rmses are due to differences in correlation lengths . as expected , for primitive pimd with langevin thermostat the correlation length depends strongly on collision frequency @xmath113 of the thermostat . the correlation length is approximately 450 - 500 @xmath36 for @xmath114 @xmath115 , falling down quickly to 120 - 150 @xmath36 for @xmath116 @xmath115 and to 5 @xmath36 for @xmath117 @xmath115 , then rising slowly again . this can be compared to correlation length of 10 - 20 @xmath36 of the primitive pimd thermostated by nos - hoover chains of 4 thermostats per degree of freedom . a smaller difference can be found between correlation lengths of the normal mode ( 3 - 8 @xmath36 ) and primitive pimd ( 10 - 20 @xmath36 ) . all pimd calculations were performed in amber 10.@xcite all mp2 and b98 calculations as well as am1 and pm3 semiempirical calculations in the ha were done in gaussian 03 revision e01.@xcite dftb and scc - dftb calculations in the ha used the dftb+ code , version 1.0.1.@xcite scc - dftb harmonic frequencies were computed numerically using analytical gradients provided by the dftb+ code . the step size for numerical differentiation was set equal to 0.01 @xmath118 . this value was also used by krger _ _ et al.__@xcite in their study validating the scc - dftb method and their frequencies differed from purely analytical frequencies of witek _ _ et al.__@xcite by at most 10 @xmath89 . to diagonalize the resulting numerical hessian , we used the ` formchk ` utility included in the gaussian program package . ( 3__z__)-penta-1,3-diene ( compound * 1 * ) is the simplest of examined molecules . its non - deuterated isotopolog has three distinguishable minima : the s-_trans _ conformer , which is the global minimum and two s-_cis _ conformers related by mirror symmetry . strictly speaking , in pentadiene the s-_cis _ species have actually _ gauche _ conformations due to sterical constraints . in their original experiments , roth and knig studied two isotopologs , * 1 - 5,5,5-__d__@xmath37 * and * 1 - 1,1-__d__@xmath57*. the eies of both isotopologs were computed using the pimd methodology of sec . [ sec : method ] at 478.45 @xmath2 . resulting reduced free energies are listed in table [ tab : f_pentadiene ] . note that the anharmonicity correction is very similar for am1 and scc - dftb methods , even though the main part of obtained in the ha ( substantially differs for the two methods . this indicates that the corrections are fairly reliable in this case and can be used to correct results of higher level methods obtained in the ha . the anharmonicity correction is as large as 20 % of the final value of the reduced free energy . unfortunately , the difference between mp2 and b98 in the ha is still approximately four times larger than the anharmonicity correction . .equilibrium ratios of [ 1,5 ] hydrogen shift reactions of * 1 - 5,5,5-__d__@xmath37 * and * 1 - 1,1-__d__@xmath57 * at 478.45 @xmath2 . [ tab : eq_ratios_pentadiene ] [ cols="<,^,^,^",options="header " , ] to conclude , the combination of higher level methods in the ha with pimd using semiempirical methods for the rigorous treatment of effects beyond the ha proved to be a viable method for accurate calculations of eies . using the generalized virial estimator for the derivative of the free energy with respect to the mass we were able to obtain accurate results at lower temperatures in reasonable time ( @xmath0 times faster than with thermodynamic estimator ) , since the statistical error is independent on the number of imaginary time slices . two semiempirical methods , am1 and scc - dftb , were used for calculation of the anharmonicity correction , both giving very similar results . calculations showed , that the anharmonicity effects account up to 30 % of the final value of the reduced free energy of considered reactions . the anharmonicity correction always decreases the absolute value of the reduced reaction free energy . this is consistent with the qualitative picture in which higher vibrational frequencies of hydrogens are lowered by the anharmonicity of a potential more than lower frequencies of deuteriums . this in turn is due to a higher amplitude of vibrations of lighter hydrogen atoms . the lower difference between frequencies of unsubstituted and deuterated species results in the lower absolute value of the reduced reaction free energy . unfortunately , the inaccuracy of the _ ab initio _ electronic structure methods used in our study is still of at least the same order as the anharmonicity corrections . for isotopologs of compound * 1 * , we predicted equilibrium ratios and free energies of the [ 1,5 ] sigmatropic hydrogen shift reaction . a comparison with experimental results was not possible due to a low precision of the original measurement . for compound * 2 * , the disagreement between theoretical and experimental data supports the suspicion by authors of the measurement that the accuracy of their results was compromised by dimerization side reactions . on the other hand , the agreement of theoretically calculated ratios with experimental observations in the case of compound * 3 * suggests that the isolation of the [ 1,5 ] hydrogen shift reaction from disturbing influences was successfully achieved and the observed eie and kie can be considered reliable . we acknowledge the start - up funding provided by epfl . we express our gratitude to daniel jana for his assistance with scc - dftb calculations , to h. witek for providing us the set of frequency optimized scc - dftb parameters , and to j. j. p. stewart for the mopac2007 code used for testing the pm6 method . to determine the suitability of semiempirical methods used in our pimd calculations , we first compared values of the eie in the ha . the b98 and mp2 methods served as a reference . as can be seen from fig . 6 in the paper , which shows the temperature dependence of @xmath66 of the first step of deuterium transfer reaction in * 1 - 5,5,5-__d__@xmath37 * , the difference between b98 and mp2 at temperatures below 500 @xmath2 is close to 0.02 kcal / mol . so is the difference between am1 and b98 methods . on the other hand , the scc - dftb method clearly overestimates the extent of eie compared to both higher level methods . a very similar trend was observed in all examined reactions . other semiempirical methods tested were pm3@xcite and scc - dftb - mwm , which are not included in fig . 6 in the paper for clarity the pm3 method overestimates the eie similarly to scc - dftb , whereas the scc - dftb - mwm curve is somewhat closer to _ ab initio _ curves than the scc - dftb one . to conclude , from this point of view am1 is the preferred semiempirical method . during simulations , the pentadiene molecule often passes two potential energy barriers . these are the barrier for the hindered rotation of the methyl group and the barrier for the rotation of the vinyl group , which connects s-_trans _ and s-_cis _ conformations . relaxed potential energy scans of these two motions were employed as the second criterion to assess the relevancy of semiempirical methods . methods tested were mp2 , b98 , am1 , scc - dftb , scc - dftb - mwm , pm3 , rm1,@xcite pm3carb-1,@xcite pddg / pm3,@xcite and pm6.@xcite potential surface scans with pm3carb1 , rm1 , pm3/pddg methods were calculated using the public domain code mopac 6 . pm6 potential surface scans were performed in mopac 2007.@xcite results for the methyl group rotation are shown in fig . [ fig : methyl_scan ] . the height of the am1 barrier is only 0.005 @xmath96 . moreover , positions of minima do not agree with b98 and mp2 . on the other hand , the scc - dftb method matches higher level methods closely . from other semiempirical methods pm3 performs best in this aspect . the height of the barrier is relatively well reproduced also by pddg / pm3 , rm1 and scc - dftb - mwm methods , but positions of extrema of the potential energy surface are incorrect . figure [ fig : vinyl_scan ] shows potential energy scans of the s-_trans_/s-_cis _ rotation of the vinyl group . again , the scc - dftb method matches higher level methods closely . all other methods ( with the exception of dftb ) give too low barrier heights as well as too low energy differences between s-_trans _ and s-_cis _ conformations . also note that the potential energy surfaces of pm3 and related methods ( pddg / pm3 and pm3carb-1 ) are not smooth in the gauche region . this peculiarity of the pm3 potential surface can be seen also in the potential surface scan performed by liu _ _ et al.__@xcite based on these results we concluded that none of the semiempirical methods except for scc - dftb is able to sufficiently improve the am1 potential energy surface . whereas the frequency optimized variant of the scc - dftb method ( scc - dftb - mwm ) improves the eie in the ha , it does not retain the scc - dftb accuracy in the potential surface scans . hence we decided to use the am1 and scc - dftb potentials for pimd calculations . to conclude , am1 performs very well in ha , but it can not properly describe potential surfaces of the two rotational motions realized during simulations . on the other hand , scc - dftb gives worse results in ha , but it reproduces both barriers very well .

the reidemeister torsion is an invariant for a cw - complex and a representation of its fundamental group . in other words , this invariant associates with the local system for a representation of the fundamental group . originally the reidemeister torsion is defined if the local system is @xmath4 , i.e. , all homology groups vanish . however we can extend the definition of the reidemeister torsion to non - acyclic cases @xcite . in this paper , we focus on the non - acyclic cases . it is known that the fox calculus plays important roles in the study of the reidemeister torsion @xcite . the many results were obtained by using the fox calculus for the acyclic reidemeister torsion . in particular , there are important results related to the alexander polynomial in the knot theory @xcite . the fox calculus is also important for non - acyclic cases @xcite . it is related to the cohomology theory of groups . this paper contributes to the study of the non - acyclic reidemeister torsion by using the fox calculus . our purpose is to apply the fox calculus for the acyclic cases to the study of the non - acyclic reidemeister torsion by using a relationship between the acyclic reidemeister torsion and the non - acyclic one . our main theorem says that the non - acyclic reidemeister torsion for a knot exterior is given by the differential coefficients of the twisted alexander invariant of the knot . the twisted alexander invariant of a knot is the acyclic reidemeister torsion and expressed as a one variable rational function @xcite . a conjecture due to j. dubois and r. kashaev @xcite will be solved in @xcite by using our main theorem . in the latter of this paper , we apply this relationship to study the reidemeister torsion for the pair of a @xmath3-bridge knot and @xmath1-representation of its knot group . we give an explicit expression of the non - acyclic reidemeister torsion associated to @xmath5 knot . this is a new example of calculation of the non - acyclic reidemeister torsion . furthermore , we investigate where the non - acyclic reidemeister torsion associated to a @xmath3-bridge knot has critical points . note that the non - acyclic reidemeister torsion is parametrized by the representations of a knot group . moreover this reidemeister torsion turns into a function on the character variety of the knot group . we will see that the critical points of the non - acyclic reidemeister torsion associated to a @xmath3-bridge knot are binary dihedral representations and these representations are related to the geometry of the character variety of a @xmath3-bridge knot group . this paper is organized as follows . in section [ review_twisted_torsion ] , we review the reidemeister torsion . in particular , we give the notion of the non - acyclic reidemeister torsion of knot exteriors @xcite . section [ main_theorem ] includes our main theorem on a relationship between the non - acyclic reidemeister torsion and the twisted alexander invariant for knot exteriors . we give a formula of the non - acyclic reidemeister torsion for a knot exterior by using a wirtinger presentation of a knot group . in section [ applications ] , we apply the results of section [ main_theorem ] to study the non - acyclic reidemeister torsion for a @xmath3-bridge knot group and @xmath1-representation of its knot group . in this paper , we use the following notations . * @xmath6 is the field @xmath7 or @xmath8 . * @xmath9 is the lie group @xmath1 ( resp . @xmath10 if @xmath6 is @xmath7 ( resp . @xmath8 ) . the symbol @xmath11 denotes the lie algebra of @xmath9 . * @xmath12 denotes the adjoint action of @xmath9 to the lie group @xmath11 . * @xmath13 is a product on the @xmath11 , which is defined by @xmath14 * @xmath15 denotes an @xmath16-dimensional vector space over @xmath6 . * for two ordered bases @xmath17 and @xmath18 in a vector space , we denote by @xmath19 the base - change matrix from @xmath20 to @xmath21 satisfying @xmath22 . we write simply @xmath23 $ ] for the determinant @xmath24 of @xmath19 . we deal with ordered bases in this paper . we recall the definition of the torsion . let @xmath25 be a chain complex over @xmath6 . for each @xmath26 let @xmath27 denote the kernel of @xmath28 , @xmath29 the image of @xmath30 and @xmath31 the homology group @xmath32 . we say that @xmath33 is _ acyclic _ if @xmath31 vanishes for every @xmath26 . let @xmath34 be a basis of @xmath35 and @xmath36 be the collection @xmath37 . we call the pair @xmath38 a _ based chain complex _ , @xmath36 the preferred basis of @xmath33 and @xmath34 the preferred basis of @xmath35 . let @xmath39 be a basis of @xmath31 . we construct another basis as follows . by the definitions of @xmath27 , @xmath29 and @xmath31 , the following two split exact sequences exist . @xmath40 @xmath41 let @xmath42 be a lift of @xmath43 to @xmath35 and @xmath44 a lift of @xmath31 to @xmath27 . then we can decompose @xmath35 as follows . @xmath45 we choose @xmath46 a basis of @xmath29 . we write @xmath47 for a lift of @xmath46 and @xmath48 for a lift of @xmath39 . by the construction , the set @xmath49 forms another ordered basis of @xmath35 . we denote simply this new basis by @xmath50 . then the definition of @xmath51 is as follows . @xmath52^{(-1)^{i+1 } } \in { { \mathbb f}}^*.\ ] ] it is well known that @xmath51 is independent of the choices of @xmath53 , the lifts @xmath54 and @xmath55 . we also define the torsion @xmath56 with the sign term @xmath57 as follows @xcite @xmath58 here @xmath59 where @xmath60 and @xmath61 . let @xmath62 be a finite connected cw - complex and @xmath63 its universal covering with the induced cw - structure . since the fundamental group @xmath64 acts on @xmath63 by the covering transformation , the chain complex @xmath65 has a natural structure of a left @xmath66$]-module . we denote by @xmath67 a homomorphism from @xmath64 to @xmath9 . we regard the lie group @xmath68 as a right @xmath66$]-module by @xmath69 . we use the notation @xmath70 for @xmath11 with the right @xmath66$]-module structure . following @xcite , we introduce the following notations . set @xmath71 where @xmath72 is @xmath73 and @xmath74 is a surjective homomorphism from @xmath64 to the multiplicative group @xmath75 . note that @xmath76 . we call @xmath77 _ the @xmath70-twisted chain complex _ and @xmath78 _ the @xmath79-twisted chain complex _ of @xmath62 . we also denote by @xmath80 the @xmath6-module consisting of the @xmath64-equivalent homomorphisms from @xmath81 to @xmath11 , i.e. , a homomorphism @xmath82 satisfies @xmath83 for @xmath84 . we call @xmath80 _ the @xmath70-twisted cochain complex _ of @xmath62 . @xmath85 and @xmath86 denote the homology and cohomology groups of the @xmath70-twisted chain and cochain complexes . we keep the notation of the previous subsection . let @xmath87 be the set of @xmath26-dimensional cells of @xmath62 . we take a lift @xmath88 of the cell @xmath89 in @xmath63 . then , for each @xmath26 , @xmath90 is a basis of the @xmath66$]-module @xmath91 . let @xmath92 be a basis of @xmath11 . then we obtain the following basis of @xmath93 : @xmath94 when @xmath95 is a basis of @xmath96 , we denote by @xmath97 the basis @xmath98 of @xmath99 . then @xmath100 is well defined . furthermore adding a sign - refinement term into @xmath101 , we define _ the reidemeister torsion _ of @xmath102 as a vector in some @xmath103-dimensional vector space as follows . let @xmath104 be the basis over @xmath7 of @xmath105 . choose an orientation @xmath106 of the real vector space @xmath107 and provide @xmath108 with a basis @xmath109 such that each @xmath39 is a basis of @xmath110 and the orientation determined by @xmath111 agrees with @xmath106 . let @xmath112 be either @xmath113 or @xmath114 according to the sign of @xmath115 . then we define the reidemeister torsion @xmath116 by @xmath117 where @xmath118 and @xmath119 here @xmath120 means the dual space of a vector space @xmath15 and the dual basis of @xmath118 is @xmath121 where @xmath122 is the dual element of @xmath123 . we made some choices in the definition of @xmath116 . however the following well - definedness is known @xcite : * the sign of @xmath116 is determined by the homology orientation @xmath106 i.e. , if we choose the other homology orientation , then the sign of @xmath116 changes ; * @xmath116 does not depend on the choice of the lift @xmath124 for each cell @xmath89 ; * @xmath116 does not depend on the choice of the basis @xmath97 in @xmath125 . we also have the following well - definedness . if the euler characteristic of @xmath62 is equal to zero , then @xmath116 does not depend on the choice of the basis of @xmath11 . this follows from the definition . similarly we define the reidemeister torsion of the twisted @xmath79-chain complex . we define @xmath126 by @xmath127 @xmath126 has the indeterminacy of @xmath128 where @xmath129 . this indeterminacy is caused by the choice of the lifts @xmath130 and the action of @xmath74 . it is also known that the sign refined torsion @xmath131 has the invariance under simple homotopy equivalences , and that it satisfies the following _ multiplicativity property_. suppose we have the following exact sequence of based chain complexes : @xmath132 where these chain complexes are based chain complexes which consist of vector spaces with bases . here we denote bases of @xmath133 by @xmath134 and a lift of @xmath135 to @xmath33 by @xmath136 . for each @xmath26 , fix the volume forms on @xmath137 by using given bases and choose volume forms on @xmath138 and @xmath139 . there exists the long exact sequence in homology associated to the short exact sequence @xmath140 : @xmath141 we denote by @xmath142 this acyclic complex . note that this acyclic complex is a based chain complex . [ m_property ] we have @xmath143 where @xmath144 \in { { \mathbb z}}/2{{\mathbb z}}.\end{aligned}\ ] ] let @xmath145 be a finitely generated group and we denote by @xmath146 the space of @xmath9-representations of @xmath145 . we define the topology of this space by compact - open topology . here we assume that @xmath145 has the discrete topology and the lie group @xmath9 has the usual one . a representation @xmath147 is called _ central _ if @xmath148 . a representation @xmath67 is called _ abelian _ if its image @xmath149 is an abelian subgroup of @xmath9 . a representation @xmath67 is called _ reducible _ if there exists a proper non - trivial subspace @xmath150 of @xmath151 such that @xmath152 for any @xmath153 . a representation @xmath67 is called _ irreducible _ if it is not reducible . we denote by @xmath154 the subset of reducible representations and by @xmath155 the subset of irreducible ones . note that all abelian representations are reducible . the lie group @xmath9 acts on @xmath146 by conjugation . we write @xmath156 $ ] for the conjugacy class of @xmath157 , and we denote by @xmath158 the quotient space @xmath159 . if @xmath9 is @xmath1 , then one can see that the reducible representations are exactly abelian ones . note that this does not hold for the case of @xmath2-representations . the action by conjugation of @xmath1 on @xmath160 factors through @xmath161 . this action is free on the @xmath162 . we set @xmath163 . if @xmath9 is @xmath2 , then the quotient space @xmath164 is not hausdorff in general . following @xcite , we will focus on the _ character variety _ @xmath165 which is the set of @xmath166 of @xmath145 . associated to the representation @xmath167 , its character @xmath168 , defined by @xmath169 . in some sense , @xmath170 is the `` algebro quotient '' of @xmath171 by @xmath172 . it is well known that @xmath171 and @xmath173 have the structure of complex algebraic affine sets and two irreducible representations of @xmath145 in @xmath2 with the same character are conjugate by an element of @xmath2 . ( for the details , see @xcite . ) in this subsection , we recall @xmath0-regular representations and how to construct distinguished bases of @xmath70-twisted homology groups of knot exteriors for a @xmath0-regular representation @xmath67 . these definitions have originally been given in @xcite . the original definitions are written in terms of the @xmath70-twisted cohomology group . we introduce the homology version by using the duality between the twisted homology and cohomology associated to _ the kronecker pairing _ @xmath174 @xcite . let @xmath175 be a knot in a homology three sphere @xmath176 . we give a knot exterior @xmath177 the canonical homology orientation defined as follows . it is well known that the @xmath7-vector space @xmath178 has the basis @xmath179 , [ \mu]\}$ ] . here $ ] is the homology class of a point and @xmath181 $ ] is the homology class of a meridian of @xmath175 . we denote by @xmath106 the orientation induced by @xmath182 , [ \mu]\}$ ] . we calculate the twisted homology groups of a circle and a @xmath3-dimensional torus before giving the definition of a natural basis of @xmath183 . here @xmath184 consists of one @xmath185-cell @xmath186 and one @xmath103-cell @xmath187 . [ lemma : homology_circle ] suppose that @xmath9 is @xmath1 . if @xmath188 is central , then @xmath189 . if @xmath67 is non - central , then we have @xmath190 , \,and \\ h_0(s^1 ; { \mathfrak{g}_\rho } ) & = { { \mathbb r}}[p_{\rho } \otimes \tilde e^{(0)}]\end{aligned}\ ] ] where @xmath191 is a vector in @xmath11 , which satisfies that @xmath192 for any @xmath193 . suppose that @xmath9 is @xmath2 . if @xmath188 is central , then @xmath194 . if @xmath67 is non - central and @xmath195 has no parabolic elements , then we have @xmath196 , \,and\\ h_0(s^1 ; { \mathfrak{g}_\rho})&={{\mathbb c}}[p_{\rho } \otimes \tilde e^{(0 ) } ] \end{aligned}\ ] ] where @xmath191 is a vector in @xmath11 , which satisfies that @xmath192 for any @xmath193 . if @xmath67 is non - central and the subgroup @xmath195 is contained in a subgroup which consists of parabolic elements , then we have @xmath197.\ ] ] this is a consequence of the following fact of homology of groups . for @xmath198 , it follows that @xmath199 and @xmath200 where @xmath9 is a group , @xmath201 is a @xmath201-module , @xmath202 is the group of invariants of @xmath201 and @xmath203 is the group of co - invariants of @xmath201 ( for the details , see @xcite ) . we denote by @xmath204 a @xmath3-dimensional torus . here @xmath204 consists of one @xmath185-cell @xmath186 , two @xmath103-cells @xmath205 and one @xmath3-cell @xmath206 . we denote each cell @xmath207 and @xmath206 by @xmath208 and @xmath204 . one can also calculate the @xmath70-twisted homology groups of @xmath209 as follows . [ lem : homology_torus ] suppose that @xmath9 is @xmath1 . if @xmath210 is central , then @xmath211 . if @xmath210 is non - central , then we have @xmath212 , \\ h_1(t^2 ; { \mathfrak{g}_\rho } ) & = { { \mathbb r}}[\ , p_{\rho } \otimes \tilde \mu\ , ] \oplus { { \mathbb r}}[\ , p_{\rho } \otimes \tilde \lambda\,],\\ h_0(t^2 ; { \mathfrak{g}_\rho } ) & = { { \mathbb r}}[\,p_{\rho } \otimes \widetilde pt\,]\end{aligned}\ ] ] where @xmath191 is a vector of @xmath11 such that @xmath213 for any @xmath214 . suppose that @xmath9 is @xmath2 . if @xmath210 is central , then @xmath215 . if @xmath210 is non - central and @xmath216 contains a non - parabolic element , then we have @xmath217,\\ h_1(t^2 ; { \mathfrak{g}_\rho } ) & = { { \mathbb c}}[\ , p_{\rho } \otimes \tilde \mu\ , ] \oplus { { \mathbb c}}[\ , p_{\rho } \otimes \tilde \lambda \,],\\ h_0(t^2 ; { \mathfrak{g}_\rho } ) & = { { \mathbb c}}[\ , p_{\rho } \otimes \widetilde pt \ , ] \end{aligned}\ ] ] where @xmath191 is a vector of @xmath11 such that @xmath213 for any @xmath214 . if @xmath210 is non - central and the subgroup @xmath216 is contained in a subgroup which consists of parabolic elements , then we have @xmath218\ ] ] and @xmath219 $ ] is a non - zero class in @xmath220 . this is a consequence of ( * ? ? ? * proposition 3.18 ) . next we give the definition of regular representations for @xmath221 in terms of the twisted @xmath70-chain complex . we say that @xmath67 is _ regular _ if @xmath67 is irreducible and @xmath222 . we let @xmath223 be a simple closed curve in @xmath224 . we say that @xmath67 is _ @xmath223-regular _ if : * @xmath67 is regular ; * an inclusion @xmath225 induces the surjective homomorphism @xmath226 * if @xmath227 , then @xmath228 . we fix an invariant vector @xmath229 as above . let @xmath223 be a simple closed curve in @xmath224 . an inclusion @xmath230 and the the kronecker pairing between homology and cohomology induce the linear form @xmath231 . by lemma [ lemma : homology_circle ] , it is explicitly described by @xmath232 ) , v ) = ( p_{\rho } , v(\tilde \gamma))_{{\mathfrak{g } } } \quad \text{for any}\,\ , v \in h^1(m_k ; { \mathfrak{g}_\rho}).\ ] ] an alternative formulation of @xmath223-regular representations is given in @xcite . similarly , we can also give the following alternative formulation of the @xmath223-regularity in our conventions . [ check_gamma_regular ] a representation @xmath233 is @xmath223-regular if and only if the linear form @xmath234 is an isomorphism . if @xmath235 is an isomorphism , then we have that @xmath236 and @xmath237)$ ] is a non - zero class in @xmath220 . it follows from the kronecker pairing between the @xmath70-twisted homology and cohomology that @xmath238 is also one . hence @xmath239 is surjective . if @xmath67 is @xmath223-regular , then we have that @xmath240 and @xmath241 is surjective . we denote a generator of @xmath220 by @xmath242 . there exists an element @xmath243 $ ] of @xmath244 such that @xmath245)=\sigma$ ] . if @xmath246 is central , then @xmath247 satisfies that @xmath248 for any @xmath249 . therefore @xmath250)$ ] induces the isomorphism @xmath235 . suppose that @xmath246 is non - central , then @xmath244 is generated by @xmath251 $ ] . there exists an element @xmath252 such that @xmath243 = c [ p_{\rho } \otimes \tilde \gamma]$ ] . hence @xmath237)$ ] is a non - zero class in @xmath220 . therefore @xmath237)$ ] induces the isomorphism @xmath235 . we define a reference generator of @xmath220 by using the above isomorphism @xmath235 . let @xmath67 be a @xmath0-regular representation of @xmath221 . by lemma [ lem : homology_torus ] , the reference generator of @xmath220 is defined by @xmath253).\ ] ] moreover the reference generator of @xmath254 is defined as follows . let @xmath255 be an inclusion map . if @xmath256 is @xmath223-regular , then we have the isomorphism @xmath257 . using this isomorphism @xmath258 , we define the reference generator of @xmath254 by @xmath259).\ ] ] the reference generators of @xmath260 and @xmath261 have been defined in @xcite by using another metric of @xmath11 . if we define reference generators of @xmath260 and @xmath261 by using our metric @xmath262 , then the resulting generators become the dual bases of @xmath263 and @xmath264 from the above propositions . ( for the details , see @xcite . ) we recall the definition of _ the twisted reidemeister torsion _ for knot exteriors . let @xmath265 be a @xmath0-regular representation . we define @xmath266 by the coefficient of the reidemeister torsion @xmath267 where we choose the reference generators @xmath268 as a basis of @xmath269 , i.e. , @xmath270 is given explicitly by @xmath271 given the reference generator of @xmath183 , the basis of the determinant line @xmath272 is also given . this means that a trivialization of the line bundle @xmath272 at @xmath67 is given . the reidemeister torsion @xmath267 is a section of the line bundle @xmath272 . we can regard @xmath273 as a section of the line bundle @xmath272 over @xmath0-regular representations with respect to the trivialization by @xmath274 . we also call @xmath270 _ the twisted reidemeister torsion_. our purpose is to express the twisted reidemeister torsion by using a limit of the acyclic reidemeister torsion . let @xmath175 be a knot in a homology three sphere @xmath176 and @xmath177 its exterior . one of the invariants which we will investigate is the twisted reidemeister torsion @xmath275 . the other is the acyclic reidemeister torsion @xmath276 . this invariant coincides with the twisted alexander invariant of @xmath221 @xcite . the twisted alexander invariant is computed by using the fox calculus @xcite . we prove that the twisted reidemeister torsion may be expressed as the differential coefficient of the twisted alexander invariant of @xmath221 . the invariant @xmath276 is only defined when the local system @xmath277 is acyclic . on the other hand , the twisted reidemeister torsion @xmath275 is defined on the set of @xmath0-regular representations of @xmath221 . we need to check whether the local system @xmath277 is acyclic for a @xmath0-regular representation @xmath67 . [ propositiona ] let @xmath67 be an @xmath1 or @xmath2-representation of a knot group . if @xmath67 is @xmath0-regular , then the twisted chain complex @xmath278 is acyclic . note that for a knot exterior in a homology @xmath279-sphere , the homomorphism @xmath74 satisfies @xmath280 where @xmath281 is the meridian of the knot . therefore @xmath275 and @xmath276 are well defined on @xmath0-regular representations . by the definitions , the twisted reidemeister torsion @xmath275 is an element of @xmath282 and the twisted alexander invariant @xmath276 is an element of @xmath283 . actually the following relation between @xmath284 and the rational function @xmath285 . [ thm : main_theorem ] if @xmath67 is a @xmath0-regular representation , then the acyclic reidemeister torsion @xmath286 for @xmath67 has a simple zero at @xmath287 . moreover the following holds : @xmath288 this says that we can compute the twisted reidemeister torsion @xmath275 algebraically by using fox calculus of the twisted alexander invariant of @xmath175 . we prove proposition [ propositiona ] by using the @xmath0-regularity of @xmath67 . it is well known that any compact connected triangulated @xmath279-manifold whose boundary is non - empty and consists of tori can be collapsed into a @xmath3-dimensional sub - complex ( see ii . cor . 11.9 in @xcite ) . moreover , by the simple - homotopy extension theorem , every cw - complex has the simple - homotopy type of a cw - complex which has only one vertex . we denote this @xmath3-dimensional cw - complex by @xmath62 and this deformation from @xmath177 to @xmath62 by @xmath289 . since two @xmath79-twisted homology groups @xmath290 and @xmath291 are isomorphic , we prove that @xmath292 vanishes in the following . the fact that @xmath293 is proved in ( * ? ? ? * proposition 3.5 ) . since the euler characteristic of @xmath62 is zero , the dimension of @xmath294 is equal to that of @xmath295 . we must prove that the dimension of @xmath296 over @xmath297 is zero . it is enough to prove that the rank over @xmath298 $ ] of the second homology group of the following local system is zero : @xmath299 ) = { \mathfrak{g}}[t , t^{-1 } ] \otimes_{\alpha \otimes ad \circ \rho } c_*(\tilde w;{{\mathbb z}})\ ] ] where @xmath300 $ ] is @xmath298 \otimes { \mathfrak{g}}$ ] . we denote the homology group of this chain complex by @xmath301)$ ] . suppose that the rank of @xmath302 ) > 0 $ ] . there exists the long exact homology sequence @xcite : @xmath303 ) \xrightarrow{(t-1)\cdot } h_2(w;{\mathfrak{g}_\rho}[t , t^{-1 } ] ) \xrightarrow{t=1 } h_2(w;{\mathfrak{g}_\rho } ) \xrightarrow{\delta } h_1(w;{\mathfrak{g}_\rho}[t , t^{-1 } ] ) \to \cdots\ ] ] associated to the short exact sequence : @xmath304 \xrightarrow{(t-1)\cdot } { \mathfrak{g}}[t , t^{-1 } ] \xrightarrow{t=1 } { \mathfrak{g}}\to 0.\ ] ] since the rank of @xmath305)$ ] is not zero , the multiplication with @xmath306 is not surjective . hence the image of the evaluation map @xmath307 is not trivial and therefore surjective since the dimension of @xmath308 is only one . this implies that @xmath309 is trivial . on the other hand the equation @xmath310 implies that @xmath311 ) = [ 1\otimes p_{\rho } \otimes \widetilde{\varphi(\lambda)}]$ ] . but @xmath312 $ ] can not be trivial since it is mapped under the evaluation map @xmath307 to @xmath313 $ ] and the chain @xmath314 represents a non - zero homology class in @xmath315 . this is a contradiction . therefore the rank of @xmath316)$ ] over @xmath298 $ ] is zero . hence we have that @xmath317 . also @xmath318 is zero . at first , we prepare some notations and an algebraic proposition . let @xmath33 is an @xmath16-dimensional chain complex which consists of left @xmath9-modules @xmath319 where @xmath9 is a group . we denote by @xmath320 the chain complex which consists of the vector spaces @xmath321 where @xmath15 is a right @xmath9-vector space over @xmath6 and @xmath67 is a homomorphism from @xmath9 to @xmath322 . let @xmath323 be the homology groups of @xmath320 , @xmath324 the subchain complex which consists of a lift of @xmath323 to @xmath320 and @xmath325 the quotient of @xmath320 by @xmath324 . we denote by @xmath326 and @xmath135 the bases of @xmath327 and @xmath325 . note that @xmath328 is a lift of @xmath329 to @xmath320 . if there exists a homomorphism @xmath74 from @xmath9 to the multiplicative group @xmath75 , we denote by @xmath330 which consists of vector spaces @xmath331 . here we denote @xmath332 by @xmath333 . moreover let @xmath334 be the subchain complex which is given by extending the coefficients of @xmath324 to @xmath297 by using @xmath74 and @xmath335 the quotient of @xmath330 by @xmath334 . [ prop : alg_preparation ] we assume that @xmath330 and @xmath334 are acyclic . the following relation holds : @xmath336 where @xmath136 is a lift of @xmath135 to @xmath320 , @xmath337 is @xmath338 in proposition [ m_property ] , and @xmath339 is given by @xmath340 . the chain complex @xmath335 is also acyclic from the long exact sequence of the pair @xmath341 . we can apply proposition [ m_property ] for the short exact sequence : @xmath342 then , we obtain the following equation of the torsions . @xmath343 note that @xmath344 because @xmath330 , @xmath334 and @xmath335 are acyclic . next we consider @xmath345 . it follows from the long exact sequence of the pair @xmath346 and the definition of @xmath324 that the chain complex @xmath325 is also acyclic . since @xmath325 is acyclic , we can choose a basis @xmath347 of @xmath348 for each @xmath26 . here @xmath348 is a lift of @xmath349 to @xmath350 . [ claim : on_lift_b ] a subset @xmath351 in @xmath352 generates a subspace on which the boundary operator @xmath30 is injective . _ proof of claim [ claim : on_lift_b ] . _ if the determinant of the boundary operator restricted on @xmath353 is zero , then substituting @xmath103 for the parameter @xmath354 we have that the determinant of the boundary operator restricted on @xmath355 is also zero . this is a contradiction to the choices of @xmath347 . ( claim [ claim : on_lift_b ] ) @xmath356 therefore @xmath357 is represented as @xmath358^{(-1)^{i+1}}.\ ] ] we denote by @xmath47 a lift @xmath351 to @xmath330 simply . note that @xmath359^{(-1)^{i+1 } } } & \\ & = \prod_{i=0}^n \left [ ( 1 \otimes c'^i ) \ , \partial_{i+1 } ( \tilde b^i ) \ , \tilde b^{i-1 } / 1 \otimes c'^i \cup 1 \otimes \bar c''^i \right]^{(-1)^{i+1}}.\end{aligned}\ ] ] we substitute these results into the equation ( [ eqn : result_m_property ] ) then we have @xmath360^{(-1)^{i+1 } } \nonumber\\ & = \prod_{i=0}^n ( -1)^{\dim_{{{\mathbb f } } } b''_i \cdot \dim_{{{\mathbb f}}}h_i(v ) } \left [ \partial_{i+1 } ( \tilde b^i)\ , ( 1 \otimes c'^i ) \ , \tilde b^{i-1 } / 1 \otimes c'^i \cup 1 \otimes \bar c''^i \right]^{(-1)^{i+1}}. \label{eqn : ratio_torsion}\end{aligned}\ ] ] the acyclicity of @xmath325 shows that @xmath361 substituting @xmath103 for @xmath354 , the right hand side ( [ eqn : ratio_torsion ] ) turns into @xmath362^{(-1)^{i+1}}.\ ] ] this is equal to @xmath363 . although the left hand side is determined up to a factor @xmath364 , the limit at @xmath287 is determined because the factor @xmath128 does not affect taking a limit at @xmath287 . we can prove theorem [ thm : main_theorem ] as an application of proposition [ prop : alg_preparation ] . as in the proof of proposition [ propositiona ] , let @xmath62 be a @xmath3-dimensional cw - complex with a single vertex which has the same simple - homotopy type as @xmath177 . we denote the deformation from @xmath177 to @xmath62 by @xmath289 . the compact @xmath279-manifold @xmath177 is simple homotopy equivalent to @xmath62 . it is enough to prove the theorem for @xmath62 because of the invariance of the simple homotopy equivalence for the reidemeister torsion . let @xmath67 be a @xmath0-regular representation of @xmath221 . we denote by the same symbols @xmath67 and @xmath106 the representation of @xmath64 and the homology orientation of @xmath108 induced from that of @xmath177 under the map @xmath289 . we define the subchain complex @xmath365 of the @xmath70-twisted chain complex @xmath77 by @xmath366 and @xmath367 where @xmath191 is an invariant vector of @xmath11 such that @xmath368 for any @xmath369 . the modules of this subchain complex are lifts of homology groups @xmath99 . by the definition , the boundary operators of @xmath365 are zero homomorphisms . let @xmath370 be the quotient of @xmath77 by @xmath365 . similarly , we define the subcomplex @xmath371 of @xmath78 to be @xmath372 and @xmath373 for @xmath374 . the boundary operators of @xmath371 is given by @xmath375 this shows that the subchain complex @xmath376 is acyclic . by proposition [ propositiona ] , the @xmath79-twisted chain complex @xmath277 is also acyclic . the twisted chain complex @xmath365 has the natural basis : @xmath377 let @xmath135 be a basis of @xmath370 and @xmath136 a lift of @xmath135 to @xmath77 . applying proposition [ prop : alg_preparation ] , we have @xmath378 [ claim : proof_main_thm ] 1 . 2 . @xmath380 . 3 . @xmath381 . _ proof of claim [ claim : proof_main_thm ] . _ @xmath382 it follows by the definition . @xmath383 if we denote the number of @xmath103-cells of @xmath62 by @xmath384 , the cw - complex @xmath62 has one @xmath185-cell , @xmath384 @xmath103-cells and @xmath385 @xmath3-cells . we have @xmath386 . @xmath387 this follows from @xmath388 and @xmath389 . ( claim [ claim : proof_main_thm])@xmath356 the equation ( [ eqn : alg_prop_w ] ) turns into @xmath390 multiplying the both sides by the alternative products of the determinants of the base - change matrices @xmath391^{(-1)^{i+1}},\ ] ] we obtain the following equation : @xmath392 finally multiplying the both sides by the sign @xmath112 gives @xmath393 summarizing the above calculation , we have shown that the rational function @xmath276 has a simple zero at @xmath287 and its differential coefficient at @xmath287 agrees with minus the twisted reidemeister torsion @xmath394 . let @xmath175 be a knot in @xmath396 and @xmath397 its exterior . we assume that @xmath398 is @xmath0-regular . from theorem [ thm : main_theorem ] we can describe @xmath394 by using the differential coefficient of @xmath399 . we will describe the differential coefficient of @xmath399 more explicitly by using a wirtinger representation of @xmath400 . for a wirtinger representation : @xmath401 we obtain a @xmath3-dimensional cw - complex @xmath62 which consists of one @xmath185-cell @xmath402 , @xmath384 @xmath103-cells @xmath403 and @xmath385 @xmath3-cells @xmath404 attached by the relation @xmath405 . this cw - complex @xmath62 is simple homotopy equivalent to @xmath397 . let @xmath406 such that @xmath280 . here @xmath281 is a meridian of @xmath175 . note that for all @xmath26 , @xmath407 is equal to @xmath354 in @xmath408 . the following calculation is due to the result of @xcite . this chain complex @xmath78 is as follows : @xmath409 where @xmath410 here we briefly denote the @xmath411-times direct sum of @xmath72 by @xmath412 . we denote by @xmath413 @xmath414 matrix : @xmath415 under this situation , the twisted alexander invariant @xmath126 is given by @xmath416 up to a factor @xmath417 . if @xmath418 is conjugate to the upper triangulate matrix @xmath419 then @xmath420 is conjugate to the upper triangulate matrix @xmath421 calculating @xmath422 , we have that @xmath423 since @xmath399 has zero at @xmath287 , @xmath424 if @xmath425 , then we have @xmath426 the function @xmath399 has a simple zero at @xmath287 and the numerator @xmath427 is an element of @xmath298 $ ] . hence @xmath428 divides @xmath427 . we write @xmath429 for @xmath427 . then the left hand side turns into @xmath430 , i.e. , @xmath431 . on the other hand , the right hand side becomes as follows . @xmath432_{t=1 } \\ & = \tau_0 f(1).\end{aligned}\ ] ] the numerator @xmath427 is called _ the first homology torsion _ of @xmath433 @xcite . we denote the first homology torsion by @xmath434 . by the above calculations , we obtain the following description of @xmath435 . [ another_form_torsion ] if @xmath436 , then we have the following expression . @xmath437 if @xmath9 is @xmath1 and @xmath67 is @xmath0-regular , then @xmath438 . we use a wirtinger representation of @xmath400 to describe @xmath399 in the above calculation . the twisted alexander invariant @xmath399 does not depend on the representation of @xmath400 @xcite . since @xmath399 is determined by the finite presentable group @xmath400 and @xmath439 , we do not necessarily need to use a wirtinger representation on calculating @xmath399 . in this section , we deal with a @xmath3-bridge knot @xmath175 in @xmath396 and @xmath1-representations of its knot group . in this case @xmath440 is irreducible if and only if @xmath441 is a non - abelian subgroup of @xmath1 . we will show the explicit calculation of @xmath1-twisted reidemeister torsion associated to @xmath5 knot and study the critical points of the twisted reidemeister torsion @xmath275 . if @xmath175 is hyperbolic and @xmath9 is @xmath2 , then some features of @xmath442 , given in this section , have appeared in ( * ? ? ? * section 4.3 ) . it is well known that @xmath400 has the representation : @xmath443 where @xmath444 is a word in @xmath445 and @xmath446 . here @xmath445 and @xmath446 represent the meridian of the knot . the method we use to describe the space of @xmath2 and @xmath1-representations is due to r. riley(@xcite ) . he shows how to parametrize conjugacy classes of irreducible @xmath2 and @xmath1-representations of any @xmath3-bridge knot group . we review his method ( @xcite ) . given @xmath447 , we consider the assignment as follows : @xmath448 let @xmath62 be the matrix obtained by replacing @xmath445 and @xmath446 by the above two matrices in the word @xmath444 . this assignment defines a @xmath449-representation if and only if @xmath450 where @xmath451 . one can obtain an @xmath2-representation from this @xmath449-representation by dividing the above two matrices by some square root of @xmath452 . if we give a path @xmath453 in @xmath151 with @xmath454 and some continuous branch of the square root along @xmath455 , then we obtain a path of @xmath2-representations . furthermore , all conjugacy classes of non - abelian @xmath2-representations arise in this way . according to proposition @xmath456 of riley s paper @xcite , a pair @xmath457 with @xmath450 corresponds to an @xmath1-representation if and only if @xmath458 , and @xmath459 is real number which lies in the interval @xmath460 = [ 2\cos \theta -2 , 0]$ ] where @xmath461 . this correspondence means that the @xmath2-representation resulting from such a pair @xmath457 and some square root of @xmath452 is conjugate to an @xmath1-representation in @xmath2 . we take the ordered basis @xmath462 of @xmath463 as follows . @xmath464 the lie algebra @xmath465 is a subspace of @xmath463 . the vectors @xmath462 also form a basis of @xmath465 . since the euler characteristic of @xmath397 is zero , the non - abelian reidemeister torsion @xmath466 does not depend on a choice of a basis of @xmath465 . we can use @xmath462 as an ordered basis of @xmath465 . we denote by @xmath467 the representation corresponding to the pair @xmath468 . the representation matrices of @xmath469 and @xmath470 for this ordered basis are given as follows . [ rep_matrix_adjoint ] @xmath471 note that even if we choose another square root of @xmath452 , we obtain the same representation matrices of the adjoint actions of @xmath472 and @xmath473 . we consider @xmath5 knot in the knot table of rolfsen @xcite . note that this knot is not fibered , since its alexander polynomial is not monic . this is the simplest example such as non - fibered in @xmath3-bridge knots . let @xmath175 be @xmath5 knot . a diagram of @xmath175 is shown as in figure [ diagram_5_2 ] . knot . ] this knot is also called @xmath279-twist knot . it follows from theorem @xmath279 of @xcite that @xmath474 consists of one circle and one open arc . the knot group @xmath475 has the following representation : @xmath476 where @xmath477 from this representation , the riley s polynomial of @xmath5 is given by @xmath478 we may take riley s polynomial @xmath479 as @xmath480 we want to know pairs @xmath457 such that @xmath481 , @xmath459 is a real number in the interval @xmath482 $ ] and @xmath450 . when we regard @xmath450 as the equation of @xmath459 , the relation between the number of solutions of @xmath450 and @xmath452 is as follows . * if @xmath483 , then @xmath450 has three different simple root in @xmath484 $ ] . * if @xmath485 , then @xmath450 has a simple root and a multiple root in @xmath484 $ ] . * if @xmath486 , then @xmath450 has a simple root in @xmath484 $ ] . the figure of @xmath474 is given as in figure [ fig : su2_rep_5_2 ] . where @xmath175 is @xmath5 knot . ] we denote the @xmath1-representation corresponding to @xmath457 by @xmath487 . then we can express @xmath488 from proposition [ another_form_torsion ] as follows . @xmath489 using a computer , we calculate a half of the differential coefficient of the second order of the numerator and simplify with the equation @xmath450 . then we have @xmath490 therefore we have @xmath491 where @xmath492 satisfies @xmath493 . from the example in the previous subsection , one can guess that the @xmath1-twisted reidemeister torsion @xmath395 associated to a @xmath3-bridge knot @xmath175 is a function for the parameter @xmath494 . indeed the following holds . [ prop : parameter_torsion ] let @xmath175 be a @xmath3-bridge knot and @xmath223 a simple closed curve in the boundary torus of @xmath397 . suppose that @xmath223-regular @xmath1-representations are parametrized by @xmath495 of riley s method . if the trace of the meridian , @xmath496 , gives a local parameter of the @xmath1-character variety , then the twisted reidemeister torsion @xmath497 is a smooth function for @xmath498 . if we denote by @xmath487 a @xmath223-regular representation corresponding to @xmath499 , then there exists some homomorphism @xmath500 such that @xmath501 is a @xmath223-regular representation corresponding to @xmath502 . by the construction of @xmath503 , @xmath504 is equal to @xmath505 . since @xmath496 is a square root of @xmath506 and regular representations are irreducible , the twisted reidemeister torsion @xmath507 is a smooth function for @xmath494 . [ cor : parameter_torsion ] if the trace of the meridian gives a local parameter of the @xmath1-character variety and the twisted reidemeister torsion @xmath275 is defined , then @xmath275 is a smooth function for @xmath498 . [ rem : parameter_dihedral ] all representations @xmath67 of @xmath3-bridge knot groups into @xmath1 such that @xmath508 are binary dihedral representations . it follows from @xcite that there exists a neighbourhood of the character of each binary dihedral representation for any @xmath3-bridge knot , which is diffeomorphic to an open interval . from @xcite , the trace of the meridian gives a local parameter on a neighbourhood of the character of each dihedral representation for @xmath3-bridge knots . we can regard the twisted reidemeister torsion @xmath270 as a smooth function on a neighbourhood of the character of each binary dihedral representation . moreover these characters can be critical points of @xmath270 as follows . [ critical_pt_torsion ] let @xmath175 be a @xmath3-bridge knot . if a @xmath0-regular component of the @xmath1-character variety of @xmath400 contains the characters of dihedral representations , then the function @xmath270 has a critical point at the character of each dihedral representation . by corollary [ cor : parameter_torsion ] and remark [ rem : parameter_dihedral ] , the twisted reidemeister torsion @xmath270 is a smooth function for @xmath494 . when we substitute @xmath509 for @xmath452 , we can describe @xmath466 as @xmath510 where @xmath511 is a smooth function for @xmath512 . this is a description of @xmath270 with respect to the local coordinate @xmath513 of @xmath474 . the derivation of this function for @xmath513 becomes @xmath514 we recall that @xmath515 . if @xmath516 , then @xmath517 . hence the derivation of @xmath270 vanishes if @xmath67 satisfies @xmath518 . from @xcite , for @xmath3-bridge knots , the character of a binary dihedral representation is a branch point of the two - fold branched cover from the @xmath1-character variety to the @xmath519-character variety . moreover , every algebraic component of the @xmath1-character variety contains the character of such a representation . by ( * theorem 10 ) , for a knot @xmath175 , the number of conjugacy class of binary dihedral representations is given by @xmath520 where @xmath521 is the alexander polynomial of @xmath175 . in particular , for a @xmath3-bridge knot @xmath522 ( schubert s notation , see for example @xcite ) , this number is given by @xmath523 . the author would like to express sincere gratitude to mikio furuta for his suggestions and helpful discussions . he is thankful to hiroshi goda , takayuki morifuji , teruaki kitano for helpful suggestions . especially the author gratefully acknowledges the many helpful suggestions of hiroshi goda during the preparation of the paper . our main theorem was written as the statement for knots in @xmath396 at first . jrme dubois pointed out that our main theorem can hold for knots in homology three spheres . the author is thankful to jrme dubois for his pointing out . he also would like to thank the referee for his / her careful reading and appropriate advices . he / she has given suggestions to improve the proofs of proposition [ propositiona ] and proposition [ prop : parameter_torsion ] . he / she also suggested the fact that critical points of reidemeister torsion are related to the dihedral representations . 99 k. s. brown , , graduate texts in mathematics 87 springer - verlag , new york ( 1994 ) x+306 pp . g. burde , , math . ann . 288 ( 1990 ) 103119 . and h. zieschang , , de gruyter studies in mathematics 5 walter de gruyter ( 2003 ) xii+559 pp . j. dubois , , ann . inst . fourier ( grenoble ) 55 ( 2005 ) 16851734 . , , canad . math . bull . 49 ( 2006 ) 5571 . and r. kashaev , , to appear in math . , arxiv : math.gt/0510607 . m. heusener and e. klassen , , proc . ( 1997 ) 30393047 . p. kirk and e. klassen , , math . ( 1990 ) 343367 . p. kirk and c. livingston , , topology 38 ( 1999 ) 635661 . t. kitano , , pacific j. math . 174 ( 1996 ) 431442 . e. klassen , , trans . ( 1991 ) 795828 . j. milnor , , bull . 72 ( 1966 ) 358426 . , , conference on the topology of manifolds @xmath524michigan state univ . 1967@xmath525 prindle weber & schmidt boston , mass . ( 1968 ) 115133 . j. w. morgan and p. b. shalen , , ann . of math . ( 2 ) 120 ( 1984 ) 401476 . j. porti , , mem . amer . math . 128 ( 612 ) ( 1997 ) x+139 pp . r. riley , , quart . oxford ser . ( 2 ) 35 ( 1984 ) 191208 . d. rolfsen , , mathematics lecture series 7 publish or perish inc . houston , tx ( 1990 ) xiv+439 pp . e. h. spanier , , springer - verlag , new york - berlin ( 1981 ) xvi+528 pp . v. turaev , , lectures in mathematics eth zrich birkhuser verlag , basel ( 2001 ) viii+123 pp . , , progress in mathematics 208 birkhuser verlag , basel ( 2002 ) x+196 pp . m. wada , , topology 33 ( 1994 ) 241256 . y. yamaguchi , , arxiv : math.gt/0512277 .

deformations of the bosonic heisenberg algebra by parameters have known successful achievements in mathematical physics @xcite-@xcite and in nonlinear physics ( e.g. nonlinear quantum optics ) @xcite-@xcite . one of the simplest deformation of the bosonic algebra , a one parameter @xmath0-deformation , was introduced by arik and coon @xcite and is defined by @xmath10 clearly , one recovers the ordinary fock algebra of the harmonic oscillator at the limit @xmath11 , with then @xmath12=\mathbb{i}$ ] . the most of studies pertaining to such deformations are made with the parameter @xmath130,1]$ ] . however , in @xcite , a study was performed even for complex values of @xmath0 . concerning the issue of convergence , infinite products and deformed exponential series require at least that the modulus @xmath14 . this leads to the consideration that @xmath15 , with @xmath16 the unit complex disc but the zero . keeping in mind these last remarks , nothing prevents to perform the following limit @xmath17 reminiscent of a fermionic algebra @xcite-@xcite . it then raises many natural questions . is it possible to understand the generators associated to this limit as fermions ? then , in the case of a positive answer , is there a mapping from the bosonic operators ( defined for @xmath18 ) to fermionic ones ( defined for @xmath19 ) , i.e. a kind of supersymmetry ? recent years , many investigations on @xmath0-deformed algebras and supersymmetry have been undertaken dealing with @xmath0-deformed supersymmetric factorization @xcite or differential representation , intertwining properties and coherent states @xcite ( and more references therein ) . nevertheless , as far as we can establish , none of them focuses on the complete study of the product of these deformed algebras for different parameters @xmath0 . so doing , one will immediately generate a full deformed universal algebra of all different deformed generators acting on a unique representation hilbert space . we propose to investigate how the notions of @xmath1 grassmann grading and supersymmetry can be extended to this multi - deformed enveloping algebra . in this paper , we introduce the notion of @xmath0-deformed grading on the enveloping algebra generated by all products in different deformed heisenberg algebras . this notion generalizes the ordinary grassmann grading and , moreover , by defining a generalized @xmath0-graded bracket , one is able to recover , in each subalgebra , the correct structure for bosonic , fermionic , @xmath1 graded and basic deformed bosonic algebras . the extension of grassmann parity affords us to understand ordinary associative superalgebras and their @xmath1 graded structure ( the usual framework of supersymmetry ) as limit algebras when the parameter @xmath20 . we then determine the modified supersymmetric hamiltonian and its deformed supercharges mapping some deformed fermions on deformed bosons . the paper s outline is as follows . the following section is dedicated to the definition and basic properties of the @xmath0-deformation of the heisenberg algebra in the sense of arik and coon , for complex parameter @xmath0 , and its representation . the limit @xmath2 is clarified . afterwards , section 3 addresses the algebraic settlement of the deformed structure producing the general @xmath0-deformed grading . the particular case of ordinary grassmann parity is discussed . section 4 investigates the extended notion of supersymmetry on the enveloping algebra . the specific limit @xmath21 , producing a modified version of the ordinary supersymmetry , is also discussed . the paper ends by some remarks in section 5 and an appendix provides useful identities and illustrations . let us consider the arik and coon deformation of the heisenberg algebra @xcite @xmath22 with parameter a complex number @xmath0 . if we regard @xmath23 as the adjoint of @xmath24 , it follows that , by hermitian conjugation of ( [ qdef0 ] ) , @xmath25 . by simple substraction of these equations , one ends with @xmath26 from the positivity of @xmath27 . hence , @xmath0 should be a real parameter . however , introducing a new operator @xmath28 , let us reconsider the same kind of deformed structure , namely @xmath29 and relax the previous condition of adjoint property between @xmath24 and @xmath28 . then , nothing can be said , _ a priori _ , on the parameter @xmath0 . we will place ourself in this general situation such that , @xmath30 , @xmath31 . all the following main equations are again valid outside the unit disc . only the notion of convergence of functions series and infinite products involved in deformed special function theory has to be reconsidered . we are not dealing with these ideas here , but we want , as much as possible , to have a theory with interesting properties for the theoretician community . ] @xmath320,1],\;\ ; \varphi_q \in [ 0,2\pi[.\end{aligned}\ ] ] in order to define the power function of @xmath0 , namely @xmath33 , one uses the complex form of the exponential function @xmath34 , where @xmath35 stands for the principal branch of complex logarithm . a realization of the algebra ( [ qdef ] ) is also well known . to construct it , one starts with the ordinary bosonic operators @xmath36 and @xmath37 , fulfilling @xmath12=\mathbb{i}$ ] with the number @xmath38 , generating the ordinary number operator in the fock hilbert space @xmath39 . then we define @xmath40_q}{n+1}}a|n\rangle,\quad a_q^\natural |n\rangle=\sqrt{\frac{[n]_q}{n}}a^\dagger|n\rangle,\quad [ n]_q = \frac{1-q^n}{1-q},\end{aligned}\ ] ] where one refers to @xmath41_q$ ] as the @xmath0-basic number of the theory . note that @xmath41_q$ ] is not necessarily self - adjoint . indeed , @xmath42_q)^\dag=[n]_{\bar q}$ ] , which is not @xmath41_q$ ] unless @xmath0 is real . the adjoint of the operator @xmath24 can be expressed as @xmath43_{\bar q}}{n}}a^\dagger \label{adjoi}\end{aligned}\ ] ] from which it appears possible to define naturally a self - adjoint deformed number operator as @xmath44_{\bar q}[n]_q}}$ ] . from ( [ adjoi ] ) , a relation between @xmath45 and @xmath28 can be inferred : @xmath46 we are then in position to define properly the unary operation @xmath47 which is the adjoint operation composed with the complex conjugation . the operator @xmath48 , viewed as a matrix , can be understood as the transpose of @xmath24 . moreover , it can be checked that @xmath49 , therefore @xmath47 is an involution ; we also have @xmath50 . for a real parameter @xmath0 , the definitions of @xmath23 and @xmath51 coincide . let us briefly mention the limit @xmath52 . the corresponding basic number @xmath41_0 $ ] proves to be the constant operator @xmath53 . this implies that @xmath54 and @xmath55 are mutually inverse in the fock space without the vacuum @xmath56 . as a result of the triviality of the @xmath57-commutator , the @xmath57-deformed algebra is again a lie algebra . then the enveloping algebra over @xmath58 becomes a division algebra ( other relations concerning division algebras built over the heisenberg generators are available in @xcite ) . let us focus now on the limit @xmath2 of the algebra ( [ qdef ] ) . this limit can be written as @xmath59 a prime remark would be that , recalling that @xmath31 , the above limit could be performed only by avoiding the forbidden value @xmath60 ; this can be done by varying continuously @xmath0 along a straight line if @xmath0 does not belong to the segment @xmath610,1]$ ] . in the case @xmath130,1]$ ] , then the same limit can be only made by choosing a contour through the complex plane . noting that , in any state @xmath62 , @xmath63_{-1}|n\rangle = \lim_{q\to -1 } [ n]_q |n\rangle = \frac{1-(-1)^n}{2 } |n\rangle = \left\{\begin{array}{cc } 0 , & { \rm if}\ ; n=2p \\ 1\,|n\rangle , & { \rm if}\ ; n=2p+1 \end{array}\right . \label{limn}\end{aligned}\ ] ] then we infer the following representation for the operators @xmath64 let us recall that a fermionic algebra is usually defined by a set of algebraic relations @xmath65 the anticommutation rule is already satisfied by the pair @xmath66 . checking , that for any @xmath67 , @xmath68_q[n]_q}|n-2\rangle$ ] , @xmath69_q[n+2]_q}|n+2\rangle$ ] , one infers from ( [ limn ] ) that , indeed , for any state , @xmath70 and @xmath71 . thus , the pair @xmath66 is close to what one usually refers to as a fermionic algebra . for this reason , we will refer henceforth to these operators to fermions and to their algebra , to a fermionic algebra . here , more rigorously , the operators @xmath66 are fermionic operators with an infinite dimensional representation space which is a direct sum of ordinary two dimensional fermionic representation spaces . the notion of @xmath1 grassmann grading for associative complex superalgebras @xcite will find , in the next lines , an extension according to the present @xmath0-deformed study . but before , for the sake of rigor , let us put in algebraic terms the definition of the deformation of the heisenberg algebra ( [ qdef ] ) . * building a @xmath0-grading on the enveloping algebra . * for all @xmath67 , we introduce the deformed complex heisenberg algebra with its three generators and deformed commutator as the pair @xmath72_q\right ) . \label{hesdef}\end{aligned}\ ] ] some remarks are in order at this stage . first , the deformed heisenberg algebra @xmath73 is not a lie algebra unless that one considers the limit points @xmath74 . the jacobi identity fails to be satisfied in the general situation when @xmath75 . note also that these algebras are not disjoint since @xmath76 , for @xmath77 . the data of the pair ( [ hesdef ] ) are equivalent to the data of a complex vector space @xmath73 and a constraint ( equivalence relation ) @xmath78_q=\mathbb{i}$ ] on the tensor algebra built out of its generators . next , let us give the definition of the @xmath0-grading of generators of any @xmath73 and find an extension for any element of the enveloping algebra spanned by all @xmath73 s , @xmath67 . this concept will be introduced by the data of two attributes related to the parameters @xmath0 : the `` degree '' , denoted by @xmath79 , and the `` radius '' , denoted by @xmath80 . by convention , elements of @xmath81 are of degree @xmath82 and we define the degree of the generators of @xmath73 as @xmath83 given a generator of @xmath73 , its degree becomes a real parameter in the segment @xmath84 which can be viewed as the normalized phase of the deformation parameter @xmath0 . for instance , the degree of an ordinary ( heisenberg ) boson is @xmath85 , while the degree of the operators @xmath86 reproducing a well defined notion of @xmath1 grassmann parity for these limit . we will characterize the generators of @xmath73 , by another quantity that we will refer to as its `` radius '' or `` length '' which is nothing but @xmath87 given the modulus @xmath88 of the deformation parameter @xmath0 . at this stage , the following deformed bracket for elementary generators can be defined @xmath89_{q , q'}:= x_{q}\,y_{q ' } - g(q , q')y_{q'}\,x_{q},\qquad g(q , q'):=e^{i\pi|x_{q}||y_{q'}|}\ell(x_{q})\ell(y_{q ' } ) . \label{gradef}\ ] ] a quick verification , using ( [ deg ] ) and ( [ leng ] ) , yields the following limits @xmath90_{1,1 } = [ a_{1},a^\dag_{1}]_{q=1}= a_{1 } a^\dag_{1 } - a^\dag_{1 } a_{1 } = \mathbb{i } , & & \label{com}\\ { \rm ( fermion)}\;\ , g(-1,-1)=-1 : \;\,[a_{-1},a^\dag_{-1}]_{-1,-1 } = [ a_{-1},a^\dag_{-1}]_{q=-1}= a_{-1 } a^\dag_{-1 } + a^\dag_{-1 } a_{-1 } = \mathbb{i } , & & \label{antic}\\ { \rm ( q - def . ) } \;\,g(q , q)=q : \;\,[a_{q},a^\dag_{q}]_{q , q } = [ a_{q},a^\dag_{q}]_{q}= a_{q } a^\dag_{q } - qa^\dag_{q } a_{q } = \mathbb{i}.&&\end{aligned}\ ] ] another interesting property of the deformed bracket ( [ gradef ] ) is that it reproduces the @xmath1 graded bracket between fermion and bosons . in other words , in addition to ( [ com ] ) and ( [ antic ] ) , the bracket of a fermion and a boson is a commutator , because of @xmath91 . having properly defined the notion of @xmath0-grading of basic generators , let us go further by defining similar ideas for more complex structures . the ( noncommutative ) product of elements of two algebras @xmath73 and @xmath92 lies in the complex vector space @xmath93 . by iteration , one can build monomials in basic generators living in a product of deformed algebras @xmath94 , @xmath95 . taking the complex span of these monomials , one forms a complex vector space . negative integer powers of generators can be defined algebraically as @xmath96 , @xmath97 , where the inverse of @xmath98 , i.e. @xmath99 , acts by representation such that @xmath100 or @xmath101 ( right or left inverse ) . further precisions on the division algebra generated by the heisenberg operators @xmath24 and @xmath23 can be found in @xcite . the overall algebra spanned by any linear combination of any kind of products of generators ( including inverse integer powers ) will be called the deformed enveloping algebra denoted by @xmath102 . we would like to give a sense to the notion of grading for any elements of the enveloping algebra @xmath102 . we start by the degree and radius of bilinear products which can be defined as @xmath103 where @xmath98 and @xmath104 are generators of @xmath73 and @xmath92 , respectively . it is remarkable that @xmath105 and @xmath106 . integer powers of elementary generators of @xmath73 belonging to @xmath107 can be also assigned with a degree and a radius as @xmath108 more generally , the following relations , valid for finite products of integer powers of elementary generators , stand for definition : @xmath109 some products of basic generators admit a spectral decomposition of the form @xmath110 ) |n \rangle \langle n|$ ] , where @xmath111)\in \mathbb{c}$ ] , any function of number operators being a typical example . for this kind of operators , rational and real powers also have a rigorous definition . for instance , one sets @xmath112))^{\alpha } |n \rangle \langle n|$ ] , @xmath113 . in this situation of a possible diagonal decomposition of an operator being a product of elementary generators , the formulas ( [ prodgen ] ) can be extended to real powers : @xmath114 in order to compute the deformed bracket of composite elements in @xmath102 , one has to perform first the decomposition in sum of monomials in the elementary generators before the computation . finally , the general bracket for any monomials @xmath115 and @xmath116 can be expressed as follows ( an explicit example is provided in the appendix ) @xmath117_{g(q_i , q_j)}:= a_{q_i}\,b_{q_j } - g(q_i , q_j)b_{q_j}\,a_{q_i } , \qquad g(q_i , q_j):=e^{i\pi|a_{q_i}||b_{q_j}|}\ell(a_{q_i}b_{q_j } ) . \label{gradef3}\ ] ] in conformity with the above definition of degree , the degree of a product of two fermions is @xmath118 ( for instance @xmath119 ) and not @xmath82 as it is customarily the case in the context of @xmath1 superalgebras . this does not lead to any contradiction and the anticommutator ( [ antic ] ) is still valid and based on the product of degrees . although one is tempted to take for definition of the degree a kind of number modulo @xmath118 ( or @xmath120 from ( [ deg ] ) ) , the study can be pursued in this general context proving that there is no need to make further assumptions in the definition ( [ deg ] ) . * @xmath121-grading and matrix representation . * as a prime interesting feature with implications in supersymmetry , we discuss the matrix representation . let us recall that the ordinary notion of @xmath1 grading applied to matrix algebras @xcite can be introduced by the data of two integers @xmath122 and @xmath123 , and the decomposition of any element of @xmath124 , the set of square matrices of dimension @xmath125 with coefficients in the field @xmath126 , into four submatrices of dimensions @xmath127 , @xmath128 , @xmath129 and @xmath130 @xcite , i.e. @xmath131 a matrix is said to be `` even '' if its entries belong either to @xmath132 or to @xmath133 and `` odd '' if its entries belong to the matrices @xmath134 or @xmath135 . one can check that any product of matrices obeys to the law @xmath136 , @xmath137 , @xmath138 such that the matrix product is stable under this grading . the same idea can be simply illustrated in ordinary matrix formulation of supersymmetry where the supersymmetric hamiltonian is diagonal ( even quantity ) and supercharges consist in off diagonal matrices ( odd elements ) . one thing remains to be clarified : if the ordinary bosonic modes @xmath139 and @xmath140 or usual fermionic operators @xmath141 and @xmath142 admit a matrix representation onto the fock hilbert space basis , it can be suggested to find the equivalent feature such that the notion of @xmath1 grading as previously discussed can be readily read from their matrix representation . at first , using the conventional matrix representation of bosonic modes , nothing can be said . however , if one organizes the states differently such that we write the fock basis in the following form @xmath143 then the ordinary boson @xmath144 and fermion @xmath145 have the following matrices with respect to the order ( [ orde ] ) @xmath146 the associated adjoint operators can be easily inferred . in this context , the notion of `` odd '' matrix can be affected to either the pair @xmath147 or to the pair @xmath148 . then , legitimately in this context , the fermions @xmath148 can be seen as `` odd '' elements while the fact that the operators @xmath147 can be also seen as `` odd '' quantities becomes confusing . nevertheless , another interesting feature emerges : multiplying `` odd '' matrices , for instance @xmath149 ( of degree @xmath82 ) , or @xmath150 ( of degree @xmath118 ) will produce `` even '' elements ( as they should be ) which are the bosonic and fermionic numbers , respectively . this section aims at defining a general notion of supersymmetry on the enveloping algebra @xmath102 . the technical difficulty comes from the fact that operators are noncommuting objects in contrast to the situation of ordinary supersymmetric quantum theory . in addition , for different @xmath0 s , all @xmath0-deformed operators act on an identical hilbert space ( the fock hilbert space ) . * @xmath151-supersymmetry . * in this paragraph , we define , as a guiding model to next discussions , a supersymmetric theory on the enveloping subalgebra spanned only by products in @xmath152 and @xmath153 . supersymmetry is realized through a set of charges commuting with a hamiltonian . operators mapping in a deformed way fermions @xmath145 and @xmath154 to bosons @xmath155 and @xmath156 are identified . the converse is essentially not true due to the deformation . simple properties allow us to investigate which kind of operators can generate a supersymmetry . we will restrict the study to the situation of a supersymmetry generated by only quadratic products of operators which is , in fact , the closest possible to the ordinary notion of supersymmetry where a hamiltonian appears as a ( supersymmetrically ) factorized by bilinear operators : the supercharges @xcite . these latter operators generically are of the form of a product of bosons and fermions . these supercharges via a graded structure maps bosonic to fermionic degrees of freedom and vice versa . let us then list the possible minimal bilinears , built from products of the fermions @xmath145 and @xmath154 by bosons @xmath155 and @xmath156 . bearing in mind that the order of operators is important , @xmath157 monomials are of interest @xmath158_{-1}}{n+1}}\,a^2,\;\;\;\ ; \tilde{\mathfrak{q}}_{1}=a_{1}a_{-1}=\sqrt{\frac{[n+2]_{-1}}{n+2}}\,a^2,\\ & & \tilde{\mathfrak{q}}_{1}^\dag = a^\dag_{-1}a^\dag_{1}=\sqrt{\frac{[n]_{-1}}{n}}(a^\dag_{1})^2 , \;\;\;\ ; \mathfrak{q}_{1}^\dag = a^\dag_{1}a^\dag_{-1}=\sqrt{\frac{[n-1]_{-1}}{n-1}}(a^\dag_{1})^2,\\ & & \mathfrak{q}_{2}=a^\dag_{1}a_{-1}=\sqrt{\frac{[n]_{-1}}{n}}n = a_{-1}^\dag a_{1},\;\;\;\ ; \tilde{\mathfrak{q}_{2 } } = a_{-1}a^\dag_{1}=\sqrt{\frac{[n+1]_{-1}}{n+1}}(n+1)=a_{1}a_{-1}^\dag,\end{aligned}\ ] ] all of degree ) . the degrees of the other operators can be derived in a similar way . ] 1 and radius @xmath159 . note that @xmath160 and @xmath161 are self - adjoint . a set of hermitian hamiltonian operators can be readily obtained from these operators @xmath162_g = \mathfrak{q}_{1}\mathfrak{q}_{1}^\dag + \mathfrak{q}_{1}^\dag\mathfrak{q}_{1}= 2[n+1]_{-1}(n+1),\\ & & \tilde{\mathfrak{h}}_{1 } = [ \tilde{\mathfrak{q}}_{1},\tilde{\mathfrak{q}}_{1}^\dag ] _ g = \tilde{\mathfrak{q}}_{1}\tilde{\mathfrak{q}}_{1}^\dag + \tilde{\mathfrak{q}}_{1}^\dag\tilde{\mathfrak{q}}_{1 } = 2[n]_{-1}n , \\ & & \mathfrak{h}_{2 } = ( \mathfrak{q}_{2})^2 = [ n]_{-1}n,\\ & & \tilde{\mathfrak{h}}_{2 } = ( \tilde{\mathfrak{q}_{2}})^2 = [ n+1]_{-1}(n+1).\end{aligned}\ ] ] we are now in position to define the basic hermitian supersymmetric hamiltonian ( up to some energy scale @xmath163 that we omit ) @xmath164_{-1}n = a^\dag_1 a_1 \ , a_{-1}^\dag a_{-1},\;\;\ ; and the following supersymmetric algebra can be verified @xmath165_{g}=0= [ { \mathcal q}^\dag,\mathfrak{h}_{\rm ss}]_{g},\;\;\ ; [ { \mathcal q},{\mathcal q}^\dag]_g= \mathfrak{h}_{\rm ss},\quad { \mathcal q } : = \frac{1}{\sqrt{2}}\mathfrak{q}_{1 } , \label{susyalg}\\ & & [ \mathfrak{q}_{2},\mathfrak{h}_{\rm ss}]_{g}=0,\quad \mathfrak{h}_{\rm ss}= ( \mathfrak{q}_{2})^2 . \label{susy2}\end{aligned}\ ] ] thus the formulation allows to generate a @xmath166 supersymmetry ( with three different symmetries ) . the other operators @xmath167 and @xmath161 have a simple meaning , in the present context . they define the partner charges allowing the construction of another supersymmetric hamiltonian @xmath168 , the so - called superpartner of @xmath169 , obtained by reversing the order of operators , namely @xmath170_g$ ] . to the question `` is this notion of @xmath151-supersymmetry equivalent to the ordinary one ? '' , the answer is no . a hint to recognize this fact is the form of the ordinary supersymmetric hamiltonian @xmath171 , where @xmath172 and @xmath173 are the bosonic and fermionic number operators , respectively . here the supersymmetric hamiltonian @xmath169 is clearly not of this form . however , the operator @xmath174 can be rebuilt as @xmath175_{-1}n + [ n+1]_{-1}(n+1)= n + [ n+1]_{-1}$ ] . from this point out view , the supersymmetries ( [ susyalg ] ) or ( [ susy2 ] ) appear therefore as the basic ones even though it is not true that all properties of the ordinary supersymmetry can be recovered . in the following , we focus on the evidence of the deformation of the ordinary supersymmetry if the symmetry is realized as ( [ susyalg ] ) or ( [ susy2 ] ) . let us check if the ordinary properties of supercharges are satisfied . first , the square of non hermitian supercharges are usually vanishing quantities , here @xmath176 this shows that the supersymmetry is actually realized in a deformed way . second , supercharges used to map bosons onto fermions and conversely . in the present situation , any @xmath0-commutation relation does not lead to interesting results . nevertheless , after scrupulous analysis ( see appendix ) , one can reach the following interesting algebras @xmath177_g = [ n]_{-1 } a_{1 } , \label{eq1}\\ & & [ \mathfrak{q}_{1}^\dag , a_{-1}^\dag ] _ g = [ n-1]_{-1 } a^\dag_{1 } , \label{eq2}\\ & & [ \mathfrak{q}_{2 } , a_{-1}]_g = [ n+1]_{-1 } a_{1 } , \label{eq3}\\ & & [ \mathfrak{q}_{2 } , a_{-1}^\dag ] _ g = [ n]_{-1 } a^\dag_{1 } , \label{eq4}\end{aligned}\ ] ] revealing that fermions are actually mapped on deformed bosons ( up to a function of the fermionic number operator ) . the converse is not true ( see appendix ) pointing out the peculiar aspects of this deformed supersymmetry . * @xmath178-supersymmetry . * in this last paragraph , we define in a more general context of @xmath0-deformation , the notion of deformed supersymmetry . we start by the simple remark that , as shown above , elements of the heisenberg algebra @xmath179 can be seen as deformed supersymmetric partners of elements of @xmath180 . the complex number @xmath181 is obtained after a rotation by @xmath182 from the complex number @xmath159 . the notion of supersymmetry could find an equivalence in @xmath102 , mapping generators of @xmath73 onto a generator of @xmath183 , with @xmath184 a transformation of @xmath0 : @xmath185 where @xmath186 and @xmath187 are real functions . the above case of @xmath151-supersymmetry corresponds to @xmath188 , a simple rotation @xmath189 in the complex plane by an angle @xmath182 . however , we will assume that @xmath190 and @xmath191 ; so doing , we write @xmath192 , thereby defining @xmath193 and @xmath194 . the second step is to define again bilinears which are of interest . it does not take long to find the following operators ( the same notation as in the previous paragraph is used but operators now refer to different quantities ) @xmath195_{\bar q}[n+2]_{q}}{(n+1)(n+2)}}\,(a_1)^2,\;\;\;\ ; \tilde{\mathfrak{q}}_{1}= a_{q } a_{\bar q}= \sqrt{\frac{[n+1]_{q}[n+2]_{\bar q}}{(n+1)(n+2)}}\,(a_1)^2,\\ & & \tilde{\mathfrak{q}}_{1}^\natural = a^\natural_{\bar q}a^\natural_{q}= \sqrt{\frac{[n]_{\bar q}[n-1]_{q}}{n(n-1)}}(a^\dag_{1})^2 , \;\;\;\ ; \mathfrak{q}_{1}^\natural = a^\natural_{q}a^\natural_{\bar q}= \sqrt{\frac{[n]_q[n-1]_{\bar q}}{n(n-1)}}(a^\dag_{1})^2,\\ & & \mathfrak{q}_{2}=a^\natural_{q}a_{\bar q}=\sqrt{[n]_{\bar q}[n]_q}= a_{\bar q}^\natural a_{q},\;\;\;\ ; \tilde{\mathfrak{q}_{2 } } = a_{\bar q}a^\natural_{q}= \sqrt{[n+1]_{\bar q}[n+1]_q}= a_{q}a_{\bar q}^\natural.\end{aligned}\ ] ] they all share the same degree @xmath196 and the same radius @xmath197 . we also notice that @xmath160 and @xmath161 remain hermitian , while @xmath198 and @xmath199 become adjoint of one another . in order to obtain a hermitian hamiltonian operator , we consider the operator generated by @xmath160 @xmath200_{\bar q}[n]_q=(\{n\}_{\bar q , q})^2,\qquad \sqrt{\frac{\varphi_{\bar q}}{\pi}}\right ) , \quad \ell(\mathfrak{h}_2 ) = r_q f(r_q ) \label{ham}\end{aligned}\ ] ] and require that @xmath160 is a symmetry of @xmath201 with respect to the deformed bracket . we are led to the following @xmath202_g= \sqrt{[n]_{\bar q}[n]_q}[n]_{\bar q}[n]_q \left[1 -e^ { 2i\pi\left(\sqrt{\frac{\varphi_{q}}{\pi } } + \sqrt{\frac{\varphi_{\bar q}}{\pi}}\right)^2}(r_q f(r_q))^{\frac{3}{2}}\right]\end{aligned}\ ] ] a set of necessary and sufficient conditions for @xmath203_g= 0 $ ] to hold is @xmath204 which can be easily solved by @xmath205 lying outside of the unit disc corresponding to @xmath206 : @xmath207 , @xmath208.,width=226,height=226 ] ( -60,125 ) @xmath209 ( -165,115 ) @xmath210 ( -100,20 ) @xmath211 ( 123,45 ) @xmath0 ( 150,82 ) @xmath212 ( 158,65 ) @xmath213 ( 160,48 ) @xmath214 ( 126,100 ) @xmath215 ( 70,85 ) @xmath216 ( 62,27 ) @xmath217 ( 143,20 ) @xmath218 ( 120,175 ) @xmath219 lying outside of the unit disc corresponding to @xmath206 : @xmath207 , @xmath208.,width=162,height=6 ] before regarding the phase @xmath220 problems , let us focus on the equation @xmath221 having a drastic consequence . indeed , if @xmath2220,1]$ ] , then @xmath223 . this is to say that if we impose @xmath224 , then obligatory we will restrict to the situation where @xmath225 in order to get relevant solutions . enlarging the scope of value of @xmath226 , another interesting feature emerges : superpartners of operators with deformation parameter @xmath0 strictly inside of the unit disc @xmath227 are operators with parameter @xmath184 lying strictly outside of the unit disc ( and vice - versa ) ; superpartners of operators labeled by a @xmath0 belonging to the unit circle @xmath228 are operators with parameter still on the circle . concerning the phase , the case @xmath229 , for @xmath230 refers to the trivial point @xmath231 . we will focus only on points @xmath232 encoding values where partners differ from one another . it turns out that for @xmath233 , there always exists a value @xmath234 since one can show that @xmath235\!]$ ] . more generally , solving the first condition of ( [ phas ] ) , for @xmath236 , one infers the following constraints on @xmath220 @xmath237\!],\;\ ; k_p(\varphi_q ) = \left(\sqrt{p\pi } - \sqrt{\varphi_{q}}\right)^2 \in [ 0,2\pi[.\end{aligned}\ ] ] these solutions imply that , regardless of their modulus which can be fixed to be equal @xmath238 , given @xmath0 fixing once for all @xmath88 and @xmath220 , there are at least @xmath118 and at most @xmath239 parameters @xmath240 , @xmath241\!]$ ] , providing good parameter candidates for different supersymmetries @xmath242 for at least @xmath118 and at most @xmath239 different models defined by @xmath243 , such that @xmath244_g=0 $ ] ( see figure 1 and figure 2 ) . in the particular situation of the so - called @xmath151-supersymmetry as built in the previous paragraph , given @xmath245 , the above solutions are consistent and reduce to the unique possibility of @xmath246 in ( [ k+ ] ) such that @xmath247 , thus implying a unique choice for @xmath248 . finally , the constraint such that @xmath249 is actually too strong and more solutions can be determined by relaxing this condition . in fact , it seems that an infinite set of inequivalent ( modulo @xmath120 ) solutions of the phase equation ( [ phas ] ) are available due to the fact that @xmath250 is a nonlinear function of @xmath220 . a careful analysis and the meaning of these solutions will be treated in a subsequent work . the meaning of supersymmetry is not clear when considering the other operators @xmath198 and @xmath199 . a hermitian hamiltonian can be easily identified using the deformed bracket @xmath251_g$ ] . however , the set of conditions in order to impose @xmath252= 0=[\tilde{\mathfrak{q}}_{1}^\natural,\mathfrak{h}_{1}]$ ] is more involved . moreover , the resulting operator @xmath253 is not equal to @xmath201 ( [ ham ] ) . other more complicated issues arise when one reverses the order of the basic generators . the former @xmath166 @xmath151-deformed supersymmetry is therefore explicitly broken in the general deformation theory , with a reduced number of supercharges equal to @xmath254 . let us turn to the properties of the mapping boson - fermions . the following relations can be obtained : @xmath255= { \textstyle\left[\sqrt{\frac{[n]_{\bar q}[n+1]_{\bar q}[n]_q}{[n+1]_q}}- e^{i\sqrt{\varphi_{\bar q}(\varphi_{\bar q } + \varphi_{q } ) } } f(r_q)\sqrt{r_q}[n+1]_{\bar q}\right ] a_q},\label{gende}\\ & & [ \mathfrak{q}_{2 } , a_{\bar q}^\natural]= { \textstyle\left[[n]_{\bar q}- e^{i\sqrt{\varphi_{\bar q}(\varphi_{\bar q } + \varphi_{q } ) } } f(r_q)\sqrt{r_q}\sqrt{\frac{[n-1]_{\bar q}[n]_{\bar q}[n-1]_q}{[n]_q } } \right ] a^\natural_q},\label{gende2},\end{aligned}\ ] ] which at the limit @xmath18 and @xmath256 , characterized by @xmath257 , @xmath258 and @xmath259 , reproduce correctly ( [ eq3 ] ) and ( [ eq4 ] ) . hence , one draws the conclusion that @xmath260 and @xmath261 are deformed partners of @xmath262 and @xmath263 , respectively . we have succeeded in setting a notion of @xmath0-grading onto the deformed enveloping algebra built from all possible products of @xmath0-deformed heisenberg algebras for different parameters @xmath67 , @xmath227 being the complex disc of radius one without @xmath82 . this notion of @xmath0-grading encompasses in specific limit the ordinary notion of @xmath1 grading of ordinary associative superalgebra . a generalized bracket is defined on the enveloping algebra which reproduces the ordinary bosonic , fermionic , @xmath1 graded and @xmath0-deformed commutators for corresponding subspaces in the total algebra . the formalism is then used to show that the notion of supersymmetry can be extended and , even , realized in the present situation where the fermions do not commute with bosons . in the specific instance such that @xmath264 , a supersymmetric hamiltonian has been defined and its @xmath166 supersymmetry properly identified with new deformed features . finally , in the full @xmath0-deformed theory , we have identified many kinds of operators ( restricting the deformed phase parameter in @xmath265 ) which are able to define different supersymmetric models . in the general context of deformation , the @xmath166 @xmath151-supersymmetry is explicitly broken and at least @xmath254 supercharge can be defined in any @xmath266-deformed supersymmetric model . m.n.h . thanks the national institute for theoretical physics ( nithep ) and its director prof . frederik g. scholtz for hospitality during a pleasant stay in stellenbosch where this work has been initiated . j.b.g . thanks pr . michal kastner , pr . jan govaerts and pr . bo - sture skargerstam for helpful discussions . this work was supported under a grant of the national research foundation of south africa and by the ictp through the oea - icmpa - prj-15 . this appendix provides useful relations and explicit illustrations to the text . we start by illustrating the kind of computations involved by the generalized @xmath0-graded bracket . let us calculate , for the specific instance , the bracket of the monomials of the form @xmath267 and @xmath268 ( as appeared in computing the bracket of the supercharge @xmath269 and supersymmetric hamiltonian @xmath270 ) . given the complex numbers @xmath271 and @xmath272 , @xmath273 , we have @xmath274_g = a_{q_1}^\natural a_{q_1 ' } a_{q_2}^\natural a_{q_2 ' } a_{q_3}^\natural a_{q_3'}\cr & & - ( -1)^{|a_{q_1}^\natural a_{q_1'}||a_{q_2}^\natural a_{q_2'}a_{q_3}^\natural a_{q_3'}| } \sqrt{r_{q_1}r_{q_1'}r_{q_2}r_{q_2'}r_{q_3}r_{q_3 ' } } a_{q_2}^\natural a_{q_2'}a_{q_3}^\natural a_{q_3 ' } a_{q_1}^\natural a_{q_1'}\cr & & = a_{q_1}^\natural a_{q_1 ' } a_{q_2}^\natural a_{q_2 ' } a_{q_3}^\natural a_{q_3 ' } \label{cal}\\ & & - ( -1)^{(|a_{q_1}|+|a_{q_1'}^\dag|)(|a_{q_2}^\natural|+| a_{q_2'}|+|a_{q_3}^\natural|+| a_{q_3'}| ) } \sqrt{r_{q_1}r_{q_1'}r_{q_2}r_{q_2'}r_{q_3}r_{q_3 ' } } a_{q_2}^\natural a_{q_2'}a_{q_3}^\natural a_{q_3'}a_{q_1}^\natural a_{q_1 ' } .\nonumber\end{aligned}\ ] ] now fixing @xmath275 , @xmath276 , @xmath277 and @xmath278 , the expression ( [ cal ] ) reduces to @xmath279&= & [ a_{1}^\dag a_{-1 } , a^\dag_1 a_1 \ , a_{-1}^\dag a_{-1}]_g\cr & = & a_{1}^\dag a_{-1}a^\dag_1 a_1 \ , a_{-1}^\dag a_{-1 } - ( -1)^{(1 + 0)\cdot(0 + 0 + 1 + 1 ) } a^\dag_1 a_1 \ , a_{-1}^\dag a_{-1 } a_{1}^\dag a_{-1}\cr & = & \sqrt{[n]_{-1}n } [ n]_{-1}n - [ n]_{-1}n \sqrt{[n]_{-1}n } = 0 , \label{cal2}\end{aligned}\ ] ] showing that @xmath160 is a symmetry of @xmath169 . of course , using an ordinary commutator this statement becomes obvious since both @xmath160 and @xmath169 are pure functions of the number operator @xmath280 . but here , the difficulty resides in the fact that we use a different commutation relation which turns out to simplify in the form of the ordinary commutator . we give the complete set of commutation relations between the supercharges @xmath198 , @xmath281 and @xmath160 and the basic degrees of freedom @xmath144 , @xmath282 , @xmath283 and @xmath284 . the following relations hold : @xmath285 @xmath286 @xmath287_g = ( n+2)a_{-1 } - \sqrt{[n]_{-1}n}a_1= { \textstyle { \left [ ( n+2)\sqrt{\frac{[n+1]_{-1}}{n+1 } } - \sqrt{[n]_{-1}n } \right]}}a_1 , \label{equa1}\\ & & [ \mathfrak{q}_{1 } , a^\dag_{-1}]_g = [ n]_{-1 } a_{-1},\\ & & [ \mathfrak{q}_{1 } , a_{1}]_g = { \textstyle { \left[\sqrt{\frac{[n+1]_{-1}}{n+1}}-\sqrt{\frac{[n+2]_{-1}}{n+2}}\right ] } } ( a_{1})^3,\\ & & [ \mathfrak{q}_{1 } , a_{-1}]_g = { \textstyle { \sqrt{\frac{[n+1]_{-1}[n+3]_{-1}}{(n+1)(n+3)}}(a_{1})^3= \frac{[n+1]_{-1}}{\sqrt{(n+1)(n+3)}}(a_{1})^3}}. \label{equa2}\end{aligned}\ ] ] @xmath285 @xmath288 @xmath289_g = { \textstyle { \left [ \sqrt{\frac{[n-1]_{-1}}{n-1 } } - \sqrt{\frac{[n-2]_{-1}}{n-2 } } \right]}}(a^\dag_1)^3 , \\ & & [ \mathfrak{q}^\dag_{1 } , a^\dag_{-1}]_g = { \textstyle { \frac{[n]_{-1}}{\sqrt{n(n-2)}}}}(a^\dag_{1})^3,\\ & & [ \mathfrak{q}^\dag_{1 } , a_{1}]_g = a^\dag_{1 } \sqrt{[n]_{-1}n } - ( n+1 ) a^\dag_{-1}= { \textstyle { \left [ \sqrt{[n-1]_{-1}(n-1 ) } - ( n+1)\sqrt{\frac{[n]_{-1}}{n}}\right ] } } a^\dag_{1},\\ & & [ \mathfrak{q}^\dag_{1 } , a_{-1}]_g = [ n-1]_{-1 } a^\dag_{1}.\end{aligned}\ ] ] @xmath285 @xmath290 @xmath291_g = \left [ \sqrt{[n]_{-1}n } - \sqrt{[n-1]_{-1}(n-1 ) } \right]a^\dag_1 , \\ & & [ \mathfrak{q}_{2 } , a^\dag_{-1}]_g = [ n]_{-1}a^\dag_{1},\\ & & [ \mathfrak{q}_{2 } , a_{1}]_g = \sqrt{[n]_{-1}n}a_{1 } - ( n+1 ) a_{-1}= \left [ \sqrt{[n]_{-1}n } - \sqrt{[n+1]_{-1}(n+1)}\right ] a_{1},\\ & & [ \mathfrak{q}_{2 } , a_{-1}]_g = \sqrt{[n+1]_{-1}(n+1 ) } a_{-1}=[n+1]_{-1 } a_{1}.\end{aligned}\ ] ] in summary , these operators map bosons on a mixed combination of bosons and fermions which can not be reduced to a deformation of a fermion . on the other hand , fermions can be exactly mapped onto bosons up to a deformation function or a nonlinearity appearing as a power of a bosonic operator .

control refers to the ability to steer a dynamical system from an initial to a final state with a desired accuracy ; optimal control does so with minimum expenditure of effort and resources . a famous example is the apollo space mission where optimal control was used to safely land the spacecraft on the moon . the essence of this control task can be stripped down to a textbook example where students calculate the change in acceleration , that is , the rate of burning fuel , required to reach the moon s surface with zero velocity . this example highlights the central role of optimal control in any type of engineering , its importance being rivaled only by feedback , a subject not covered in this review . in _ quantum _ optimal control @xcite , newton s equations governing the motion of the spacecraft are replaced by the quantum mechanical laws of motion , of course . in contrast , the control , corresponding for example to a radio - frequency ( rf ) amplitude or the electric field of a laser , is assumed to be classical . quantum optimal control represents one variant of quantum control @xcite and is closely related to coherent control @xcite . the latter requires exploitation of quantum coherence , i.e. , matter wave interference . in contrast , quantum control could also refer to inducing a desired dynamics , for example by amplitude modulations that avoid driving certain transitions , without matter wave interference . despite of its prominence in mathematics and engineering @xcite , optimal control was introduced to nmr spectroscopy @xcite and to the realm of matter wave dynamics @xcite only in the 1980s . in the latter case , the idea was to calculate , via numerical optimization , laser fields that would steer a photoinduced chemical reaction in the desired way @xcite . it was triggered by the advent of femtosecond lasers and pulse shaping capabilities that opened up seemingly endless possibilities to create intricate laser pulse trains . while a controlled breaking of chemical bonds was indeed demonstrated soon after @xcite , the pulses were obtained by closed - loop optimizations in the experiments @xcite rather than from theoretical calculations . in experimental closed - loop optimization , a shaped pulse is applied to the sample , and the outcome is measured . based on the outcome , the pulse shape is modified , typically by a genetic algorithm . however , even for a chemical reaction as simple as breaking the bond in a diatomic alkali molecule , the calculated optimized laser field can not directly be used in the experiment @xcite . the reason for this is two - fold : the way how the optimized laser fields are obtained is rather different in theory and experiment . whereas the field in calculations is shaped as a function of time @xcite , experiments employ spectral shaping @xcite . as a result , calculated pulses are often incompatible with experimental pulse shaping capabilities . second , the theoretical modeling is simply not accurate enough . this results in pulses which are optimal for the wrong dynamics and which can therefore not directly be applied in the experiments . these obstacles are not present , or at least much less severe , in other fields of application @xcite . once the timescale of the relevant dynamics is nanoseconds or slower , pulse shaping in the experiment is also done in time domain @xcite . while device response might still be an issue @xcite , the overall approaches in theory and experiment are similar in spirit . moreover , hamiltonians and relaxation parameters may be known much more accurately than is currently the case in photoinduced chemical reactions . a prominent example is nmr where the development of optimal control in theory and experiment went hand in hand , yielding beautiful results , for example on arbitrary excitation profiles @xcite , or robust broadband excitation @xcite . given these observations , quantum information processing ( qip ) and related technologies offer themselves as an obvious playground for quantum optimal control : in these applications , typically the quantum system to be controlled is well characterized , and timescales are sufficiently slow to use electronics for pulse shaping . not surprisingly , quantum optimal control has attracted much interest in these fields over the last decade . this included the adaptation of optimal control tools , for example to gate optimization @xcite , creation of entanglement @xcite , or measurement @xcite . gate optimizations were carried out for almost all qip platforms , notably comprising ions @xcite , atoms @xcite , nitrogen vacancy ( nv ) centers in diamond @xcite , and superconducting qubits @xcite . other qip tasks , such as state preparation @xcite , transport @xcite , and storage @xcite , have also been the subject of optimal control studies . these tasks are not only relevant for quantum computing and communication but also for related applications that exploit coherence and entanglement , for example quantum sensing or quantum simulation . protocols derived with optimal control have by now reached a maturity that allows them to be tested in experiments . examples include the crossing of a phase transition studied with trapped , cold atoms @xcite ; the improvement of the imaging capabilities of a single nv center @xcite ; and the creation of spin entanglement @xcite , quantum error correction @xcite and matter wave interferometry @xcite . all of these examples share a generic feature that is typical for quantum engineering : control over the system , which inevitably also brings about noise , needs to be balanced with sufficient isolation of the desired quantum features . this sets the theme for controlling open quantum systems . traditionally , a quantum system is defined to be open when it interacts with its environment @xcite . this interaction results in loss of energy and phase information . it can be modeled phenomenologically within the semigroup approach or microscopically , by embedding the system in a bath . besides coupling to a bath , the dynamics of a quantum system becomes effectively dissipative also when the system is subject to measurements or noisy controls . dissipative processes pose a challenge to quantum control . at the same time , desired dissipation may act as an enabler for control . we will review control strategies in both cases and then explain how optimal control theory can be used to adapt them to more complex quantum systems . this topical review is organized as follows : section [ sec : oqs ] briefly recalls the basic concepts in the theory of open quantum systems , introducing the distinction between markovian and non - markovian dynamics in sec . [ subsec : marknonmark ] and addressing the issue of gauging success of control for open quantum systems in sec . [ subsec : success ] . the problem of analyzing controllability of open quantum systems , an important prerequisiste to synthesizing control fields , is introduced in sec . [ subsec : controllability ] . progress in the control of open quantum systems is reviewed in sections [ subsec : strategies ] and [ sec : oc - oqs ] with sec . [ subsec : strategies ] dedicated to control strategies that were constructed with analytical methods and sec . [ sec : oc - oqs ] covering numerical optimal control . in sec . [ subsec : oct ] , the numerical methodology is explained in detail for a simple example , followed by a discussion of the modifications required to adapt it to more advanced control targets . the remainder of sec . [ sec : oc - oqs ] reviews applications of numerical optimal control to open quantum systems , starting with examples for fighting or avoiding decoherence in sec . [ subsec : fighting ] . control strategies that rely on the presence of the environment are discussed in secs . [ subsec : cooling ] and [ subsec : exploiting ] . section [ sec : concl ] concludes . the state of an open quantum system is described by the density operator @xmath0 which is an element of liouville space . any theory that aims at the control of an open quantum system is faced with two basic prerequisites the ability to calculate the system s dynamics , @xmath1 , and the ability to quantify success of control . formally , the time evolution of any open quantum system can be described by a dynamical map , @xmath2 which is completely positive and trace preserving ( cptp ) @xcite . the dynamical map is divisible if it can be written as the composition of two cptp maps @xmath3 @xmath4 . if the dynamical map is divisible , the open system s time evolution is memoryless and called markovian . various scenarios can lead to markovian dynamics , weak coupling between system and environment together with a decay of environmental correlations much faster than the timescales of the system dynamics being the most common case @xcite . however , open systems often exhibit pronounced memory effects , in particular in condensed matter experiments , which reflect characteristic features of the environment . the dynamics are then called non - markovian . memory effects are caused by structured spectral densities , nonlocal correlations between environmental degrees of freedom and correlations in the initial system - environment state @xcite . in the markovian case , the dynamics can be described by a master equation in lindblad form @xcite . in general , it needs to be solved numerically to determine @xmath5 . this can be done with arbitrarily high precision @xcite . however , the computational effort may quickly become challenging due to the exponential scaling of the size of hilbert and liouville space . to date , room for improvement seems to be limited @xcite . the situation is worse for non - markovian dynamics , where a unified framework such as the master equation in lindblad form does not exist . a variety of methods has been developed @xcite , each with different assumptions and hence a different range of applicability . they include time - local non - markovian master equations @xcite , stochastic unravellings @xcite , and an auxiliary density matrix approach @xcite . a common feature of these methods is their ability to correctly describe thermalization of the system . slightly different in philosophy are methods which attempt to solve the dynamics of both system and environment @xcite . key is to account only for that part of the environment that is relevant for the system s dynamics , i.e. , for the effective modes , which can be spins or harmonic oscillators . the underlying idea is that of quantum simulation on a classical computer @xcite , where the true environment is replaced by a surrogate one that generates the same dynamics . if one is interested in short times , the number of modes in the surrogate hamiltonian can be truncated with a prespecified error @xcite . longer propagation times than those computationally affordable with exact dynamics of system and environment become possible by separating the environment into two baths , one that is responsible for the memory effects and that is modeled by effective modes as explained above , and a second one that by itself would lead to markovian dynamics only . the secondary bath can be accounted for in terms of a markovian master equation in lindblad form @xcite or via a stochastic unravelling using a single secondary bath mode @xcite . a more comprehensive overview over methods to tackle non - markovian dynamics is found in ref . @xcite . understanding the influence of memory effects requires the ability to quantify them . an obvious way to define a measure of non - markovianity is to quantify deviation from divisibility @xcite . interestingly , this corresponds to an increase of correlations if the system is bi- or multipartite @xcite . other measures to capture memory effects focus on specific features such as recoherence and information backflow . these can be characterized in terms of distinguishability between quantum states @xcite , re - expansion of the volume of accessible states @xcite , or the capacity to reliably transmit quantum information @xcite . a comprehensive overview over the different measures is found in ref . @xcite , and an illustrative comparison for a toy model is presented in ref . @xcite . the proposed non - markovianity measures can be classified in a hierarchy by generalizing the notion of divisibility @xcite . while this is gratifying from a theoretical perspective , it is still an open question how non - markovianity can be measured in a condensed phase experiment . although some of the measures have been evaluated in experiment @xcite , dissipation in these examples was either engineered or artificial , in the sense that different degrees of freedom within one particle were considered to play the role of system and bath . a true condensed phase setting is more challenging due to limited control and thus limited access to measurable quantities . current interest in non - markovian dynamics is fueled by the revival of genuine quantum properties such as quantum coherence and correlations that non - markovianity entails . it has sparked the hope to exploit non - markovianity as a resource . quantum control in particular , which relies on properties such as coherence , should be more powerful in the non - markovian compared to the markovian regime . when the goal is to control an open quantum system , the ability to gauge success of control is even more important than that to measure the degree of non - markovianity . any suitable figure of merit needs to fulfill two conditions : ( i ) it should take its optimum value if and only if the control target is reached . ( ii ) it needs to be computable . an obvious control target is to drive a given initial state to a desired target state . the corresponding figure of merit is the projection onto the target state . for open quantum systems , this is given in terms of the hilbert schmidt product of the state of the system at the final time and the target state , @xmath6 this figure of merit has been used for example in control studies of cooling where the timescales of the dissipative process and the coherent system dynamics are comparable @xcite . typically , the target for cooling is a pure state . sometimes the timescale of dissipation is much slower than that of the coherent dynamics . this situation is encountered when using femtosecond lasers for laser cooling @xcite . in order to avoid the very long propagation times for repeated cooling cycles consisting of coherent excitation and spontaneous emission as well as the expensive description in liouville space , one can expand @xmath7 in a basis of hilbert space vectors and tailor the dynamics of the hilbert space vectors such that the target will be reached irrespective of the initial state @xcite . the construction of the proper figure of merit in that case will be discussed below in section [ subsec : cooling ] . an important control target in the context of quantum information processing is the implementation of unitary operations , or quantum gates . this corresponds to simultaneous state - to - state transitions for all states in the logical basis @xcite . a straightforward way to express this control target in liouville space is given by @xcite @xmath8 where @xmath9 denotes the desired target operation , defined on the logical subspace of dimension @xmath10 . the set of @xmath11 forms a suitable orthonormal basis of the @xmath12-dimensional liouville ( sub)space or , more simply , all @xmath12 matrices for which one entry is equal to one and all other zero . the hilbert schmidt product in eq . checks how well the actual evolution , @xmath13 , matches the desired one , @xmath14 . the scaling of eq . with system size @xmath10 restricts its use to examples with very few qubits . if the target is the implementation of a unitary operation , and not an arbitrary dynamical map , much less resources are required to gauge success of control . this observation is at the basis of all current proposals to estimate the average gate fidelity @xcite which forego the full knowledge of @xmath15 that is obtained in quantum process tomography in favor of efficiency . one way to understand the reduction in effort is to start from eq . and ask how many states are required in the sum to have a well - defined figure of merit , i.e. , a figure of merit that takes its optimum value if and only if the target operation is realized . surprisingly , the answer to this question is three , independent of system size @xcite : @xmath16 one state in eq . measures the departure from unitarity or , more precisely , from unitality in the logical subspace , and two more states are necessary to distinguish any two unitaries . the latter requires two states because one needs to determine the basis in which the unitary is diagonal and then check whether the phases on the diagonal are identical . only two states are required because it is possible to construct one density operator that fixes the complete hilbert space basis , @xmath17 , using one - dimensional orthogonal projectors , @xmath18 , with non - degenerate eigenvalues , @xmath19 . a variant of eq . is obtained by replacing the two states for the basis and phases by @xmath10 states @xmath20 . this is still a reduction compared to the @xmath12 states in eq . and was found to lead to faster convergence in control calculations than eq . evaluation of both eq . and its @xmath21 state variant are much more efficient than that of eq . @xcite . both eqs . and target implementation of a specific unitary @xmath9 . for difficult control problems , where a numerical search can easily get stuck , it is desirable to formulate the control target in the most flexible way . for example , instead of implementing a specific two - qubit gate such as cnot , it may be sufficient to realize a gate that is locally equivalent to cnot , i.e. , that differs from cnot by local operations . the corresponding figure of merit is based on the so - called local invariants @xcite . similarly , one can formulate a figure of merit for targeting an arbitrary perfect entangler @xcite . since these figures of merit are based on the local invariants which in turn are calculated from the unitary evolution , extension to non - unitary dynamics requires to first determine the unitary part of the overall evolution . this is possible , using the same mathematical concepts that have led to eq . the theory of controlling open quantum system can be divided into two main questions analysis of controllability and synthesis of controls ( called motion planning in the classical automatic control community ) : when the goal is to realize a certain desired dynamics , it is worthwhile to check first whether performance of the task is possible at all , before starting to search for controls that lead to the target . section [ subsec : controllability ] briefly reviews the current state of the art in controllability analysis of open quantum systems , whereas sec . [ subsec : strategies ] summarizes strategies for control synthesis that are based on certain properties of the system s interaction with its environment , and optimal control theory as a tool for control synthesis will be presented in sec . [ sec : oc - oqs ] . controllability analysis provides the mathematical tools for answering the question whether the target is reachable @xcite . in particular , a complete framework exists for finite - dimensional systems undergoing unitary dynamics : separating the hamiltonian into drift and control terms , @xmath22 the system is controllable provided the lie algebra spanned by the nested commutators of @xmath23 and @xmath24 is full rank @xcite . the elements of the lie algrebra can be interpreted as tangential vectors , i.e. , as the directions , of the unitary evolution ( which is an element of a lie group ) . if evolution into all directions can be generated , the system is controllable . controllability can also be viewed in terms of connectivity between hilbert space basis states @xcite ; it then corresponds to presence of the respective matrix elements . for open quantum systems , the evolution is not unitary anymore , and analysis of the hamiltonian alone is not sufficient to decide controllability : the dissipative part of the evolution may prevent or enable certain states to be reached . for example , even if the full rank condition for the lie algebra of the hamiltonian is fulfilled , fast decoherence will inhibit transitions between pure states by turning any pure state into a mixed one . merely the presence of non - vanishing matrix elements in the hamiltonian is thus not sufficient to decide controllability . their magnitude matters as well , in particular of those in the drift hamiltonian @xmath23 that can not be tuned by external controls @xmath25 . to date , rigorous controllability analysis does not take such a dependence on operator norms , or competing time scales into account . as a result , one needs to turn to numerical search for most open quantum systems , even for an example as simple as the central spin model @xcite . this is rather unsatisfactory since a numerical search can not provide rigorous answers to controllability , in particular lack thereof ( in the sense of reachability of the target with a predetermined error @xmath26 ) , due to its local character . an extension of controllability analysis to the needs of open quantum systems would address this issue but remains an open problem to date . it would be particularly relevant for open quantum systems with almost unitary dynamics , which are often encountered e.g. in quantum technology applications . as an example where the dissipative part of the overall evolution enables certain states to be reached , consider cooling which turns mixed states into pure ones . obviously , controllability analysis based on the lie rank condition for the hamiltonian alone can not provide any information on time evolutions which change the purity of the system s state , @xmath27 $ ] . by and large , controllability of open quantum systems still remains uncharted territory to date @xcite . this refers in particular to dynamic controllability where the analysis accounts for available dynamical resources such as coupling to external fields and environmental degrees of freedom , or measurements , in contrast to kinematic controllability . the latter implies existence of a dynamical map that transforms any initial state into the target state . while this existence can be proven for finite - dimensional systems @xcite , it is of limited relevance for practical applications since , in general , one can not derive any dynamical information from the proof . to tackle controllability of open quantum systems , two routes can be followed : either one starts from a complete description of the system and its environment , or one considers the reduced description of the system alone . in the first case , the tools of controllability analysis for unitary dynamics can be employed . this way it was possible to show , for example , that even for completely controllable system - environment dynamics , cooling is possible only if the environment contains a sufficiently large virtual subsystem which is in a state with the desired purity @xcite . while exact solution of the combined system - environment dynamics is extremely challenging ( and impossible in many cases ) , controllability analysis from this perspective is expected to significantly advance our understanding already for simple models and for generic questions , as in the example of ref . moreover , it appears to be promising for two reasons it does not rely on _ a priori _ assumptions on the reduced dynamics , and most likely it will benefit from recent progress in the controllability of infinite - dimensional systems @xcite . rigorous controllability analyses for reduced dynamics have been limited to date to the markovian case @xcite . in particular , the sets of reachable states were characterized @xcite , and the lindblad operators were shown to form a lie wedge @xcite . while the lie wedge provides a sufficient , but not necessary condition for controllability , necessary but not sufficient criteria can be identified by considering isotropy of the generator of the dissipative motion @xcite . numerically , non - markovian dynamics were shown to lead to @xmath28 controllability ( in the sense of reachability of the target with a predetermined error @xmath26 ) for a system whose hamiltonian allows for realization of @xmath29 operations only @xcite . however , no rigorous analysis of controllability for non - markovian dynamics has been performed to date ; and it is not yet clear whether and under which conditions non - markovian effects can improve controllability of an open quantum system . control strategies that are obtained by analytical methods can be roughly divided into two classes those that exploit symmetries in the system - bath interaction and those that make assumptions on the timescale of this interaction . in the first case , protection from decoherence is achieved by keeping the system s state in a so - called decoherence - free subspace @xcite : if the system - bath interaction contains a symmetry , for example qubits couple indistinguishably to their environment , it is possible to construct a set of system states that are invariant under the system - bath interaction . these states form a decoherence - free subspace in the system s total hilbert space @xcite . stimulated raman adiabatic passage ( stirap ) and electromagnetically induced transparency represent examples for decoherence - free subspaces @xcite . in the context of quantum information , two or more physical qubits , carried for example by atoms , can be used to encode one logical qubit that is decoherence - free @xcite . decoherence - free subspaces have been demonstrated , for example , in liquid - state nuclear magnetic resonance @xcite and with trapped ions @xcite . numerical methods can be employed to identify ( approximate ) decoherence - free subspaces @xcite and to find an external control that drives the system dynamics into a decoherence - free subspace @xcite . more generally , in physics the presence of a symmetry implies existence of a conserved quantity . in the context of decoherence , a symmetry in the system - bath interaction leads to a quantum number which is preserved under this interaction . the eigenstates belonging to the preserved quantum number define a noiseless subsystem , i.e. , a logical subsystem that is intrinsically protected from noise @xcite . the main limitation of this set of approaches is imposed by the existence of a suitable symmetry which is not necessarily available . in the second case of control strategies that are built on assumptions on the timescale of the system - bath interaction , a trivial strategy is obtained for slow decoherence : one simply needs to perform the desired operation on a time scale much faster than that of decoherence . but this may not always be possible , and slow decoherence also allows for eliminating the effects of decoherence based on average hamiltonian theory @xcite or , in more intuitive terms , spin echo techniques from nuclear magnetic resonance and generalizations thereof . this set of strategies is often referred to as dynamical decoupling @xcite . it relies on many quasi - instantaneous actions of control fields that do not allow the system to interact with its environment , creating an effective dynamics of the system alone . extensive work over the last two decades has allowed to account for e.g. pulse imperfections @xcite and extend the technique beyond spin dynamics for which it was originally developed , for example to ions @xcite , and superconducting circuits @xcite . the main limitation of the dynamical decoupling approach is the finite duration of any control field which can not always be made sufficiently short . by now dynamical decoupling has grown into a field of its own which has been covered by several reviews @xcite , and the reader is referred to these for a more in - depth analysis . the time - dependent perspective used in dynamical decoupling can be translated into a frequency - space picture using the generalized transfer filter function approach @xcite . this allows to efficiently predict fidelities , at least for weak noise @xcite , and provides a connection to control strategies that decouple the system from its environment by engineering a spectral separation @xcite . in particular , it was shown that the system - bath interaction can be cancelled , at least to second order , by choosing the time - dependence of the control field such that its spectrum becomes orthogonal to the bath or noise spectrum @xcite . such an approach requires weak coupling and negligible initial system - bath correlations . moreover , the bath spectrum needs to be known . another example of engineering a spectral separation is given by the protection of so - called edge states in topological insulators via band liouvillians @xcite . in addition to clarifying the relation between time - domain based and frequency - domain based control strategies , the transfer filter function perspective also provides a practical tool for characterizing the noise spectral density @xcite . its applicability to a variety of physical platforms has already been demonstrated @xcite . noise spectroscopy provides an excellent starting point for deriving microscopic models for the system - environment interaction and thus a more thorough understanding of noise at the quantum mechanical level . it also allows for tailoring control synthesis to specific spectral features of the noise , either , if possible , by using the transfer filter function approach directly @xcite , or by exploiting knowledge of these features using optimal control theory , in sec . [ sec : oc - oqs ] below . quantum optimal control theory refers to a set of methods that synthesize external control fields from knowledge of the control target , including constraints , and the time evolution of the quantum system @xcite . it is based on the calculus of variations , i.e. , on formulating the control target as a functional of external controls that realize the desired dynamics . knowledge of the system s time evolution enters implicitly via evaluation of the target functional and , possibly , its derivatives . for a few exceptional cases , for example one or two spins ( or qubits ) @xcite , a harmonic oscillator @xcite , or a sequence of @xmath30-systems subject to decay @xcite , the external controls can be determined using geometric techniques based on pontryagin s maximum principle @xcite . typically , however , the control problem can not be solved in closed form , and one needs to resort to numerical optimization . most often , controlling open quantum systems is a difficult control task such that efficiency of the optimization algorithm is important . therefore , mainly algorithms based on the target functional s gradient have been employed for open quantum systems to date . they will be reviewed in sec . [ subsec : oct ] . the remainder of this section is dedicated to control strategies that were developed using numerical optimization , starting with strategies avoiding decoherence in sec . [ subsec : fighting ] , followed by strategies exploiting presence of the environment in sec . [ subsec : cooling ] and [ subsec : exploiting ] . conceptually the simplest control problem is represented by state - to - state transfer : given a known initial state , @xmath31 , at time @xmath32 , find the external field that drives this state at final time @xmath33 into the target state , @xmath34 , with prespecified error , @xmath26 . the corresponding target functional is found in eq . where dependence on the set of external controls @xmath35 is implicit in the time evolution , @xmath36 $ ] . this control problem was first stated in the context of laser cooling molecular vibrations @xcite . the time evolution was modeled by a markovian master equation in lindblad form , and later extended to a non - markovian example @xcite . for the sake of conceptual clarity , we present here the algorithm for a markovian master equation , @xmath37,{\boldsymbol{\mathsf{\hat{\rho}}}}\right ] + \mathcal{l}_d({\boldsymbol{\mathsf{\hat{\rho}}}})\,,\ ] ] with the hamiltonian of the form . for simplicity , we assume a single control @xmath38 in eq . . in the example of laser cooling molecular vibrations , @xmath39 would be the electric field of a short laser pulse , and @xmath40 in eq . the transition dipole moment of the molecule . the dissipative part of eq . is given by @xcite @xmath41 where the lindblad operators @xmath42 model the various dissipative channels . for example , @xmath43 for spontaneous decay from a level @xmath44 to the ground state with rate @xmath45 , inversely proportional to the level s lifetime . optimization algorithms are obtained by seeking an extremum of @xmath46 , cf . eq . , with respect to the control , @xmath39 . this can be done by direct evaluation of the extremum condition @xcite or by building in monotonic convergence _ a priori _ using krotov s method @xcite . the resulting set of coupled equations are , surprisingly , rather similar . the main difference is in the update of the control which can be performed concurrently @xcite , i.e. , for all times @xmath47 at once , or sequentially in time @xcite . guaranteed monotonic convergence is only obtained with a sequential update of the control . in this case , the equation for determining the control reads @xmath48 \right\}\,.\end{aligned}\ ] ] @xmath49 denotes a shape function that can be used to switch the control on and off smoothly @xcite or to impose an initial or final ramp @xcite ; and @xmath50 is a parameter of the algorithm whose choice determines the step size in the change of the control . its optimal value can be estimated in an automated way , similarly to a line search in quasi - newton methods @xcite . @xmath51 denotes the so - called co - state or adjoint state , and the derivative with respect to the control is given by the commutator @xmath52 } { \partial u}\big|_{u^{(i+1)}},{\boldsymbol{\hat{\rho}}}^{(i+1)}\right]\,.\ ] ] for the common case of linear coupling to the control , as in eq . , the explicit dependence on @xmath53 vanishes and the commutator simply becomes @xmath54 $ ] . nonetheless , the right - hand side of eq . depends on @xmath53 via @xmath55 , i.e. , it is an implicit equation . solution of the implicit equation can usually be avoided by a low order approximation in the iterative algorithm , employing two shifted time discretizations to represent the time dependence of states and control , @xmath56 and @xmath57 @xcite . equation also depends on the state of system at time @xmath47 , @xmath5 , and the co - state , @xmath51 . these are obtained by solving [ eq : forward ] @xmath58,{\boldsymbol{\mathsf{\hat{\rho}}}}^{(i)}\right ] + \mathcal{l}_d({\boldsymbol{\mathsf{\hat{\rho}}}}^{(i)})\end{aligned}\ ] ] with initial condition @xmath59 and [ eq : backward ] @xmath60,{\boldsymbol{\mathsf{\hat{\sigma}}}}^{(i)}\right ] -\mathcal{l}_d({\boldsymbol{\mathsf{\hat{\sigma}}}}^{(i)})\ , , \end{aligned}\ ] ] which is solved backward in time . the initial condition is derived from the target functional at final time @xmath46 , @xmath61 this coupled set of equations needs to be solved iteratively , starting with some guess for the control , @xmath62 , where the index @xmath63 denotes iteration . the algorithm represented by eqs . , and is straightforwardly extended from targeting a single state to targeting a unitary operation @xcite : a unitary operation @xmath64 can be viewed as several simultaneous state - to - state transfers @xcite which are all driven by the same control . consequently , eq . becomes @xmath65 \right\}\,,\ ] ] and eqs . and need to be solved for @xmath66 states @xmath67 and @xmath66 co - states @xmath68 simultaneously . as explained in sec . [ subsec : success ] , it was first believed that the sum in eq . has to run over @xmath69 states where @xmath10 is the dimension of the space on which the desired operation is defined @xcite . the initial conditions @xmath70 are then given by orthogonal basis states spanning this space . recently it was shown that @xmath66 can be reduced all the way down to 3 in which case the states discussed in sec . [ subsec : success ] need to be taken as initial conditions @xmath70 @xcite . the initial conditions for the co - states are always given in terms of the desired target operation , @xmath71 , up to a suitable normalization @xcite . moreover , weights may be introduced in @xmath72 to speed up convergence by attaching different importance to different basis states @xmath70 @xcite . extension of the optimization algorithm to more flexible control targets , such as an arbitrary perfect entangler @xcite instead of a specific unitary , requires two steps . first , a modified target functional results in a modification of eq . : the right - hand side of eq . will be replaced by the derivative of the new target functional with respect to the states , @xmath11 , evaluated at time @xmath33 . for a target functional based on the local invariants , this requires , in particular , to determine the unitary part of the overall actual evolution , as explained in sec . [ subsec : success ] . second , dependence of the functional on the states @xmath11 may be non - convex . in this case , eq . needs to be amended by a second term in its right - hand side @xcite which depends additionally on the change in the states , @xmath73 . while the additional computational effort for evaluating the control update is negligible , storage of all @xmath74 is necessary and may become a limiting factor when scaling up the system size . such extensions of the optimization algorithm to control targets other than a specific state or unitary have been applied to coherent dynamics @xcite . for open quantum systems , they are still under exploration . typically , more than one solution exists to a quantum control problem . when using the basic optimization algorithm presented above , it will then depend on the initial guess @xmath75 which solution @xmath76 the algorithm identifies . two strategies can be employed in order to fine - tune the iterative search and guide it toward a solution with certain desired properties careful selection of the initial guess by scanning or preoptimization @xcite or use of additional constraints @xcite . in the first case , the initial guess needs to be parametrized in a suitable form , for example in terms of amplitudes and phases of fourier components , or amplitudes , widths and positions of gaussians . these parameters can easily be pre - optimized within a prespecified range employing a standard gradient - free optimization method @xcite . when the result is used as initial guess in the optimization algorithm presented above , the ensuing fine - tuning will usually not lead to drastic changes in the field , keeping it close to the parametrized form @xcite . a more stringent way to enforce certain desired properties of the control solution is obtained by employing additional constraints @xcite . this comes at the expense of a modified optimization algorithm . an explicit description for deriving the modified algorithms is available when using krotov s method @xcite . it allows for formulating constraints as a functional of the control , with the only condition that the functional be positive ( or negative ) semidefinite @xcite . this requirement is necessary to ensure monotonic convergence . as an example , consider a spectral constraint on the control @xcite , @xmath77 = \frac{1}{2\pi } \int_0^t \int_0^t \delta u(t ) k(t - t^\prime ) \delta u(t^\prime ) dt^\prime dt\,,\ ] ] where @xmath78 is the fourier transform of a spectral filter @xmath79 . as a result , the update equation for the control becomes a fredholm equation of the second kind @xcite . a judicious choice of the shape function @xmath49 allows for analytically solving the fredholm equation in frequency domain such that the additional numerical effort for including the spectral constraint consists merely in two additional fourier transforms @xcite . more than one constraint may be employed at a time , with different weights allowing to emphasize importance of one compared to another @xcite . while additional constraints provide information that guides the optimization algorithm , they also restrict the space of admissible solutions @xcite . therefore care needs to be taken to balance their benefits and their disadvantages . while it is probably the dream of every control engineer to discover unthought of control schemes , a more realistic scenario starts from known a control strategy , as those described in sec . [ subsec : strategies ] , and extends it to a wider range of conditions , new types of systems , or new types of dissipative processes using the optimization techniques described above . one of the most popular control strategy in the area of quantum technologies currently is dynamical decoupling @xcite . while already powerful in itself , dynamical decoupling can be made more robust by numerical optimization that targets specific noise features that were previously unaccounted for , using , for example , the gradient - ascent technique @xcite or genetic algorithms @xcite . in these examples , optimization did not compromise feasibility of the pulse sequences . moreover , the length of dynamical decoupling sequences can be minimized @xcite . dynamical decoupling and numerically optimized pulses can also complement each other , as recently demonstrated for entanglement generation and distribution in nv centers in diamonds @xcite . a second successful control strategy described in sec . [ subsec : strategies ] is based on decoherence - free subspaces and noiseless subsystems . these are somewhat less often used than dynamical decoupling , mainly due to the difficulty of identifying them for more complex systems . while direct identification of decoherence - free subspaces is hampered by presence of numerous traps in the search space @xcite , quantum optimal control may be used to dynamically identify them . indeed , optimization of an open system s dynamics for targets that rely on quantum coherence is intrinsically biased toward those subspaces in hilbert ( or liouville ) space that are least affected by decoherence @xcite . for example , transfer of coherence and polarization between coupled heteronuclear spins was improved by cross - correlated relaxation optimized pulse ( crop ) sequences and relaxation optimized pulse elements ( rope ) @xcite . optimization can take both longitudinal and transveral relaxation into account @xcite . the underlying mechanism was revealed to consist in tuning cross - correlated to autocorrelated relaxation rates @xcite . counterintuitively , maximum polarization transfer between coupled spins was achieved with sequences that are longer than conventional ones @xcite , highlighting the importance to include the dissipative dynamics in the optimization . finally , if all regions in hilbert space are similarly affected by decoherence , an obvious control strategy consists in beating decoherence by the fastest possible operation . this strategy is faced , however , with the so - called quantum speed limit , i.e. , a fundamental bound on the shortest operation time @xcite . for a two - level system , it can be estimated analytically @xcite . for more complex systems , optimal control theory can be used as a tool to both identify the quantum speed limit and determine controls that drive the system at the quantum speed limit @xcite . for example , the shortest possible duration of entangling quantum gates was determined for cold , trapped atoms @xcite and for superconducting qubits @xcite . in quantum dots , phonon - assisted decoherence was minimized @xcite . interestingly , presence of the environment may improve the quantum speed limit @xcite . this has not yet been explored systematically but could be done , using quantum optimal control . the example of cooling @xcite was already taken as reference to introduce optimal control of open quantum systems in sec . [ subsec : oct ] . since cooling changes the purity of the state , it relies on the presence of the environment . when using the algorithm outlined in sec . [ subsec : oct ] , the control that drives the cooling process is determined for each initial state separately . alternatively , one may seek a control that will lead to cooling irrespective of the initial state @xcite . such an approach is particularly useful if the timescales of the coherent and the dissipative dynamics is different , as in the case of optical pumping @xcite which consists of many repeated cycles of excitation and spontaneous emission . the theoretical framework lends itself to generalization to quantum reservoir engineering which is why it is outlined in more detail in the following . the idea is to start from an orthogonal basis for the space on which the initial states are defined and ensure transitions favorable to cooling for every basis state @xcite . in the case of optical pumping , these are transitions into states from which spontaneous emission preferrably occurs to the cooling target . as is usual in quantum optimal control , this control task is stated in terms of a yield functional . additionally , the control should not excite any population which has already been accumulated in the cooling target state . in other words , the cooling target should remain a steady state of the evolution . this requirement is translated into a second term in the optimization functional , besides the yield . moreover , it is often not possible to cool arbitrary initial states , due to limitations on control bandwidth . instead , one can seek to cool states which are defined on a certain subspace . this results in an additional term in the optimization functional that suppresses leakage from this subspace in order to keep the cooling cycle closed . finally , one needs to guarantee that for all basis states cooling occurs with the same efficiency ; otherwise the cooling might get stuck . this can be achieved either by imposing symmetric excitation of all basis states or by having all basis states form an assembly line , i.e. , enforce one specific excitation pathway for all states . all requirements need to be met simultanously and , consequently , the optimization functional consists of four terms one for the yield , one of the steady state , one to suppress leakage and one to ensure symmetric excitation or enforce a specific excitation pathway for all states @xcite . application of this optimization framework showed that laser cooling of molecular vibrations is possible even in cases where the molecular structure favors heating rather than cooling @xcite . it also answers the question about the minimal requirement on the molecular structure to realize , with shaped pulses , cooling instead of heating , assuming no constraints on the control existence of one state which undergoes spontaneous emission with moderate probability into the cooling target . if the molecular structure is favorable to cooling vibrations , an optimized laser pulse results in a substantially smaller number of cooling cycles than an unshaped pulse @xcite . a similar speed up of the cooling due to pulse sequences obtained from quantum optimal control theory has also been reported for an optomechanical resonator @xcite and for trapped , quasi - condensed cold atoms @xcite . laser cooling can be viewed as a particular example of quantum reservoir engineering @xcite , where a desired state becomes the ground state of a driven dissipative system . it holds the promise of a particularly robust control strategy . however , applications of quantum reservoir engineering have been limited to quantum optics to date . in a condensed phase settings , both desired and undesired dissipative channels come into play , and non - markovian effects may occur . thus , quantum reservoir engineering in the condensed phase represents a challenging control problem . as with any control problem , two questions need to be tackled that of controllability and that of control synthesis , i.e. , what states are attainable and how can the necessary driving be realized . the first question has been answered for generic models , such as a two- and a four - level system and a harmonic oscillator , that undergo markovian dynamics @xcite . the obtained understanding of controllability can be exploited to construct dissipative channels that allow for the robust generation of long - distance entanglement @xcite . the question which states are attainable in the presence of additional undesired dissipative channels , a generic feature of any condensed phase setting , has not yet been tackled to date . a possible influence of non - markovian effects on the reachable states has also not yet been addressed . control synthesis may be achieved in several ways . first , incoherent control by the environment , for example , via certain population distributions in the environmental modes , may be used to control the system @xcite . however , this contradicts the assumption that the environment by definition is uncontrollable . alternatively , measurements effectively lead to dissipative dynamics and may thus be used to generate desired dissipation @xcite . they may be augmented by suitably tailored coherent excitation for more effective control @xcite . quantum reservoir engineering may also be formulated as an optimization problem where the target is a certain desired steady state . an optimization algorithm is obtained by generalizing the theoretical framework for laser cooling @xcite outlined above . however , the search space is even larger than for a standard quantum control problem ; and the efficient numerical implementation is an open challenge . meeting this challenge would allow for exploring quantum reservoir engineering in the presence of undesired dissipation typical for condensed phase settings and in the non - markovian regime . it was shown already several years ago that non - markovian evolution may ease control @xcite , and cooperative effects of dissipation and driving were reported @xcite . however , a more thorough understanding of the nature of non - markovianity was required to understand its interplay with quantum control . as described in sec . [ subsec : marknonmark ] , non - markovianity has in the meantime been characterized in terms of information flow from the environment to the system @xcite , increase of correlations in a bipartite system @xcite , or re - expansion of the volume of accessible states @xcite , among others . the important point in the context of quantum control is that each of these measures holds a promise for better control : correlations between system and environment may improve fidelities of single qubit gates @xcite , cooperative effects of control and dissipation may allow for entropy export and thus cooling @xcite ; and harnessing non - markovianity may enhance the efficiency of quantum information processing and communication @xcite . a first example of exploiting non - markovianity for quantum control was reported in ref . @xcite , showing that , for an anharmonic ladder system , the environment may be utilized to extend possible operations from @xmath29 to @xmath28 . presence of at least one two - level defect in the environment that is sufficiently isolated and sufficiently strongly coupled to the system was identified as prerequisite for the observed controllability enhancement . such conditions are found in current experiments with superconducting circuits and other systems which are immersed in a small , natural spin bath , for example color centers in diamond . the limited number of optimal control studies of open quantum systems with non - markovian dynamics @xcite testifies to the fact that control of these systems remains largely uncharted territory . the full potential of the specific features of non - markovian dynamics for quantum control remains yet to be explored . open questions include , for example , how the build - up of memory influences control ; whether specific features of the spectral density can be exploited for control , and if so , how . the tools for performing these studies , both in terms of simulating non - markovian dynamics @xcite and carrying out optimal control calculations , cf . [ subsec : oct ] , have been developed and are there to be used . the present review has been focused on control of open quantum systems as they are encountered in the field of quantum technologies . it will be concluded by briefly mentioning examples from other fields of current interest : quantum optimal control for open systems has been employed in the context of quantum thermodynamics , in order to determine the optimal efficiency of a noisy heat engine @xcite ; biological chromophore complexes , in order to maximize exciton transfer @xcite ; molecules immersed in dissipative media , in order to maximally align them with respect to a laboratory axis @xcite ; molecular junctions , in order to control the current , shot noise and fano factors @xcite ; as well as chemical reaction dynamics @xcite , including charge transfer in molecules @xcite , and surface photochemistry @xcite . the numerous applications attest to the maturity as well as versatility of the quantum control toolbox @xcite . 1 . if the desired operation shall keep pure states pure , markovian dynamics are unwanted . the effect of the environment in this case is detrimental . a suitable strategy is then to perform any desired operation as fast as possible . quantum optimal control theory is a viable tool to determine both the shortest operation time and the control that drives the desired dynamics . the actual dissipative processes may be neglected during the optimization for computational simplicity . explicit account of the dissipative processes comes at a significantly larger numerical cost but allows for identifying subspaces which are less affected by or even immune to decoherence . if the desired operation changes the purity of the system s state , presence of the environment is necessary for realizing the control target . in this case , markovian dynamics may be desired : the control target is reachable if it is a fixed state of the liouvillian . external control fields may be used to ensure that this is the case . the corresponding control strategy is referred to as quantum reservoir engineering . if the liouvillian has several fixed points , external fields may also be used to drive the dynamics to the desired one . the role of non - markovian effects in this type of desired dissipation has not been explored to date . 3 . non - markovian dynamics in general may have both beneficial and detrimental effects on controlled quantum dynamics . improved controllability is a first example of a benefit . it requires presence of a few strongly coupled and sufficiently isolated environmental modes which can effectively act as ancillas , and use of quantum optimal control for properly exploiting these modes . an improved quantum speed limit is a second example . while the number of examples for successful control of open quantum systems is growing , our current understanding of controllability and the most promising control strategies for open quantum systems is still rather limited . in particular , a thorough understanding of the role of non - markovian effects is lacking to date , and it is currently still unknown which features of non - markovianity can be exploited for quantum control . investigation of a larger range of models , with both small and large baths , consisting of harmonic modes and spins , and a systematic analysis of non - markovianity may elucidate this question . such an improved understanding would not only be crucial for advancing quantum technologies but would also be beneficial for adjacent fields such as condensed matter physics or chemical reaction dynamics . i would like to thank ugo boscain , steffen glaser , dominique sugny , and david tannor for their comments on the manuscript . recent advances in the control of open quantum systems in my own group were made possible by the work of michael goerz , giulia gualdi and daniel reich whom i would like to thank for their dedication and perserverance .

video streaming applications have become over the past few years the dominant applications in the internet and generate the prevalent part of traffic in today s ip networks ; see for instance guillemin et al . @xcite for an illustration of the application breakdown in a commercial ip backbone network . video files are currently downloaded by customers from large data centers , like google s data centers for youtube files . in the future , it is very likely that video files will be delivered by smaller data centers located closer to end users , for instance cache servers disseminated in a national network . it is worth noting that as shown in guillemin et al . @xcite , caching is a very efficient solution for youtube traffic . while this solution can improve performances by reducing delays , the limited capacity of those servers in terms of bandwidth and computing can cause overload . one possibility to reduce overload is to use bit rate adaptation . video files can indeed be encoded at various bit rates ( e.g , small and high definition video ) . if a node can not serve a file at a high bit rate , then the video can be transmitted at a smaller rate . it is remarkable that video bit rate adaptation has become very popular in the past few years with the specification of mpeg - dash standard where it is possible to downgrade the quality of a given transmission , see schwarz et al . @xcite , sieber et al . @xcite , aorga et al . @xcite , vadlakonda et al . @xcite and fricker et al . adaptive streaming is also frequently used in mobile networks where bandwidth is highly varying . in this paper , we investigate the effect of bit rate adaptation in a node under saturation . we assume that customers request video files encoded at various rates , say , @xmath3 for @xmath4 , with @xmath5 . jobs of class @xmath6 require bit rate @xmath3 . the total capacity of the communication link is @xmath0 . if @xmath7 is the state of the network at some moment , with @xmath8 being the number of class @xmath9 jobs , the quantity @xmath10 has to be less than @xmath0 . the quantity @xmath11 is defined as the _ occupancy _ of the link . the algorithm has a parameter @xmath1 and works as follows : if there is an arrival of a job of class @xmath12 , * if @xmath13 then the job is accepted ; * if @xmath14 then the job is accepted but as a class @xmath15 job , i.e. it has an allocated bit rate of @xmath16 and service rate @xmath17 ; * if @xmath18 , the job is rejected . for @xmath19 , jobs of class @xmath9 arrive according to a poisson process with rate @xmath20 and have an exponentially distributed transmission time with rate @xmath21 additionally , it is assumed that @xmath22 to study this allocation scheme , a scaling approach is used . it is assumed that the server capacity is very large , namely scaled up by a factor @xmath23 . the bit rate adaptation threshold and the request arrival rates are scaled up accordingly , i.e. @xmath24 _ performances of the algorithm . _ our main result shows that , for the downgrading policy and if @xmath25 is chosen conveniently , then 1 . the equilibrium probability of rejecting a job converges to @xmath26 as @xmath23 goes to infinity ; 2 . the equilibrium probability of accepting a job without downgrading it converges to @xmath27 as @xmath23 goes to infinity . see theorem [ theoeq ] and corollary [ corol ] . the above formula gives an explicit expression of the success rate of this allocation mechanism . the quantity @xmath28 , the probability of downgrading requests , can be seen as the `` price '' of the algorithm to avoid rejecting jobs . the scaling has been introduced by kelly to study loss networks . see kelly @xcite . the transient behavior of these networks under this scaling has been analyzed by hunt and kurtz @xcite . this last reference provides essentially a framework to establish convenient convergence theorems involving stochastic averaging principles . this line of research has been developed in the 1990 s to study uncontrolled loss networks where a request is rejected as soon as its demand can not be accepted . when the demand can be adapted to the state of the network , for controlled loss networks , several ( scarce ) examples have been also analyzed during that period of time . one can mention bean et al . @xcite , zachary and ziedins @xcite and zachary @xcite for example . our model can be seen as a `` controlled '' loss networks instead of a pure loss network . controlled loss networks may have mechanisms such as trunk reservation or may allocate requests according to some complicated schemes depending on the state of the network . in our case , the capacity requirements of requests are modified when the network is in a `` congested '' state . contrary to classical uncontrolled loss networks , as it will be seen , the markov process associated to the evolution of the vector of the number of jobs for each class is not reversible . additionally , the invariant distribution of this process does not seem to have a closed form expression . kelly s approach @xcite is based on an optimization problem , it can not be used in our case to get an asymptotic expression of some characteristics at equilibrium . for this reason , the equilibrium behavior of these policies is investigated in a two step process : 1 . transient analysis . we investigate the asymptotic behavior of some characteristics of the process on a finite time interval when the scaling parameter @xmath23 goes to infinity . 2 . equilibrium . the stability properties of the limiting process are analyzed , we prove that the equilibrium of the system for a fixed @xmath23 converges to the equilibrium of the limiting process . for our model , the transient analysis involves the _ explicit _ representation of the invariant distribution of a specific class of markov processes . it is obtained with complex analysis arguments . as it will be seen , this representation plays an important role in the analysis of the asymptotic behavior at equilibrium . it should be noted that related models have recently been introduced to investigate resource allocation in a cloud computing environment where the nodes receive requests of several types of resources . we believe that this domain will receive a renewed attention in the coming years . see stolyar @xcite and fricker et al . @xcite for example . in some way one could say that the loss networks are back and this is also a motivation of this paper to shed some light on the methods that can be used to study these systems . we consider a system in overload . because of bit rate adaptation , requests may be downgraded but not systematically rejected as in a pure loss system . as it will be seen , the stability properties of this algorithm are linked to the behavior of a markov process associated to the occupation of the link . under exponential assumptions for inter - arrival and service times , this process turns out to be , after rescaling by a large parameter @xmath23 , a bilateral random walk instead of a reflected random walk as in the case of loss networks . using complex analysis methods , an explicit expression of the invariant distribution of this random walk is obtained . with this result , the asymptotic expression of the probability that , at equilibrium , a job is transmitted at its requested rate ( and therefore does not experience a bit rate adaptation ) is derived . this paper is organized as follows : in section [ model ] , we present the model used to study the network under some saturation condition . convergence results when the scaling factor @xmath23 tends to infinity are proved in section [ secmod ] . the invariant distribution of a limiting process associated to the occupation of the link is computed in section [ secinv ] by means of complex analysis techniques . applications are discussed in section [ app ] . * acknowledgments * + the authors are very grateful to an anonymous referee for pointing out a gap in the proof of theorem [ theoeq ] in the first version of this work . one considers a service system where @xmath29 classes of requests arrive at a server with bandwidth / capacity @xmath0 . requests of class @xmath9 , @xmath30 , arrive according to a poisson process @xmath31 with rate @xmath20 . a class @xmath9 request has a bandwidth requirement of @xmath3 units for a duration of time which is exponentially distributed with parameter @xmath21 . for the systems investigated in this paper , there is no buffering , requests have to be processed at their arrival otherwise they are rejected . without any flexibility on the resource allocation , this is a classical loss network with one link . see kelly @xcite for example . this paper investigates allocation schemes which consist of reducing the bandwidth allocation of arriving requests to a minimal value when the link has a high level of congestion . in other words the service is downgraded for new requests arriving during a saturation phase . if the system is correctly designed , it will reduce significantly the fraction of rejected transmissions and , hopefully , few jobs will in fact experience downgrading . we introduce @xmath1 , the parameter @xmath2 will indicate the level of congestion of the link . it is assumed that the vector of integers @xmath33 is such that @xmath34 . the condition @xmath16 is used to simplify the presentation of the results and to avoid problems of irreducibility in particular but this is not essential . if the network is in state @xmath7 and if the occupancy @xmath11 is less than @xmath2 , then any arriving request is accepted . if the occupancy is between @xmath2 and @xmath35 , it is accepted but with a minimal allocation , as a class @xmath15 job . finally it is rejected if the link is fully occupied , i.e. @xmath36 . it is assumed that @xmath37 , for @xmath19 , i.e. class 1 jobs are served with the smallest service rate . mathematically , the stochastic model is close to a loss network with the restriction that a job may change its requirements depending on the state of the network . this is a controlled loss network , see zachary and ziedins @xcite . it does not seem that , like in uncontrolled loss networks , the associated markov process giving the evolution of the vector @xmath38 has reversibility properties , or that its invariant distribution has a product form expression . related schemes with product form are trunk reservation policies for which requests of a subset of classes are systematically rejected when the level of congestion of the link is above some threshold . see bean et al . @xcite and zachary and ziedins @xcite for example . concerning controlled loss networks , mathematical results are more scarce . one can mention networks where jobs requiring congested links are redirected to less loaded links . several mathematical approximations have been proposed to study these models . see the surveys kelly @xcite and zachary and ziedins @xcite . in our model , in the language of loss networks , the control is on the change of capacity requirements instead of a change of link . the invariant distribution being , in general , not known , a scaling approach is used . the network is investigated under kelly s regime , i.e. under heavy traffic regime with a scaling factor @xmath23 . it has been introduced in kelly @xcite to study the equilibrium of uncontrolled networks . the arrival rates are scaled by @xmath23 : @xmath20 is replaced by @xmath39 as well as the capacity @xmath0 by @xmath40 and the threshold @xmath2 by @xmath41 which are such that @xmath42 for @xmath43 . for @xmath30 and @xmath44 , @xmath45 denotes the number of class @xmath9 jobs at time @xmath46 in this system and @xmath47 . it will be assumed that the system is overloaded when the jobs have their initial bandwidth requirements @xmath48 with @xmath49 and @xmath50 , @xmath19 . the first condition gives that , without any change on the bandwidth requirement of jobs , the system will reject jobs . the second condition implies that the network could accommodate all jobs without losses ( with high probability ) if all of them would require the reduced bit rate @xmath16 and service rate @xmath17 . it should be noted that , from the point of view of the design of algorithms , the constant @xmath25 has to be defined . if one takes @xmath51 then , @xmath52 hold . if @xmath53 , it is not difficult to see that the system is equivalent to a classical underloaded loss network with one link and multiple classes of jobs . there is , of course , no need to use downgrading policies since the system can accommodate incoming requests without any loss when @xmath23 is large . see kelly @xcite or section 7 of chapter 6 of robert @xcite for example . in this section , we prove convergence results when the scaling parameter @xmath23 goes to infinity . these results are obtained by studying the asymptotic behavior of the occupation of the link around @xmath41 , @xmath54 in the context of loss networks , the analogue of such quantity is the number of empty places . the following proposition shows that , for the downgrading policy , the boundary @xmath40 does not play a role after some time if condition holds . [ propbound ] under condition and if the initial state is such that @xmath55 then , for @xmath56 , there exists @xmath57 such that , for @xmath58 , @xmath59 define @xmath60 where @xmath61 is the process of the number of jobs of an independent @xmath62 queue with @xmath63 , service rate @xmath17 and arrival rate @xmath49 and , for @xmath30 , @xmath64 where @xmath65 is a sequence of i.i.d . exponentially distributed random variables with rate @xmath21.the quantity @xmath66 is the number of initial class @xmath9 jobs still present at time @xmath46 . using theorem 6.13 of robert @xcite , one gets the convergence in distribution @xmath67 and , consequently , @xmath68 since @xmath69 for @xmath19 , @xmath70 by condition . note that the asymptotic occupancy , when @xmath23 is large , remains below the initial occupancy . if @xmath71 and @xmath72 such that @xmath73 , let @xmath74 then , on the event @xmath75 , the downgrading policy gives that the identity in distribution @xmath76 holds . condition gives the existence of @xmath77 such that @xmath78 convergence shows that the sequence @xmath79 converges in distribution to @xmath77 . note that , if @xmath80 , as long as the process @xmath81 stays above @xmath41 on @xmath82 , a relation similar to holds . by using again convergence , one gets that , as @xmath23 goes to infinity , the process @xmath83 remains below @xmath84 with probability close to @xmath15 on @xmath85 . the proposition is proved . we are now investigating the asymptotic behavior of the process @xmath86 defined by relation . the variable indicates if the network is operating in saturation at time @xmath46 , @xmath87 , or not , @xmath88 . in pure loss networks , when @xmath23 is large , up to a change of time scale , the analogue of this process , the process of the number of empty places converges to a reflected random walk in @xmath89 . in our case , as it will be seen , the corresponding process is in fact a random walk on @xmath90 . [ def1 ] for @xmath91 , let @xmath92 be the markov process on @xmath90 whose @xmath93-matrix @xmath94 is defined by , for @xmath95 and @xmath30 , @xmath96 with @xmath97 . the following proposition summarizes the stability properties of the markov process @xmath98 . [ propm ] if @xmath91 , then the markov process @xmath92 is ergodic if @xmath99 with @xmath100 @xmath101 denotes the corresponding invariant distribution . the markov process @xmath92 on @xmath90 behaves like a random walk on each of the two half - lines @xmath89 and @xmath102 . definition implies that if @xmath103 , then the drift of the random walk is positive when in @xmath102 and negative when in @xmath89 . this property implies the ergodicity of the markov process by using the lyapounov function @xmath104 , for example . see corollary 8.7 of robert @xcite for example . one now extends the expression @xmath101 for the values @xmath105 . this will be helpful to describe the asymptotic dynamic of the system . see theorem [ thlds ] further . [ def2 ] one denotes @xmath106 , the dirac measure at @xmath107 when @xmath108 , with @xmath109 and @xmath110 if @xmath111 , with @xmath112 for @xmath113 , denote by @xmath114 a poisson process on @xmath115 with rate @xmath116 and @xmath117 an i.i.d . sequence of such processes . all poisson processes are assumed to be independent . classically , the process @xmath118 can be seen as the unique solution to the following stochastic differential equations ( sde ) , @xmath119 with initial condition @xmath120 such that @xmath121 . [ thlds ] under condition , if the initial conditions are such that @xmath122 and @xmath123 then there exists continuous process @xmath124 such that the convergence in distribution @xmath125 holds for any function @xmath126 with finite support on @xmath90 . furthermore , there exists @xmath127 such that @xmath128 satisfies the differential equations @xmath129 where @xmath101 , for @xmath130 , is the distribution of proposition [ def1 ] and definition [ def2 ] . it should be noted that , since the convergence holds for the convergence in distribution of processes , the limit @xmath131 is a priori a _ random _ process . by using the same method as hunt and kurtz @xcite , one gets the analogue of theorem 3 of this reference . fix @xmath56 such that @xmath132 , from proposition [ propbound ] , one gets that the existence of @xmath133 such that @xmath134 which implies that the boundary condition @xmath135 in the evolution equations can be removed . consequently , only the boundary condition of @xmath86 at @xmath26 plays a role which gives relation as in hunt and kurtz @xcite . note that , contrary to the general situation described in this reference , we have indeed a convergence in distribution because , for any @xmath136 , @xmath92 has exactly one invariant distribution ( which may be a dirac mass at infinity ) by proposition [ propm ] . see conjecture 5 of hunt and kurtz @xcite . the following proposition gives a characterization of the equilibrium point of the dynamical system @xmath131 . [ fpprop ] under conditions and , there exists a unique equilibrium point @xmath137 of the process @xmath138 defined by equation given by @xmath139 where @xmath140 with @xmath141 . the process @xmath142 is ergodic in this case . assume that there exists an equilibrium point @xmath143 of @xmath138 defined by equation , it is also an equilibrium point of the dynamical system defined by equation , then @xmath144 with @xmath145 . one gets @xmath146 we now show that the vector @xmath147 is on the boundary , i.e. @xmath148 if we assume that @xmath149 from theorem [ thlds ] and the definition of @xmath86 , we know that , for the convergence of processes , the following relation holds @xmath150 for @xmath151 , @xmath152 and @xmath153 , @xmath154 by using again theorem [ thlds ] and the fact that @xmath147 is an equilibrium point of the dynamical system , we have , for the convergence in distribution @xmath155).\ ] ] the left - hand side of the above expression can be arbitrarily close to @xmath15 when @xmath156 is large . by convergence of the sequence @xmath157 to @xmath158 , one gets that , for the convergence in distribution , the relation @xmath159 holds for @xmath56 , which implies that @xmath160 . thus relation holds . finally , relations and give relation . one concludes therefore that @xmath161 , the associated process @xmath142 is necessarily ergodic by proposition [ propm ] and relations . to prove that the @xmath147 defined by relations and is indeed an equilibrium point of the dynamical system defined by equation , one has to show that the right - hand side of equation is indeed equal to @xmath162 . this is proved in proposition [ propfp2 ] of section [ secinv ] . in this section our main result establishes the convergence of the invariant distribution of the process @xmath86 as @xmath23 gets large . this will give in particular the convergence with respect to @xmath23 of the probability of not downgrading a request at equilibrium . [ lem22 ] if the process @xmath163 is the process @xmath164 at equilibrium then , for any @xmath56 and @xmath165 , @xmath166 let @xmath164 be the process with initial state empty , then one can easily construct a coupling such that the relation @xmath167 holds almost surely , where @xmath168 is the @xmath62 queue associated to class @xmath9 requests . one deduces that , @xmath169 where @xmath170 is a poisson random variable with parameter @xmath171 and @xmath172 is the stochastic ordering of random variables . one can therefore construct another coupling such that @xmath173 where @xmath174 is a stationary version of the @xmath62 queue associated to class @xmath9 requests . the lemma is then a consequence of the following convergence in distribution of processes , @xmath175 for @xmath176 , see theorem 6.13 pp . 159 of robert @xcite for example . [ defla ] let @xmath177 be the dynamical system on @xmath178 satisfying @xmath179 with @xmath180 [ lem3 ] if @xmath181 and if there exists an instant @xmath165 such that @xmath182 for @xmath183 $ ] then @xmath177 and @xmath184 coincide on the time interval @xmath185 $ ] , where @xmath131 is the solution of equations with @xmath186 . the proposition is a simple consequence of the representation of the differential equations defining the dynamical system @xmath131 and of the explicit expression of the quantity @xmath187 given by relation when @xmath103 , see relation . the next proposition investigates the stability properties of @xmath177 . let @xmath188 be the hyperplane @xmath189 if @xmath190 then @xmath191 for all @xmath44 and @xmath177 is converging exponentially fast to @xmath147 defined in proposition [ fpprop ] . it is easily checked that @xmath192 so that if @xmath193 , then the function @xmath194 is constant and equal to @xmath25 , hence @xmath195 for all @xmath196 . for @xmath176 , @xmath197 with @xmath198 in matrix form , if @xmath199 , it can be expressed as @xmath200 with @xmath201 and @xmath202 with @xmath203 if @xmath204 is an eigenvector for the eigenvalue @xmath205 of @xmath206 , then @xmath207 hence , @xmath205 is an eigenvalue if and only if it is a solution of the equation @xmath208 if @xmath209 is the number of distinct values of @xmath21 , @xmath210 , such that @xmath211 , then the above equation shows that an eigenvalue is a zero of a polynomial of degree at most @xmath209 . using conditions , it is easy to check that the relation @xmath212 holds . in particular @xmath26 is not an eigenvalue and , consequently @xmath206 is invertible . due to the poles of @xmath213 at the @xmath214 , @xmath210 and the relations @xmath212 and @xmath215 for @xmath210 , one has already @xmath209 negative solutions of the equation @xmath216 . all eigenvalues of @xmath206 are thus negative , consequently , @xmath217 converges to @xmath26 . ( see corollary 2 of chapter 25 of arnold @xcite for example . ) equation can be solved as @xmath218 therefore the function @xmath219 has a limit at infinity given by @xmath220 which is clearly @xmath221 . the proposition is proved . one can now prove the main result of this section . [ theoeq ] if @xmath147 is the quantity defined in proposition [ fpprop ] , then the equilibrium distribution of @xmath86 converges to @xmath222 when @xmath23 goes to infinity . recall that @xmath223 and let @xmath224 be the invariant distribution of @xmath118 . it is assumed that the distribution of @xmath225 is @xmath224 for the rest of the proof . in particular @xmath86 is a stationary process . one first proves that @xmath226 converges in distribution to @xmath147 . the boundary condition @xmath227 gives that the sequence of random variables @xmath226 is tight . if @xmath228 is a convergent subsequence to some random variable @xmath229 , by theorem [ thlds ] , one gets that , for the convergence in distribution , the relation @xmath230 holds , where @xmath131 is a solution of equation with initial point at @xmath231 . note that @xmath131 is a stationary process , its distribution is invariant under any time shift . by lemma [ lem22 ] one has that the relation @xmath232 , for @xmath176 , holds almost surely on any finite time interval and , by proposition [ propbound ] , @xmath233 also holds almost surely on finite time intervals . assume that @xmath234 holds . the odes defining the limiting dynamical system are given by @xmath235 as long as the condition @xmath236 holds , hence on the corresponding time interval , one has @xmath237 so that @xmath238 since @xmath239 , there exists some @xmath240 such that @xmath241 . hence , by stationarity in distribution of @xmath131 , one can shift time at @xmath133 and assume that @xmath242 . on this event @xmath243 similarly , since @xmath244 for all @xmath176 , @xmath245 and the last quantity is independent of @xmath246 . relations and show that @xmath247 and , by equations and , they also hold for @xmath46 in a small neighborhood @xmath85 of @xmath26 independent of @xmath246 so that @xmath248 for @xmath249 . consequently , the dynamical system @xmath131 never leaves @xmath250 . lemma [ lem3 ] shows that the two dynamical systems @xmath131 and @xmath177 ( with @xmath251 ) coincide . hence , on one hand @xmath131 is a stationary process and , on the other hand , it is a dynamical system converging to @xmath147 , one deduces that it is constant and equal to @xmath147 . we have thus proved that the sequence @xmath226 converges in distribution to @xmath147 . using again theorem [ thlds ] , one gets that , for the convergence in distribution , @xmath252 holds for any function @xmath126 with finite support on @xmath90 . by using the stationarity of @xmath86 and lebesgue s theorem , one obtains @xmath253 the theorem is proved . since a job arriving at time @xmath46 is not downgraded if @xmath254 , one obtains the following corollary . [ corol ] as @xmath23 goes to infinity , the probability that , at equilibrium , a job is not downgraded in this allocation scheme is converging to @xmath255 defined in proposition [ fp ] , @xmath256 we assume in this section that @xmath99 , as defined in proposition [ propm ] , so that @xmath92 is an ergodic markov process . the goal of this section is to derive an explicit expression of the invariant distribution @xmath101 on @xmath90 of @xmath92 . at the same time , proposition [ propfp2 ] below gives the required argument to complete the proof of proposition [ fpprop ] on the characterization of the fixed point of the dynamical system . in the following we denote by @xmath257 a random variable with distribution @xmath258 . for @xmath259 , we will use the notation @xmath260 for sake of simplicity , we will use @xmath261 and @xmath262 . with the notation @xmath263 the random variable @xmath257 is such that @xmath264 where @xmath265 and @xmath266 are polynomials defined by @xmath267,\\ \displaystyle p_2(z ) = \sum\limits_{j=1}^j \left[\rule{0mm}{4 mm } \lambda_j z^{{a_j } + a_j } + \mu_j \ell_j z^{{a_j}-a_j } -(\lambda_j + \mu_j \ell_j ) z^{a_j } \right ] . \end{cases}\ ] ] for @xmath268 define @xmath269 such that @xmath270 , for @xmath271 . equilibrium equations for @xmath92 give the identity @xmath272 where @xmath94 is the @xmath93-matrix of @xmath92 given by equation . after some simple reordering , one gets the relation @xmath273 by using the definition of @xmath274 and @xmath275 , equation can be rewritten as equation . [ propfp2 ] if @xmath103 then @xmath276 in particular if @xmath277 is given by relation then @xmath278 note that the right - hand side of the last relation is precisely @xmath255 of relation which is the result necessary to complete the proof of proposition [ fpprop ] . with the same notations as before , from relation , @xmath279 holds for @xmath280 , with @xmath281 . by definition of @xmath282 and @xmath283 , @xmath284 since @xmath15 is a zero of @xmath265 and @xmath266 , this gives the relation @xmath285 using the expression of @xmath286 , with some algebra , one gets @xmath287 the proposition is proved . relation is valid on the unit circle , however the function @xmath288 ( resp . @xmath289 ) is defined on @xmath290 ( resp . @xmath291 ) . this can then be expressed as a wiener - hopf factorization problem analogous to the one used in the analysis of reflected random walks on @xmath89 . this is used in the analysis of the @xmath292 queue , see chapter viii of asmussen @xcite or chapter 3 of robert @xcite for example . in a functional context , this is a special case of a riemann s problem , see gakhov @xcite . in our case , this is a random walk in @xmath90 , with a drift depending on the half - space where it is located . the first ( resp . second ) condition in the definition of the set @xmath250 in definition implies that the drift of the random walk in @xmath293 ( resp . in @xmath89 ) is positive ( resp . negative ) . the first step in the analysis of equation is to determine the locations of the zeros of @xmath265 and @xmath266 . this is the purpose of the following lemma . [ lemma1][lemma2](location of the zeros of @xmath265 and @xmath266 ) let @xmath38 be in @xmath250 . 1 . polynomial @xmath266 has exactly two positive real roots @xmath15 and @xmath2940,1[$ ] . there are @xmath295 roots in @xmath296 and @xmath295 roots whose modulus are strictly greater than @xmath15 . polynomial @xmath265 has exactly two positive real roots @xmath15 and @xmath297 . the @xmath295 remaining roots have a modulus strictly smaller than @xmath15 . one first notes that @xmath266 is a polynomial with the same form as the @xmath126 defined by equation ( 13 ) in bean et al . @xcite ( with @xmath298 , @xmath299 and @xmath300 ) . the roots of @xmath93 are exactly the roots of @xmath126 . lemma 2.2 of bean et al . @xcite gives assertion ( _ i _ ) of our lemma . the proof of assertion _ ( ii ) _ uses an adaptation of the argument for the proof of lemma 2.2 of bean et al . define the function @xmath301 . recall that @xmath265 is a polynomial with degree @xmath302 . there are exactly two real positive roots for @xmath265 . indeed , @xmath303 and it is easily checked that @xmath126 is strictly concave with @xmath304 since @xmath99 , by the second condition in definition . hence @xmath265 has a real zero @xmath305 greater than @xmath15 . let @xmath306 be fixed , note that @xmath307 . define @xmath308 so that @xmath309 . fix some @xmath310 . by expressing these functions in terms of real and imaginary parts , @xmath311 one gets @xmath312 with @xmath313 cauchy - schwarz s inequality gives the relation @xmath314 since @xmath315 for @xmath316 , @xmath317 . thus , @xmath318 since @xmath319 , @xmath320 can be chosen so that @xmath321 . from the above relation and equation , one gets that for @xmath322 , the relation @xmath323 holds . by rouch s theorem , one obtains that , for any @xmath324 , @xmath265 has exactly @xmath325 roots in @xmath326 . one concludes that @xmath265 has exactly @xmath325 roots in @xmath327 . it is easily checked that if @xmath281 and @xmath328 then the real part of @xmath329 is positive , hence @xmath330 can not be a root of the polynomial @xmath265 . consequently , @xmath265 has exactly @xmath295 roots in @xmath290 . the lemma is proved . [ def2 ] for @xmath331 , denote by @xmath332 the set of the zeros of @xmath333 different from @xmath15 . define @xmath334 with @xmath49 and the same notations as before . by definition , function @xmath335 is holomorphic in @xmath290 and @xmath291 and , from relation , is continuous on @xmath336 . the analytic continuation theorem , theorem 16.8 of rudin @xcite for example , gives that @xmath335 is holomorphic on @xmath337 . for @xmath338 , @xmath339 since the cardinality of @xmath340 ( resp . @xmath341 ) is @xmath342 ( resp . @xmath343 ) , the holomorphic function @xmath335 is therefore bounded on @xmath337 . by liouville s theorem , @xmath335 is constant , equal to @xmath344 . therefore @xmath345 recall that @xmath346 is a generating function , in particular @xmath347 . plugging the previous expressions for @xmath288 and @xmath289 in @xmath348 , one gets the relation @xmath349 hence , using equation , @xmath350 where @xmath351 is introduced in definition [ defla ] . note that @xmath352 is positive . we can now state the main result of this section . [ invariant ] if @xmath99 defined by relation , then the invariant measure @xmath353 can be expressed , for @xmath354 , as @xmath355 where @xmath305 is defined in lemma [ lemma1 ] , and @xmath265 and @xmath266 by relation , @xmath356 for @xmath357 , @xmath358 is the coefficient of degree @xmath359 of the polynomial @xmath360 note that , for @xmath361 , @xmath362 for @xmath338 , @xmath363 since @xmath364 by lemma [ lemma1 ] , @xmath289 has the following partial fraction decomposition @xmath365 denote @xmath366 then @xmath367 one concludes by using the expression of @xmath352 obtained before . using the probability generating function @xmath369 of @xmath370 from equation , one can derive an explicit expression of the mean , the variance and the skewness of such distribution . the skewness of a random variable @xmath371 is a measure of the asymmetry of the distribution of @xmath371 , @xmath372 ^ 3)}\ ] ] see doane and seward @xcite for example . [ mean_variance_skew ] if @xmath373 is a random variable with distribution @xmath370 then @xmath374 where , for @xmath375 , @xmath376 and @xmath377 with @xmath378 defined in proposition [ invariant ] . the proof is straightforward , modulo some tedious calculations of the successive derivatives of @xmath369 evaluated at @xmath15 . figure [ figdist ] shows that the distribution of @xmath379 is significantly asymmetrical . for this example @xmath380 , @xmath381 and @xmath382 . with the parameters @xmath383 , @xmath384 , @xmath385 , @xmath386 and @xmath387 . in this case , a request which can not be accommodated is rejected right away . recall that , with probability @xmath15 , our algorithm does not reject any request . the purpose of this section is to discuss the price of such a policy . intuitively , at equilibrium the probability @xmath388 of accepting a job at requested capacity in a pure loss system is greater that the corresponding quantity @xmath389 for the downgrading algorithm . see proposition [ propcmp ] below . a further question is to assess the impact of such policy , i.e. the order of magnitude of the difference @xmath390 . under the same assumptions about the arrivals and under the condition @xmath391 with @xmath392 , then , as @xmath23 gets large , the equilibrium probability that a request of class @xmath19 is accepted in the pure loss system is converging to @xmath393 , where @xmath394 is the unique solution of the equation @xmath395 see kelly @xcite . consequently , the asymptotic load of accepted requests is given by @xmath396 under the downgrading policy , the equilibrium probability that a job is accepted without degradation is given by @xmath255 , the asymptotic load of requests accepted without degradation is @xmath397 for @xmath398 . note that , when the service rates are constant equal to @xmath15 , then @xmath388 ( resp . @xmath389 ) is the asymptotic throughput of accepted requests ( resp . of non - degraded requests ) . the representation of these quantities gives that the relation to prove is equivalent to the inequality @xmath400 by using the fact that @xmath401 and equation , it is enough to show that the quantity @xmath402 is positive . but this is clear since @xmath403 and the terms of both series of the right hand side of this relation are non - negative due to the fact that @xmath404 . numerical experiments have been done to estimate the difference @xmath390 , see figure [ figccmp ] . the general conclusion is that , at moderate load under condition ( @xmath405 ) , the downgrading algorithm performs quite well with only a small fraction of downgraded jobs . as it can be seen this is not anymore true for high load where , as expected , most of requests are downgraded but nobody is lost . we consider now a link with large bandwidth , @xmath406 gbps , in charge of video streaming . requests that can not be immediately served are lost . video transmission is offered in two standard qualities , namely , _ low quality _ ( lq ) and _ high quality _ ( hq ) . from aorga _ et al . _ @xcite , the bandwidth requirement for youtube s videos at 240p is 1485 kbps , and for 720p it is 2737.27 kbps . using the values above , after renormalization , one takes @xmath16 , @xmath407 and @xmath408 , @xmath409 . jobs arrive at rate @xmath410 in this system asking for hq transmission , but clients accept to watch the video in lq . in particular @xmath411 . service times are assumed to be the same for both qualities and taken as the unity , @xmath412 . condition is satisfied when @xmath413 we define @xmath414 , with @xmath415 . the quantity @xmath416 is defined as the largest value of @xmath417 such that the loss probability of a job is less than @xmath56 . with the notations of section [ secinv ] , we write @xmath418 note that this is an approximation , since the variable @xmath379 corresponds to the case when the scaling parameter @xmath23 goes to infinity . by using the explicit expression of the distribution of @xmath379 of proposition [ invariant ] , figure [ fig_alpha ] plots the threshold @xmath419 that ensures a loss rate less than @xmath420 as a function of @xmath420 , for several values of @xmath410 . in the numerical example , taking @xmath421 is sufficient to get a loss probability less than @xmath422 . now let @xmath423 be the value of @xmath255 defined by corollary [ corol ] for @xmath424 . recall that @xmath425 is the asymptotic equilibrium probability that a job is not downgraded is given by relation , @xmath426 for comparison , @xmath427 is defined as the corresponding acceptance probability when no control is used in the system . we show in figure [ fig_down ] the relation between these quantities and the workload @xmath410 , for fixed loss rates of @xmath428 , @xmath429 and @xmath430 . we have @xmath431 , see robert ( * ? ? ? * proposition 6.19 ) . the difference @xmath432 can be seen as the fraction of jobs which are downgraded for our policy but lost in the uncontrolled policy . intuitively it can be seen as the price of not rejecting any job . notice also that the curves plotting @xmath423 for @xmath433 , @xmath429 , @xmath430 are close and that @xmath427 is larger than @xmath255 . one remarks nevertheless that , for high loads , the system can not hold these demands , because our policy is no longer effective . javier aorga , saioa arrizabalaga , beatriz sedano , maykel alonso - arce , and jaizki mendizabal . s dash implementation analysis . in _ proceedings of the 19th international conference on communications _ , volume 50 of _ recent advances in electrical engineering series _ , pages 6166 , zakynthos , ionion , greece , 06 2015 . fabrice guillemin , thierry houdoin , and stphanie moteau . volatility of youtube content in orange networks and consequences . in _ proceedings of ieee international conference on communications , icc 2013 , budapest , hungary , june 9 - 13 , 2013 _ , pages 23812385 , 2013 . fabrice guillemin , bruno kauffmann , stphanie moteau , and alain simonian . experimental analysis of caching efficiency for youtube traffic in an isp network . in _ 25th international teletraffic congress , itc 2013 , shanghai , china , september 10 - 12 , 2013 _ , pages 19 , 2013 . christian sieber , tobias hofeld , thomas zinner , phuoc tran - gia , and christian timmerer . implementation and user - centric comparison of a novel adaptation logic for dash with svc . in _ im13 _ , pages 13181323 , 2013 . stan zachary and ilze ziedins . loss networks . in richard j. boucherie and nico m. van dijk , editors , _ queueing networks _ , volume 154 of _ international series in operations research & management science _ , pages 701728 . springer us , 2011 .

it is well known that radio waves propagating in the interstellar medium ( ism ) are scattered by the irregularities in the galactic electron density . the scattering in turn gives rise to a number of observable phenomena . among others , these include angular broadening and intensity fluctuations ( both in time and frequency ) of compact radio sources . while a nuisance in many radio astronomical observations , these phenomena can be used to investigate the nature of the irregularities in the interstellar plasma density . these density irregularities are in turn believed to follow the fluctuations in the interstellar kinetic and magnetic energies . ideally one would like to invert observations of radio scintillation and scattering to determine the statistics of the plasma density . as noted by narayan ( 1992 ) this inverse problem is not well posed and one must rely on modeling methods . a complete prediction of scintillation observables requires an _ a priori _ knowledge of both the form of the spatial power spectrum of the electron density fluctuations and the distribution of the scattering material along the line of sight . for a given profile of the distribution of the scattering material , one may compare observations and predictions and so constrain the functional form of the spectrum . the power spectrum provides useful insight into the physics of the plasma irregularities . hence a knowledge of the form of the density spectrum becomes central in both predicting scintillation phenomena and understanding the physics of the interstellar plasma . in this paper , we revisit the investigation of the form of the density spectrum . a commonly used model for the density spectrum has been based on a power - law model with a large range between `` inner '' and `` outer '' scales ( e.g. rickett 1977 ) : @xmath1 \ , \mbox{. } \label{eq : extpowerlaw } \ ] ] here @xmath2 is the magnitude of the three - dimensional wavenumber @xmath3 . @xmath4 denotes the strength of fluctuations ( with a weak dependence on distance @xmath5 ) . @xmath6 is the spectral exponent , and @xmath7 and @xmath8 are the inner and outer scales respectively . for @xmath9 , we obtain the _ simple _ power - law model : @xmath10 . armstrong , rickett , & spangler ( 1995 ) have constructed an empirical density spectrum by combining radio scintillation observations in the local ism ( @xmath11 1 kpc ) with measurements of the differential faraday rotation angles and large - scale electron density gradients . they have shown the power spectrum to be consistent with a simple kolmogorov power - law model ( @xmath12 ) over an astronomical 10 orders of magnitude in wavenumber scale ( @xmath13m@xmath14m@xmath15 ) . the kolmogorov spectrum in density suggests a turbulent cascade in the magnetic and kinetic energies . this has lead to several theoretical investigations of the generation and maintenance of hydromagnetic turbulence in the ism ( e.g. pouquet , 1978 ; higdon 1984 & 1986 ; biskamp , 1993 : sridhar & goldreich 1994 ; goldreich & sridhar 1995 and 1997 ) . however , the armstrong et al . study combined observations from many lines of sight and the scatter among them leaves a substantial uncertainty in the exponent @xmath6 . a list of symbols is given in table 3 . in spite of the positive evidence for the simple kolmogorov spectrum , substantial observational inconsistencies remain . for instance , long - term refractive intensity scintillations of some pulsars have modulation indices as much as a factor of 2 larger than predicted by the simple kolmogorov model ( cf.gupta , rickett , & coles 1993 ) . other discrepancies are revealed in the diffractive dynamic spectra of pulsars . on some occasions , periodic fringes are observed , which are not predicted by the simple kolmogorov model ( cf . roberts & ables 1982 ; cordes & wolszczan 1986 ; rickett , lyne , & gupta 1997 ; gupta , bhat & rao , 1999 ) ; in addition , sloping bands in the dynamic spectra often persist longer than predicted by the model ( cf.gupta , rickett , & lyne 1994 ; bhat , rao & gupta , 1999b bhat , gupta & rao , 1999c ) . further for some pulsars , the decorrelation bandwidth has larger amplitude variations than predicted for the kolmogorov spectrum ( bhat , gupta & rao , 1999c ) . such anomalies suggest the presence of large refractive structures giving rise to the focusing and defocusing of the scattered ray bundles . the interference of the ray bundles can also explain the occasional fringes observed in the dynamic spectra of some pulsars . given the relatively frequent occurrence of such events , one can ask whether they can be considered as mere occasional anomalies or should be considered as a widespread phenomenon intrinsic to the spectrum on a grand scale . the observational inconsistencies suggest the need for an enhancement in the power on the large `` refractive '' ( 10@xmath16 m to 10@xmath17 m ) spatial scales relative to the power on the small `` diffractive '' ( 10@xmath18 m to 10@xmath19 m ) scales . there are several means by which this ratio may be enhanced . one is to include the inner scale cut - off in the density spectrum , which reduces the power at small scales ( coles et al . 1987 ) ; these authors proposed inner scales of @xmath20 m. , though this does not correspond to any obvious physical scale . physically , the inner scale corresponds to the scale at which the turbulent cascade dissipates and becomes a source of heating for the plasma ( spangler , 1991 ) . the value of the inner scale is largely unknown . using different methods , spangler & gwinn ( 1990 ) , kaspi & stinebring ( 1992 ) , and gupta et al . ( 1993 ) have reported values for the inner scale ranging from @xmath21 to @xmath22 meters . in proposing the smaller values , spangler & gwinn ( 1990 ) argued that the inner scale is the larger of the ion inertial length , @xmath23 ( where @xmath24 is the alfvn speed , and @xmath25 is the ion cyclotron frequency ) , and the ion larmor radius , @xmath26 ( where @xmath27 is the ion thermal speed ) ; they obtained parameters for the warm ionized medium in reasonable agreement with observations . in a recent discussion , minter & spangler ( 1997 ) have suggested ion - neutral collisional damping and wave - packet steepening as possible dissipation mechanisms for the turbulence in the diffuse ionized gas , which would make the mean - free path for ion - neutral collisions a possible value for the inner scale . however , this is thought to be larger than the maximum values proposed to explain the observations , making it a less convincing dissipation mechanism . observationally , the kolmogorov model with a large inner scale predicts refractive modulation indices consistent with pulsar measurements ( gupta et al . it has also been proposed to explain the occasional periodic fringes , with an inner scale on the order of the fresnel scale ( cordes , pidwerbetsky , & lovelace 1986 ; goodman et al . . however , rickett et al . ( 1997 ) reported a fringe event for the pulsar b0834 + 06 that could not be explained as the effect of a large inner - scale spectrum . the event requires similar conditions to those needed to explain the extreme scattering events ( fiedler et al . 1987 ) . another way to enhance the ratio of the power between the refractive and diffractive scales in the spectrum is to steepen the spectrum with spectral exponents @xmath28 ( blandford & narayan 1985 ; goodman & narayan 1985 ; romani , narayan , & blandford 1986 ) . while power - law spectra with @xmath29 11/3 have a turbulence connotation , spectra with @xmath30 might involve some forms of turbulence , but are also consistent with a distribution of non - turbulent structures with a range of spatial scales . such steep spectra with @xmath31 predict refractive modulation indices close to unity ( goodman & narayan 1985 ) , which is substantially larger than the range 30 to 40 % observed from the nearby pulsars . on this basis rickett & lyne ( 1990 ) and armstrong et al . ( 1995 ) have rejected spectra steeper than 4 for the interstellar plasma . however , the special case of @xmath32 has been given little attention . we can conceive of a power - law model with spectral exponent @xmath32 given by : @xmath33 hereafter , we refer to this model as the `` @xmath32 model . '' blandford & narayan ( 1985 ) briefly discussed this special case without including a cut - off at low wavenumbers . it is interesting to note that even though the kolmogorov spectral exponent @xmath12 is very close to 4 , the @xmath32 model has a very different physical implication . its physical origin has been rarely discussed , and it has not been formally compared with observations . physically , this spectrum suggests the random distribution in location and orientation of discrete discontinuous objects across the line of sight . an `` outer '' scale , @xmath34 , is included to account for the typical size of such objects . the `` inner '' scale here would correspond to the scale of the sharpness of a typical discontinuity . we assume the `` inner '' scale to be smaller than the diffractive scale of scintillations , and hence has no significant effect on the scintillations . the @xmath32 model could characterize stellar wind boundaries , supernova shock fronts , sharp boundaries of hii regions at the strmgren radius , or any plasma `` cloud '' with sufficiently sharp boundaries ( transition regions shorter than the diffractive scintillation scale ) which may cross the line of sight . note that , though turbulence is not necessarily implied by the model , strong turbulence which has steepened to form shocks would also be described by the @xmath32 model . in these models a single discrete object crossing the line of sight could also explain the `` extreme scattering events '' observed in the flux density variations of extra - galactic sources ( e.g. fiedler et al . we note that the analysis of such events have been primarily based on geometrical optics involving a single `` cloud '' ( goodman et al . 1987 ; cordes et al . 1986 ; cordes & wolszczan 1986 ; roberts & ables 1982 ; ewing et al . 1970 ) . our analysis of the @xmath32 model includes the `` wave optics '' effects when the line of sight passes through very many such clouds . the ism is assumed to consist of a random assembly of discrete structures with abrupt density steps which may be independent of each other . if the @xmath32 model were compatible with all of the scintillation observations , it would remove the implication of interstellar plasma turbulence with an inertial range spanning as much as 10 orders of magnitude in scale , which has become the canonical model for iss phenomena . it would , however , still be consistent with turbulence that has steepened into shocks . we start with a simple derivation of the @xmath32 model in section [ sec : deriv ] . in section [ sec : structfn ] , we give equations for the wave structure function for the @xmath32 model and compare predictions in section [ sec : decbw ] , the variation of the diffractive decorrelation bandwidth with frequency is used to test the @xmath32 model against the simple kolmogorov model and the kolmogorov model with an inner - scale ( hereafter , the `` inner - scale '' model ) . in section [ sec : mr ] , the observed variation of the refractive scintillation index with the normalized diffractive decorrelation bandwidth is compared with theoretical predictions for the simple kolmogorov , inner - scale , and @xmath32 models . we use theoretical results from a previous paper ( lambert & rickett 1999 ; hereafter , lr99 ) , in which we developed the theory of diffractive scintillations in a medium modeled by these spectra , and we add details of the theory for refractive scintillation in appendix a. in appendix b , we discuss the use of the square of the second moment to approximate the intensity correlation function for diffractive scintillation with @xmath32 . in section [ sec : conclusion ] we give a discussion and our conclusions . in this section , we give a simple derivation of the @xmath32 model . consider first a random distribution in space of identical plasma structures ( blobs ) . the resulting electron density may be described as the convolution of a three - dimensional poisson point process with the density profile of one individual blob . this is the spatial analog of shot noise , in which each charge carrier contributes the same temporal profile of current arriving at randomly distributed times . in the wavenumber domain , the power spectrum of the electron density is then given by the spectrum of the poisson point process multiplied by the squared magnitude of the fourier transform of one blob . the spectrum of the poisson point process is a constant equal to the number density , @xmath35 , of the blobs in space ( papoulis 1991 ) . thus the power spectrum of the medium follows the same shape as that of an individual blob . a similar description applies to water droplets in a fog and many naturally occurring media , as discussed by ratcliffe ( 1956 ) . if the blobs are asymmetrical and randomly oriented , one must also average the spectrum over the possible angles of orientation . the basic feature of structures with discontinuous boundaries is that their power spectrum has an asymptotic behavior at large wavenumbers that varies as ( wavenumber)@xmath36 . this is the spatial analog of the idea that the temporal spectrum of any pulse shape with an abrupt rise or fall has a high - frequency asymptote as ( frequency)@xmath37 . the prime example is a rectangular pulse , for which the spectrum is a sinc - squared function , whose high - frequency envelope follows this law . the simplest spatial example is spherical blobs of radius @xmath38 with uniform density @xmath39 inside and zero outside ; clearly , an added uniform background density does not affect the power spectrum . the three - dimensional spatial fourier transform of an _ isotropic function @xmath40 of radial distance @xmath41 is given by ( tatarskii 1961 ) : @xmath42 where @xmath2 is the magnitude of the wavenumber . for any function @xmath40 that falls to zero beyond a radius @xmath38 , the integral will vary as @xmath43 in the high - wavenumber limit @xmath44 , making @xmath45 . for the spherical blobs of size @xmath38 we have : @xmath46 for the squared magnitude of the fourier transform , for very small and large values of @xmath47 : @xmath48 { \displaystyle q^{-4 } \ ; a^2 f_0 ^ 2 /(8\pi^4 ) \ , \mbox { , } } & q a > > 1 \ , \mbox{. } \\ \end{array } \right . \label{eq : magsqspherefourier } \ ] ] the spectrum of the electron density is then @xmath49 . we can approximate this by equation ( [ eq : beta4 ] ) , which exactly matches equation ( [ eq : magsqspherefourier ] ) at small and large values of @xmath50 , with @xmath51 and @xmath52 . _ there are evident generalizations to make the model more like a real medium . the spheres could have a probability distribution for their radii , which would weight the average of the square of equation ( [ eq : spherefourier ] ) . the spheres could have smoothly varying density inside and an abrupt boundary at radius @xmath38 . as with the pulse example , the high - wavenumber behavior would not change . more realistic models would include anisotropic structures with random orientations and a distribution of scales . for example , consider a circular cylinder as a model for an anisotropic blob , with a particular radius and thickness . its fourier transform can be computed with respect to its major axes . this can then be transformed to tilted axes and the result averaged over an isotropic distribution of angles of tilt . evidently the result will also be isotropic ; we have solved this approximately and find a spectrum with the same high - wavenumber _ asymptotic form as equation ( [ eq : beta4 ] ) , where the `` outer scale , '' @xmath53 , is the smaller of the thickness and radius . this result could be averaged over a probability distribution of radii and thicknesses and a similar conclusion reached , with @xmath53 as a weighted average of the smallest dimension . planar sheets at random angles are modeled as very flat cylinders and provide a crude representation of a distribution of shocks . another simplified model would be ellipsoids of different sizes , eccentricities , and orientations , and again we expect a similar result . _ we conclude that the @xmath32 spectrum is an approximate description of the power spectrum for a medium with a random distribution of discontinuous structures , in the limit for wavenumbers greater than the reciprocal of the smallest dimension across the structures . we note that for wavenumbers at or below about @xmath54 the functional form depends on the detailed shapes , though it will be much less steep than @xmath55 . thus equation ( [ eq : beta4 ] ) will not be generally applicable at the lowest wavenumbers . similarly , in practice there is a finite scale over which the density jump occurs ( e.g. shock thickness ) . this can be modeled by convolving the idealized discontinuous structures with a suitable narrow , say , gaussian function which thus provides a very high - wavenumber cut - off beyond which the spectrum falls rapidly to zero . consequently , we consider a spectrum with an inverse fourth power law between inner and outer scales , though these do not have the same connotations as inner and outer scales in a turbulent cascade . we do not discuss the physical origin of the supposed discontinuous plasma structures beyond the ideas formulated in the introduction , such as supernova shock fronts , stellar wind boundaries or sharp boundaries of hii regions at the strmgren radius . there needs to be no physical coupling between the individual blob structures , and there is no turbulent cascade implied by the @xmath32 spectrum . however , we note that shocks developing from the steepening of very strong turbulence could also be described by the model , and so if successful , the model would not rule out such strong turbulence , though in this case the @xmath32 range does not correspond to an inertial cascade as does the 11/3 spectrum . in the following sections , we give the necessary scintillation theory needed to develop equations for the scintillation observables corresponding to the @xmath32 model , which we then compare with iss observations . we concentrate on measurements which are sensitive to the form of the density spectrum and are relatively insensitive to the distribution of scattering material along the line of sight . central to the description of the second moment and the coherence function for intensity is the phase structure function . the longitudinal gradient of the phase structure function is related to the electron density spectrum through ( cf . coles et al . 1987 ) : @xmath56 \kappa d\kappa \ , \mbox{. } \label{eq : long_grad_phase_strucfunc } \ ] ] here @xmath57 is the speed of light , @xmath58 is the classical electron radius , and @xmath59 is the radio frequency . for a plane wave incident on a scattering medium , the line of sight integral of @xmath60 gives the structure function of the geometric optics phase , also called the _ wave _ structure function , @xmath61 . for the spherical wave geometry applicable to a pulsar at distance @xmath62 , the separation @xmath63 at the observer s plane is projected back to the point of integration , giving @xmath64 . the integrals are evaluated explicitly in equations ( 13 ) , ( 17 ) , and ( 21 ) of lr99 for a uniform scattering medium with the simple power - law , inner - scale , and @xmath32 spectral models , respectively . in the extended scattering geometry , the field coherence length at the observer , @xmath65 , is defined by : @xmath66 . for a screen of thickness @xmath67 at distance @xmath68 from the pulsar , the wave structure function at the observer becomes @xmath69 , which has an explicit dependence on the screen location . in lr99 , we found it useful to define a coherence scale @xmath70 for the screen by its phase structure function @xmath71 , which is independent of the screen location . note that in applying equation ( [ eq : long_grad_phase_strucfunc ] ) , we are assuming that the screen thickness , @xmath67 , is much greater than the outer scale @xmath53 and small compared to total distance @xmath62 . here , we repeat the equation for the phase structure function of the @xmath32 model in a thin layer ( screen ) . substituting the @xmath32 model for the density spectrum into equation ( [ eq : long_grad_phase_strucfunc ] ) , integrating , and multiplying by the screen thickness , @xmath67 , we obtain : @xmath72 \ , \mbox { , } \label{eq : struc4exact } \ ] ] where @xmath73 is the first order modified bessel function of the second kind , and @xmath74 is the scattering measure . equation ( [ eq : struc4exact ] ) may be approximated by the following logarithmic expression : @xmath75 \ , \mbox { , } \label{eq : struc4 } \ ] ] which matches the full bessel expression at both large and small values of the argument . the detailed shape , near where the structure function flattens , is governed by the shape of the low - frequency turnover in our model spectrum , equation ( [ eq : beta4 ] ) ; however , as noted above , the turnover shape depends on details of the density profiles in the plasma blobs which are not constrained in our density model . the logarithmic structure function for the @xmath32 model has far - reaching consequences . as the outer scale becomes larger , the structure function approaches a square - law . however , due to the presence of the logarithm , this occurs only slowly ; that is , only at the limit as @xmath63 goes to zero does the @xmath32 model structure function become exactly square - law . a square - law structure function for the medium is a convenient mathematical model , but as noted in lr99 , in a real medium the structure function must eventually saturate at very large separations . nevertheless , the square - law structure function has been widely used since it provides a valid approximation for scales smaller than the reciprocal of the wavenumber at which the spectrum is cut - off . the fact that the @xmath32 model structure function deviates from a pure square - law , even for very large values of the outer scale , makes this model interesting and an independent investigation worthwhile . in figure [ fig : slope ] , we have plotted the structure function of the @xmath32 model and its effective local logarithmic slope , @xmath76 , versus the transverse spatial lag , @xmath63 , for various values of the outer scale , @xmath53 ( see equation 20 of lr99 ) . to illustrate the shapes , we have chosen @xmath77 km for all curves . the thin dashed lines correspond to the simple kolmogorov model with logarithmic slope of 5/3 , and the thin dotted lines correspond to the square - law structure function . the plot shows how slowly the local slope approaches 2 ; even for @xmath78 , the slope is 1.95distinctly less than 2 . the points in the upper panel correspond to measured local exponent of the wave structure function obtained from vlbi observations , as tabulated by spangler & gwinn ( 1990 ) . in principle , this method of comparison provides a good test for the form of the density spectrum . these authors presented a similar plot corresponding to the inner - scale model , for which @xmath76 is close to 2.0 for @xmath63 less than the inner scale and close to 5/3 for @xmath63 greater than the inner scale , with a transition occurring over about a decade in @xmath63 . they derived estimates of the inner scale ranging from 50,000 to 200,000 meters , for the highly scattered sources that they studied . whereas there is some uncertainty in some of the estimates of @xmath79 , there are several examples in which values greater than 5/3 are reliably observed ( e.g. trotter et al . , 1998 ) . as can be seen , there is also reasonable agreement with the @xmath32 model . assuming the model to be correct , a separate model - fitting to each observation yields an outer scale in the range @xmath80 to @xmath81 meters . these values are much smaller than the parsec scales of supernova remnants and interstellar clouds . however based on this comparison alone , we are unable to discriminate between the inner - scale and @xmath32 models since they both provide equally satisfactory agreement with the observations . hence further independent tests are needed , and in the following sections , we present comparisons with two other scintillation observables which in the end argue against the @xmath32 model as a universal description of the ionized ism . the diffractive decorrelation bandwidth , @xmath82 , is perhaps the easiest iss observable to measure . it is the frequency - difference for a decorrelation to 50% of the correlation function for diffractive intensity scintillations . these are usually recorded as a dynamic spectrum centered on a given radio frequency , @xmath59 , for a pulsar . observations of @xmath82 for one pulsar at a wide range of radio frequencies have provided an important test of power - law models for the interstellar density spectrum on that particular line of sight . when the diffractive scale is far from the inner and outer scales , one expects @xmath83 , which for the simple kolmogorov model is @xmath84 , in both screen and extended medium geometries . in this section we re - examine the published measurements and compare them with theory for the three different spectral models . since the scaling laws are all so steep , in figures [ fig : vela ] and [ fig : other ] , we have plotted the observations logarithmically as @xmath85 . overplotted are lines giving the theoretical predictions for our three spectral models , with the simple kolmogorov model as a line of slope 0.4 . before giving our conclusions , we must discuss the error bars and various corrections to the data and also the method for deriving the theoretical predictions . decorrelation bandwidth data , collected together by cordes , weisberg , & boriakoff ( 1985 ; hereafter , cwb ) , for pulsars psr b0833@xmath8645 ( vela ) , psr b0329@xmath8754 , psr b1642@xmath8803 , psr b1749@xmath8628 , and psr b1933@xmath8716 are shown in figures [ fig : vela ] and [ fig : other ] . we have also included recently measured points by johnston et al . ( 1998 ) for the vela pulsar . these are @xmath89 and @xmath90 . the data are shown as solid circles if derived from dynamic spectra and as solid stars if derived from pulse broadening measurements . the latter become easier to estimate at low frequencies , where the resolution bandwidth required for the former becomes too narrow for an adequate signal - to - noise ratio . the conversion of an estimate of a pulse broadening time to a decorrelation bandwidth relies on the uncertainty relation @xmath91 . unfortunately , the `` constant '' @xmath92 takes on different values for different geometries and spectral models . for the @xmath32 model , it is also very weakly dependent on frequency . in table 1 of lr99 , we gave numerical values for @xmath92 for the three spectral models with spherical waves in both a screen and an extended medium geometry . for a given geometry , the smallest @xmath92 is for the kolmogorov spectrum and the largest ( by about 50% ) is for the square - law structure function . thus in figures [ fig : vela ] and [ fig : other ] we plot , as solid and open stars , the values converted using @xmath92 for the kolmogorov and square - law structure function models , respectively . we display the predictions for extended medium and note that the screen values are only about 20% less for each spectrum . thus in comparing data with the theory for the @xmath0 and inner - scale models , the data should lie between the extremes of the open and solid stars , depending on the outer and inner scales , respectively . note that the value of @xmath92 in equation ( 7 ) of the taylor et al . ( 1993 ) catalog is more than 50% greater than values computed by lr99 , since it concerns the mean pulse delay rather than the 1/e decay time of the pulse . the second correction to the data concerns the effects of refractive scintillation at the higher radio frequencies , which are closer to the transition to weak scintillation . gupta et al . ( 1994 ) gave a heuristic theory of the effect and expressions for the bias to the diffractive @xmath82 due to refractive shifts ; lr99 discussed the effect from the work of codona et al . ( 1986 ; hereafter , ccffh ) . in figures [ fig : vela ] and [ fig : other ] , we have plotted as open circles @xmath82 corrected for this effect using equation ( d6 ) of gupta et al . ( 1994 ) , who also estimated the variability in estimates of @xmath82 due to the refractive modulation . we used this to estimate an error bar on the open circles , which is typically larger than the error bars quoted by the original observers , plotted on the solid circles . in summary , the open circles and their error bars provide the best direct estimates of @xmath82 and the indirect estimates lie between the open and solid stars . the theory of diffractive scintillations has been discussed by many authors ; we will use the results of section 6 in lr99 for a point source ( spherical waves ) either in an extended scattering medium or with a screen at @xmath68 from the pulsar and @xmath93 from the observer ( with @xmath94 ) . following usual practice , we assume the strong scattering limit , in which the frequency decorrelation function for intensity is the squared magnitude of the diffractive component of the two - frequency second moment of the field . the validity of this approximation is examined in appendix b , where we find that it introduces a small error for the @xmath32 model , which however is negligible compared to the errors in the observations . the decorrelation bandwidth , @xmath82 , can be defined in terms of a normalized bandwidth , @xmath95 : @xmath96 here @xmath59 is the ( geometric ) mean of the two frequencies , and the parameter @xmath97 determines the strength of scattering ; @xmath57 is the speed of light and @xmath98 is the effective scattering distance . for a uniform scattering medium , @xmath99 , and for a screen geometry , @xmath101 is the field coherence scale where the appropriate phase structure function equals unity ; for the uniform scattering medium , it is defined at the observer , and for a screen , it is @xmath70 which is independent of the distances to the pulsar and screen , and leads to @xmath102 as the appropriate strength of scattering parameter . the computations presented in lr99 include a tabulation of the normalized bandwidth @xmath95 for the various models under discussion , and associated plots of the intensity decorrelation function itself . for the simple kolmogorov model , a constant value for @xmath95 is obtained regardless of the frequency or distance ( 0.773 and 0.654 for the extended medium and screen geometries , respectively ) . thus the frequency dependence of @xmath82 reflects the frequency dependence of @xmath65 , giving @xmath83 . cwb used this to estimate @xmath6 from observations of @xmath82 . we now compare the observations with theoretical scaling laws for the two other spectral models . there are , however , some complications . the quantity @xmath95 is no longer exactly independent of frequency , and @xmath65 depends on both frequency and the outer scale , @xmath103 , or the inner scale , @xmath104 , that parameterize the other two models . consider the details for the discontinuity spectrum . in figure [ fig : vela ] , we plot @xmath105 versus @xmath106 as observed for the vela pulsar . in figure [ fig : vela]a , we have plotted theoretical curves for @xmath32 model with various outer scales . we use equation ( [ eq : delta_nu_d ] ) for the models and so need the variation of both @xmath65 and @xmath95 with @xmath59 . for a screen , we determine @xmath70 from equation ( [ eq : struc4 ] ) for the @xmath32 model , and eliminate the scattering measure @xmath107 using the same equation for @xmath108 at a reference frequency of 1 ghz . this gives a relation between @xmath109 and @xmath106 , with @xmath110 as a parameter . for an extended scattering medium , we use equation ( 21 ) of lr99 to determine @xmath65 at the observer , and the same method as for the screen to eliminate @xmath107 . in figure [ fig : vd ] , we show the variation of @xmath95 with @xmath111 at a fixed frequency ( computed by lambert 1998 and described in lr99 ) . this is combined with the @xmath65@xmath106 relation to obtain the relation of @xmath95 to @xmath106 . for the curves in figure [ fig : vela ] , @xmath108 is determined so that , for each value of @xmath53 , the model @xmath112 fits the measured values near 1ghz . for most of the models of interest @xmath113 , and then the logarithm functions vary much more slowly than the @xmath114 term in equation ( [ eq : struc4 ] ) . in which case there is nearly a linear relation @xmath115 . furthermore , figure [ fig : vd ] shows that as @xmath65 changes , the parameter @xmath95 changes very slowly . consequently , for a very wide range of the outer scale , a good approximation is @xmath116 , which reflects the fact that the underlying structure function then approaches a square - law behavior . consider the plot for the vela pulsar psr b0833@xmath8645 ( figure [ fig : vela ] ) for which the data are most extensive . although we have computed curves and data corrections for both the screen and extended scattering geometries , we only show the extended medium results in figure [ fig : vela ] because the two plots are so similar . in the comparison with the screen model , the star points are lowered about 0.08 vertical units ( 20% ) , and the @xmath32 model curves are very slightly steeper functions of frequency . however , the scatter among the observations is greater than the differences in the models between screen and extended geometries . in the vela plot , we see that the @xmath32 model agrees somewhat better with the observations than does the simple kolmogorov model . this pulsar is known to lie in a highly scattered region , and hence the presence of discontinuities , as incorporated in the @xmath32 model , is perhaps reasonable . however , the conclusion is not strong , since the data are also reasonably fitted by an inner scale in the range @xmath117 km to @xmath118 km , as can be found from figure [ fig : vela]b . for the other pulsars ( figure [ fig : other ] ) , the data show a stronger frequency dependence than predicted by both the simple kolmogorov and @xmath32 models . however , the inconsistencies between the measurements at different frequencies are even greater , and better observations are needed , before such comparisons could discriminate between the models . a recent series of measurements at 327 mhz ( bhat et al . 1999a , b , c ) documents the variability of @xmath82 , and a similar long sequence of such measurements at other frequencies is needed on the same pulsars , before reliable conclusions can be reached from the frequency scaling observations . slow variations in the flux of pulsars are caused by refractive interstellar scintillation ( riss ) , and can be characterized by the rms deviation in flux density normalized by its mean , or scintillation index , @xmath119 . riss is due to inhomogeneities in the interstellar electron density on scales much larger than those responsible for diffractive scintillation , as characterized by the decorrelation bandwidth . there are now measurements of both phenomena on a substantial number of pulsars , from which we can constrain the density spectrum on each line of sight . the relation of the refractive scintillation index , @xmath119 , to the normalized diffractive decorrelation bandwidth , @xmath120 , depends on the ratio of the power in the density spectrum at the large refractive scales ( @xmath121 m to @xmath122 m ) to the power at the smaller diffractive scales ( @xmath123 m to @xmath81 m ) . we compare pulsar measurements gathered from the literature with theoretical predictions by plotting @xmath119 versus @xmath120 ( cf . rickett & lyne 1990 and gupta et al . related tests have been made by armstrong et al . ( 1995 ) , bhat et al . ( 1999a ) and smirnova , shishov , & stinebring ( 1998 ) . @xmath119 measurements at 610 mhz are listed in table [ table : mr610 ] and include the results of long - term monitoring observations by smirnova et al . ( 1998 ) , and older measurements near 100 mhz are listed in table [ table : mr100 ] . the tables also include the decorrelation bandwidth obtained from diffractive scintillation , typically observed at a different radio frequency , scaled to the observing frequency for @xmath119 . we used the kolmogorov scaling law @xmath124 , and we note that the minor differences that result from changes in the scaling law for other spectral models are insignificant in the comparison . the 100 mhz data consist of @xmath119 measurements at 73.8 mhz , 81.5 mhz , and 156 mhz . the pulsars observed at 610 mhz are primarily located at distances @xmath125 kpc , whereas the pulsars observed at 100 mhz are nearer at distances @xmath126 kpc . figures [ fig : mr4 ] and [ fig : mrin ] show plots of the theoretical and measured refractive scintillation index versus the normalized decorrelation bandwidth in separate panels for the two sets of measurements . theoretical curves are also shown in the figures for different spectral models in both the screen and extended medium geometries . the dashed line corresponds to the simple kolmogorov model ( @xmath127 ) , and the solid lines in figures [ fig : mr4 ] and [ fig : mrin ] correspond to the @xmath32 and inner - scale models , respectively . in section [ sec : decbw ] we noted that variable refraction tends to decrease the measured decorrelation bandwidth and used an expression from gupta et al . ( 1994 ) to apply a nominal correction to measured values . in the table of data from near 100 mhz , we have made use of the recently published measurements of bhat et al . ( 1999a ) . they monitored the apparent decorrelation bandwidth near 327 mhz for 20 nearby pulsars , many of which have @xmath119 measurements in table [ table : mr100 ] . they found the bandwidth to vary by factors 3 - 5 and discussed the influence of varying refraction as a refractive bias . from their measurements , they derived a corrected decorrelation bandwidth , which we have scaled ( using the @xmath84 scaling law ) to the frequency at which @xmath119 was observed . these gave values typically 25 times bigger than obtained from earlier measurements ( e.g. cordes 1986 ) . the pulsar b0809 + 74 was not observed by bhat et al . ( 1999a ) ; however , the recent weak scintillation observations of this pulsar by rickett et al . ( 1999 ) similarly suggests that earlier decorrelation bandwidth measurements overestimated the strength of scattering . the effect of these changes is to shift the plots to the right in figures [ fig : mr4 ] and [ fig : mrin ] by about half a decade , compared to figure 5 of gupta et al . there is only one pulsar ( b0329 + 54 ) in table [ table : mr610 ] common to the bhat et al . ( 1999a ) observations , and its decorrelation bandwidth has also been corrected for the refractive bias . since the pulsars observed at 610 mhz were mostly more heavily scattered , the refractive bias correction is substantially smaller ( see gupta et al . 1994 ) and has been ignored . the theory of refractive scintillations has been known since the 1970s and was applied to pulsar flux variations by rickett et al . for example , prokhorov et al . ( 1975 ) described how the modulation index for intensity can exceed unity in strong scattering in their equations ( 4.53 ) _ et . we use the notation of coles et al . ( 1987 ) and confine our discussion to spherical wave sources , propagating in a scattering plasma which is either concentrated in a screen or extended uniformly between source and observer . the `` low - frequency '' approximation for the intensity covariance function is given by equation ( 10 ) of coles et al . ( 1987 ) , from which we obtain the normalized refractive variance , @xmath128 , by setting the spatial offset equal to zero : @xmath129 dy \right\ } \times \nonumber \\ & & \sin^2\left [ 0.5 \kappa^2 r_{{\rm f},l}^2 x(1-x)\right ] \ , \kappa d\kappa \ , dx \ , \mbox { , } \label{eq : crextmed } \ ] ] where : @xmath130 \\ r_{{\rm f},l } & = & \sqrt{lc/2 \pi \nu } \mbox{. } \nonumber \label{eq : h } \ ] ] here the transverse wavenumber is @xmath131 , and @xmath132 , where @xmath5 is the distance along the line of sight measured from the source . for the screen geometry , both the density spectrum and the gradient in the phase structure function @xmath133 are concentrated in a thin layer of thickness @xmath67 at distance @xmath68 from the source and @xmath93 from the observer . thus the @xmath134 and @xmath135 integrations in equation ( [ eq : crextmed ] ) give @xmath136 , and we find it useful to define an equivalent fresnel scale @xmath137 : @xmath138 the @xmath139 integration in equation ( [ eq : crextmed ] ) involves the product of the density spectrum , decreasing as a steep power of @xmath139 , times the high - pass fresnel filter @xmath140 up to @xmath141 , times the low - pass exponential term that cuts off wavenumbers above @xmath142 , where @xmath143 is the radius of the scattering disc . hence in strong scattering ( @xmath144 ) we only consider the @xmath145 part of the fresnel filter . consequently , the integration depends on the ratio @xmath146 or @xmath147 . for an extended scattering medium the line - of - sight integration softens the exponential cut - off in the @xmath139 integration , but the basic relationships remain the same . as others have noted the level of refractive scintillation is greater in an extended medium than in a screen with the same observed diffractive scintillation . _ the canonical spectral model in iss studies of pulsars has been the simple kolmogorov spectrum @xmath148 . by fitting this model to diffractive scintillation observations , observers have estimated the scattering measure , @xmath149 , toward many pulsars . when divided by the pulsar distance , this gives a line - of - sight average of @xmath4 , which is found to vary greatly from one direction to another and to increase dramatically for distant pulsars seen at low galactic latitudes ( e.g. cwb ) . taylor & cordes ( 1993 ) have developed a smoothed model for the galactic plasma density distribution , which includes enhanced scattering in spiral arms and toward the galactic center . however , there are also large random variations on much finer spatial scales , which would produce a scatter in a plot of @xmath119 against distance of dispersion measure . for a simple power - law spectrum both @xmath119 and @xmath120 depend on @xmath107 through a single strength of scattering parameter . thus for the kolmogorov spectrum the variation of @xmath119 with @xmath120 is independent of distance or frequency and is given by a single dashed line in each plot in figures [ fig : mr4 ] and [ fig : mrin ] . if the scattering medium is uniform , the strength of scattering is @xmath150 . for the screen it becomes @xmath151 , and in that case , the dashed theoretical line is independent of the location of the screen between the source and the observer . this point is substantiated in appendix a , where the theory is laid out in more detail . according to this screen model , along each line of sight there is a scattering layer with a certain @xmath107 , which determines both @xmath119 and @xmath120 , but @xmath107 is not necessarily related to the pulsar distance . the theoretical curve with @xmath107 as the variable is independent of where the layer is located along the line of sight . for the @xmath32 and inner - scale models , the theoretical curves for @xmath119 versus @xmath120 depend on the extra parameter @xmath53 or @xmath104 , respectively . details of the theory are given in appendix a , where the relevant parameters are shown to be @xmath152 and @xmath153 . since @xmath70 depends on frequency , @xmath107 , and distance , then @xmath119 also depends somewhat on frequency and distance . in order to fix the frequency dependence of the theoretical @xmath119 values , we have separated the measurements into two groups ( at 610 mhz and near 100 mhz ) . the distance dependence is dealt with by assuming that @xmath154 distance . whereas this is clearly appropriate for the extended scattering medium , it is less clear for the screen model , since the screen model supposes that @xmath107 is not necessarily related to distance . however , it is reasonable that , if on a long line of sight there is a single region that dominates the scattering , its @xmath107 value will statistically increase with line - of - sight distance . indeed the experimentally derived scattering measure increases faster with distance than if the medium were uniform ( see figure 1 of cordes et al . ( 1991 ) . in figure [ fig : mr4 ] we show the theoretical curves for @xmath119 in the @xmath32 model , which are relatively flat where @xmath155 . as @xmath97 increases ( @xmath120 decreases ) , @xmath156 increases , and when @xmath157 the scintillations are suppressed . in a screen this occurs when @xmath158 . inspection of figure [ fig : mr4 ] shows that the measured @xmath119 values at 100 mhz are above the prediction of the simple kolmogorov model for the extended medium geometry . these measurements can , however , be modeled with the @xmath32 spectrum with suitable specific choices of outer scale , @xmath53 . the screen geometry is inconsistent with the measurements at 100 mhz for both the simple kolmogorov and @xmath32 models . for the 610 mhz data , the measured @xmath119 values are mostly in agreement with the prediction of the simple kolmogorov model for the extended medium geometry ; however , they lie below the predictions of the @xmath32 model . for the screen geometry , the measured @xmath119 values are above the simple kolmogorov model curve . however , for the @xmath32 model , though less convincingly , an agreement with the measurements can be found by suitable choices of the outer scale , @xmath53 . the most striking feature of the figure is the good agreement with the uniform extended kolmogorov model for most of the pulsars . this result , relies most heavily on the excellent data from smirnova et al . ( 1998 ) , who also note this agreement . we are persuaded by this figure that the @xmath32 model can not be viewed as a global alternative to the kolmogorov spectrum . we also conclude that for the medium - distance pulsars measured at 610 mhz , @xmath119 does not agree with the screen geometry . it appears that even if the scattering medium is not uniform on scales of a few kpc , there is not a single region that dominates the scattering . computations of @xmath119 due to a patchy distribution along the line of sight could test what distribution would start to approximate the uniformly extended medium . turning to a comparison with the alternative inner - scale spectrum in figure [ fig : mrin ] , the 100 mhz observations could be explained by very large values of @xmath104 ( @xmath159 m to @xmath121 m ) in a screen configuration or by more modest , but still large , values of @xmath104 ( @xmath81 m ) in an extended medium . the latter was essentially the proposal made by coles et al . ( 1987 ) for the nearby lines of sight . the 610 mhz points are relatively lower than those at 100 mhz . for an inner scale spectrum in a screen , large values of @xmath104 ( @xmath81 m to @xmath159 m ) would be necessary ; and in an extended medium 60% of the points lie near the simple kolmogorov model , with the rest requiring inner scale values ( @xmath123 m to @xmath81 m ) . the 610 mhz data in our analysis come from riss observations of 21 pulsars made by stinebring and are described in more detail by smirnova et al . ( 1998 ) , who also compare the results with various models for the spectrum and spatial distribution of the electron density . they note that 4 pulsars which are seen through known hii regions or supernovae remnants show relatively elevated values for @xmath119 , and find 3 other pulsars with similar behavior . these are the points which lie above the kolmogorov line in figure [ fig : mrin]b . their interpretation is that for these objects the scattering is concentrated into regions either near the pulsar or near the earth , and that these regions are characterized by an inner scale near @xmath160 m . their spectrum model is very similar to our inner - scale model , except that the cut - off is characterized by a steep power - law rather than by an exponential function . we find a somewhat larger numerical value for the inner scale required to match those objects . we also note that the theoretical @xmath119 values are consistently higher for an extended medium , and so require a more modest inner scale . however , an important result of our analysis is an alternative explanation for the relatively high @xmath119 values seen for some pulsars . we suggest that pulsar lines of sight can pass through discrete clouds with increased plasma density on large scales , that steepen the low - wavenumber spectrum as opposed to cutting off the high - wavenumbers . a discontinuity spectrum ( @xmath32 ) is one way that the spectrum can be steepened , but smoother structures on scales larger than @xmath121 m ( @xmath161 au ) would also boost the low - wavenumbers and could cause similar enhancements of refractive compared to diffractive scintillations . such enhancements are likely to be associated with hii regions or supernova remnants . a similar idea was also discussed by lestrade , rickett , & cognard ( 1998 ) in the context of extreme scattering events in pulsar timing measurements . to take this idea further , a composite between the kolmogorov and the @xmath32 or steeper spectra should be investigated , particularly one in which the line of sight is not uniformly weighted . smirnova et al . ( 1998 ) also note that the strength of scattering increases very much faster with distance and dispersion measure than if the medium were statistically uniform . the distant pulsars in their sample are mostly observed at low galactic latitudes and so are subject to the enhanced density and turbulence described by the taylor & cordes ( 1993 ) model . it appears that most distant pulsars follow the uniform extended kolmogorov model with inner scale smaller than @xmath162 m . thus although these lines of sight are subject to enhanced scattering in the inner galactic plane , the spectrum effectively follows the kolmogorov law and the plasma is dispersed enough to approximate a uniform scattering medium . there are presumably discrete refracting structures present along these lines of sight , but their contribution to the density spectrum is masked by the higher densities of the inner galaxy , which still follow an apparently turbulent spectral form . llr@.lr@.lr@.lr@.lr@.l rickett & lyne ( 1990 ) & b0531 + 21 & 2&0 & 610&0 & 2&96@xmath163 & 0&327 & 0&052 & b0531 + 21 & 2&0 & 610&0 & 2&96@xmath163 & 0&402 & 0&095 kaspi & stinebring ( 1992 ) & b0329 + 54 & 1&4 & 610&0 & 0&44 & 0&39 & 0&07 & b0833@xmath8645 & 0&55 & 610&0 & 1&29@xmath164 & 0&11 & 0&01 & b1749@xmath8628 & 1&2 & 610&0 & 5&13@xmath163 & 0&26 & 0&05 & b1911@xmath8604 & 2&29 & 610&0 & 1&91@xmath165 & 0&20 & 0&03 & b1933 + 16 & 7&8 & 610&0 & 1&82@xmath165 & 0&18 & 0&04 & b2111 + 46 & 5&22 & 610&0 & 2&69@xmath165 & 0&16 & 0&05 & b2217 + 47 & 2&31 & 610&0 & 0&26 & 0&21 & 0&02 stinebring et al . ( 1999 ) & b0136@xmath16657 & 2&9 & 610&0 & 5&7@xmath165 & 0&15 & 0&02 & b0329@xmath16654 & 1&4 & 610&0 & 0&350 & 0&37 & 0&02 & b0525@xmath16621 & 2&3 & 610&0 & 0&30 & 0&31 & 0&01 & b0531@xmath16621 & 2&0 & 610&0 & 4&0@xmath163 & 0&32 & 0&01 & b0736@xmath8640 & 2&1 & 610&0 & 9&0@xmath167 & 0&03 & 0&01 & b0740@xmath8628 & 1&9 & 610&0 & 3&5@xmath165 & 0&13 & 0&01 & b0818@xmath8613 & 2&5 & 610&0 & 0&061 & 0&23 & 0&01 & b0833@xmath8645 & 0&5 & 610&0 & 1&50@xmath164 & 0&24 & 0&01 & b0835@xmath8641 & 4&2 & 610&0 & 7&0@xmath164 & 0&21 & 0&02 & b1641@xmath8645 & 4&6 & 610&0 & 7&0@xmath168 & @xmath1690&1 & 0&05 & b1642@xmath8603 & 0&5 & 610&0 & 0&770 & 0&46 & 0&04 & b1749@xmath8628 & 1&5 & 610&0 & 6&0@xmath163 & 0&25 & 0&02 & b1818@xmath8604 & 1&6 & 610&0 & 4&0@xmath164 & @xmath1690&1 & 0&05 & b1859@xmath8603 & 8&1 & 610&0 & 5&0@xmath167 & @xmath1690&05 & 0&03 & b1911@xmath8604 & 3&2 & 610&0 & 8&0@xmath164 & 0&22 & 0&01 & b1933@xmath16616 & 3&5 & 610&0 & 2&0@xmath165 & 0&24 & 0&02 & b1946@xmath16635 & 7&9 & 610&0 & 3&0@xmath170 & @xmath1690&05 & 0&03 & b2111@xmath16646 & 5&0 & 610&0 & 3&0@xmath165 & 0&15 & 0&01 & b2217@xmath16647 & 2&5 & 610&0 & 0&20 & 0&30 & 0&01 llr@.lr@.lr@.lr@.lr@.l cole et al . ( 1970 ) & b0809 + 74 & 0&31 & 81&5 & 1&38@xmath165 & 0&45 & 0&18 & b0834 + 06 & 0&72 & 81&5 & 1&3@xmath165 & 0&39 & 0&11 & b1919 + 21 & 0&66 & 81&5 & 1&2@xmath165 & 0&53 & 0&18 helfand et al . ( 1977 ) & b0329 + 54 & 1&4 & 156&0 & 5&7@xmath165 & 0&35 & 0&08 & b0823 + 26 & 0&38 & 156&0 & 1&0@xmath163 & 0&46 & 0&11 & b1133 + 16 & 0&27 & 156&0 & 2&9@xmath163 & 0&50 & 0&08 & b1508 + 55 & 1&93 & 156&0 & 7&0@xmath165 & 0&41 & 0&07 & b1919 + 21 & 0&66 & 156&0 & 1&4@xmath163 & 0&50 & 0&07 & b2217 + 47 & 2&31 & 156&0 & 6&45@xmath164 & 0&41 & 0&11 gupta et al . ( 1993 ) & b0329 + 54 & 1&4 & 73&8 & 3&1@xmath164 & @xmath1710&15 & & b0809 + 74 & 0&31 & 73&8 & 8&90@xmath164 & 0&34 & 0&06 & b0834 + 06 & 0&72 & 73&8 & 8&6@xmath164 & 0&16 & 0&03 & b0950 + 08 & 0&12 & 73&8 & 0&17 & 0&45 & 0&05 & b1133 + 16 & 0&27 & 73&8 & 1&6@xmath165 & 0&18 & 0&02 & b1237 + 25 & 0&56 & 73&8 & 1&8@xmath165 & 0&30 & 0&06 & b1508 + 55 & 1&93 & 73&8 & 3&8@xmath164 & 0&28 & 0&09 & b1919 + 21 & 0&66 & 73&8 & 7&5@xmath164 & @xmath1710&21 & in this paper , the theory of the @xmath32 model for the electron density spectrum was derived for discontinuous density structures and compared with pulsar observations . a new feature of our analysis is the inclusion of an `` outer scale '' needed in any realistic model . the model is characterized by an effective exponent @xmath76 of the structure function , which remains between 1.95 and 1.6 over a very wide range of @xmath53 values ( cf . figure [ fig : slope ] ) . this at first seems a promising explanation for the spread in the estimates of @xmath79 derived from vlbi observations of the angular broadening profile , as observed on heavily scattered lines of sight . as discussed in section [ sec : decbw ] , from figures [ fig : vela ] and [ fig : other ] , we find that the @xmath32 model provides a somewhat better agreement with the measurements of the diffractive decorrelation bandwidth versus frequency for pulsar psr b0833@xmath8645 ( vela ) than does the simple kolmogorov model . this might arise from refractive scattering effects caused in the supernova remnant associated with the vela pulsar . four other pulsars with decorrelation bandwidths measured against frequency show an appreciably stronger frequency dependence than the predictions of both the simple kolmogorov and @xmath32 models . however , there are substantial inconsistencies among the measurements and better observations are clearly needed , especially in view of the variability in @xmath112 documented by bhat at al . ( 1999c ) . the predictions of the @xmath32 model for the variation of the refractive scintillation index with the diffractive decorrelation bandwidth are in partial agreement with the observations . as discussed in section [ sec : mr ] , the values of @xmath119 measured near 100 mhz are above the prediction of the simple kolmogorov model and had previously been explained as the effect of an inner scale , substantially larger than the values invoked by spangler & gwinn 1990 . however , the measurements are also consistent with the @xmath32 model with suitable choices of the outer scale . for the 610 mhz data , most of the measured @xmath119 values are in good agreement with the prediction of the simple kolmogorov model for the extended medium , and they lie below the curves of the @xmath32 model . however , there are a significant number of good - quality observations , which lie somewhat above the kolmogorov line . we suggest an alternative to a large inner scale on those lines of sight , that they pass through regions of enhanced density , which cause enhanced refractive scattering ; these regions must have less small - scale substructure than in a turbulent medium and could include discontinuities . based on the above considerations , we reject the @xmath32 model as a universal spectral model for the interstellar electron density fluctuations . the corollary is to strengthen the evidence for the kolmogorov density spectrum , which in turn suggests a turbulent process in the interstellar plasma . however , the simple kolmogorov spectrum is not a universal model either , since it disagrees with several of the @xmath119 observations . since the @xmath32 model provides reasonable agreement for many of these discrepant observations , we propose that enhancements in the large scale part of the spectrum ( which need not be described by discontinuities in density ) occur on these lines of sight . with such enhancements causing the increase in refractive scintillation , there is no need to invoke the relatively large inner scales proposed by coles et al . ( 1987 ) . as proposed by spangler & gwinn ( 1990 ) , a relatively small inner scale is then likely , controlled by the ion inertial length or larmor radius . [ cols= " < , < " , ] it appears that different spectral models need to be considered for different lines of sight . a widely distributed turbulent plasma with occasional large ionized structures that increase the effective average power density , @xmath172 , at low wavenumbers ( large scales : @xmath121 m to @xmath173 m ) is thus a model that needs further formal investigation . very similar conclusions have been reached by lestrade et al . ( 1998 ) and by bhat et al . ( 1999b , c ) . this model could also explain the occasional `` extreme scattering events '' and episodes of fringes in dynamic spectra when a line of sight passes through a particular discrete density enhancement . new theoretical work is needed to quantify the expected statistics of these propagation events . it is likely that numerical modeling will be necessary to model non - stationary scattering media . * acknowledgements : this work was supported by the nsf under grant ast-9414144 . * here we derive expressions for the refractive modulation index , @xmath119 , as a function of the normalized diffractive decorrelation bandwidth , @xmath120 , for the inner - scale and @xmath32 models . the simple kolmogorov model is included as a special case where the inner scale goes to zero . a general integral expression for the refractive scintillation index is given by equation ( [ eq : crextmed ] ) . with the medium concentrated into a thin layer ( thickness @xmath67 at @xmath68 from the source ) and an assumption of isotropy in the density spectrum , the equation becomes : @xmath174 \nonumber \\ & & \times \sin^2\left(\frac{r_{\rm f , scr}^2 \kappa^2}{2}\right ) \kappa \ , d\kappa \ , \mbox{. } \label{eq : app : mrscreen } \ ] ] for both the inner - scale and @xmath32 models , the exponential function in equation ( [ eq : app : mrscreen ] ) may be approximated by the gaussian ( coles et al . 1987 ) : @xmath175 \simeq \exp\left [ -(r_{\rm f , scr } u_{\rm scr})^2 \kappa^2 \right ] \ , \mbox { , } \label{eq : app : gauss } \ ] ] where @xmath137 is the fresnel scale and @xmath102 is the strength of scattering , as defined in section [ sec : mr ] . we now substitute models for the density spectrum @xmath176 , given by equations ( [ eq : extpowerlaw ] ) ( with @xmath177 ) and ( [ eq : beta4 ] ) and obtain relations between @xmath119 and the scattering measure @xmath74 . to these we add the connections between @xmath107 and @xmath70 , for each spectral model , from the equations for the phase structure functions ( 17 ) , ( 11 ) , and ( 21 ) of lr99 . for the inner - scale model , we obtain : @xmath178^{(2-\alpha)/\alpha } \ , \mbox { , } \label{eq : app : sminscale } \ ] ] where @xmath179^{\alpha/(\alpha-2 ) } \ , \mbox{. } \label{eq : app : mualpha } \ ] ] then substituting for @xmath107 we obtain an integral for @xmath128 , which can be solved analytically using standard techniques ( see e.g. appendix 2 of rickett 1973 ) : @xmath180^{(2-\alpha)/\alpha } \times \nonumber \\ & & \left [ ( \cos\psi)^{-\alpha/2 } \cos(\psi\alpha/2 ) - 1 \right ] \ , \times \nonumber \\ & & \left [ ( l_i/2s_{\rm 0,scr})^2 + u_{\rm scr}^4 \right]^{\alpha /2 } \ , \mbox { , } \label{eq : app : mrscreeninscale } \ ] ] where @xmath151 , and @xmath181 is given by : @xmath182 \ , \mbox{. } \label{eq : app : thetae } \ ] ] we see that @xmath119 depends on @xmath102 with @xmath153 as a parameter . similarly the diffractive decorrelation bandwidth , @xmath120 , depends on @xmath102 through equation ( [ eq : delta_nu_d ] ) with @xmath95 as a parameter , which is a very slow function of @xmath153 as discussed in section [ sec : decbw ] . for the simple power - law model we set @xmath183 , and in the strong scattering limit ( @xmath184 ) , equation ( [ eq : app : thetae ] ) gives @xmath185 , and we obtain the asymptotic expression : @xmath186 putting this together for the simple kolmogorov model with @xmath127 , we have the simple relation : @xmath187 , plotted as the thick dashed lines in the screen plots in figures [ fig : mr4 ] and [ fig : mrin ] . for the general inner - scale model , with @xmath104 non - zero , we show computed curves in figure [ fig : mrin ] . we can recognize two regimes . consider first the case of small @xmath104 . with sufficiently strong scattering , @xmath188 , @xmath189 . however , there is a complication in portraying this behavior in a plot versus @xmath190 for fixed inner scale , because of the variation of @xmath137 with distance , which is not specified by the horizontal variable @xmath190 . we deal with this by obtaining an approximate scaling of @xmath137 with @xmath190 . when @xmath191 , equation ( [ eq : app : sminscale ] ) relates @xmath192 , or @xmath193 . we now argue that on average @xmath154 pulsar distance @xmath194 , and eliminating @xmath107 , we see that , @xmath195 . with these scalings and @xmath188 , we see : @xmath196 here @xmath197 is the fresnel scale for a `` reference '' pulsar , and the last version in equation ( [ eq : mr.asymptote1 ] ) comes from using the kolmogorov exponent @xmath198 , which gives the asymptotic slope of 1/8 for curves at the lower left of figure [ fig : mrin ] . now with @xmath199 and @xmath137 fixed , let @xmath70 increase ( i.e. less scattering ) until @xmath200 . the inner scale is no longer important , and we get the same relation as for the simple kolmogorov spectrum : @xmath201 ; visible where the curves steepen with increasing @xmath190 . for larger inner scales , the curves in figure [ fig : mrin ] show a pronounced peak . these occur for inner scales greater than the fresnel scale and will be associated with focusing and caustics . since our treatment only includes the first order term in the low wavenumber expansion , it is not reliable in the region of the peak . goodman et al . ( 1987 ) have discussed caustics at length for the same inner - scale spectrum . they note that when @xmath202 , the scintillation power spectrum starts to fill in at scales intermediate between the diffractive and refractive wavenumbers . their equation ( 2.5.7 ) gives an estimate of the variance in this extra term as : @xmath203 with fixed @xmath104 , @xmath204 increases much more steeply with decreasing @xmath102 than does @xmath128 . thus as @xmath205 decreases , there are fewer and fewer independent phase perturbations across the scattering disc . when @xmath206 , focusing represented by @xmath207 occurs . we note that the theoretical @xmath119 versus @xmath208 plots in gupta et al . ( 1993 ) omit the focusing condition and are wrong for inner scales greater than the fresnel scale . the effect of higher order terms in the low - wavenumber expansion has been studied by dashen & wang ( 1993 ) . they obtain a more efficient expansion scheme that gives improved accuracy near the peak in scintillation index . nevertheless , it seems that a reliable prediction for the behavior near the peak in scintillations requires numerical evaluation . this becomes even more necessary in treating an extended scattering medium . we also note that the drop in @xmath119 as scattering gets weaker past the peak in figure [ fig : mrin ] is real . it represents the fact that when a square - law structure function applies for scales from @xmath70 up to the scattering disc size , there is insufficient phase curvature and the scintillations remain weak even though @xmath144 . in such circumstances the `` scattering disc '' is a misnomer , since an observer would see only a single angle of arrival , that could wander over a region of scale @xmath205 . turning to the screen analysis of the @xmath32 model , we relate @xmath70 to @xmath107 using equation ( 21 ) from lr99 , where the phase structure function equals one . this gives the following analog of equation ( [ eq : app : sminscale ] ) for the inner - scale model : @xmath209 \right\}^{-1 } \mbox{. } \label{eq : app : smbeta4 } \ ] ] substituting the @xmath32 model for the density spectrum in equation ( [ eq : app : mrscreen ] ) and letting @xmath210 , we obtain : @xmath211 \ ; \sin^2 ( r_{\rm f , scr}^2 \eta/2 ) \ ; d\eta \label{eq : app : mrscreenb4_1 } \ ] ] in order to evaluate this integral , we let @xmath212 represent the integrand , and we let @xmath213 be the same as @xmath212 but with @xmath214 set equal to zero . we can then rewrite the integral in equation ( [ eq : app : mrscreenb4_1 ] ) as : @xmath215 \right\}^{-1 } \times \nonumber \\ & & \left\ { \int_0^{\infty } g_1(\eta ) d\eta - \int_0^{\infty } [ g_1(\eta ) - g(\eta ) ] d\eta \right\ } \label{eq : app : mrscreenb4_2 } \ ] ] the first integral can be evaluated analytically ( see e.g. & 1965 ) . in strong scattering , the exponential term cuts off the oscillations of the sine term , which can be approximated by its argument , and the @xmath216 becomes negligible for values of @xmath217 larger than @xmath218 . with these approximations we can also do the second integral . putting these together , we obtain : @xmath219 \right\}^{-1 } \times \nonumber \\ & & \left\ { 4 u_{\rm scr}^2 \tan^{-1}(u_{\rm scr}^{-2 } ) - 2 u_{\rm scr}^4\ln ( 1 + u_{\rm scr}^{-4 } ) - \right . \nonumber \\ & & \left . 2 \zeta \ , [ ( 2 + \zeta ) \ , e^{\zeta } \ , { \rm e_1}(\zeta ) - 1 ] \right\ } \ , \mbox { , } \label{eq : app : mrscreenb4 } \ ] ] where @xmath220 is the exponential integral , @xmath221 . in the latter form @xmath222 is the refractive scale ( equal to scattering disk radius ) . again , @xmath119 is related to @xmath120 through : @xmath223 . the resulting curves are shown in the screen panels of figure [ fig : mr4 ] . consider the asymptotic behavior for @xmath128 in equation ( [ eq : app : mrscreenb4 ] ) as a function of @xmath102 . as the strength of scattering @xmath102 increases , @xmath120 decreases ( @xmath224 ) . in equation ( [ eq : app : mrscreenb4_1 ] ) , the exponential term cuts off the integral ( at @xmath225 ) before the oscillations of the sin@xmath226 fresnel filter , which then approximates @xmath227 . if also @xmath228 , we can ignore @xmath218 in the denominator and the remaining @xmath229 cancels the @xmath230 from the fresnel filter , and the integral depends only on @xmath231 . in this approximation , @xmath128 is then simply proportional to the slowly varying logarithmic term , which explains the relatively flat part of the curves in figure [ fig : mr4 ] ; under these conditions , in equation ( [ eq : app : mrscreenb4 ] ) the first two terms in the curly brackets sum to 2 and the last term is negligible . with @xmath53 fixed , now let @xmath102 increase , making @xmath156 increase . eventually @xmath232 becomes greater than one when the scattering disk becomes greater than the outer scale . at this point the exponential term cuts off the integral below @xmath218 , where the spectrum flattens . as @xmath102 increases even further , the integral decreases steeply , causing the down - turn at very small @xmath120 . we again note that our expressions rely on the first order of an expansion and will not be reliable near the peak in the scintillation index . however , there is not the same focusing condition that applied for very large inner scales . in order to obtain expressions for @xmath119 for the two spectrum models in the _ uniform extended medium geometry , we must complete the line - of - sight integrals in equation ( [ eq : crextmed ] ) in addition to following the steps used in the screen geometry . for each distance @xmath233 in the line of sight , there is also an integration over variable @xmath135 in the exponential cut - off . if @xmath234 , this @xmath135-integration yields @xmath235 , where @xmath236 . this again provides a low - pass cut - off at the reciprocal of the radius of the effective scattering disk , where @xmath237 . we define the scattering strength by @xmath238 , with the field coherence scale @xmath239 , as in lr99 , defined where the spherical wave structure function equals unity , measured in the observing plane . for the other spectrum models there is not such a simple relation for the @xmath135-integration , but there is still an effective cutoff given by a similar equation . when the @xmath134-integration is completed , the effective fresnel scale is actually smaller than @xmath240 due to averaging over the @xmath241 . _ in analogy with the screen geometry , we make use of identities similar to those given by equations ( [ eq : app : sminscale ] ) and ( [ eq : app : smbeta4 ] ) . for the extended medium , these identities are in turn derived from the _ wave _ structure functions for the inner - scale and @xmath32 models ( cf . the identities obtained thus will be similar to those for the screen geometry , except that for the inner - scale model , ( 1 ) there will be a factor of 3 on the right side of equation ( [ eq : app : sminscale ] ) , and ( 2 ) the 1 in square brackets is replaced by @xmath242^{\alpha/(\alpha-2 ) } \approx 1.8 $ ] for the kolmogorov exponent . for the @xmath32 model , the only change will be a factor of 3 on the right side of equation ( [ eq : app : smbeta4 ] ) . we use all of the aforementioned identities for the inner - scale and @xmath32 models and compute the @xmath134 integral numerically , since it can not be carried out analytically . the shapes of the curves bear a close relationship to the screen results , though the extended medium values generally lie above the associated screen values at the same @xmath190 . the second moment of intensity is needed to describe the fluctuations of intensity . under strong scintillation conditions , separate forms can be used for refractive and diffractive fluctuations , since their spatial scales differ by several orders of magnitude . in lr99 , as elsewhere in the iss literature , the correlation of diffractive scintillations is approximated by the squared magnitude of the second moment of the field , leading to the simple result that the spatial scale of the diffractive scintillations is equal to the scale where the phase structure function equals unity ( @xmath65 ) . however , goodman and narayan ( 1985 ) showed that for steep spectra ( @xmath243 ) this is no longer the case and the diffractive scale can be larger than @xmath65 . here we examine this question for the @xmath32 spectrum . we give the details for a phase screen with plane wave source , which are readily generalized to a spherical wave source . the two - frequency intensity cross - spectrum at wavenumber @xmath244 for a screen at distance @xmath62 is given by the fourier - like integral equation ( 17 ) of ccffh . this depends on the combination of structure functions @xmath245 , which for a plasma screen can be written as : @xmath246 where @xmath247 and @xmath248 are the two radio wavenumbers , @xmath249 is their geometric mean , @xmath250 is their arithmetic mean , @xmath251 , @xmath252 , and @xmath253 is evaluated at @xmath249 ; @xmath254 is a spatial offset which is the variable of integration . consider first the single frequency case @xmath255 ) . in the limit of very large @xmath139 , the first four structure functions saturate and sum to zero . the last two are equal and @xmath256 , which gives the simple diffractive limit mentioned above . this is the zero - order term of an expansion , which is obtained in terms of the sum of the first four terms as a small quantity . the zero order result requires full saturation , which requires @xmath257 . in diffractive scintillation @xmath258 ; hence , the condition becomes that the refractive scale @xmath259 . our concern here is to consider what happens when the diffractive @xmath139 is not large enough for saturation of @xmath253 . for shallow density spectra ( @xmath260 ) , small argument approximations to the structure function follow an exponent @xmath261 , and the zero - order term gives a good approximation even when @xmath262 . however , for steep spectra goodman and narayan ( 1985 ) showed that the leading term in the structure function follows a square law , which exactly cancels in @xmath245 ; the result is that the high wavenumber limit depends on the next term in the structure function expansion , which yields a diffractive scale that is greater than the scale @xmath65 ( defined by the square law term ) . now we consider the case for @xmath32 model , when the scattering disc @xmath156 is smaller than the outer scale @xmath53 . here we can approximate equation ( [ eq : struc4 ] ) by : @xmath263 as for the steep spectra , the leading term in the structure function follows a square law , which cancels when substituted into equation ( [ eq : v4 ] ) . @xmath245 can then be approximated for large @xmath139 by expanding in @xmath264 . the result is : @xmath265 } { s_0 ^ 2 \ln(4/s_0 ^ 2\kappa_o^2 ) } \ , \mbox{. } \label{eq : v4approx}\ ] ] here @xmath97 is the strength of scattering defined in equation ( [ eq : delta_nu_d ] ) ; @xmath266 is the angle between vectors @xmath139 and @xmath267 . we note that for a kolmogorov spectrum in the high - wavenumber limit , @xmath245 also includes terms in @xmath268 , which would be accounted for in the higher order terms of the expansion . a result similar to equation ( [ eq : v4approx ] ) is given by dashen & wang ( 1993 ) , though for a 1-dimensional phase screen . in considering the spectrum of intensity fluctuations ( eq . 17 of ccffh ) , one can show that the dominant wavenumber is approximately @xmath269 with these substitutions in @xmath245 , which we then set @xmath270 , we solve for @xmath271 ; this gives an approximate equation for the diffractive spatial scale @xmath272 : @xmath273 under the condition assumed in this approximation , @xmath274 , we find @xmath275 . however , in practice the ratio @xmath276 never becomes large . with a large outer scale , say @xmath277 pc , and typical observing conditions @xmath278 m and @xmath279 , we find @xmath280 . thus the diffractive scale could be 70% greater than @xmath65 and would slowly approach @xmath65 for smaller outer scales . turning to the two - frequency intensity correlation ( @xmath281 ) in the diffractive limit of large @xmath139 , the results of ccffh still apply . namely , that the last two terms of equation ( [ eq : v4 ] ) largely control the decorrelation versus frequency . they group the remaining terms into a filter that depends only on @xmath139 and a smaller term that becomes the basis of the expansion . the filter term was discussed by lr99 and shown to be important only as the strength of scattering decreases . it is the last two terms in @xmath245 that determine the zero - order result , so we looked at the effect of the higher order terms . the quantity that we are ultimately concerned with is the cross - correlation of intensity at offset frequencies at the same observing point . this comes from the integral of the cross - spectrum . equations ( 31 ) through ( 34 ) of ccffh give the zero and first order terms of the cross spectrum in terms of the spectrum of refractive index fluctuations in the layer . for the @xmath32 spectrum we reduced these to a sum of confluent hypergeometric functions , which can be explicitly computed . for a sample observing condition we found that the higher order terms for the cross - spectrum itself were significant compared to the zero - order term ; however when integrated to give intensity cross - correlation , they only had a minor effect on the decorrelation bandwidth itself ( @xmath282% increase ) . the reason for this appears to be the dominant effect of the last two terms in the @xmath245 summation with unequal frequencies . to summarize we find a modest ( logarithmic ) increase in the diffractive scale relative to the field coherence scale @xmath65 , but that this remains less than a factor of 1.7 for the likely iss parameters . this is accompanied by a smaller increase in the decorrelation bandwidth relative to the calculations of lr99 , which relied on the normal zero - order expansion at high wavenumbers . this small offset in the decorrelation bandwidth is negligible compared to the measurement errors for the observations under consideration . we assume that the conclusions reached here for a screen would also apply for an extended scattering medium . stinebring , d. r. , smirnova , t. v. , hovis , j. , kempner , j. c. , myers , e. b. , hankins , t. h. , kaspi , v. m. , & nice , d. j. 1996 , in `` pulsars : problems and progress '' , ed . johnston , s. , walker , m. a. , & bailes , m. , proceedings of iau colloquium 160 , asp conference series , 105 , 455

relative dispersion of passive particles in turbulent flows is one of the fundamental problems in turbulence research . it characterizes the transport and mixing properties of turbulence and is important from both theoretical and practical points of view . reflecting universal behavior of turbulent fluctuations , relative dispersion also has some universal properties because of its locality - in - scale nature @xcite . in particular , the dispersion process exhibits anomalous dispersion in the inertial range . this is first observed by richardson ( 1926 ) @xcite , and since then , a number of theoretical , experimental , and numerical investigations have been devoted to understand and model relative dispersion process @xcite . however , comprehensive understanding has not been obtained yet . recently , a few works focusing on underlying mechanism of the anomalous dispersion were reported . in the inertial range , the mean free path , @xmath2 , the mean length for persistent expansion of relative separation without changing its moving direction , is an order of relative separation itself , @xmath0 : @xmath3 @xcite . here , @xmath4 is a non - dimensional constant called the persistent parameter . in two - dimensional inverse cascade turbulence ( 2d - ic ) , @xmath4 is estimated as @xmath5 @xcite . this means separating motions are not purely diffusive but composed of an appreciable amount of persistent motions . in addition , it was also reported that there is a relation between stagnation - point structures and richardson s law @xcite , and that dispersion process is described by persistent streamline topology @xcite . from these results , it is expected that coherence of turbulent field , which must share its origin with fine coherent vortical structures such as worms in three - dimensional navier - stokes ( 3d - ns ) turbulence , has a significant role in turbulent relative dispersion . the correlations in turbulence are characterized in scale - space due to their self - similarity , and are not made disappear by coarse graining in real space . reflecting these correlations , relative separation moves persistently to some extent , so that , unlike the brownian motion , relative separation process should not be described only by random collision motions even as an approximation@xcite . in other words , the characteristic length can not be defined because the mean free path , @xmath2 , varies depending on the spatial scale . whereas there are several experiments and numerical simulations of which results are rather close to the prediction of richardson s diffusion equation @xcite that closely relates to random collision motions . thus , these results raise a question ; how are the effects of persistent motions wiped out ? in the previous paper @xcite , we have introduced a self - similar telegraph model of turbulent relative dispersion , and showed that the separation pdf can be close to the prediction of richardson s diffusion equation for slowly separating particle pairs even in the presence of persistent motions . in the present paper , we check the consistency of the physical picture of the self - similar telegraph model by carrying out direct numerical simulations ( dns ) of 2d free convection ( 2d - fc ) turbulence instead of 3d - ns turbulence . this is because ( i ) dns of 3d - ns turbulence requires extremely large computer resources , so that it is difficult to track particles for a long time , which is necessary to investigate dynamical properties of relative dispersion process , and ( ii ) 2d - fc turbulence has both statistical and dynamical characteristics similar to those of 3d - ns turbulence ( see appendix a ) @xcite . among them , the existence of coherent structures , which are approximated by the burgers t - vortex layer , is notable . this is a crucial difference from coherent structures in 2d - ic turbulence that are nested cat s eye vortices @xcite . therefore , comparing the results of the 2d - fc case with those of the 2d - ic case , we can investigate the effects of coherent structure on turbulent relative dispersion . this comparison gives a physical meaning of the drift term of the self - similar telegraph model . the inertial range achieved by our dns is not so wide that the relative motions of particle - pairs in the dissipation , the inertial , and the energy containing scales are not sufficiently resolved by usual fixed time statistics . in order to investigate the scaling natures of relative dispersion in such a narrow and limited inertial range , we utilize exit - time statistics introduced into research of turbulent relative dispersion by boffetta and sokolov ( see appendix b ) @xcite . by detailed investigation of the pdf of exit - time , we show that the pdf is divided into two region , the region - i and -ii , corresponding to persistent expansion and random transition between expansion and compression of relative separation , respectively . this result agrees with the picture of the self - similar telegraph model . in addition , we provide a method for estimation of the parameters of the self - similar telegraph model by using exit - time pdf . in the following sections , first we provide a brief review of the self - similar telegraph model in 2 . section 3 presents a summary of the numerical scheme and parameters of our dns of 2d - fc turbulence . some of the results by the fixed - time and the exit - time statistics are provided and discussed in 4 . concluding remarks are made in 5 . in addition , some properties of 2d - fc turbulence and exit - time statistics are presented in appendix a and b , respectively . in the previous paper @xcite , we have introduced a self - similar telegraph model for turbulent relative dispersion , which is a model describing the evolution of the pdf of relative separation of particle pairs in the inertial range . the relative separation of two particles , @xmath6 is defined as follows : @xmath7 where @xmath8 and @xmath9 are the lagrangian positions of the particles . the model is based on sokolov s model @xcite and consists of persistent expansion and compression of relative separation , @xmath0 , according to the relative velocity , @xmath10 , where @xmath11 and @xmath12 are a dimensional constant and a scaling exponent , respectively : @xmath13 for kolmogorov scaling and @xmath14 for bolgiano - obukhov scaling . the transition rate from expansion to compression and that from compression to expansion are given by @xmath15 and @xmath16 , respectively . here @xmath17 is a characteristic time scale at a spatial scale @xmath0 , @xmath18 , and @xmath19 are the inverses of persistent parameters , @xmath20 for expansion and @xmath21 for compression . introducing @xmath22 and @xmath23 , the evolution equation of the separation pdf , @xmath24 , is derived as follows : @xmath25 \\ + \sigma\frac{{\partial}}{{\partial}r } \left[v(r)p\right ] , \label{eq : t - model } \end{aligned}\ ] ] where @xmath26 is richardson s diffusion coefficient , @xmath27 , and @xmath28 . the parameters of the model are @xmath29 and @xmath30 : @xmath29 represents the strength of persistency of moving directions , and @xmath30 does the difference in persistency between expansion and compression . therefore , @xmath29 and @xmath30 reflect the strength and structure of coherence of the flow , respectively . for slowly - separating particle pairs , i.e. , in the case of @xmath31 , the first term of the l.h.s . ( [ eq : t - model ] ) can be neglected and the approximated equation is given by @xmath32 + \sigma\frac{{\partial}}{{\partial}r } \left[v(r)p\right ] . \label{eq : palm - eq}\ ] ] this form of the equation was first derived by palm @xcite and is also the same as the diffusion equation of goto - vassilicos model @xcite . we call eq . ( [ eq : palm - eq ] ) palm s equation . the similarity solution of eq . ( [ eq : palm - eq ] ) is @xmath33 where @xmath34 is the normalization factor . because the tail of the exit - time pdf , @xmath35 , consists of slowly - separating particle pairs , it is calculated from palm s equation ( [ eq : palm - eq ] ) with the method used by boffetta and sokolov @xcite . the asymptotic form of @xmath35 is given by @xmath36 where @xmath37 is the @xmath38-th zero of the @xmath39-th order bessel function and @xmath40 is the mean exit - time from @xmath0 to @xmath41 : @xmath42 note that the mean exit - time calculated from the self - similar telegraph model , eq . ( [ eq : t - model ] ) , is the same as eq . ( [ eq : avetime - palm ] ) . this is because the mean exit - time is calculated from a steady solution of the equation @xcite , and the solution of eq . ( [ eq : t - model ] ) is the same as that of eq . ( [ eq : palm - eq ] ) . by comparing the tail of the exit - time pdf obtained by dns and eq . ( [ eq : palm - etime ] ) , we can estimate the value of @xmath30 . the last term of the r.h.s . of eq . ( [ eq : t - model ] ) is a drift term consistent with the scaling law , and the direction of the drift is determined by @xmath43 . the parameter @xmath43 consists of two parts , the `` scaling - determined '' one , @xmath44 , and the `` dynamics - determined '' one , @xmath30 . in order for eq . ( [ eq : palm - eq ] ) to recover richardson s equation , the drift term has to disappear , which means @xmath45 . we call this case the richardson case or the zero - drift case , where the parameters of the model reduce to one , @xmath46 ; @xmath47 is determined by the relation @xmath48 . rescaled energy and entropy spectrum @xmath49 and @xmath50 obtained by dns at resolution @xmath51 ( nv10 ) and @xmath52 ( nv11 ) . the straight lines refer to the bolgiano - obukhov scaling : @xmath53 , @xmath54 . , width=332 ] .[table : dnsparms ] parameters used in the present dns . [ cols="^,^,^,^,^,^",options="header " , ] in this section , we explain the method of dns used in the present work and show basic properties of turbulent fields produced by our simulation . we generate turbulent field by dns of the vorticity equation with a large - scale friction term @xmath55 and the temperature equation with a large - scale forcing term @xmath56 : @xmath57 where @xmath58 , @xmath59 , @xmath60 , @xmath39 , @xmath61 , @xmath62 , and @xmath63 represent the vorticity field , the temperature field , the velocity field , the kinematic viscosity , the thermal diffusivity , the thermal expansion coefficient , and the gravitational acceleration , respectively . the large - scale forcing term used here is @xmath64 where @xmath65 is a constant . the large - scale friction term is written in the fourier space as @xmath66 where @xmath67 , @xmath68 , @xmath69 , and @xmath70 are a constant , the wave number vector , the fourier mode of the friction term , and the fourier mode of the vorticity field , respectively . our dns is performed on a @xmath71 domain with the doubly periodic boundary conditions at resolutions @xmath72 : @xmath51 ( nv10 ) and @xmath73 ( nv11 ) . we employ a pseudo - spectral method for accurately calculating convolutions and spatial derivatives , and 4-th order runge - kutta method for time integration . aliasing error is removed by adopting the phase - shift method ( nv10 ) and the @xmath74 method ( nv11 ) . all results presented in this paper are obtained for statistically stationary and locally isotropic turbulence . we summarize the parameters used in our simulation in table [ table : dnsparms ] and characteristic quantities of generated turbulence in table [ table : char_quantities ] . figure [ fig : spectrum ] shows the entropy and energy spectra obtained by our dns . the spectra are rescaled with entropy dissipation scales and the entropy dissipation rate , and there is a region consistent with the bolgiano - obukhov scaling , eqs . ( [ eq : bo - ek ] ) and ( [ eq : bo - sk ] ) , ( see appendix a ) . we call this region the inertial range . most investigations in the present paper are carried out in this range . in the velocity field generated by dns , we track a number of particle pairs ( @xmath75 pairs ) according to the advection equation : @xmath76 where @xmath77 is the position of the @xmath78-th particle at time @xmath79 . we employ the 4-th order runge - kutta method for numerical integration , and the linear interpolation to obtain the velocity of each particles . particle pairs are distributed homogeneously with relative separation of the grid scale @xmath80 at the initial time . temporal evolution of mean relative separation @xmath81 obtained in the case of nv11 . solid , dotted and dashed line refer to @xmath82 and @xmath83 , respectively . straight lines indicate richardson s law . the inset shows the local slope of @xmath81 . , width=332 ] first , we briefly discuss the results obtained by standard fixed - time statistics , which concerns the distribution of relative separation at a certain ( fixed ) time . in fig . [ fig : ftime ] we plot temporal evolutions of the mean relative separation of particle pairs , @xmath84 , for different powers , @xmath85 , @xmath86 , and @xmath83 . in the cases of @xmath85 and @xmath86 , the generalized richardson s relations , @xmath87 , are realized , though the times at which they start and the time intervals in which they hold differ . these differences result from the limited width of the inertial range in dns . for example , in the case of nv11 , the inertial range is roughly estimated as @xmath88 . from fig . [ fig : fpdf](a ) , it is clear that the separation pdf broadens rapidly , so that @xmath84 is contributed to by a quite broad range of relative separation including the dissipation , the inertial , and the energy containing ranges . as a result , the weight of each range varies with the power of the moment of the relative separation and thus , the range of time in which the contribution of the inertial range dominates differs accordingly . the slope of @xmath89 is less steep than richardson s law . this is because , in the case of @xmath90 , the contribution of particle pairs in the energy containing range to @xmath84 is much larger than that in the cases of @xmath91 and @xmath92 . in the energy containing range , relative separation process is described by the brownian motion , @xmath93 , which is less steep than richardson s law . hence , the larger the contribution of the range is , the less steep the slope of @xmath84 is . even though the richardson s law is observed for @xmath85 and @xmath86 , the fact does nt necessarily support the validity of the richardson s law because it is reported that the temporal evolution of @xmath84 strongly depends on the initial separation of particle pairs @xcite . hence we have to check the scaling law by adopting a different method that is independent of the initial separation . in order to check self - similarity of the pdf of relative separation , we plot the pdf rescaled with @xmath94 in fig . [ fig : fpdf](b ) . the values of @xmath94 at @xmath95 , @xmath96 , @xmath97 , and @xmath98 are @xmath99 , @xmath100 , @xmath101 , and @xmath102 , respectively . all pdfs collapse well for @xmath103 and they are in good agreement with the similarity solution of palm s equation with @xmath104 . however , we can not conclude that the temporal evolution of the separation pdf is self - similar and governed by palm s equation because fig . [ fig : fpdf](b ) is obtained not from separations only in the inertial range but in the much wider range , @xmath105 . although the collapse in fig . [ fig : fpdf](b ) implies existence of a self - similar stage governed by palm s equation , we have not had any reasonable explanation of the collapse . residual ratio of particle pairs . dotted lines with open and closed circles refer to data obtained by dns at resolution @xmath51 ( nv10 ) and @xmath52 ( nv11 ) respectively . the inset is the log - log plot of the ratio . , width=332 ] scale dependence of the mean exit - time @xmath106 for @xmath107 rescaled with the dissipation time scale , @xmath108 . open and closed circles refer to the results obtained by the present dns at resolution @xmath51 ( nv10 ) and @xmath52 ( nv11 ) , respectively . the inset is the compensated plot with @xmath109 . the dashed line represents estimated values of the coefficients @xmath110 . , width=332 ] the scale dependence of the mean exit - time obtained by our dns is shown in fig . [ fig : etime ] . it is observed that , although the width is narrow , there is a region consistent with the bolgiano - obukhov scaling , @xmath111 . the exit - time statistics is independent of initial separation of particle pairs if spatial scale @xmath0 is large enough for them to forget information of their initial conditions @xcite . in our simulation , the initial separation of particle pairs is @xmath80 that is much smaller than the scales in which the scaling law holds . therefore , this result indicates that richardson s law is valid in the 2d - fc turbulence . according to the scaling law , the form of the mean exit - time is given by @xmath112 where @xmath113 is considered to be a universal constant . the inset of fig . [ fig : etime ] shows a compensated plot of the mean exit - time . although the weak @xmath114 dependence of @xmath113 seen in fig . [ fig : etime ] requires higher resolution of dns to more accurate estimation , @xmath113 is estimated as @xmath115 in the present dns . in order to obtain statistically reliable data of exit - time , it is important that the residual ratio of particle pairs must be small enough to take in slowly separating particle pairs . the residual ratio at a scale @xmath0 and a time @xmath79 is defined as the ratio of particle pairs of which first passage times at @xmath0 are less than @xmath79 . if the residual ratio is not small enough at the time when statistics of exit - time are calculated , the statistics can not reflect slowly separating particle pairs . figure [ fig : erate ] shows the residual ratio of particle pairs used in the present work at the termination time of dns , @xmath116 . it is obvious that the ratio is almost zero in the inertial range , so that our exit - time data is reliable in the sense mentioned above . figure [ fig : marginal - etime](b ) shows the scale dependence of the mean exit - time obtained from insufficient data and illustrates the importance of the residual ratio . although the results at @xmath117 seem to have the wider inertial range than others , it is a fake . therefore , to obtain statistically reliable data , we have to track particle pairs until the residual ratio becomes at least less than @xmath118 . in contrast to the inertial range , the mean exit - time is almost constant in the dissipation range ( @xmath119 in fig . [ fig : etime ] ) except in the initial transient region , because the relative velocity is proportional to relative separation : @xmath120 . the exit - time , then , is given by @xmath121 note that the mean exit - times rescaled by @xmath108 collapse each other in fig . [ fig : etime ] . this is because there is only one characteristic time scale , @xmath108 , in the dissipation range @xcite . we plot exit - time pdf , @xmath35 , in the inertial range in fig . [ fig : pdf - et](a ) , and that rescaled by the mean exit - time , @xmath122 , in fig . [ fig : pdf - et](b ) . obviously the pdfs of different scales collapse onto one curve in fig [ fig : pdf - et](b ) . this means exit - time pdf is self - similar in the inertial range and indicates that relative dispersion process is self - similar . as is clear from fig . [ fig : pdf - et](b ) , the exit - time pdf consists of two regions : the sharp peak and the long exponential tail . we call these two regions the region - i and the region - ii , respectively . the qualitative difference in form between these indicates that the pdf reflects two different types of motions . in order to clarify the difference , snapshots of typical distribution of particle pairs in the region - i and -ii are shown in figs . [ fig : pp - dist - i ] and [ fig : pp - dist - ii ] , respectively , with a snapshot of the magnitude of t - vorticity field @xmath123 . temporal evolution of relative separation for several particles in the case of nv11 . different lines represent different particle pairs . , width=332 ] in fig . [ fig : pp - dist - i](a ) , on the snapshot of @xmath123 at @xmath124 ( @xmath125 ) , we plot line segments representing particle pairs of which exit - time @xmath126 for @xmath127 and @xmath128 satisfies @xmath129 , that is , @xmath126 is in the shaded region in fig . [ fig : pp - dist - i](b ) . in order to select such particle pairs , we first extract pairs satisfying the condition @xmath130 , and then pick out ones of which first passage times at @xmath131 are smaller than @xmath132 and those at @xmath133 are larger than @xmath132 . figure [ fig : pp - dist - ii](a ) is drawn by the same procedure . to draw figs . [ fig : pp - dist - i ] and [ fig : pp - dist - ii ] , we used the data of nv10 . in 2d - fc turbulence , fine coherent structures are well approximated by the burgers t - vortex layer , so that shear layers are formed around the structures @xcite . in addition , such structures are persistent in time to some extent . hence , particles around the coherent structures are advected along them and relative separations of the particles expand or compress persistently . we call this type of motion the persistent separation . as is shown in fig . [ fig : pp - dist - i](a ) , particle pairs in the region - i appear to be along the fine coherent structures . moreover relative separations of particle pairs in fig . [ fig : pp - dist - i](a ) expand rapidly as can be known by their short exit - time . therefore , fig . [ fig : pp - dist - i ] supports the picture of persistent separations . in contrast to the region - i , the distribution of particle pairs in the region - ii is in disorder ( see fig . [ fig : pp - dist - ii](a ) ) . because the particle pairs contained in the region - ii have long exit - time , the longitudinal relative velocity of them remains small positive during the passage from @xmath131 to @xmath133 or moving direction of relative separation of the pairs changes from expansion to compression at least once before they reach @xmath133 . however it is quite unlikely that relative velocity remains small positive for a long time , so that the latter is probably the main mechanism of the formation of the region - ii . hence , it is expected that both of expanding and compressing pairs are contained in fig . [ fig : pp - dist - ii](a ) . in fact , there are several pairs placed along coherent structures , which probably are expanding . note that their relative separations are not confined to be larger than @xmath131 . in addition , there are also pairs across the structures , which are presumably compressing because the structures are approximated by the burgers t - vortex layer . moreover , some pairs are located at positions where the fine coherent structures twist . the twisted regions are generated when well - stretched structures lose their activities and are folded , or when the structures are generated by plumes . therefore the particle pairs in such regions are changing their moving directions from expansion to compression or the opposite . figure [ fig : pp ] shows five typical evolutions of particle - pair separation . sample 3 expands persistently without any strong compression but sample 4 experiences both persistent expansion and compression several times . the former corresponds to motions along structures and belongs to the region - i ; the latter does across structures or twisted region and belongs to the region - ii . note that even a single evolution contains motions that are categorized into the region - i or into the region - ii . these facts are consistent with the assumptions of the self - similar telegraph model : relative separation process consists of persistent expansion and compression with some random transition mechanisms . estimated values of @xmath30 in the case of nv11 . dotted lines with open box , closed box , open circle , closed circle , open triangle , closed triangle , and reverse open triangle denote @xmath134 , @xmath135 , @xmath136 , @xmath137 , @xmath138 , @xmath139 , and @xmath140 , respectively . solid and dashed lines indicate the mean value , @xmath141 , and the value of the richardson s case , @xmath142 , respectively . the shaded region represents the standard deviation . , width=332 ] it is expected that expansion and compression of relative separation differ in persistency ; the former is experienced mainly by particle pairs along coherent structures and the latter does by those across the structures . since the auto - correlations of strain and temperature along the structures have longer characteristic lengths than those across them in 2d - fc turbulence @xcite , expanding motions must be more persistent than compressing ones . so as to confirm this consideration , we investigate the slope of the tail of the exit - time pdf . then , with eq . ( [ eq : palm - etime ] ) , we estimate @xmath30 that describes the difference in persistency between expansion and compression of relative separation . note that using the asymptotic form of the exit - time pdf calculated from palm s equation , eq . ( [ eq : palm - etime ] ) , may be justified as follows . introducing a time scale @xmath143 , the l.h.s . of the self - similar telegraph model , eq . ( [ eq : t - model ] ) , is rewritten as follows : @xmath144 where @xmath145 . in the case that @xmath146 at a certain spatial scale @xmath0 , the first term of the l.h.s . can be neglected , and thus , the model is reduced to palm s equation . hence , the tail of the exit - time pdf by the telegraph model , which consists of slowly - separating particle pairs , must coincide with that by palm s equation . it is also easy to show that the mean exit - time is the same between eqs . ( [ eq : t - model ] ) and ( [ eq : palm - eq ] ) . we , therefore , adopt eq . ( [ eq : palm - etime ] ) for the estimation of @xmath30 . figure [ fig : delta ] shows the estimated values of @xmath30 in the inertial range . the mean value of @xmath30 is @xmath147 and the standard deviation is @xmath148 . this negative value denotes that expanding motions are more persistent than compressing ones , which supports the above expectation . in addition , @xmath30 is larger than that of the richardson case ( the zero - drift ) , @xmath149 . because @xmath23 , @xmath150 means @xmath151 , that is , the compressing motion of relative separation is more persistent than that of the richardson case . this fact results in the negative drift in the self - similar telegraph model , eq . ( [ eq : t - model ] ) . the negativity of the drift can be accepted considering that coherent structures in 2d - fc turbulence are string - like and scale - transversal ones @xcite . that is , because the coherent structures in 2d - fc turbulence are not nested in scales , there are few obstacles to compressing motions . on the other hand , the typical structures in 2d - ic turbulence are nested vortices called `` cat s eye in a cat s eye '' . a separation process of particle pairs in 2d - ic turbulence , thus , is a step - by - step one each step of which consists of a trapping by one of nested vortices and sudden separation into a next larger vortex @xcite ; compressing motions are probably blocked by the nested vortices . in fact , the estimated values of @xmath30 from dns of 2d - ic turbulence by boffetta & sokolov @xcite and goto & vassilicos @xcite are smaller than that of the richardson case @xcite ; the compressing motion of relative separation is less persistent than the richardson case . this results in the positive drift in eq . ( [ eq : t - model ] ) . we , therefore , conclude that the drift term in the self - similar telegraph model , eq . ( [ eq : t - model ] ) , reflects the characteristics of coherent structures of the flow . rescaled pdf of exit - time ( @xmath152 ) in the case of nv11 and the prediction of palm s equation ( @xmath141 ) for several values of @xmath153 . solid and dashed lines refer to the results of dns and that obtained from palm s equation through the relation eq . ( [ eq : palm - epdf ] ) . , width=332 ] in fig . [ fig : epdf - palm ] , we compare the pdf of exit - time obtained by dns with that calculated from palm s equation by eq . ( [ eq : palm - epdf ] ) . in the region - i , the form of the pdf by dns is totally different from that by palm s equation . in the region - ii , however , the two pdfs collapse onto a single curve even when @xmath153 is very small , and thus , the separation process of particle pairs in the region - ii can be described by palm s equation , eq . ( [ eq : palm - eq ] ) . this indicates that the motion of particle pair is diffusive in the region - ii ; the separating process from @xmath154 to @xmath0 does nt affect that from @xmath0 to @xmath155 . it is also observed that the form of the exit - time pdf varies depending on the value of @xmath153 in fig . [ fig : epdf - palm ] . the larger the value of @xmath153 is , the larger the proportion of the region - ii becomes . if @xmath153 is sufficiently large , the whole pdf seems to be occupied by the region - ii and the relative separation process is substantially described by palm s equation , eq . ( [ eq : palm - eq ] ) . as mentioned in the previous subsection , the tail of the exit - time pdf calculated from the self - similar telegraph model , @xmath156 , must agree with that calculated from palm s equation , @xmath157 . in addition , as the value of @xmath153 gets larger , the region of @xmath156 overlapping with @xmath157 expands . this is because , for large values of @xmath153 , most of exit - times of particle pairs are longer than the characteristic time @xmath158 . hence , the self - similar telegraph model probably has the same properties as the results shown in fig . [ fig : epdf - palm ] . rescaled pdf of exit - time ( @xmath152 ) in the case of nv11 . the peak value is normalized to be unity . different lines represent different values of @xmath153 : @xmath159 , @xmath160 , @xmath161 , @xmath162 , @xmath137 , @xmath163 , @xmath164 , and @xmath165 from the curve with the lowest cut - off rescaled exit - time to that with the highest . the inset is the same plot in linear scale . , width=332 ] figure [ fig : epdf - eqpeak ] shows the exit - time pdf for nv11 rescaled with @xmath40 and normalized by their peak values . it is clear that the shape of the region - i is independent of @xmath153 . that is , there exists a @xmath153-independent function representing the shape of the exit - time pdf in the region - i , @xmath166 . here , @xmath167 denotes rescaling by the mean exit - time , @xmath40 . we assume that @xmath168 is normalized as @xmath169 . then , in the region - i , the exit - time pdf is written as @xmath170 where @xmath171 is a normalization factor depending on @xmath153 . the exit - time pdf , @xmath172 , takes the peak value at @xmath173 . it is estimated that @xmath174 in fig . [ fig : epdf - eqpeak ] , which can be regarded as a characteristic time scale of the region - i . as shown in the inset of fig . [ fig : epdf - eqpeak ] , @xmath166 distributes sharply around @xmath173 . if the region - i is formed by particle pairs of which relative separations expand persistently according to @xmath10 , the exit - time in the region - i , @xmath175 , is given by the pass through time from @xmath0 to @xmath41 : @xmath176 then , using the general form of the mean exit - time , eq . ( [ eq : etime - general ] ) , the exit - time rescaled by @xmath40 in the region - i , @xmath177 , is @xmath178 which is independent of @xmath153 . thus , @xmath166 can be connected to the pdf of @xmath11 , @xmath179 , which is the pdf of lagrangian relative velocity , @xmath180 , rescaled with @xmath181 : @xmath182 note that @xmath179 is expected to be independent both of @xmath0 and @xmath153 . since , in the self - similar telegraph model , the distribution of the coefficient , @xmath11 , of the relative velocity is not considered , @xmath179 is a @xmath30-function : @xmath183 . accordingly , the region - i of the exit - time pdf is approximated by a @xmath30-function at @xmath184 in the model . combining eqs . ( [ eq : avetime - palm ] ) and ( [ eq : t - model_region - i ] ) , we can estimate @xmath29 from @xmath184 . from eq . ( [ eq : avetime - palm ] ) , @xmath185 in the case of the self - similar telegraph model is given by @xmath186 then , @xmath29 is calculated as follows : @xmath187 if we assume that @xmath188 , then , @xmath29 is estimated as @xmath189 . however , the value of @xmath184 is not necessarily well defined . roughly speaking , it may take a value satisfying @xmath190 , if assuming @xmath184 is in the half - value width of the region - i around @xmath173 . then , the estimated value of @xmath29 is in the range , @xmath191 . we have investigated relative dispersion in 2d free convection turbulence by direct numerical simulation . in the inertial range , where the entropy cascade dominates , we have confirmed with exit - time statistics that relative dispersion satisfies the bolgiano - obukhov scaling and , therefore , is self - similar . it was also shown that the exit - time pdf , @xmath192 , is divided into two parts , the region - i and -ii , and that both of them satisfy the scaling - law and the self - similarity . @xmath192 is written as the following form : @xmath193 where @xmath194 is a division time - scale ( exit - time ) between the region - i and -ii , and @xmath195 is a smoothed step function such that @xmath196 if @xmath197 and @xmath198 if @xmath199 . @xmath200 and @xmath201 correspond to the exit - time pdf of the region - i and -ii , respectively . the investigation of the distribution of particle pairs in the real space indicates that the region - i and -ii are formed , respectively , by particle pairs expanding along coherent structures and by those experiencing turns between expansion and compression ( figs . [ fig : pp - dist - i ] and [ fig : pp - dist - ii ] ) . figures [ fig : epdf - eqpeak ] and [ fig : epdf - palm ] show the following characteristics of @xmath200 and @xmath201 . the form of @xmath200 is independent of @xmath153 if it is rescaled with the mean exit - time . moreover , if we assume that the region - i is constituted by persistently - separating particle pairs , the form of @xmath200 is related to the pdf of lagrangian relative velocity , @xmath10 . that is , @xmath202 where @xmath171 is a @xmath153-dependent normalization factor , and @xmath179 is the pdf of @xmath180 rescaled with @xmath181 . on the other hand , @xmath201 agrees with the exit - time pdf calculated from palm s equation , @xmath157 , the form of which varies depending on @xmath153 even when it is rescaled with @xmath40 . hence @xmath172 is written as @xmath203 similarity solution ( fixed - time pdf of relative separation ) of the self - similar telegraph model in the case of 2d - fc turbulence . the parameters used in this figure are estimated from dns data for nv11 by using the exit - time pdf . different lines refer to different values of @xmath29 . @xmath204 corresponds to the similarity solution of palm s equation . the inset is a blowup of the head region in the linear scale . , width=332 ] these results support the self - similar telegraph model . in the model , @xmath205 is approximated by a @xmath30-function . on the other hand , for slowly - separating particle pairs , the model is approximated by palm s equation . that is , the head region of the fixed - time pdf as well as the tail of the exit - time pdf is approximately described by palm s equation . by taking advantage of these characteristics of the exit - time pdf , we can estimate the parameters of the model , @xmath29 and @xmath43 ( @xmath30 ) : @xmath29 is estimated in the region - i and @xmath30 is in the region - ii . by our dns of 2d - fc turbulence , these parameters are estimated as @xmath206 and @xmath207 ( @xmath208 ) . the estimated value of @xmath43 corresponds to the negative drift in the model . this is the opposite to the 2d - ic turbulence case @xcite . we speculate that the difference in the sign of @xmath209 is caused by the difference in coherent structures . with these parameters , we can numerically calculate the similarity solution of the self - similar telegraph model , which is shown in fig . [ fig : tm - fpdf ] @xcite . however we can not compare the solution directly with the results of dns because the inertial range achieved by the dns is too narrow . we need dns with higher resolutions to the comparison . although the self - similar telegraph model can capture the essential characteristics of the relative separation process , the distribution of the relative velocity is crucial to understand further details of separation processes and their relation to the coherent structures , as suggested by the exit - time pdf obtained by dns . in fact , we have obtained preliminary results showing that the distribution of the relative velocity in the inertial range is tightly connected to that in the dissipation range @xcite . this strongly indicates that the distribution directly relates to the coherent structures because their length are of the order of the inertial range and their width of the order of the dissipation range @xcite . an extension of the model to include the distribution of relative velocity , as well as an extension to the dissipation range , will be done in the future work . this work was supported by the grant - in - aid for the 21st century coe `` center for diversity and universality in physics '' from the ministry of education , culture , sports , science and technology ( mext ) of japan . numerical computation in this work was carried out on a nec sx-5 at the yukawa institute computer facility . the governing equations of the 2d - fc turbulence are @xmath210 where @xmath60 , @xmath59 , and @xmath211 represent velocity , temperature , and pressure field , respectively . @xmath212 is the unit vector in the direction of the gravity . @xmath39 , @xmath61 , @xmath213 , @xmath62 , and @xmath63 are the kinematic viscosity , the molecular diffusivity , the mean density of the fluid , the thermal expansion coefficient , and the gravitational acceleration , respectively . the 2d - fc turbulence has two important properties : the bolgiano - obukhov scaling and the fine coherent structures . in this system , the integral of the squared temperature , @xmath214 is a conserved quantity in the ideal case , where @xmath215 is the volume of the system . we call this quantity entropy for convenience . the entropy cascades from large to small scales similar to the energy cascade in the 3d - ns turbulence , which leads scaling laws of the energy and entropy spectra , @xmath216 and @xmath217 : @xmath218 where @xmath219 and @xmath220 are considered to be universal constants , and @xmath221 is the entropy dissipation rate defined as follows : @xmath222 in this paper , @xmath223 denotes an ensemble average . these scaling laws , eqs . ( [ eq : bo - ek ] ) and ( [ eq : bo - sk ] ) , are called the bolgiano - obukhov scaling @xcite . entropy dissipation length scale @xmath224 and time scale @xmath108 is estimated by dimensional analysis with @xmath221 , @xmath225 , and @xmath61 : @xmath226 these scales correspond to the kolmogorov scales in the 3d - ns turbulence . there are several other quantities characterizing 2d - fc turbulence : the thermal taylor microscale @xmath29 , and the rayleigh number at @xmath29 , @xmath227 . these are defined as @xmath228 and @xmath229 , where @xmath230 according to the bolgiano - obukhov scaling , richardson s law in the 3d - ns turbulence is modified as follows : @xmath231 where @xmath6 is relative separation of passive particles at time @xmath79 and @xmath232 is considered to be a universal constant . in addition , richardson s diffusion equation of the separation pdf @xmath24 is modified as follows : @xmath233 , \label{eq:2dfc - richardson } \end{aligned}\ ] ] where @xmath24 is the probability density of relative separations at @xmath0 and @xmath79 , and @xmath234 is considered to be a universal constant . this equation has a self similar solution if the initial condition is the delta function . for @xmath235 , @xmath236 , \label{eq:2dfc - richardson - solution } \end{aligned}\ ] ] where @xmath237 is the normalization constant . we introduce a vector quantity called t - vorticity , @xmath238 , as follows : @xmath239 figure [ fig : snap_tv_a ] shows a snapshot of the magnitude of the t - vorticity and the strain field . it is observed that there are linearly concentrated area of t - vorticity . its width and length are an order of the entropy dissipation scale and the integral scale , respectively @xcite . we call these fine coherent structures . at the same location of the structures , we can observe linearly straining regions in fig . [ fig : snap_tv_a](b ) . t - vorticity has similar properties to vorticity in the 3d - ns turbulence . the evolution equation of the t - vorticity has the stretching term and is the same as that of vorticity in 3d - ns system : @xmath240 accordingly , there is a solution corresponding to the burgers vortex in the 3d - ns system , which is called the burgers t - vortex layer . it is numerically confirmed that fine coherent structures in the 2d - fc turbulence are well approximated by the burgers t - vortex layer @xcite . this is similar to fine coherent structures such as worms in the 3d - ns turbulence which are well approximated by the burgers vortex . the exit - time statistics is one of the scale - fixed statistics . the exit - time , @xmath241 , is defined as @xmath242 where @xmath243 is the time when a relative separation @xmath6 reach a threshold @xmath244 for the first time ( first - passage time ) @xcite . according to the scaling law of the characteristic time , @xmath245 , and the additivity of the mean , the form of the mean exit - time is expected as follows : @xmath246 where @xmath185 is a constant depending on the system . in the case of the bolgiano - obukhov scaling , @xmath247 , where @xmath113 is considered to be a non - dimensional universal constant . 1 . the exit - time statistics can specify a spatial scale by choosing a threshold @xmath244 . therefore we can extract information of the inertial range if both @xmath244 and @xmath248 are in the inertial range . the interval between thresholds ( the width for averaging ) can be controlled by @xmath153 . this means that we can control the degree of coarse graining of dispersion process . in order to calculate exit - times , we prepare a set of thresholds @xmath249 , where @xmath38 is a positive integer , and record the time at which @xmath6 reach @xmath250 for the first time for every particle pairs and every @xmath38 . in the present work , we set @xmath153 to @xmath251 @xmath252 and @xmath253 to the grid size @xmath80 . if palm s equation , eq . ( [ eq : palm - eq ] ) , describes relative dispersion well , from the equation we can calculate the probability density function ( pdf ) of exit - time @xmath254 , which represents the probability density for a particle pair of which exit - time from @xmath244 to @xmath248 is @xmath255 , @xcite . in the present case , to calculate the pdf , first we solve eq . ( [ eq : palm - eq ] ) with the initial condition @xmath256 where the boundary conditions are the reflecting condition at @xmath257 and the absorbing condition at @xmath258 . and then , the pdf is obtained as the time derivative of the probability of @xmath259 : @xmath260 we have numerically calculated the pdf of exit - time by the above procedure and compared with the pdf obtained by our dns .

in recent years , uv and optical photometry and spectroscopy of nearby elliptical galaxies has suggested that these galaxies , which have a reputation for being old , red , and dead , may not be quite as dead as previously assumed . between a few percent to 30% of local ellipticals appear to be experiencing low levels of ongoing star formation activity @xcite . the star formation is not intense enough to affect the galaxies morphological classification , as it only amounts to a few percent of the total stellar mass . however , this disk growth inside spheroidal galaxies may be a faint remnant of a process that was more vigorous in the past and may have played a role in establishing the range of galaxy morphologies we observe today . star formation , of course , requires cold gas , so interpreting the uv and optical data in terms of star formation activity has important implications both for the early - type galaxies and for a general understanding of the star formation process . it is not obvious that star formation should `` work '' the same way inside spheroidal galaxies as it does inside disks , with the same efficiency , the same dependence on the gas surface density , or the same regulatory mechanisms . for example , it has been hypothesized that even if there is a molecular disk inside an elliptical or lenticular galaxy , the disk would probably be stabilized by the galaxy s steep gravitational potential @xcite . thus it is of interest to probe the relationships between molecular gas and star formation activity in early - type galaxies . it is not as straightforward , however , to measure star formation rates in early - type galaxies as it is in spirals . nebular line emission is common in ellipticals @xcite , but it is usually not thought to be associated with star formation . its distribution is generally smooth , centrally peaked , and sometimes filamentary @xcite ; its morphology and line ratios suggest ionization sources such as agn activity , evolved stars , or cooling from the hot gas phase rather than star formation . far - ir ( fir ) and cm - wave radio continuum emission are also commonly used as tracers of star formation activity in gas - rich spirals , but agns and the evolved stellar population have been identified as the sources of mid - ir ( mir ) and fir emission in most elliptical galaxies @xcite . here we investigate evidence for and against star formation activity in a sample of elliptical and lenticular galaxies that have unusually large molecular gas contents . we make use of matched - resolution images of the molecular gas distribution , the cm - wave radio continuum and the 24 intensity . the properties of more typical , co - poor early - type galaxies are reviewed so that they can provide a comparison sample for the co - rich early - type galaxies . we then present the morphology of the 24 emission , with simple parametric fits and comparisons to molecular gas , radio continuum , and optical images that show dust disks in silhouette . the mid - ir and far - ir flux densities of the co - rich galaxies are presented along with discussion of the 24 to 2 flux density ratios and the fir / radio flux density ratios as diagnostics of whether the mid- to far - ir emission has a circumstellar or star formation origin . the results are mixed ; in some cases it is clear that the 24 emission is primarily associated with star formation , and in other cases heating by the evolved stellar population or even by an agn are inferred . thus , the results are at least qualitatively consistent with the suggestions that present - day star formation activity may be occurring in substantial numbers of local early - type galaxies . the isocam and isophot instruments aboard the _ infrared space observatory _ provided some insight into the variety of processes that contribute to the mid - ir and far - ir emission of early - type galaxies . for example , a handful of elliptical and lenticular galaxies found their way into the _ iso _ atlas of bright spiral galaxies and are discussed by @xcite with particular reference to the rate and distribution of star formation activity . the 12 images show very little emission beyond their galaxies nuclei . in @xcite , enhanced mir and fir to @xmath1 flux density ratios are used as star formation indicators . it is not at all obvious that this interpretation is accurate for early - type galaxies , but in most of these cases the ratios are consistent with interstellar radiation fields due to the old stellar population so little if any star formation activity is inferred . there are a minority of e - s0/a galaxies whose fir/@xmath1 flux density ratios ( within a 15 aperture ) are as high as the median values for sb - scd spirals , suggesting the possibility of star formation activity . @xcite fit the spectral energy distributions ( seds ) and compared 15 images to optical and near - ir images for a sample of 18 ellipticals , dwarf ellipticals , and lenticulars . in two cases they found no evidence for excess 15 emission over the stellar photospheric emission ; in most of the rest of the sample there is such an excess and it is smoothly distributed , more or less following the stellar distribution . a few targets show thermal emission at 15 from the dust that is visible in silhouette in the optical images , and two show agn emission ( a nuclear point source and even some synchrotron radiation from the jet in m87 ) . thus , in early - type galaxies the stellar photospheres , circumstellar dust , silhouette dust lanes or disks , agn , and ( possibly ) star formation all contribute in the mid - ir images . fourteen galaxies classified as e , s0 , or s0/a were observed as part of the spitzer infrared nearby galaxies survey ( sings ; * ? ? ? eleven of these were detected up to 160 and the 24 morphologies of these galaxies are discussed by bendo et al . ( 2007 ) . with a couple of exceptions , such as ngc 1316 ( which has asymmetric extended emission over 2 ) and ngc 5866 ( which contains an edge - on disk ) , these early - type galaxies tended to have only poorly resolved nuclear emission at 24 . six of the 11 also have published co observations @xcite , and four of the six are detected . ngc 5866 is notably co - rich , as a large amount of molecular gas ( 4.4 ) is very clearly detected by @xcite . however , maps of the molecular gas distributions in most of these galaxies are not currently available . mips observations of more typically co - poor elliptical galaxies have also been published by @xcite and @xcite . of the 19 galaxies analyzed by @xcite , 13 have been searched for co emission and none have been detected @xcite . @xcite show that the 24 emission from their ellipticals follows the @xmath2 near - ir surface brightness profiles very closely . the 24 emission even has the same effective radius as in @xmath1 ; the ratio of two radii is found to be 0.96 with a dispersion of 0.20 . in addition , @xcite have shown that the 24 emission globally tracks the optical luminosity in elliptical galaxies as there is a tight linear correlation between the 24flux density and the @xmath3-band flux density . this 24 emission is interpreted to be circumstellar dust from the mass loss of post main sequence stars . in short , several processes may contribute to mir emission from early - type galaxies . in the majority of the elliptical galaxies that are not co - rich the 24 emission seems to either follow the stellar photospheric emission or a nuclear source . extended 15 and 24 emission has been observed from a few early - type galaxies that are very rich in molecular gas or that exhibit silhouetted dust . until now , however , it has been rare to be able to compare the distribution of possible star formation activity in early - type galaxies to that of the molecular gas , the raw material for the star formation . most early surveys for molecular gas in early - type galaxies were strongly biased towards fir - bright targets , with a typical selection criterion having an iras 100 flux density @xmath4 1 jy @xcite . more recent co searches are not fir - biased , but they still find significant molecular gas contents . @xcite reached a surprisingly high co detection rate of @xmath5 in a volume - limited sample of nearby field lenticular galaxies , and @xcite detected co emission in @xmath6 of a similar sample of field ellipticals . @xcite also detected co emission in @xmath7 of the early - type galaxies in the sauron survey @xcite , a representative sample that uniformly fills an optical magnitude apparent axis ratio space . thus , the co detection rates in ellipticals and lenticulars are high enough to support the uv - inferred incidence of star formation activity ( if that gas does indeed engage in star formation ) . the cold gas masses are highly variable in these detections , with @xmath8 in the range @xmath9 to @xmath10 and lower . since we are interested in morphology as a means of distinguishing the origin of the mir , fir , and radio emission , we have selected for this project some of the relatively few elliptical and lenticular galaxies with maps resolving their molecular gas distribution @xcite . because of the way galaxies were selected for co mapping , the targets were already known to be fir - bright and to have concentrations of molecular gas in their centers ( as opposed to their outskirts ) . we also apply a criterion on the angular extent of the molecular gas to be able to test the correspondence between the molecular gas and 24 emission . if the 24 emission arises in star formation activity , we expect it to trace the molecular gas . if the 24 emission is related to agn activity , we expect it to be a point source , and if it comes from circumstellar dust , it should trace the stellar distribution . thus , we required the targets to have molecular gas in structures on the order of 20 to 30 or more in diameter . corresponding dust emission , if present , should be resolved in the 24 images . the targets distances range up to 80 mpc and optical luminosities are in the range @xmath11 ( table [ sampletable ] ) . observations of ugc 1503 , ngc 807 , ngc 2320 , ngc 3032 , ngc 3656 , ngc 4476 , and ngc 5666 were made with the mips instrument at 24 , 70 , and 160 in project 20780 of cycle go-2 . the data were taken in photometry mode using the large field size in all cases . relatively short exposures were used to avoid saturation on these bright sources in medium - high background regions . exposures were made in four cycles of 10 seconds at 24 , with sky offsets of 300 , and in 8 cycles of 3 seconds at 160 . observations at 70 were made in the fixed cluster - offsets mode with offsets @xmath12 80 , and 8 cycles of 3 seconds . to maximize the morphological information recoverable at 70 the fine pixel scale was used . however , most sources were not expected to be usefully resolved at 160 so the single pointing ( rather than a raster map ) was used at the longer wavelength . data for ngc 4526 were obtained from gto project 69 ( pi : fazio ) . those observations consist of a 10s exposure at 24 using the small field with a 300 sky offset , 3 cycles of 3 seconds at 70 using the default pixel scale and small field , and 3 cycles of 3 seconds at 160 in the default pixel scale in a 3-by-1 raster map . data for ngc 4459 were obtained from go project 3649 ( pi : cte ) ; the observing modes are similar to those for ngc 4526 except that the exposure times are 10 cycles of 4 seconds at 24 , 10 cycles of 5 seconds at 70 , and 10 cycles of 6 seconds at 160 . the 24 , 70 , and 160 @xmath13 m images were created from raw data frames using the mips data analysis tools ( mips dat ; * ? ? ? * ) version 3.10 along with additional processing steps . the processing steps for the 70 and 160 @xmath13 m data are similar , but the steps for the 24 @xmath13 m data are significantly different from these other two bands . therefore , the 24 @xmath13 m data processing is described first followed by descriptions of the 70 and 160 @xmath13 m data processing . the individual 24 @xmath13 m frames were first processed through a droop correction ( removing an excess signal in each pixel that is proportional to the signal in the entire array ) and were corrected for non - linearity in the ramps . the dark current was then subtracted . next , scan - mirror - position dependent flats were created from the data in each astronomical observation request ( aor ) and were applied to the data . detector pixels that had measured signals of 2500 dn s@xmath14 in any frame were masked out in the following three frames so as to avoid having latent images appear in the data . next , a scan - mirror - position independent flat was created from the data in each aor and were applied to the data . following this , planes were fit to the zodiacal light emission in the background regions in each frame ( regions falling outside the optical disks of the galaxies that also did not contain any other bright sources ) , and these planes were subtracted from the data . next , a robust statistical analysis was applied in which the values of cospatial pixels from different frames were compared to each other and statistical outliers ( e.g. probable cosmic rays ) are masked out . after this , a final mosaic was made with pixel sizes of @xmath15 , any residual background in the image was subtracted , and the data were calibrated into astronomical units . the calibration factor for the 24 @xmath13 m data is given by @xcite as @xmath16 mjy sr@xmath14 [ mips instrumental unit]@xmath14 . in the 70 and 160 @xmath13 m data processing , the first step was to fit ramps to the reads to derive slopes . in this step , readout jumps and cosmic ray hits were also removed , and an electronic nonlinearity correction was applied . next , the stim flash frames taken by the instrument were used as responsivity corrections . the dark current was subtracted from the data , and an illumination correction was applied . short term variations in the the signal ( often referred to as drift ) were removed from the 70 @xmath13 m data ; this also subtracted the background from the data . next , a robust statistical analysis was applied to cospatial pixels from different frames in which statistical outliers ( e.g. pixels affected by cosmic rays ) were masked out . once this was done , final mosaics were made using square pixels of 4.5 for the 70 @xmath13 m data and 9 for the 160 @xmath13 m data . the backgrounds in the 160 @xmath13 m data and the residual backgrounds in the 70 @xmath13 m data were measured in regions outside the optical disks of the galaxies and subtracted , and then flux calibration factors were applied to the data . the 70 @xmath13 m calibration factors given by @xcite are @xmath17 mjy sr@xmath14 [ mips instrumental unit]@xmath14 for coarse - scale imaging and @xmath18 mjy sr@xmath14 [ mips instrumental unit]@xmath14 for fine - scale imaging . the 160 @xmath13 m calibration factor is given by @xcite as @xmath19 mjy sr@xmath14 [ mips instrumental unit]@xmath14 . an additional 70 @xmath13 m nonlinearity correction given as @xmath20 by @xcite was applied to coarse - scale imaging data where the surface brightness exceeded 66 mjy sr@xmath14 . for analyzing the radial profiles of the 70 and 160 @xmath13 m data and for creating accurate models of the spatial distribution of 24 @xmath13 m emission , we needed point spread functions ( psfs ) that accurately represent the observed psfs . while the stinytim model @xcite can be used to model the psf , the model output does not match the observed psf of point sources . this is because the model assumes that the psf is infinitesimally subsampled , whereas the observed psf is measured with pixels that effectively blur some of the fine features . one approach to circumvent this problem is to simply smooth the model psf @xcite . as an alternative approach , we created a set of empirical psfs from archival data . the 24 and 70 @xmath13 m psfs were constructed using archival mips photometry mode data from program 20496 ( pi : marscher ) of 3c 273 , 3c 279 , and bl lac , which are relatively point - like infrared sources at these wavelengths . data from a total of 25 aors were used . these data were processed using the same techniques described in section [ reduction ] except that the final mosaics were made for each aor , and the images axes were left in the native instrumental rotation instead of being aligned with the j2000 coordinate system . the final psfs were created by normalizing the flux densities of every mosaic to 1 and then median combining the frames from all aors , which filtered out extended emission such as a tail - like feature extending north of 3c 273 . mips 160 photometry mode observations of compact sources typically do not completely sample the psf , so we needed to use scan map data to create a psf at this wavelength . in this case , we used sings data of mrk 33 , ngc 1266 , ngc 1377 , ngc 3265 , and ngc 3773 to create the psf . these galaxies were selected because they are galaxies from sings that are unresolved or marginally resolved at 24 @xmath13 m , so we can safely assume that the 160 @xmath13 m counterparts are also unresolved . these data were processed in the same way as the 160 @xmath13 m photometry data for the elliptical galaxies in this paper except that an additional drift removal step was applied to the data , separate final mosaics were made for each aor , and the images axes were left in the native instrumental rotation . as with the 24 and 70 @xmath13 m psfs , the final psf was created by normalizing the flux densities of every mosaic to 1 and then median combining the frames from all aors . the 70 and 160 emission from our co - rich targets is very poorly resolved , so no detailed morphological analysis was done in these bands . however , the 24 emission is resolved , and simple model fits are made to parametrize the 24 morphologies . we used the image fitting techniques described by @xcite to characterize the spatial distribution of the 24 @xmath13 m dust emission and to measure the flux densities of the galaxies . these techniques have been improved in several ways compared to the models that @xcite applied to the sombrero galaxy . first , we are now using the empirical psfs described above instead of the stinytim theoretical psfs . with these psfs , we can more accurately model the region near the centers of these galaxies . second , we now treat the central coordinates of each model component as free parameters . the models used in this paper include unresolved point sources , inclined exponential disks , de vaucouleurs ( @xmath21 ) profiles , and rings with exponential profiles as well as combinations of these , all of which are convolved with the empirical 24 @xmath13 m psf . in addition to the co observations cited above , we also compare the mid - ir morphology to that of the 1.4 ghz radio continuum , the stellar distributions from optical and nir images , and the dust seen in silhouette against the optical continuum . the 1.4 ghz radio continuum emission from galaxies originates in both agn activity and star formation @xcite . active nuclei should be distinguishable as nuclear point sources , perhaps with a jet , whereas kpc - scale extended emission more likely originates from star formation @xcite . radio continuum images ( and one nondetection ) for ugc 1503 , ngc 807 , ngc 3656 , ngc 4476 , and ngc 5666 have been published by @xcite . the first survey @xcite provides radio continuum images of ngc 2320 , ngc 3032 , ngc 4459 , and ngc 4526 . all of these 1.4 ghz images have resolutions on the order of 5 , which facilitates comparisons to the mips 24 images at 6 resolution . table [ radioims ] lists the beam sizes and rms noise levels of the continuum images . it also shows that while the first images have typically a factor of 3 to 4 higher noise levels than our own , they do not have systematically lower signal - to - noise ratios . thus we do not expect significant biases in the detectability of extended structures , for example , between the first images and our own 1.4 ghz data . broadband optical @xmath22 and @xmath23 images of ngc 3032 , ngc 4459 , and ngc 4526 were obtained from the sloan digitized sky survey ( sdss ) . @xmath24 and @xmath25 images of ugc 1503 , ngc 807 , ngc 2320 , and ngc 3656 were obtained by l. van zee in november 2002 with the mini - mosaic imager ( minimo ) on the wiyn 3.5 m telescope and were reduced as described by @xcite . the seeing in those images ranged from 0.9 to 1.3 . @xmath24 and @xmath26 images of ngc 4476 and ngc 5666 were obtained with the kitt peak 2.1 m telescope and t2ka ccd in april 2003 . exposure times for those images were 1200 s in @xmath24 and 1800 s in @xmath26 ; the seeing varied from 1.1 to 1.6 in those images and they were reduced in the same manner as for the wiyn minimo images . finally , wfpc2 images of ngc 3032 , ngc 4459 , and ngc 4526 in the f606w or f555w filters were retrieved from the hst archive along with acs observations of ngc 4476 . the 24 emission from ngc 807 is also compared with the distribution of hi in the galaxy . hi emission was mapped with the national radio astronomy observatory s very large array in its c configuration for 6 hours on 2002 dec 10 and 6 hours on 2002 dec 16 in program ay135 . the total bandwidth was 3.125 mhz centered at 4650 , giving 63 channels of 21.3 . the absolute flux scale , bandpass calibration , and time dependent gain corrections were determined from observations of the nearby source j0137 + 331 . all data calibration and image formation were done using standard calibration tasks in the aips package ; continuum emission was subtracted directly from the visibility data by making first order fits to the line - free channels . the calibrated data were fourier transformed using briggs robust weighting scheme with a robustness parameter of 0.0 , which gave a resolution of @xmath27 and a rms noise level of 0.2 mjy beam@xmath14 . the dirty images were cleaned down to a residual level approximately equal to the rms noise fluctuations . the integrated intensity image was made by smoothing , clipping , and then summing the cleaned data cube in a manner similar to that used by @xcite . none of the co - rich early - type galaxies are pure point sources at 24 . all are resolved , though they are significantly less extended than the stellar distributions in the nir and optical . total flux densities at 24 are thus derived from large aperture photometry in the sky - subtracted images . unrelated point sources were first cleaned from the vicinity of the target using the task _ imedit _ in iraf , and the total flux densities were summed in a series of circular apertures up to 60 pixels ( 90 ) in radius . the total flux densities converge to within @xmath28 for radii @xmath29 75 , and therefore table [ fluxes ] quotes the flux density within the 75 aperture . a color correction is derived based on the power law index @xmath30 ( @xmath31 between the 24 and 70 flux densities ; the power law indices range from @xmath32 to @xmath33 . these indices are consistent with the mid - ir spectra of nearby galaxies presented by @xcite . the color corrections tabulated by @xcite are 0.960 to 0.967 . the dominant uncertainty in the flux densities is that of the absolute calibration , roughly 4% @xcite . the 70 and 160 emission generally appear unresolved ( section [ 70 - 160 ] ) , so we treated these as point sources when measuring flux densities at these wavelengths . to select aperture and color corrections for the 70 and 160 @xmath13 m data , we first fit the 60 - 160 @xmath13 m iras and _ spitzer _ data with unmodified blackbody functions to roughly characterize the seds . the typical color temperatures derived from these fits were @xmath34 - 50 k. we choose to measure flux densities in the 70 data within an aperture of 81 radius , as it is the maximum aperture that stays entirely within the field - of - view of our 70 images . @xcite do not specifically quote an aperture correction for these parameters , so we derived that correction factor . the stinytim model @xcite was used to create 60-wide model psfs for blackbodies with temperatures of 30 , 40 , and 50 k. after estimating the `` background '' level in an annulus of radii 81 to 100 , we measured the fraction of the total flux density in a circle of radius 81 . this procedure gives an aperture correction of @xmath35 . we also verified that we were able to reproduce the aperture corrections quoted in @xcite to within 1% . we take the color correction for a black body of temperature 40 10 k as 0.886 0.015 @xcite . in most of our targets , the dominant contribution to the uncertainty in the 70 flux density is that of the absolute calibration scale ( 10% for fine scale data and 5% for coarse scale data , * ? ? ? . however , the images of ngc 2320 and ngc 4476 still suffer from some negative artifacts just outside the first airy ring . initial exploration suggests that the 70 flux densities of these galaxies may be 20% and 10% low , respectively . flux densities in the 160 images were measured in an aperture of radius 64 with sky background determined in an annulus of radii 80 to 160 . for this aperture and a psf approximately represented by a 40 k blackbody , we take an aperture correction from @xcite as an interpolation between a 30 k and a 50 k blackbody , which gives 1.3575 0.0035 . the color correction , assuming a blackbody of 40 10 k , is 0.9640.010 @xcite . for ngc 4459 and ngc 4526 , the small fields of view made it difficult to estimate the background level and flux densities for these two should be treated with greater caution . aside from the background level and the nonlinearity effect discussed below , the dominant contribution to uncertainty in the 160 flux densities is the absolute calibration scale ( 12% , * ? ? ? additional uncertainties in the flux densities come from the fact that the targets are bright at 70 and 160 . @xcite have shown that aperture photometry gives systematically lower flux densities than a psf fitting technique , for sources brighter than about 1 jy at 70 . the effect is roughly 5% at 2 jy . the nonlinearity correction of @xcite ( section [ reduction ] ) should mitigate this effect for the coarse scale data of ngc 4526 and ngc 4459 , but the corresponding correction is not known for fine - scale data and the effect could be important for ngc 3032 , ngc 3656 , and ngc 5666 . @xcite also show that sources brighter than 2 jy at 160 are underestimated , with the effect being as large as 20% at 4 jy . finally , we note that since the 160 emission for ngc 807 is slightly resolved its aperture correction may be low . ngc 807 and ngc 5666 also have fir flux density measurements from the iso satellite @xcite . in the case of ngc 807 , the 160flux density of 2.390.29 jy that we derive from the mips data is consistent with the 2.8 0.8 jy measured by temi et al . at 150 . for ngc 5666 , our mips 160 flux density ( 2.480.30 jy ) seems low in comparison with the iso data ( 3.9 1.2 jy at 150 and 2.70.8 jy at 200 ) , but given the sizes of the uncertainties we can not conclude that those three values are inconsistent with each other . we note that the analysis and processing of the mips 160 data is routine , whereas the iso data for ngc 807 and ngc 5666 were obtained with the pht 32 mode , which is known to suffer from strong transient effects that make it difficult to calibrate @xcite . far - ir spectral energy distributions for our targets are shown in figure [ sedplot ] , where we have included the 60 and 100 iras flux densities and 350 observations of @xcite ( see table [ q ] ) . the seds are similar to those of the `` typical '' spiral galaxy from the sings survey @xcite , but with 70/160 colors slightly warmer than that of a typical spiral and 24/70 colors slightly cooler than the spiral . the iras 60 and 100 flux densities , mips 70 and 160 flux densities and ( where available ) 350 flux densities have been fitted with a blackbody modified with an emissivity @xmath36 to characterize the dust temperatures in these galaxies . these color temperatures are given in table [ fluxes ] . we find that the modified blackbody temperatures of 25 - 28 k for most of these galaxies are a few degrees warmer than the average color temperatures for spiral galaxies found by @xcite , @xcite , and @xcite , which range from @xmath37 k to @xmath38 k depending on the number of thermal components and combinations of wave bands used . however , the color temperatures of the early - type galaxies in this paper are similar to the color temperatures measured in early - type galaxies by @xcite and @xcite . in contrast , the color temperatures of 22 - 23 k measured for ngc 807 and ngc 2320 are closer to those of spiral galaxies than to other elliptical and so galaxies . if the dust opacities in the diffuse ism of most early - type galaxies are low , the dust producing the fir cirrus emission could be bathed in a somewhat more intense or harder radiation field than is typical for spirals . this scenario could explain the higher color temperatures measured in the 60350 range . however , since 24 emisson primarily traces hot dust associated with star - forming regions in spiral galaxies , the 24 emission in these early - type galaxies could be relatively faint compared to the cirrus emission if these galaxies contain relatively little star formation compared to their dust content . in all of the targets , the 24 emission is resolved . a variety of structures ( point sources , rings , disks , and @xmath2 profiles ) are evident in the images and are discussed in greater detail below . the interpretation of these structures is complicated by the fact that the 24 intensity is a function of both the dust surface density and the illuminating radiation field . possible dust heating sources include the post - main sequence stellar population , star formation regions , and agn . ( none of our targets have strong enough radio jets for spatially resolved synchrotron emission to be important at 24 . ) thus , the 24 emission by itself may not necessarily indicate the presence of star formation . however , evidence from the co and radio continuum helps to resolve some of these ambiguities . for example , star formation is expected to be accompanied by cm - wave radio continuum emission @xcite , whereas dust heated by the radiation from evolved stars would not be . therefore the comparisons with the distribution of the molecular gas ( the raw material for star formation ) and the radio continuum provide constraints on the origin of the 24 emission . for each galaxy we consider in detail the evidence for star formation activity , with a particular view to distinguishing how much of the 24 emission is due to star formation . we take the molecular gas to be the raw material , so that star formation should only be found within the co disks . we ask whether the radio continuum emission is resolved on scales similar to the molecular gas , which ( if true ) suggests that the radio continuum does not arise in agn activity . we investigate whether the 24 emission or its morphological sub - components are distributed like the radio continuum and molecular gas , which would strongly suggest star formation activity . a schematic summary is found in table [ schematic ] with brief comments and additional evidence from the fir / radio flux density ratios , optical emission lines , stellar populations or galex colors where those are available . the fir / radio flux density ratios are discussed in greater detail in section [ radiofir ] . figure [ u1503 ] presents optical , unsharp - masked , 24 , co , and radio images of ugc 1503 . in addition , a smooth galaxy model was constructed using the multi - gaussian expansion technique of @xcite and the top left panel of the figure shows the ratio of the original @xmath24 image to the mge model . the unsharp - masked @xmath24 image shows some fairly subtle mottling in the interior of the galaxy , with perhaps some hint of flocculent arm segments at radii less than about 10 . this is the same region where we find a regularly rotating molecular gas disk ( radius 15 = 5.2 kpc ) . the radio continuum emission is also found in an asymmetric ring of similar size . the 24 image shows a central dip , which we have verified is not due to saturation , and this image is well fit with a ring of radius 4.8 @xmath39 3.5 ( 1.6 @xmath39 1.2 kpc ) . table [ fittable ] gives the best - fit parameters describing the model of ugc 1503 and the other galaxies . a dust distribution following the stellar photospheric emission would be centrally peaked , unlike the 24 emission of ugc 1503 . in addition , if an annulus of dust in ugc 1503 were illuminated by an old stellar population it could conceivably reproduce the 24image , but as discussed above one would have to appeal to some other process for the origin of the radio continuum ring . for these reasons we argue that the simplest explanation for the co , radio and fir morphologies of ugc 1503 is that star formation is taking place in an annulus of the molecular / dust disk . the fir / radio flux density ratios support this interpretation , as discussed further in section [ discussion ] . while the global similarities between radio continuum and 24morphology indicate that star formation activity in a @xmath40 1.6 kpc ring powers them both , the details of their morphologies have implications for the propagation of the star formation through the galaxy . as figure [ u1503 ] suggests , the ridgeline of the radio continuum ring is clearly outside the ridgeline of the 24ring , so that the two most prominent radio peaks are 3 to 4 away from corresponding 24 peaks . the effect is not caused by a difference in spatial resolution of the two bands ; the radio continuum image has somewhat better resolution ( 5 @xmath39 4.5 @xmath40 1.7 kpc ) than the 24 image ( 6 ) , and the absolute registration ought to be good to 1 or better . neither is this the effect described in some detail by @xcite , in which the radio continuum emission looks like a smoothed version of the mir or fir image because the diffusion length of cosmic ray electrons is larger than the mean free path of the photons that heat the dust . instead , as the radio continuum emission has a longer rise and decay timescale than fir emission after a burst of star formation activity @xcite , this offset could indicate that the star formation activity is propagating inward or that the star formation rate is decreasing more strongly in the outer parts of the ring than the inner parts . figure [ n807 ] shows the 24 emission from ngc 807 along with an optical image , optical dust maps , co line emission and 20 cm radio continuum emission . the dust distribution traced in the @xmath41 color map and in the mge residuals suggests a symmetric , dynamically relaxed disk . the 24 emission from ngc 807 shows an unresolved peak on the nucleus of the galaxy , an elongated plateau and a surrounding envelope . the structure of the emission is well fit by a superposition of a nuclear point source ( 3.4 mjy ) , an exponential disk of scale length 17 = 5.4 kpc , and a plateau or a flat - topped inner disk whose flat region has a semimajor axis 11= 3.5 kpc . the two disks each contain roughly 50% of the total 24 flux density and the point source only about 5% of the total . at surface brightness levels of 1.2 @xmath42 and higher ( 20% of the peak ) , the 24 emission shows a high degree of reflection symmetry . that fact is significant because both the molecular gas and radio continuum emission are notably stronger in the southeast and their peaks are 6 southeast of the optical/24nucleus . ( it is worth noting that the radio continuum emission from ngc 807 is quite faint and data of higher sensitivity would be beneficial . ) the nuclear point source component of the 24 image might be related to either an agn or to unresolved nuclear star formation . it is curious , though , that no corresponding nuclear peak is found in the radio continuum emission . the plateau or flat - topped disk component is too extended to be attributed to an agn , and as it does not follow the stellar distribution either , it is unlikely to be the type of circumstellar dust emission discussed by @xcite . this component could conceivably originate in star formation in the molecular disk since its size scale is similar to that of the molecular gas . ( the co dist extends to radii @xmath43 20 , and the plateau is flat to a semimajor axis 11 and declines with a scale length of 4 beyond . ) the third component in ngc 807 , the exponential component with a scale length of 17 , could also conceivably originate in star formation in the molecular gas or it could trace circumstellar emission . if star formation activity strictly depends on the kpc - scale local gas surface density , as in a kennicutt - schmidt relation @xcite , we would expect it to be asymmetrically distributed like the molecular gas . but in the region where the molecular gas is found ( radii less than 20 ) the 24 emission lacks this asymmetry , and this observation suggests that either the 24 emission is not primarily dust heated by star formation or that the local star formation rate is determined by something else in addition to the local gas density . the radio continuum emission does seem consistent with star formation in the molecular gas in both morphology and in total flux density ( section [ discussion ] ) . in linear size this is quite a large star - forming disk ; the radius of the molecular disk is 20= 6 kpc . thus , it seems likely that the 24 emission may arise partly from star formation activity , but it must also include a substantial contribution from more symmetrically distributed heat sources such as an evolved stellar population and a nuclear source . a unique feature of the 24 emission in ngc 807 is the very low level , smooth emission extending to the west and northwest sides of the galaxy at radii of 20 to 60 ( 6 to 19 kpc ; figures [ n807 ] and [ 807 - 24 - 160 ] ) . this emission is at surface brightness levels of 0.1 @xmath42 to 0.3 @xmath42 ( 3@xmath44 to 10@xmath44 ) , and it is not accounted for in the parametric fit described above . figure [ 807 - 24 - 160 ] shows that the 160emission is also extended to the northwest over the same spatial region . similar asymmetries are found in the broadband optical and hi images and are undoubtedly caused by a gravitational interaction that has left large - scale disturbances in the outer parts of the galaxy . for example , in the northeast and southwest corners of the hi image in figure [ 807 - 24 - 160 ] there are sections of tidal arms that stretch beyond the frame to radii of 5 ( 91 kpc ) . hi column densities northwest of the galaxy nucleus peak at 8.6 ( 9.4 pc@xmath45 , including helium ) at a resolution of 14 ( 4.2 kpc ) ; this hi column density occurs 50 from the nucleus of the galaxy , and is the brightest hi emission at radii @xmath46 by a factor of two . thus , the low level asymmetries in the 24 and 160 images of ngc 807 are probably consistent with an interstellar dust component that is distributed like the atomic gas . @xcite showed that the @xmath24 morphology of ngc 2320 is well described by an @xmath2 profile out to a semimajor axis of at least 90 , which bolsters its classification as an elliptical even though it has quite a large molecular gas mass of 4 . the @xmath41 image of figure [ n2320 ] clearly shows the inner dust disk to be 20 in diameter , with somewhat enhanced reddening 10 southeast of the nucleus . the unsharp - masked image also shows a bright ring of 35 diameter . molecular gas in ngc 2320 is distributed rather like the dust in the @xmath41 image and it also shows a `` tail '' of gas to the southeast of the nucleus . in contrast , the radio continuum emission is not well resolved , with fwhm @xmath47 1.7 @xcite , and it is believed to be powered by agn activity as described in greater detail by @xcite . unlike the co distribution in ngc 2320 , the 24 emission is notably symmetric . it is well described by a @xmath2 model whose half - light ellipse has semimajor and semiminor axes 4 @xmath39 2 ( 1.6 @xmath39 0.8 kpc ; table [ fittable ] ) . the 24@xmath2 model is much more compact than the optical emission from the galaxy , though . measured values for the effective or half - light radius of this galaxy range from 13 ( @xmath48 , 2mass extended source catalog ) to 29.5 ( @xmath24 , * ? ? ? * ) to 37(@xmath49 , * ? ? ? * ) , but all are significantly larger than our fitted half - light radii at 24 . we have confirmed the discrepancy in effective radii by convolving the 2mass @xmath48 image with the 24 psf and additionally by running our model fitting software on the original 2mass @xmath48 image to recover the effective radius in the same way as was done for the 24 image . we find an effective radius of 23.4 0.07 at @xmath48 , more than five times larger than the 24 effective radius . in this respect ngc 2320 is different from the ellipticals studied by @xcite , which had very similar effective radii at optical / nir and 24 . the difference in scale sizes implies that the 24 emission from ngc 2320 is not primarily circumstellar dust around evolved stars , unless there is a strong age gradient in the stellar population such that the inner arcseconds host a larger proportion of intermediate - age stars that are relatively bright at 24 . it is possible that the 24 emission in ngc 2320 is attributable to star formation activity that is restricted to the central portions of the molecular disk . the 24 emission does not show the same `` tail '' to the southeast that is seen in the molecular gas , but these might still be self - consistent if the molecular tail " is locally gravitationally stable ( perhaps not yet settled into its equilibrium orbit ) or if the local star formation rate has a dependence on gas surface density that is steeper than linear . the co column densities 10 southeast of the nucleus are at least a factor of 3 lower than the peak in the center of the galaxy , and if the star formation rate ( traced by 24 emission ) were proportional to the square of the local molecular surface density then the 24 emission 10 southeast of the nucleus would be a factor of 10 fainter than the nuclear value . but since the pointlike radio continuum emission can not be attributed to star formation activity the current data do not necessarily favor star formation as the heating source for the 24 emission . more detailed analysis of the ionized gas and the stellar population itself would provide important circumstantial evidence about the origin of the 24 emission in this case . co observations of this lenticular galaxy have recently been published by @xcite , and figure [ n3032 ] shows the comparison between optical , mid - ir , co and radio morphology . a clearly defined dusty disk of diameter 28 is viewed at rather low inclination . the molecular disk is coincident with this dust disk , but it also has a tail stretching to the southeast , and the tail has no obvious dust counterpart . since the molecular gas is nearly exactly counterrotating with respect to the stellar rotation @xcite , it is most likely to have been accreted from another galaxy or from the intergalactic medium into its retrograde orbit , and the asymmetric tail may be the remnant of the settling process . radio continuum emission from ngc 3032 is very marginally resolved but is elongated in the same direction as the dust and stellar disks . the 24emission from the galaxy is also very marginally resolved , being well fit by an exponential disk of scale length 1.8 ( 0.19 kpc ; table [ fittable ] ) . we find a marginal detection of an off - nuclear point source that may or may not be physically related to ngc 3032 . similarly to ngc 2320 , and unlike the cases described in @xcite , the 24 emission is a poor match to the optical / nir structure of the galaxy . the 2mass @xmath48 image of ngc 3032 was analyzed in the same way as we have done for the 24images , and it was found best described by a model with 15% of the total luminosity in a compact nuclear source ( probably a bright star cluster ) and 85% in a @xmath2 spheroid with an effective semimajor axis of 87 2 . thus , again , the 24 emission is not primarily circumstellar dust heated by evolved stars . the east - west elongations of the dust disk , the radio continuum emission and the bright part of the co emission suggest that indeed star formation activity could be powering the radio and 24emission from ngc 3032 . radio and fir flux densities are also consistent with star formation ( section [ discussion ] ) , and there is other evidence for star formation activity in the optical line ratios @xcite and stellar balmer line absorption @xcite . interestingly , the young counterrotating stellar core found by @xcite has a radius of roughly 2 , similar to the scale radius of the 24emission , and this young stellar core is rotating in the same direction as the molecular gas . thus the optical data are entirely consistent with our interpretation that radio and 24 emission in ngc 3032 arise in star formation . this star formation activity is most vigorous in the inner part of the molecular disk , which could be due to a nonlinear dependence of the star formation rate on the local gas density or to the gas disk being gravitationally stable in its outer regions . ngc 3656 is the most disturbed galaxy of this sample , being a clear example of a fairly recent merger remnant with a dramatic blue ring and shells . figure [ n3656 ] shows that the molecular gas is located in a nearly edge - on disk of radius @xmath43 10 = 1.9 kpc , also clearly visible as a dark dust lane bisecting the nucleus . the structure of the dust lane is somewhat irregular , suggesting a warped or folded lane rather than the very thin , flat , relaxed disks seen in ngc 4526 and ngc 4459 ( sections [ 4459 ] and [ 4526 ] ) . the radio continuum emission is clearly elongated in the same direction as the gas / dust disk , with a deconvolved gaussian source size ( hwhm ) of 3.0 @xmath39 1.4 @xcite . the 24 emission is only modestly resolved but is also clearly elongated north - south , and is adequately fit by an exponential disk of scale length 2 0.4 = 0.39 0.08 kpc and axis ratio 0.3:1.0 ( table [ fittable ] ) . thus , the morphologies of radio continuum , 24 , and co emission are all consistent with the interpretation that star formation powers the bulk of the radio and 24 emission . the scale sizes of the radio and 24 disks are substantially smaller than the molecular disk , as is also the case for ngc 3032 . this result suggests that the local star formation rate or efficiency is higher in the inner part of the gas disk . the virgo cluster lenticular ngc 4459 clearly shows a relaxed , thin dust disk of semimajor axis 9 = 0.7 kpc ( figure [ n4459 ] ) , and the unsharp - masked hst image resolves the dust into a broad , flocculent outer annulus plus an inner ring of semimajor axis 2 . the molecular gas lies in this regularly rotating disk , as shown in the channel maps presented by @xcite . the radio continuum emission from ngc 4459 is weak and unresolved in fact , it is unusually weak , and is discussed further in section [ discussion ] . the 24 emission is very symmetric and also centrally concentrated but clearly resolved ; it is well described by the sum of two exponential disks of scale radii 2.1 ( 0.16 kpc ) and 33(2.6 kpc ) . the more compact disk has 78% of the total 24flux density . the more extended ( 33 ) 24 component in ngc 4459 is unlikely to be driven by star formation activity because its scale length is so much larger than the size of the molecular gas and dust disk . it may in fact trace circumstellar dust in an evolved stellar population as discussed by @xcite . it is suggestive , though , that the size of the more compact 24 component , 2 , is a good match to the size of the inner dust ring in the top right panel of figure [ n4459 ] . the radio continuum source is also of a similar size . thus it is possible that the compact 24component and the radio continuum trace star formation activity in the central part of the gas disk ( though we can not rule out agn activity on morphological grounds alone ) . optical spectroscopy provides abundant evidence for active star formation in ngc 4459 ; the sauron data show that the molecular disk coincides with an ionized gas disk of low [ o iii]/h@xmath30 ratio @xcite , strong balmer absorption @xcite , and a dynamically cold stellar subpopulation @xcite . if , then , 80% of the 24and all of the radio continuum in ngc 4459 trace star formation activity , this is another case in which the efficiency must be much higher in the inner part of the gas disk than in the outer part . ngc 4476 is the second of our three virgo cluster early - type galaxies with well - developed dust disks ( figure [ n4476 ] ) . the f475w image also shows numerous bright point sources embedded in the dust . as for ngc 4459 and ngc 4526 , the molecular gas is confined to this disk , which has a semimajor axis of 11 = 0.9 kpc . the bulk of the 24emission can also be attributed to this dusty disk , as an excellent model of the image is achieved from a ring with its maximum intensity at a semimajor axis of 6 ( 0.5 kpc ) and gradually decreasing intensities on the interior and exterior ( table [ fittable ] ) . thus , there is extremely good agreement between 24 morphology and co morphology . no radio continuum emission is detected from ngc 4476 , and @xcite hypothesize that the magnetic field and/or the relativistic electrons may have been stripped by interactions with the intracluster medium . circumstantial evidence for this idea comes from the fact that no atomic gas is detected either @xcite . we can not , therefore , use the radio continuum morphology to argue that the 24 emission indicates star formation activity in this case . however , galex images of ngc 4476 show significantly bluer colors at radii @xmath47 12 , becoming as blue as ( fuv - nuv ) @xmath40 1.2 mag in the central resolution element compared to ( fuv - nuv ) @xmath40 2.5 mag in the outskirts of the galaxy @xcite . in the absence of the radio continuum detection it is not obvious that the 24 emission indicates star formation activity , but the optical point sources and the galex colors strongly suggest this interpretation . ngc 4526 is the third of our virgo cluster early - type galaxies with well - developed molecular and dust disks ( figure [ n4526 ] ) . the gas / dust disk is seen at high inclination and has a sharp outer edge at a semimajor axis of 15 = 1.2 kpc . the radio continuum morphology closely matches that of the molecular gas ; the 24emission is described by a plateau or a flat - topped disk of semimajor axis 7.5 ( 0.6 kpc ) and a more extended , rounder exponential disk of scale length 28 ( 2.4 kpc ) . the plateau contains 82% of the total flux density ; given its size scale and orientation it is highly likely that this 24 component is associated with the radio continuum emission via star formation in the molecular gas . the more extended , fainter and rounder 24 component could well be attributed to circumstellar dust as in @xcite , but it only contributes 18% of the 24 flux density . again , similar to the case of ngc 4459 , the optical spectroscopy traces star formation activity and its after - effects in an ionized gas disk @xcite and a young , dynamically cold stellar disk @xcite that have similar sizes to the molecular disk . ngc 5666 was originally classified as an elliptical , but upon closer inspection , its morphological status is uncertain . color images ( e.g. figure [ n5666 ] ) clearly show the red nucleus and a blue ring of radius approximately 5 , which are also shown by @xcite ( dd03 ) . the ratio of the v image to its best - fit mge model also shows one spiral arm at radii of 15 20 , and dd03 find blue star - forming knots in the arm . the arm would tend to suggest a spiral classification but it is not obvious that the bulge + exponential disk surface brightness fit of dd03 is well constrained or is a better fit to the optical surface brightness profile than a @xmath2 law . in addition , dd03 comment that their bulge + disk decomposition has an unusually extended bulge and compact disk , with the effective radius of the bulge being larger than the scale length of the disk . the galaxy therefore does not seem a particularly close match to either a prototypical elliptical or a prototypical spiral . measurements of the stellar velocity dispersion and rotation velocity @xmath50 would give better physical insight into the structure of this galaxy . the molecular gas in ngc 5666 ( figure [ n5666 ] ) is found in an inclined disk of radius roughly 78 . radio continuum emission is distributed in an asymmetric ring , brightest in the southeast and the northwest , with a central dip . local maxima in the radio continuum emission are at radii of 3.5 . the best fit model to the 24 image is a flat - topped disk or plateau of radius 5.9= 0.9 kpc , but the outer scale length is so small that the model is essentially an inclined top hat . the observed 24peak is offset a few arcseconds to the southeast of the galaxy nucleus and it coincides with the peak of the radio continuum emission . in this case the very close matches between the size of the 24plateau , the extent of the molecular disk and the radio continuum ring ( as well as the blue ring in optical images ) strongly suggest that both the radio and 24 emission are driven by star formation activity . models in which the 24 emission comes from a point source or is distributed like the stars are strongly ruled out . the non - rotated images at 70 and 160 were fit with the isophotal analysis program _ ellipse _ in stsdas , and the resulting surface brightness profiles are shown in figures [ 70sb ] and [ 160sb ] . from these comparisons it is evident that , with the exception of ngc 807 , the targets are not resolved at 70 and 160 . ngc 807 is modestly resolved at 160 and shows an elongation in the same sense as the 24 image , as discussed already in section [ morph807 ] . the 24 morphologies in these co - rich early - type galaxies show a variety of structures including disks , rings , point sources , and very faint smooth emission . we have argued that , at least in the cases of ugc 1503 , ngc 3032 , ngc 3656 , ngc 4526 , and ngc 5666 , and possibly also in ngc 4476 , the bulk of the 24 emission must arise from dust heated by star - forming regions . in ngc 4526 , we attribute the 24 emission from the elongated , flat - top disk component ( 82% of the total flux density ) to star formation and the rest to circumstellar emission . in ngc 4459 , we have argued that 78% of the 24 emission is in a modestly - resolved component that is probably heated by star formation though it could conceivably also be powered by an active nucleus ; the remaining 22% we attribute to circumstellar dust . in ngc 807 , approximately half to 95% of the 24 emission may be related to star formation activity . in ngc 2320 the origin of the 24 emission is not clear , so the most that can be said is that studies of the stellar populations and the ionized gas distribution and line ratios would provide useful clues . our morphological work also indicated that in our co - rich early type galaxies the 24 emission usually traces the molecular gas and the silhouette dust disks more closely than it does the stellar distribution . in some cases this is fairly obvious , as the 24emission is best modeled by a ring or a structure that is flat at radii 6 to 11 . in other cases the 24 model is centrally peaked , but it has a radial scale length many times smaller than that of the stellar distribution as we have verified by analyzing near - ir @xmath48 images in the same way as the 24images . this result is an obvious contrast to the findings of @xcite that the 24 emission in their elliptical galaxies closely traced the optical / nir star light , with very similar size scales and fairly tight correlations between @xmath51 and @xmath52 . however , that circumstellar dust should also be present in all of our targets . it must simply be outshone by the star formation . as a test of this hypothesis , figure [ temifig ] compares the 24 flux densities and @xmath53 apparent magnitudes of our sample galaxies to those of @xcite . integrated @xmath53 total magnitudes are taken from the two micron all sky survey and its extended objects final release as tabulated in the nasa extragalactic database . the median 24 / 2.2 flux density ratio in the co - rich early - type galaxies is a factor of 15 higher than that in the sample of @xcite ; in other words , the 24 flux densities are 15 times larger than would be expected if the 24 emission were entirely circumstellar in origin . in figure [ temifig ] we also show the flux densities of the extended 24 components of ngc 4459 and ngc 4526 . on the basis of morphology we had previously argued that those components would be consistent with the circumstellar emission ; here we show that their flux densities also support this interpretation because they fall on the @xmath54 correlation defined by the co - poor ellipticals . radio to fir flux density ratios confirm the suggestions of star formation activity in all cases except ngc 2320 . for the computation of these ratios we follow @xcite ( yrc ) , who have used iras 60 and 100 flux densities ; the choice is made because yrc provide the most complete comparison set of fir and radio continuum flux densities for many types of galaxies . the logarithmic fir / radio ratio @xmath55 is presented in figure [ qplot ] for our sample of co - rich early type galaxies as well as for yrc s sample ( which includes all galaxies having 100 flux densities greater than 2 jy ) and the early - type galaxies of @xcite . according to the definitions of yrc , ngc 2320 is a radio excess galaxy whose radio continuum is likely powered by an agn ; this classification is consistent with its point source radio morphology . ngc 4476 and ngc 4459 are fir - excess galaxies . the remainder of the galaxies have fir / radio ratios @xmath55 that are consistent with those of star forming galaxies and that confirm our inferences based on the 24morphology . for comparison , the majority of the elliptical galaxies studied by @xcite are radio excess galaxies hosting agn emission , though some of them do also sit near the star formation fir - radio correlation . the fir - excess galaxies are rare , comprising 9 of 1809 galaxies in the sample of yrc , although they are somewhat more common in other samples of spirals @xcite . being rare , they are valuable for the insights they give into the physical processes underlying the fir / radio correlation . for example , both yrc and @xcite note that some of the fir - excess galaxies have unusually high dust temperatures as indicated by iras 60/100 flux density ratios @xmath56 . others have `` normal '' dust temperatures . yrc have argued that the `` hot dust '' fir excess galaxies may be very densely enshrouded agn or compact nuclear starbursts . @xcite have argued that a few of the hot dust fir excess galaxies are likely to be cases of recently initiated starbursts , in which the starburst began so recently that there has not yet been time for appreciable numbers of supernovae to accelerate the cosmic ray population and produce the cm - wave synchrotron emission . they have also argued that `` cool dust '' fir - excess galaxies are dominated by heating from the evolved stellar population , not star formation ( c.f . * ? ? ? all of the co - rich targets studied here fall into the `` cool dust '' category , and our fir excess galaxies ngc 4476 and ngc 4459 have iras 60/100 flux density ratios of 0.36 and 0.39 respectively . thus , of the three preceding explanations for the fir excess phenomenon , we reject the idea of highly enshrouded agn in our targets . we have also used morphology to argue that the bulk of the 24 emission does not originate from dust heated by the evolved stellar population in ngc 4459 . it is possible that recently initiated star formation could be responsible for the relatively low level of radio continuum emission in ngc 4476 and ngc 4459 ( and perhaps also in ngc 0807 and ngc 4526 , which have relatively high @xmath55 values ) . close inspection of their stellar populations could test that idea . an alternative , possibly viable explanation is that the weak radio continuum flux densities may be related to unusually weak magnetic fields . in the case of the virgo cluster members ngc 4459 , ngc 4476 , and ngc 4526 , the interactions with the intracluster medium could be responsible for stripping the plasma and/or the magnetic fields in a more advanced version of the stripping that is now evident in the virgo spirals studied by @xcite . high quality radio polarization imaging would help to answer this question , and such work would also be valuable because rather little is known about the strengths of interstellar magnetic fields in early - type galaxies . @xcite have made careful models of the mid- to far - ir emission in nearby spirals and have concluded that the fir emission ( specifically , emission at wavelengths longer than 100 ) originates in the diffuse ism , not star forming regions . if this is true , then the fir emission in spirals may be a better tracer of the total dust mass than of the total light emitted from star - forming regions . fir emission in spirals may still be associated with star formation , but as a cause rather than an effect ; the association could be through the schmidt law in the same way that molecular clouds are associated with star formation . in this case , the tightness of the radio - fir correlation is still puzzling . galaxies that are rich in cold neutral gas but are not forming stars ( such as ngc 2320 ? ) should be particularly valuable test cases for disentangling the physical processes driving these emissions , and it would be helpful to have fir images at spatial resolutions matching those of the molecular gas maps . aside from the slight extension in the 160 image of ngc 807 there is still very little information on the resolved fir emission of early - type galaxies . one notable exception is that the ground - based observations of @xcite have clearly resolved the 350 emission in at least ugc 1503 , ngc 807 , ngc 3656 , and ngc 4476 , and they show it to be elongated at the same orientation as our 24 and co images . @xcite find an empirical relation between the 24luminosity of a star - forming region or galaxy and its star formation rate : @xmath57 here @xmath58 is the monochromatic luminosity at 24 in units of erg s@xmath14 , given by @xmath59 with @xmath60 being the flux density at 24 and @xmath61 the central frequency of the 24 band . if this relation is accurate in early - type galaxies as well ( e.g. if the initial mass function is the same in both types of galaxies , among other issues ) , the star formation rates in our co - rich targets are on the order of a few tenths of a solar mass per year ( table [ sfrs ] ) . gas depletion timescales can be estimated from the total molecular gas mass ( molecular hydrogen with a correction for helium ) divided by the star formation rate . those timescales are a few gyr , with one exception . the 24 luminosity of ngc 2320 is so low compared to its total molecular mass that even if all of its 24 luminosity were powered by star formation the gas depletion timescale would be longer than the hubble time . since the molecular gas in the early - type galaxies settles into kpc - scale disks @xcite , the young stars will also be dynamically cold populations . the co - rich early - type galaxies , with the exceptions already discussed , are building disky stellar components on timescales of a few gyr . indeed , in ngc 4526 the dynamically cold stellar disk is already visible in the stellar kinematics @xcite . however , since the total molecular gas masses are only a few to a few , the young stellar populations will not make up more than a few percent of the total stellar mass . the gas depletion timescales in table [ sfrs ] are similar to the corresponding timescales in the disks of spiral galaxies . depletion timescales are , of course , very sensitive to assumptions about initial mass functions and co - to - h@xmath62 conversion factors , but estimates for spirals tend to be in the range of a few gyr @xcite and the values in table [ sfrs ] are 1 to 4 gyr in seven of nine cases . in other words , while the elliptical and lenticular galaxies have smaller molecular masses than spirals , their star formation efficiencies are similar to within factors of a few . @xcite has also pointed out that recycled matter from stellar evolution ( especially from the more massive stars ) may make the gas reservoir last a factor of 2 to 3 times longer than the simple depletion timescale estimate . thus , we can expect the molecular disks in the early - type galaxies to remain for many gyr , perhaps as long as another hubble time , unless there are significant destructive processes such as stripping , interactions with hot gas , and/or possible agn feedback . ram pressure stripping will be more intense in clusters than in the field , and tidal interactions more frequent , so the environments of our co - rich early - type galaxies are also of interest . ngc 4476 , ngc 4459 , and ngc 4526 are members of the virgo cluster and ngc 2320 is a member of the abell 569 cluster ; ngc 4476 is already known to have suffered some ram pressure stripping @xcite , but it still retains its molecular gas at least for the time being . to put this disk growth into the broader context of galaxy formation and evolution , it is important to know what ( if anything ) the molecular content of an early - type galaxy reveals about its history . however , the origins of the molecular gas in early - type galaxies are not yet known in most cases , and it is not yet clear why some early - type galaxies are rich in molecular gas while others are not . for many years there has been speculation that the molecular gas could originate from the mass loss of evolved stars ( e.g. * ? ? ? * and references therein ) . in other cases such as ngc 3032 @xcite and ngc 2768 @xcite , the clear kinematic misalignments between molecular gas and stars indicate that the molecular gas was accreted from some external source or possibly it is leftover from a major merger . it will be necessary to obtain larger numbers of molecular gas maps and better statistics on the co content of the early - type galaxies before we will understand what fraction of their molecular gas could have come from internal and external origins , or whether the co - rich and the co - poor galaxies have had systematically different formation histories . for the majority of the co - rich early - type galaxies , the close agreements between co , 24 , and radio continuum morphologies suggest that the bulk of the 24 emission should be attributed to star formation activity . in our sample , these galaxies include ugc 1503 , ngc 3032 , ngc 3656 , ngc 4459 , ngc 4526 , and ngc 5666 . the 24 emission in ngc 4476 is likely due to star formation . a portion of the 24 emission from ngc 807 may also trace star formation activity , but the origin of the 24emission in ngc 2320 is not yet clear . radio and fir flux density ratios are consistent with this interpretation , as are the increased @xmath63 flux denstiy ratios of the co - rich over the co - poor early - type galaxies . thus , one of the major implications of this work is that the co , radio , and mips data are roughly consistent with the uv results implying star formation activity in a few tens of percent of the nearby early - type galaxies . the necessary raw material is present , more or less , and the molecular gas is often being processed into stars . in the cases of ngc 3032 , ngc 3656 , and ngc 4459 , the star formation activity seems to be taking place on more compact spatial scales than the distribution of the gas itself . this situation could arise if the star formation rate is a strong ( nonlinear ) function of the local gas surface density , or if the molecular disks are only unstable to star formation in their inner portions . however , the origin of the 24 emission in the co - rich early - type galaxies is not entirely clear - cut . an extended component in ngc 807 ( detected both at 24 and at 160 ) may trace dust in the diffuse atomic ism . extended 24 components in ngc 4459 and ngc 4526 may arise from dust around the evolved stars , just as @xcite find in the co - poor early - type galaxies . likewise , though ngc 2320 is very co - rich , the mismatches between co , radio continuum and 24 morphology make it unclear how much ( if any ) star formation is occurring there . in these cases , detailed comparisons with the uv morphologies and the stellar populations would provide firmer information about the distribution of star formation activity ( or the lack of it ) within the galaxies molecular disks . this work also opens the way for more quantitative tests of theoretical and phenomenological models of the star formation process . for example , with a model of the gravitational potential ( from the stellar distribution and a mass - to - light ratio ) , it would be possible to test whether the toomre - type local gravitational instability is consistent with the locations of star formation activity , as discussed by kawata et al . the gas - rich early - type galaxies could also provide useful perspective on the workings of the kennicutt - schmidt relationship between the star formation rate and the gas surface density . if these models have truly captured some underlying physics of the star formation process they ought to work in the ellipticals and lenticulars as well as in the spirals . lmy thanks the university of oxford sub - 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@xmath69 & & 14 & 180 & 37 ( 5 ) & 8 + ngc 4459 & 12 29 00.0 & + 13 58 43 & 16.1 ( 0.4 ) & s0@xmath70 & @xmath71 & 0.97 & 25 & 174 & 1.7 ( 0.3 ) & 6 + ngc 4476 & 12 29 59.2 & + 12 20 55 & 17.6 ( 0.6 ) & s0@xmath72 & @xmath73 & 0.85 & 10 & 53 & 1.0 ( 0.1 ) & 6 + ngc 4526 & 12 34 03.0 & + 07 41 57 & 17.3 ( 1.5 ) & s0@xmath67 & @xmath65 & 0.98 & 44 & 256 & 6.3 ( 1.1 ) & 2 , 4 , 7 + ngc 5666 & 14 33 09.1 & + 10 30 37 & 31 ( 2 ) & ? & @xmath74 & 0.86 & 6 & & 4.5 ( 0.6 ) & 8 + lccccc ugc 1503 & 5.0@xmath394.5 & 0.038 & 0.33 & 8.7 & ly07 + ngc 0807 & 10.9@xmath398.8 & 0.056 & 0.27 & 4.8 & ly07 + ngc 2320 & 5.4@xmath395.4 & 0.17 & 13.1 & 77 & first + ngc 3032 & 5.4@xmath395.4 & 0.14 & 1.22 & 8.7 & first + ngc 3656 & 4.5@xmath393.8 & 0.037 & 9.4 & 254 & ly07 + ngc 4459 & 5.4@xmath395.4 & 0.15 & 0.86 & 5.7 & first + ngc 4476 & 4.5@xmath394.5 & 0.17 & @xmath75 0.5 & @xmath75 3 & ly07 + ngc 4526 & 5.4@xmath395.4 & 0.15 & 3.75 & 25 & first + ngc 5666 & 4.6@xmath394.4 & 0.050 & 1.9 & 38 & ly07 + lrrrr ugc 1503 & 0.0482 0.0019 & 0.793 0.079 & 1.49 0.18 & 25.0 0.7 + ngc 0807 & 0.0689 0.0028 & 0.608 0.061 & 2.39 0.29 & 22.2 0.5 + ngc 2320 & 0.0220 0.0009 & 0.385 0.077 & 1.12 0.13 & 22.9 0.5 + ngc 3032 & 0.165 0.007 & 2.66 0.266 & 3.32 0.40 & 27.6 0.6 + ngc 3656 & 0.169 0.007 & 3.63 0.36 & 3.81 0.46 & 27.5 0.8 + ngc 4459 & 0.129 0.005 & 3.10 0.19 & 3.76 0.45 & 27.1 0.6 + ngc 4476 & 0.0400 0.0016 & 0.886 0.133 & 1.38 0.17 & 25.6 0.7 + ngc 4526 & 0.314 0.013 & 10.2 0.61 & 12.7 1.5 & 26.5 0.5 + ngc 5666 & 0.162 0.006 & 3.00 0.30 & 2.48 0.30 & 28.0 0.8 + lccccc ugc 1503 & 0.24 0.05 & 1.44 0.15 & 0.50 0.04 & 2.5 0.5 & 2.57 0.09 + ngc 0807 & 0.45 0.09 & 1.83 0.12 & 0.41 0.03 & 1.2 0.4 & 2.91 0.15 + ngc 2320 & & 1.60 0.16 & 0.26 0.02 & 19.3 0.7 & 1.60 0.03 + ngc 3032 & & 4.70 0.47 & 1.94 0.10 & 7.2 0.5 & 2.66 0.04 + ngc 3656 & 0.69 0.14 & 6.58 0.66 & 2.54 0.13 & 19.8 0.7 & 2.35 0.03 + ngc 4459 & & 4.82 0.48 & 1.87 0.09 & 1.8 0.2 & 3.25 0.05 + ngc 4476 & 0.28 0.06 & 1.84 0.18 & 0.66 0.05 & @xmath75 0.5 & @xmath4 3.38 + ngc 4526 & & 17.1 1.7 & 5.56 0.05 & 12.0 0.5 & 2.94 0.03 + ngc 5666 & 0.52 0.10 & 3.98 0.40 & 1.99 0.10 & 17.5 0.7 & 2.24 0.03 + lcccccl ugc 1503 & y & y & y & & y & sf ring . + ngc 0807 & y & plateau / disk : y & y & & y & some sf . + & & pt src : no & no & & & + ngc 2320 & no & no & no & & no & no sf ? + ngc 3032 & y & y & y & 1 , 2 , 3 & y & sf inner disk . + ngc 3656 & y & y & y & & y & sf inner disk . + ngc 4459 & no & 2 disk : y & y & 1,2 & y & sf inner disk . + & & 33 disk : no & no & & & + ngc 4476 & & & y & 4 , 5 & & sf ring . + ngc 4526 & y & plateau : y & y & 1 , 2 & y & sf inner disk . + & & 28 disk : no & no & & & + ngc 5666 & y & y & y & 6 & y & sf ring . + lc total flux density & 46.6 1.3 mjy + radius of maximum & 4.80.4 ( 1.65 0.14 kpc ) + outer scale length & 3.20.2 ( 1.10 0.07 kpc ) + inner scale length & 2.20.6 ( 0.8 0.2 kpc ) + axis ratio & 0.73 0.03 + position angle & 582 + plateau outer scale & 3.770.12 ( 1.21 0.04 kpc ) + plateau axis ratio & 0.423 0.002 + plateau pa & @xmath76 0.06 + + exp . disk flux density & 31.23 0.07 mjy + exp . disk scale length & 16.990.14 ( 5.44 0.04 kpc ) + exp . disk axis ratio & 0.577 0.004 + exp . disk pa & @xmath77 0.2 + + nuclear point source & 3.36 0.08 mjy + total flux density & 20 2 mjy + effective radius ( major ) & 4.060.06 ( 1.65 0.02 kpc ) + effective radius ( minor ) & 1.880.06 ( 0.77 0.2 kpc ) + pa & @xmath78 2 + exp . disk flux density & 140 10 mjy + exp . disk scale length & 1.830.04 ( 0.188 0.004 kpc ) + exp . disk axis ratio & 0.808 0.006 + exp . disk pa & 636 + + off - nuclear point source & 6 14 mjy + exp . disk flux density & 146 4 mjy + exp . disk scale length & 2.00.4 ( 0.39 0.08 kpc ) + exp . disk axis ratio & 0.30 0.05 + exp . disk pa & @xmath79 1.4 + disk 1 flux density & 100 1 mjy + disk 1 scale length & 2.080.02 ( 0.162 0.001 kpc ) + disk 1 axis ratio & 0.706 0.006 + disk 1 pa & @xmath80 0.7 + + disk 2 flux density & 29 5 mjy + disk 2 scale length & 338 ( 2.6 0.6 kpc ) + disk 2 axis ratio & 0.85 0.03 + disk 2 pa & @xmath81 2 + ring total flux density & 31.6 0.2 mjy + radius of maximum & 6.00.2 ( 0.51 0.02 kpc ) + outer scale length & 1.290.07 ( 0.110 0.006 kpc ) + inner scale length & 3.70.6 ( 0.32 0.05 kpc ) + axis ratio & 0.447 0.009 + pa & 24.50.6 + + off - nuclear point source & 5 2 mjy + plateau total flux density & 254 1 mjy + plateau radius & 7.50.1 ( 0.629 0.003 kpc ) + outer scale length & 1.91 0.06 ( 0.160 0.005 kpc ) + axis ratio & 0.20099 0.00017 + pa & @xmath82 0.14 + + exp . disk flux density & 54.0 0.6 mjy + scale length & 28.41.2 ( 2.4 0.1 kpc ) + axis ratio & 0.397 0.004 + pa & @xmath83 0.4 + total flux density & 1443 mjy + plateau radius & 5.870.12 ( 0.88 0.02 kpc ) + outer scale length & 0.50.4 ( 0.08 0.06 kpc ) + axis ratio & 0.86 0.26 + pa & @xmath84 28 + lccccr ugc 1503 & 42.57 & 0.60 & 4.3 + ngc 0807 & 42.66 & 0.69 & 2.7 + ngc 2320 & 42.37 & @xmath75 0.40 & @xmath4 17 + ngc 3032 & 42.05 & 0.21 & 3.2 + ngc 3656 & 42.61 & 0.66 & 7.7 + ngc 4459 & 41.71 & 0.083 & 2.8 + ngc 4476 & 41.27 & 0.043 & 3.2 + ngc 4526 & 42.15 & 0.22 & 4.0 + ngc 5666 & 42.37 & 0.40 & 1.5 + optical , ir , co and radio continuum morphology of ugc 1503 . optical contours are spaced by a factor of two . a @xmath41 image shows no discernible structure but the unsharp masking and model division techniques do show some faint structure . contour levels in the 24 image are 0.12 , 0.31 , 0.61 , 1.22 , 1.83 , 3.05 , 4.27 , and 4.70 mjy sr@xmath14 ( the lowest three are 4.4@xmath44 , 11@xmath44 , and 22@xmath44 ) . contour levels in the co image are @xmath85 , 0.63 , 1.27 , 1.90 , 3.12 , 4.43 , and 5.70 @xmath86 . contours in the 1.4 ghz radio continuum image are 0.09 , 0.12 , 0.15 , 0.18 , and 0.24 @xmath87 ( 2.4 , 3.1 , 3.9 , 4.7 , and 6.3@xmath44 ) . in this and subsequent figures , green circles in the lower right corners of panels indicate the resolution ( fwhm ) of their respective data . contours are colored either white or black as necessary to enhance their visibility . ] optical , ir , co and radio continuum morphology of ngc 807 . optical contours are spaced by a factor of two . the top row , middle panel is a @xmath41 image and the top row , right panel is a @xmath24 image divided by a smooth mge model constructed using the software of @xcite . contours in the 24 image are 0.12 , 0.29 , 0.58 , 1.16 , 1.74 , 2.91 , 4.07 , and 5.23 mjy sr@xmath14 ( the lowest three are 4.3@xmath44 , 10@xmath44 , and 20@xmath44 ) . contours in the co image are @xmath88 , 0.77 , 1.53 , 2.30 , 3.83 , 5.33 , and 6.88 @xmath86 . contours in the 1.4 ghz radio continuum image are @xmath89 , @xmath90 , @xmath91 , 0.1 , 0.15 , 0.2 , and 0.25 @xmath87(@xmath92 , @xmath93 , @xmath94 , 1.8 , 2.7 , 3.6 , and 4.5@xmath44 ) . ] large - scale optical , ir , and hi morphology of ngc 807 . optical contours are spaced by a factor of two . contour levels in the 24 image are the same as in figure [ n807 ] . contour levels in the 160 image are 1.14 , 2.28 , 4.56 , 6.84 , 11.4 , 16.0 , and 20.5 mjy sr@xmath14 ( the lowest three are 2.0@xmath44 , 4.1@xmath44 , and 8.2@xmath44 ) . the hi integrated intensity image has a resolution of 14 @xmath39 13 and contour levels are at 0.01 , 0.02 , 0.04 , 0.06 , 0.10 , 0.14 , and 0.18 @xmath86 , where the peak in the hi intensity is 0.20 @xmath86 = 1.2 . ] optical , ir , co and radio continuum morphology of ngc 2320 . optical contours are spaced by a factor of two . contours in the 24 image are 0.11 , 0.28 , 0.56 , 1.13 , 1.69 , 2.82 , 3.94 , and 5.08 mjy sr@xmath14 ( the lowest three are 3.9@xmath44 , 9.8@xmath44 , and 20@xmath44 ) . contours in the co image are @xmath95 , 1.46 , 2.91 , 5.82 , 8.73 , 14.5 , 20.4 , and 26.2 @xmath86 . contours in the 1.4 ghz radio continuum image are @xmath96 , 0.39 , 1.31 , 2.62 , 6.55 , and 11.8 @xmath87 ( @xmath97 , 2.3 , 7.7 , 15 , 39 , and 69@xmath44 ) . ] optical , ir , co and radio continuum morphology of ngc 3032 . optical contours are spaced by a factor of two . the unsharp - masked image is from a wfpc2 f606w image . contours in the 24 image are 0.51 , 1.01 , 2.53 , 5.06 , 10.1 , 15.2 , 25.3 , 35.4 , and 45.5 mjy sr@xmath14 ( the lowest three are 16@xmath44 , 32@xmath44 , and 81@xmath44 ) . contours in the co image are @xmath98 , 1.81 , 3.62 , 5.43 , 9.05 , 12.7 , and 16.3 @xmath86 . contours in the 1.4 ghz radio continuum image are 3.2 , 5.4 , and 7.5 times the rms noise level ( 0.14 @xmath87 ) . ] optical ( @xmath24 ) , ir , co and radio continuum morphology of ngc 3656 . optical contours are spaced by a factor of two . the unsharp - masked image is made from the @xmath24 image . contours in the 24 image are 0.57 , 1.15 , 2.87 , 5.75 , 11.5 , 17.2 , 28.7 , 40.2 , and 51.7 mjy sr@xmath14 ( the lowest two are 23@xmath44 and 26@xmath44 ) . contours in the co image are 4.05 , 8.09 , 16.2 , 24.3 , 40.5 , 56.6 , and 72.8 @xmath86 . contours in the 1.4 ghz radio continuum image are @xmath99 , 0.09 , 0.3 , 0.9 , 2.4 , and 6.0 @xmath87 ( @xmath100 , 2.4 , 8.1 , 24 , 64 , and 162@xmath44 ) . ] optical , ir , co and radio continuum morphology of ngc 4459 . optical contours are spaced by a factor of two . contours in the 24 image are 0.33 , 0.66 , 1.65 , 3.30 , 6.59 , 9.89 , 16.5 , 23.1 , and 29.7 mjy sr@xmath14 ( the lowest three are 8@xmath44 , 15@xmath44 , and 39@xmath44 ) . contours in the co image are @xmath101 , 2.14 , 4.28 , 6.42 , 10.7 , 15.0 , and 19.3 @xmath86 . contours in the 1.4 ghz radio continuum image are @xmath102 , 3 , 4 , and 5 times the rms noise level ( 0.15 @xmath87 ) . ] optical , ir , co and radio continuum morphology of ngc 4476 . optical contours are spaced by a factor of two . contours in the 24 image are 0.40 , 0.80 , 1.60 , 2.40 , 4.0 , 5.6 , and 7.2 mjy sr@xmath14 ( the lowest three are 13@xmath44 , 25@xmath44 , and 50@xmath44 ) . contours in the co image are @xmath103 , @xmath104 , 1.24 , 2.48 , 3.72 , 6.20 , 8.68 , and 11.2 @xmath86 . radio continuum emission from ngc 4476 is undetected @xcite . ] optical , ir , co and radio continuum morphology of ngc 4526 . optical contours are spaced by a factor of two . contours in the 24 image are 0.69 , 1.38 , 3.45 , 6.90 , 13.8 , 20.7 , 34.5 , 48.3 , and 62.1 mjy sr@xmath14 ( the lowest three are 7@xmath44 , 15@xmath44 , and 37@xmath44 ) . contours in the co image are @xmath105 , 3.71 , 7.42 , 11.1 , 18.6 , 26.0 , and 33.4 @xmath86 . contours in the 1.4 ghz radio continuum image are @xmath106 , 0.38 , 0.75 , 1.88 , and 3.37 @xmath87 ( @xmath107 , 2.5 , 5.0 , 12.5 , and 22@xmath44 ) . ] optical , ir , co and radio continuum morphology of ngc 5666 . optical contours are spaced by a factor of two . the top right panel is constructed by dividing the @xmath24 image with a smooth elliptical mge model . contours in the 24 image are 0.33 , 0.67 , 1.66 , 3.33 , 6.66 , 9.99 , 16.7 , 23.3 , and 30.0 mjy sr@xmath14 ( the lowest three are 12@xmath44 , 24@xmath44 , and 58@xmath44 ) . contours in the co image are @xmath108 , 1.68 , 3.35 , 5.03 , 8.38 , 11.7 , and 15.1 @xmath86 . contours in the radio continuum image are @xmath109 , 0.12 , 0.4 , 0.8 , 1.6 , and 1.8 @xmath87 ( @xmath100 , 2.4 , 8 , 16 , 32 , and 36@xmath44 ) . ] the 24 flux density and @xmath53 apparent magnitude for the co - rich early - type galaxies ( circles ) . for comparison , the sample of ellipticals studied by @xcite are shown as crosses and spirals from the sings sample are shown as squares @xcite . in addition , for ngc 4459 and ngc 4526 we show both the total 24 flux density and the more extended component by itself , with dotted lines connecting the symbols . uncertainties are comparable to or smaller than the symbol sizes . the solid line is not a fit but is instead a representative linear relationship between the 24 and @xmath53 flux density . in the generally co - poor ellipticals of @xcite , the 24 emission is thought to be primarily associated with mass loss from the evolved stars . the extended 24 components in ngc 4459 and ngc 4526 have 24/@xmath53 ratios consistent with this behavior . ] . the circles and the large arrow are the co - rich early - type galaxies studied here , and crosses are the data of @xcite . as defined by those authors , the flux ratio @xmath55 has a mean value of 2.34 for star - forming galaxies ( mostly spirals ) ; the lines at @xmath110 and @xmath111 indicate the ir excess and radio excess boundaries , respectively , at roughly @xmath112 from the mean . uncertainties are comparable to or smaller than the symbol sizes , though we have not included the effect of the distance uncertainty on the 60 luminosity . the ratio @xmath55 is , of course , independent of distance . evidently the co - rich early - type galaxies follow the same radio / fir relation as the star forming spirals . another sample of 48 elliptical galaxies from @xcite are shown in triangles and small arrows . [ qplot ] ]

from the atmospheric and solar neutrino data , there is increasing evidence for neutrino oscillations@xcite . if this is a correct interpretation , the standard model ( sm ) has to be extended to incorporate the small masses of the neutrinos suggested by data . there have been several ideas proposed in literature to generate small neutrino masses . the zee - model is one of such attempts@xcite . in this model , all flavor neutrinos are massless at the tree level , and their small masses are induced radiatively through one - loop diagrams . for such a mass - generation mechanism to work , it is necessary to extend the higgs sector of the sm to contain at least two weak - doublet fields and one weak - singlet charged scalar field . although some studies have been done to examine the interaction of the leptons and the higgs bosons in the zee - model@xcite , the scalar ( higgs ) sector of the model remains unexplored in detail . in this paper we study the higgs sector of the zee - model to clarify its impact on the higgs search experiments , either at the cern lep - ii , the run - ii of the fermilab tevatron , the cern large hadron collider ( lhc ) , or future linear colliders ( lc s ) . experimental search for the higgs boson has been continued at the cern lep and the fermilab tevatron experiments . in the lep - ii experiments , the higgs boson with the mass less than about 110 gev has been excluded if its production cross section and decay modes are similar to those of the sm higgs boson@xcite . run - ii of the tevatron can be sensitive to a sm - like higgs boson with the mass up to about 180 gev , provided that the integrated luminosity of the collider is large enough ( about @xmath2)@xcite . furthermore , the primary goal of the cern lhc experiments is to guarantee the discovery of a sm - like higgs boson for its mass as large as about 1 tev@xcite , which is the upper bound of the sm higgs boson mass . ( for a higgs boson mass beyond this value , the sm is no longer a consistent low energy theory . ) when the higgs boson is discovered , its mass and various decay properties will be measured to test the sm and to distinguish models of new physics at high energy scales . for example , the allowed mass range of the lightest cp - even higgs boson ( @xmath0 ) can be determined by demanding the considered theory to be a valid effective theory all the way up to some cut - off energy scale ( @xmath3 ) . for @xmath4 gev ( i.e. , the planck scale ) , the lower and upper bounds of the sm higgs boson masses are 137 gev and 175 gev , respectively @xcite . the higgs mass bounds for the two - higgs - doublet - model ( thdm ) were also investigated@xcite with and without including the soft - breaking term with respect to the discrete symmetry that protects the natural flavor conservation . it was found in ref . @xcite that the lower bound of the lightest cp - even higgs boson is about 100 gev in the decoupling regime where only one neutral higgs boson is light as compared to the other physical states of higgs bosons . the higgs sector of the zee - model is similar to that of the thdm except for the existence of an additional weak - singlet charged higgs field , so that the physical scalar bosons include two cp - even , one cp - odd and two pairs of charged higgs bosons . in this paper , we shall first determine the upper and lower bounds for the lightest cp - even higgs boson mass ( @xmath5 ) as a function of the cut - off scale @xmath6 of the zee - model , using renormalization group equations ( rge s ) . we show that the upper and lower mass bounds for @xmath0 are almost the same as those in the thdm . we also study the possible range of the higgs - boson self - coupling constants at the electroweak scale as a function of @xmath6 . by using these results , we examine effects of the additional loop contribution of the singlet charged higgs boson to the partial decay width of @xmath7 . we show that , by taking @xmath8 gev , the deviation of the decay width from the sm prediction can be about @xmath920% or nearly @xmath1010% for @xmath11 between 125 gev and 140 gev when the mass of the isospin singlet charged higgs boson is taken to be around 100 gev . the magnitude of the deviation becomes larger for lower cutoff scales and smaller masses of the singlet charged higgs boson . if we choose @xmath12 gev and the singlet charged higgs boson mass to be 100 gev , the positive deviation can be greater than @xmath1030% ( @xmath1040% ) for @xmath13 gev ( @xmath14 gev ) . such a deviation from the sm prediction could be tested at the lhc , the @xmath15 lc and the @xmath16 option of lc@xcite . we also discuss phenomenology of the singlet charged higgs boson at present and future collider experiments , which is found to be completely different from that of the ordinary thdm - like charged higgs bosons . to detect such a charged higgs boson at lep - ii experiments , experimentalists have to search for their data sample with @xmath17 or @xmath18 plus missing energy , in contrast to the usual detection channels : either @xmath19 or @xmath20 decay modes . this paper is organized as follows . in sec . [ sec : zeemodel ] , we introduce the higgs sector of the zee - model and review the neutrino masses and mixings in this model which are consistent with the atmospheric and solar neutrino observations . numerical results on the possible range of the mass and coupling constants of the higgs bosons are given in sec . [ sec : rge_analysis ] . in sec [ sec : h-2gamma ] , we discuss the one - loop effect of the extra - higgs bosons in the zee - model to the partial decay width of @xmath7 and its impacts on the neutral higgs - boson search at high - energy colliders . the phenomenology of the charged higgs boson that comes from the additional singlet field is discussed in sec . [ sec : charged ] . in sec . [ sec : conclusion ] , we present additional discussions and conclusion . relevant rge s for the zee - model are given in the appendix . to generate small neutrino mass radiatively , the zee - model contains a @xmath21 singlet charged scalar field @xmath22 , in addition to two @xmath21 doublet fields @xmath23 , and @xmath24 . the zee - model lagrangian is written as : @xmath25 where @xmath26 @xmath27 where @xmath28 ( @xmath29 are the generation indices , and @xmath30 \nonumber \\ & & + \sigma _ { 1}\left| \omega^{-}\right| ^{2}\left| \phi _ { 1}\right| ^{2}+ \sigma _ { 2}\left| \omega^{-}\right| ^{2}\left| \phi _ { 2}\right| ^{2}+ \frac{1}{4}\sigma _ { 3}\left| \omega^{-}\right| ^{4 } \ , . \label{higgs_potential } \end{aligned}\ ] ] in the above equations , @xmath31 is the left - handed quark doublet with an implicit generation index while @xmath32 and @xmath33 denote the right - handed singlet quarks . similarly , @xmath34 and @xmath35 denote the left - handed and right - handed leptons in three generations . the charge conjugation of a fermion field is defined as @xmath36 , where @xmath37 is the charge conjugation matrix ( @xmath38 ) with the super index @xmath39 indicating the transpose of a matrix . also , @xmath40 and @xmath41 with @xmath42 . without loss of generality , we have taken the anti - symmetric matrix @xmath43 and the coupling @xmath44 to be real in the equations ( [ fij_int ] ) and ( [ higgs_potential ] ) . in order to suppress flavor changing neutral current ( fcnc ) at the tree level , a discrete symmetry , with @xmath45 , @xmath46 , @xmath47 , is imposed to the higgs sector of the lagrangian , which is only broken softly by the @xmath48 term and the @xmath49 term . under the discrete symmetry there are two possible yukawa - interactions ; that is , for type - i @xmath50 and for type - ii , @xmath51 where @xmath52,@xmath53,@xmath54 are diagonal yukawa matrices and @xmath55 is the cabibbo - kobayashi - maskawa ( ckm ) matrix . later , we shall only keep the top yukawa coupling constants @xmath56 in our numerical evaluation of the rge s . ] . in that case , there is no difference between the yukawa couplings of the type - i and the type - ii models . finally , for simplicity , we assume that all @xmath57 and @xmath58 are real parameters . let us now discuss the higgs sector . the @xmath59 symmetry is broken to @xmath60 by @xmath61 and @xmath62 , the vacuum expectation values of @xmath63 and @xmath64 . ( they are assumed to be real so that there is no spontaneous cp violation . ) the number of physical higgs bosons are two cp - even higgs bosons ( @xmath65,@xmath66 ) , one cp - odd higgs boson ( @xmath67 ) and two pairs of charged higgs boson ( @xmath68 , @xmath69 ) . we take a convention of @xmath70 and @xmath71 . in the basis where two higgs doublets are rotated by the angle @xmath72 , with @xmath73 , the mass matrices for the physical states of higgs bosons are given by @xmath74 \,,\ ] ] for cp - even higgs bosons , @xmath75 for cp - odd higgs boson , and @xmath76 \,,\ ] ] for charged higgs bosons . here , @xmath77 and @xmath78 . the vacuum expectation value @xmath79 ( @xmath80 gev ) is equal to @xmath81 . mass eigenstates for the cp - even and the charged higgs bosons are obtained by diagonalizing the mass matrices ( [ neutralmassmatrix ] ) and ( [ chargedmassmatrix ] ) , respectively . the original higgs boson fields , @xmath63 , @xmath64 , @xmath22 , can be expressed in terms of the physical states and the nambu - goldstone modes ( @xmath82 and @xmath83 ) as @xmath84 where the angle @xmath85 and @xmath86 are defined from the matrices which diagonalize the @xmath87 matrices @xmath88 and @xmath89 , respectively . namely , we have @xmath90 where @xmath91 and @xmath92 . the mixing angles @xmath93 and @xmath94 then satisfy @xmath95 which show that @xmath93 and @xmath94 approaches to @xmath96 and zero , respectively . ] , when @xmath97 is much greater than @xmath98 , @xmath99 and @xmath100 ; i.e. , in the decoupling regime . in this limit , the massive higgs bosons from the extra weak - doublet are very heavy due to the large @xmath101 so that they are decoupled from the low energy observable . although neutrinos in this model are massless at the tree level , the loop diagrams involving charged higgs bosons , as shown in fig . [ fig : neutrino_mass ] , can generate the majorana mass terms for all three - flavors of neutrinos . it was shown @xcite that at the one - loop order , the neutrino mass matrix , defined in the basis where the charged lepton yukawa - coupling constants are diagonal in the lepton flavor space , is real and symmetric with vanishing diagonal elements . more explicitly , we have @xmath102 with @xmath103 where @xmath104 @xmath105 is the charged lepton mass for type - i . for type - ii , @xmath106 should be replaced by @xmath107 . note that eq . ( [ nmass2fij ] ) is valid for @xmath108 . the phenomenological analysis of the above mass matrix was given in ref . it was concluded that , in the zee - model , the bi - maximal mixing solution is the only possibility to reconcile the atmospheric and the solar neutrino data . here we give a brief summary of their results , for completeness . let us denote the three eigenvalues for the neutrino mass matrix , cf . ( [ neutrino3x3 ] ) , as @xmath109 , @xmath110 and @xmath111 , which satisfy @xmath112 . the possible pattern of the neutrino mass spectrum which is allowed in the zee - model is @xmath113 , with @xmath114 , and @xmath115 , where @xmath116 ev@xmath117 from the atmospheric neutrino data , and @xmath118 ev@xmath117 ( msw large angle solution ) or @xmath119 ev@xmath117 ( vacuum oscillation solution ) from the solar neutrino data ) , only the hierarchy pattern @xmath120 , rather than @xmath121 , is realized in the zee - model@xcite ] . thus , we have @xmath122 ( @xmath123 ) and @xmath124 . the approximate form of the neutrino mass matrix is given by @xmath125 where the upper ( lower ) sign corresponds to @xmath126 ( @xmath127 ) case , and the corresponding maki - nakagawa - sakata ( mns ) matrix@xcite which diagonalizes the neutrino mass matrix is @xmath128 in the above equations , we took the limiting case where @xmath129 and @xmath130 and @xmath131 in the notation of ref . @xcite . ] . from eqs ( [ nmass2fij ] ) and ( [ bimaximal_neutrino_mass ] ) , we obtain @xmath132 therefore , the magnitudes of the three coupling constants should satisfy the relation @xmath133 . this hierarchy among the couplings @xmath134 is crucial for our later discussion on the phenomenology of the singlet charged higgs bosons . for a given value of the parameters @xmath135 , @xmath136 , @xmath107 and @xmath49 , the coupling constants @xmath134 can be calculated from eq . ( [ nmass2fij ] ) . for example , for @xmath137gev , @xmath138gev , @xmath139 , @xmath140gev and @xmath141ev , we obtain @xmath142 . as in this example , when @xmath143 is rather heavy and the lighter charged higgs boson @xmath144 is almost a weak singlet , i.e. the mixing angle @xmath94 approaches to zero , it is unlikely that there are observable effects to the low energy data@xcite ; e.g. , the muon life - time , the universality of tau decay into electron or muon , the rare decay of @xmath145 , the universality of @xmath146-boson decay into electron , muon or tau , and the decay width of @xmath147 boson . when @xmath148 are small , we do not expect a large rate in the lepton flavor violation decay of a light neutral higgs boson , such as @xmath149 ( the largest one ) , @xmath150 , or @xmath151 ( the smallest one ) . on the contrary , as to be discussed in section iv , the decay width of @xmath1 can significantly deviate from the sm value . finally , the phenomenological constraints on @xmath152 were derived in ref . @xcite . from the consistency of the muon decay rate and electroweak precision test it was found that @xmath153 where @xmath154 is the fermi constant , and @xmath155 this means that the @xmath134 can not be @xmath156 unless the charged higgs boson masses are at the order of 10 tev . in this section , we determine the bounds on the mass of the lightest cp - even higgs boson as a function of the cut - off scale of the zee - model by analyzing the set of renormalization group equations ( rge s ) . we also study the allowed ranges of the coupling constants , especially @xmath157 and @xmath158 in eq . ( [ higgs_potential ] ) . in sec . iv , they will be used to evaluate how much the partial decay width of @xmath159 can deviate from its sm value due to the one - loop contribution from the singlet charged higgs boson . the mass bounds are determined in the following manner . for each set of parameters defined at the electroweak scale , the running coupling constants are calculated numerically through rge s at the one - loop level . we require that all the dimensionless coupling constants do not blow up below a given cut - off scale @xmath3 , and the coupling constants satisfy the vacuum stability condition . we vary the input parameters at the electroweak scale and determine the possible range of the lightest cp - even higgs boson mass as a function of @xmath3 . in a similar manner , we also study the allowed ranges of various higgs boson self - coupling constants at the electroweak scale as well as a function of the lightest cp - even higgs boson mass . we derived the one - loop rge s for the zee - model , and listed them in the appendix for reference . for simplicity , in the rge s , we have neglected all the yukawa coupling constants ( @xmath160 , @xmath161 , @xmath162 ) but the top yukawa coupling @xmath163 .. ] although we kept the new coupling constants @xmath43 in the rge s listed in the appendix , we have neglected @xmath43 in the numerical calculation . this is because the magnitudes of these coupling constants are numerically too small to affect the final results unless the singlet - charged scalar - boson mass is larger than a few tev [ cf . ( [ f12value ] ) ] . the dimensionless coupling constants relevant to our numerical analysis are the three gauge - coupling constants @xmath164 , @xmath165 , @xmath166 , the top yukawa - coupling constant @xmath163 , and eight scalar self - coupling constants , @xmath57 ( @xmath167 ) and @xmath168 ( @xmath169 ) . there are five dimensionful parameters in the higgs potential , namely @xmath170 , @xmath171 , @xmath48 , @xmath172 and @xmath44 . instead of @xmath170 , @xmath171 , @xmath48 , we take @xmath173 , @xmath174 , and @xmath78 , as independent parameters , where @xmath79 ( @xmath80 gev ) characterizes the weak scale and @xmath101 the soft - breaking scale of the discrete symmetry . in the actual numerical calculation we first fix @xmath174 and @xmath175 . for a given mass ( @xmath176 ) of the lightest cp - even higgs boson , we solve one of the @xmath57 , which is chosen to be @xmath177 here , in terms of other @xmath57 . we then numerically evaluate all dimensionless coupling constants according to the rge s . from @xmath176 to @xmath175 we use the sm rge s , which are matched to the zee - model rge s at the soft - breaking scale @xmath175 . and @xmath49 are only relevant to the charged scalar mass matrix . in principle , our numerical results also depend on these parameters through the renormalization of various coupling constants from the scale of @xmath5 to the charged scalar mass . since these effects are expected to be small , we calculate the rge s as if all the scalar - bosons except @xmath0 decouple at the scale @xmath101 . ] we requires the following two conditions to be satisfied for each scale @xmath178 up to a given cut - off scale @xmath3 . 1 . applicability of the perturbation theory implies + @xmath179 2 . the vacuum stability conditions must be satisfied . the requirement that quartic coupling terms of the scalar potential do not have a negative coefficient in any direction leads to the following conditions at each renormalization scale @xmath178 : 1 . @xmath180 2 . @xmath181 + @xmath182 + @xmath183 + where @xmath184 . 3 . if @xmath185 and @xmath186 , then @xmath187 if @xmath185 and @xmath188 , then @xmath189 if @xmath186 and @xmath188 , then @xmath190 [ when @xmath191 , @xmath192 and @xmath193 are all negative , the above three conditions are equivalent . ] in addition to the above conditions , we also demand local stability of the potential at the electroweak scale , namely , we calculate the mass spectrum of all scalar fields at the extremum of the potential and demand that all eigenvalues of the squared scalar mass are positive . we scan the remaining seven - dimensional space of @xmath57 and @xmath168 and examine whether a given mass of the lightest cp - even higgs boson is allowed under the above conditions . in this way we obtain the allowed range of @xmath5 as a function of @xmath174 and @xmath175 , for each value of the cut - off scale @xmath3 . the allowed mass range of the lightest cp - even higgs boson for @xmath194 gev . @xmath6 is the cut - off scale . ] first , we discuss our result in the decoupling case , in which the soft - breaking scale @xmath101 is much larger than the electroweak scale @xmath195 , and the masses of all the higgs bosons but @xmath0 ( and @xmath196 ) are at the order of @xmath197 , which leads to @xmath198 and @xmath199 ) , the masses of @xmath0 and @xmath196 are dominated by the @xmath200 component of the mass matrix in eq . ( [ neutralmassmatrix ] ) and the @xmath201 component of that in eq . ( [ chargedmassmatrix ] ) , respectively . the mass of @xmath0 is determined by the self - coupling constants @xmath202 , while that of @xmath196 depends not only on the self - couplings constants @xmath203 but also on the free mass parameter @xmath204 . as noticed in the footnote 7 , from @xmath5 to @xmath101 , the sm rge are used in our analysis , even if the mass of @xmath196 is smaller than @xmath101 . the effect of @xmath196 on the mass bound of @xmath0 is expected to be small , because at the one - loop level the primary effect is through the running of @xmath205 , whose contribution to the right - handed side of the rge for the higgs - self coupling constant is small . ] . in fig . [ fig : z1000 ] , the allowed range of @xmath176 is shown as a function of @xmath174 for @xmath206 gev . ( we take the pole mass of top quark @xmath207 gev , @xmath208 for numerical calculation . ) the allowed ranges are shown as contours for six different values of @xmath3 , i.e. @xmath4 , @xmath209,@xmath210,@xmath211,@xmath212 and @xmath213 gev . for most values of @xmath174 , except for small @xmath174 region , the upper bound of @xmath176 is about @xmath214 gev and the lower bound is between @xmath215 gev and @xmath216 gev for the cut - off scale @xmath6 to be near the planck scale . the numerical values in this figure are very close to those in the corresponding figure for the thdm discussed in ref . @xcite . compared to the corresponding lower mass bound in the sm , which is @xmath217 gev when using the one - loop rge s , the lower mass bound in this model is reduced by about @xmath218 gev to @xmath219 gev . the reason is similar to the thdm case : the lightest cp - even higgs boson mass is essentially determined by the value of @xmath220 for @xmath221 to be larger than about 2@xmath222 , where @xmath220 plays the role of the self - coupling constant of the higgs potential in the sm can not be too large to ignore the contribution of the bottom quark in the case with the type - ii yukawa interaction . ] . on the right - hand side of the rge for @xmath220 , cf ( [ rge_lambda2 ] ) , there are additional positive - definite terms @xmath223 as compared to the rge for the higgs self - coupling constant in the sm . these additional terms can improve vacuum stability , and allow lower values of @xmath176 . therefore , one of the features of the model is to have a different mass range for the lightest cp - even higgs boson as compared to the sm higgs boson , for a given cut - off scale . the allowed mass range of the lightest cp - even higgs boson for @xmath224 gev . ] next , we show our result for @xmath101 to be around @xmath79 . in fig . [ fig : z100 ] , we present the @xmath5 bound for @xmath225 gev . in this case , the allowed range of @xmath5 is reduced as compared to that in the decoupling case , and lies around @xmath226 for large @xmath107 . notice that we have not included phenomenological constraints from the @xmath227 , @xmath228 parameter and the direct higgs boson search experiment at lep . as mentioned before , the mass bounds obtained from the rge analysis are the same for the type - i and type - ii models without these phenomenological constraints . however , it was shown in ref . @xcite that the @xmath227 data can put a strong constraint on the allowed range of the higgs boson mass for @xmath229@xmath230 gev in the type - ii thdm , whereas there is no appreciable effect in the type - i model . this is because a small @xmath101 implies a light charged higgs boson in the thdm which can induce a large decay branching ratio for @xmath227 in the type - ii model@xcite data also give strong constraints on the charged higgs bosons in the type - ii thdm@xcite . ] . we expect a similar constraint from the @xmath227 data on the type - ii zee - model , when @xmath101 is small . in fig . [ fig : zeem ] , we show the upper and lower bounds of @xmath5 as a function of @xmath101 for various values of @xmath6 . for given @xmath101 , we scan the range of @xmath107 for @xmath231 . we find that the obtained @xmath5 bounds are almost the same as those for the thdm . the primary reason for this is that the new coupling constants @xmath157 , @xmath158 and @xmath232 do not appear directly in the mass formula for @xmath176 , and therefore , do not induce large effects on the bounds of @xmath5 . the allowed range of @xmath233 and @xmath5 for various @xmath6 values . ] the allowed range of @xmath234 and @xmath5 for various @xmath6 values . ] the allowed range of @xmath233 and @xmath234 for @xmath235 gev . ] the allowed range of @xmath233 and @xmath234 for @xmath236 gev . ] the allowed range of @xmath237 and @xmath5 for various @xmath6 values . ] we also investigate the allowed range of coupling constants @xmath157 , @xmath158 and @xmath232 . for this purpose , we fix @xmath157 ( or @xmath158 , @xmath232 ) as well as @xmath174 and @xmath175 to evaluate the upper and the lower bounds of @xmath176 for each @xmath3 value . in this way , we determine the possible range of @xmath157 ( or @xmath158 , @xmath232 ) under the condition that the theory does not break down below the cut - off scale @xmath3 . in fig . [ fig : s1range ] , we present the allowed range of @xmath157 and @xmath176 for different choice of @xmath3 in the case of @xmath206 gev and @xmath238 or @xmath239 . a similar figure is shown for the possible range of @xmath158 in fig . [ fig : s2range ] . we see that the maximal value of @xmath157 and @xmath158 is around @xmath240 for @xmath241 gev if we take the cut - off scale to be @xmath242 gev . for smaller value of @xmath3 the allowed ranges of @xmath168 becomes larger . for example , @xmath157 can exceed @xmath243 for @xmath244 gev . we have calculated for other value of @xmath174 and checked that these figures does not change greatly between @xmath245 and @xmath239 . we also present the allowed range in the @xmath157 and @xmath158 plane for a fixed value of @xmath176 in figs . [ fig : s12range_m1000_mh125 ] and [ fig : s12range_m1000_mh140 ] for @xmath246 gev and @xmath247 gev , respectively . for either value of @xmath176 with @xmath248 , both @xmath157 and @xmath158 can be as large as 0.5 ( 2 ) for @xmath4 ( @xmath212 ) gev . the allowed range of @xmath249 and @xmath5 for various values of @xmath6 is given in fig . [ fig : s3range ] . it is shown that , @xmath249 has to be larger than zero , due to the vacuum stability condition . the maximal value of @xmath232 is about @xmath243 ( 3 ) for @xmath4 ( @xmath212 ) gev and @xmath206 gev . the impact of these new coupling constants on the collider phenomenology is discussed in the next section . in this section , we study the phenomenological consequences of the higgs boson mass and the higgs - boson - coupling constants derived in the previous section . the important feature of the higgs sector of the zee - model is that there are an additional weak doublet and a singlet charged higgs boson . the physical states of the higgs particles are two cp - even higgs bosons , one cp - odd higgs boson and two pairs of charged higgs bosons . therefore , the higgs phenomenology is quite close to the ordinary two - higgs - doublet model . one unique difference is the existence of the additional weak - singlet charged higgs boson . the effect of this extra charged higgs boson is especially important when @xmath175 is much larger than the @xmath250 boson mass , i.e. in the decoupling regime . in such a case , the heavier cp - even higgs boson , the cp - odd higgs boson as well as one of the charged higgs bosons have masses approximately equal to @xmath175 , and these heavy states are decoupled from low energy observables . ( note that the condition on the applicability of the perturbation theory forbids too large self - couplings among the higgs bosons . hence , in the limit of large @xmath101 , the heavy higgs bosons decouple from the low energy effective theory . ) the remaining light states are the lighter cp - even higgs boson @xmath0 and the lighter charged higgs boson @xmath196 which mainly comes from the weak - singlet . in the previous section , we show that , even in the decoupling case , there can be large difference in the allowed range of @xmath5 between the zee - model and the sm . similarly , we expect that , even in the decoupling case , the presence of the additional weak - singlet charged higgs boson can give rise to interesting higgs phenomenology . since the lighter charged higgs boson @xmath196 can couple to higgs bosons and leptons , it can affect the decay and the production of the neutral higgs bosons at colliders through radiative corrections . in the following , we consider the decay width of @xmath251 as an example . for a sm higgs boson , the partial decay width ( or branching ratio ) of @xmath7 is small : @xmath252 kev ( or @xmath253 ) for @xmath246 gev , and @xmath254 kev ( or @xmath255 ) for @xmath247 gev , with a 175 gev top quark . nevertheless , it is an important discovery mode of the higgs boson at the lhc experiments for @xmath5 less than twice of the @xmath146-boson mass . needless to say that a change in the branching ratio of @xmath256 would lead to a different production rate of @xmath257 . at future @xmath15 lc s , the branching ratio of @xmath159 can be determined via the reaction @xmath258 and @xmath259 with a 16 - 22% accuracy@xcite . at the photon - photon collision option of the future lc s , the partial decay width of @xmath7 can be precisely tested within a 2 % accuracy@xcite by measuring the inclusive production rate of the higgs boson @xmath0 . clearly , a change in the partial decay width of @xmath7 will lead to a different production rate for @xmath0 . in the zee - model , such a change is expected after taking into account the loop contribution of the extra charged higgs boson . we find that the deviation from the sm prediction can be sizable , and therefore testable at the lhc and future lc s . the partial decay width of @xmath7 is calculated at the one - loop order . similar to our previous discussion , we limit ourselves to the parameter space in which @xmath260 , and keep only the top quark contribution from the fermionic loop diagrams . including the loop contributions from the @xmath146 boson and the charged higgs bosons @xmath261 and @xmath196 together with the top quark loop contribution , we obtain@xcite @xmath262 with @xmath263 where @xmath264 and @xmath265 is the mass of the internal lines in the loop diagram . @xmath266 and @xmath267 are given by @xmath268 \ , , \end{aligned}\ ] ] @xmath269 \ , , \end{aligned}\ ] ] and @xmath270 with @xmath271 ^{2 } & \mbox { if}\ ; r\geq 1\\ -\frac{1}{4}\left [ \ln \frac{1+\sqrt{1-r}}{1-\sqrt{1-r}}-i\pi \right ] ^{2 } & \mbox { if}\ ; r<1 \end{array}\right . .\ ] ] in the decoupling case of the model , namely @xmath272 , the above formulae are greatly simplified . this limit corresponds to @xmath273 and @xmath274 , so that the light charged higgs boson @xmath69 is identical to the weak - singlet higgs boson @xmath275 . thus , we have @xmath276 and both the top - quark and the @xmath146 boson loop contributions reduce to their sm values . we like to stress that the weak - singlet higgs boson does not directly couple to the quark fields in the limit of @xmath277 . therefore , it does not affect the decay rate of @xmath278 at the one - loop order . similarly , being a weak singlet , it also gives no contribution to the @xmath279 parameter . hence the low - energy constraint from either the @xmath278 decay or the @xmath279 parameter on the zee - model in the limit of @xmath280 is similar to effects of that on the thdm . let us examine at the one - loop effect of the weak - singlet charged higgs boson on the decay width of @xmath159 in the decoupling limit . let us recall that in fig . [ fig : s12range_m1000_mh140 ] , the size of the new couplings @xmath281 and @xmath282 can be as large as 2 simultaneously , if the cut - off scale is at the order of @xmath283 gev . for the zee - model to be a valid low energy effective theory up to @xmath284 gev , @xmath281 and @xmath282 can not be much larger than @xmath285 . to illustrate the implications of this result , we show in figs . [ fig : ratio125_140 ] ( a ) and [ fig : ratio125_140 ] ( b ) the ratio ( @xmath286 ) of the @xmath287 width predicted in the zee - model to that in the sm , @xmath288 , as a function of the coupling constant @xmath282 and the charged higgs boson mass @xmath136 . here , for simplicity , we have set @xmath289 so that the @xmath107 dependence drops in the decoupling case , cf . ( [ eq : decoup ] ) . for illustrations , we consider two cases for the mass of the lighter cp - even higgs boson : @xmath246 gev and @xmath247 gev . as shown in the figures , the ratio @xmath290 can be around 0.8 for @xmath291 and @xmath292 gev . this reduction is due to the cancellation between the contribution from the @xmath196-boson loop and the @xmath146-boson loop contributions . to have a similar reduction rate in @xmath293 for a heavier @xmath196 , the coupling constant @xmath282 ( and @xmath281 ) has to be larger next , as shown in figs . 7 and 8 , @xmath281 and @xmath282 do not have to take the same values in general , and they can be less than zero . in the case where both @xmath281 and @xmath282 are negative , the contribution of the @xmath196-loop diagram and that of the @xmath146-loop diagram have the same sign , so that @xmath286 can be larger than 1 . such an example is shown in fig . [ fig : ratio125_140_negatives2 ] ( a ) , where the ratio @xmath286 for @xmath246 gev is shown as a function of @xmath294 at various negative @xmath282 values with @xmath295 and @xmath296 . we consider the case with @xmath297 , @xmath298 or @xmath299 , which is consistent with the cut - off scale @xmath8 , @xmath300 or @xmath301 gev , respectively . in the case of @xmath8 gev ( @xmath301 gev ) , the deviation from the sm prediction can be about + 6% ( + 30% ) for @xmath302 gev . in fig . [ fig : ratio125_140_negatives2 ] ( b ) , the similar plot of the ratio @xmath286 is shown for @xmath247 gev with @xmath295 and @xmath296 . each case with @xmath303 , @xmath304 or @xmath305 is consistent with @xmath8 , @xmath300 or @xmath301 gev , respectively . the correction is larger in the case with @xmath247 gev than in the case with @xmath246 gev for a given @xmath6 . the deviation from the sm prediction can amount to about + 8% ( + 40% ) for @xmath8 gev ( @xmath301 gev ) when @xmath306 gev . larger positive corrections are obtained for smaller @xmath294 values . such a deviation from the sm prediction can be tested at the lhc , the @xmath15 lc and the @xmath16 option of lc . before concluding this section , we remark that if @xmath5 is larger than @xmath307 such that the decay mode @xmath308 is open , the total decay width of @xmath0 can be largely modified from the sm prediction for large @xmath309 . in terms of @xmath310 , the partial decay width of @xmath311 is given by @xmath312 where @xmath313 . in fig . [ fig : hsswidth ] ( a ) , we show the partial decay width @xmath314 for @xmath315 gev with @xmath316 , cf . ( [ eq : decoup ] ) , for the allowed range of @xmath5 from 100 gev to 500 gev . in fig . [ fig : hsswidth ] ( b ) , the ratio of @xmath314 to the total width of the sm higgs boson ( @xmath317 ) is shown as a function of @xmath5 for each value of @xmath318 . this is to illustrate the possible size of the difference between the total width of the lightest cp - even higgs boson @xmath0 in the zee - model and that of the sm higgs boson gev , which allows a wide range of values for @xmath319 s , @xmath136 and @xmath5 . ] . clearly , the impact of the @xmath320 decay channel is especially large in the small @xmath5 region . we note that @xmath317 can be determined to the accuracy of 10 - 20% at the lhc and the lc if @xmath321 , and to that of a couple of per cents if @xmath322@xcite . ( @xmath323 is the mass of the @xmath147-boson . ) hence , measuring the total width of the lightest neutral higgs boson can provide a further test of the zee - model for @xmath324 . the change in the total width also modifies the decay branching ratio of @xmath325 , hence yields a different rate of @xmath326 for a given @xmath5 . ( in the sm , the branching ratio of @xmath325 is about 1/3 for @xmath327 gev . ) needless to say that for @xmath328 , the production mode of @xmath329 is also useful to test the zee - model . further discussion on this possibility will be given in sec . in the zee - model , two kinds of charged higgs bosons appear . if there is no mixing between them ( @xmath330 ) , the mass eigenstates @xmath331 and @xmath332 correspond to the thdm - like charged higgs field and the singlet higgs field @xmath275 , respectively . the case with @xmath330 occurs in the limit of @xmath333 and @xmath100 ; i.e. in the decoupling limit . the detection of @xmath332 can be a clear indication of the zee - model . as to be shown later , its phenomenology is found to be drastically different from that of the thdm - like charged higgs bosons @xmath331@xcite . here , we discuss how the effects of this extra charged boson can be explored experimentally . we first consider the case with @xmath330 , and then extend the discussion to the case with a non - zero @xmath94 . the @xmath334 boson decays into a lepton pair @xmath335 with the coupling constant @xmath134 . the partial decay rate , @xmath336 , is calculated as @xmath337 and the total decay width of @xmath334 is given by @xmath338 by taking into account the hierarchy pattern of @xmath134 , cf . ( [ eq21 ] ) and ( [ eq22 ] ) , and by assuming @xmath302 gev and @xmath339 , the total decay width and the life time ( @xmath340 ) is estimated to be . if we take @xmath341 gev or @xmath342 gev , @xmath343 can become one order of magnitude larger than @xmath344 , while still being consistent with the phenomenological bounds discussed in sec . @xmath345 this implies that @xmath196 decays after traveling a distance of @xmath346 m , which is significantly shorter than the typical detector scale . therefore , @xmath332 decays promptly after its production , and can be detected at collider experiments . the main production channel at the lep - ii experiment may be the pair production process @xmath347 , similar to the production of the thdm - like charged higgs boson @xmath348 . the matrix - element squares for the @xmath349 production ( @xmath350 ) are given by @xmath351 where @xmath352 and @xmath353 ( @xmath354 ) for the incoming electron @xmath355 ( @xmath356 ) ; @xmath357 and @xmath358 ( @xmath354 ) for @xmath359 ( @xmath360 ) ; @xmath361 , @xmath362 , @xmath363 , and @xmath364 is the scattering angle of @xmath365 in the @xmath15 center - of - mass ( cm ) frame whose energy is @xmath366 . for the other electron - positron helicity configuration ( @xmath367 and @xmath368 ) , the cross sections are zero . thus the total cross section for the @xmath320 pair production is given by @xmath369 . \end{aligned}\ ] ] hence , the production rates of @xmath143 and @xmath334 are different . we note that the ratio of cross sections for @xmath370 and @xmath371 production , @xmath372 , is @xmath373 at @xmath374 gev assuming that the masses of @xmath331 and @xmath332 are the same . this ratio is independent of the masses of @xmath375 and @xmath376 for a fixed cm energy . ( only the difference between @xmath377 and @xmath378 coupling constants determines this ratio . ) the lower mass bound of the thdm - like charged boson @xmath379 can be obtained by studying its @xmath19 and @xmath380 decay modes , completely in the same way as the charged higgs boson search in the minimal supersymmetric standard model ( mssm)@xcite . similar experimental constraints may be obtained for the extra charged bosons @xmath332 . the situation , however , turns out to be fairly different from the @xmath331 case . first of all , decays of @xmath332 are all leptonic . secondly , the branching ratios of various @xmath332 decay modes are estimated as @xmath381 where we have used the relations given in eqs . ( [ eq21 ] ) and ( [ eq22 ] ) . clearly , the branching ratio into the @xmath382 mode is very small , so that it is not useful for detecting @xmath332 at all . this is different from the case of detecting the ordinary thdm - like charged higgs boson , which preferentially decays into heavy fermion pairs ( e.g. @xmath19 and @xmath20 ) . instead of studying the @xmath383 mode , the @xmath384 and @xmath385 modes can provide a strong constraint on the mass of @xmath332 . in fact , the branching ratio of @xmath386 or @xmath387 is almost 100 % , so that we have @xmath388 , where @xmath389 and @xmath390 represent @xmath391 or @xmath392 ( not @xmath393 ) . let us compare this with the cross section @xmath394 , where @xmath395 . as seen in fig . [ fig : zee_pair ] , the cross section @xmath396 is comparable with @xmath397 . therefore , by examining the lep - ii data for @xmath398 ( where @xmath399 , @xmath400 or @xmath401 , in contrast to @xmath402 for the @xmath331 case ) , the experimental lower bound on the mass of @xmath332 can be determined . such a bound can be induced from the smuon search results at the lep experiments@xcite in the case that neutralinos are assumed to be massless . from the @xmath403 data accumulated up to @xmath404 gev@xcite , we find that the lower mass bound of @xmath332 is likely to be 80 - 85 gev for the @xmath330 cases . [ we note that the right - handed smuon ( @xmath405 ) in the mssm carries the same @xmath406 quantum number as the weak - singlet charged higgs boson ( @xmath332 for @xmath407 ) . ] we next comment on @xmath332-production processes at hadron colliders and future lc s . at hadron colliders , the dominant production mode is the pair production through the drell - yan - type process . the cross sections for @xmath408 at the tevatron run - ii energy ( @xmath409 tev ) and @xmath410 at the lhc energy ( @xmath411 tev ) are shown as a function of @xmath136 in fig . [ fig : zee_hadron ] for @xmath412 . at future lc s , the @xmath332 boson may be discovered through the above - discussed pair - production process from the electron - positron annihilation if @xmath413 . in fig . [ fig : zee_lc ] , we show the total cross section of @xmath414 for @xmath330 as a function of @xmath136 for @xmath415 , @xmath416 , and @xmath417 gev . finally , we like to discuss the case with a non - zero @xmath94 , in which @xmath334 is a mixture of the singlet charged higgs boson state ( @xmath418 ) and the doublet charged higgs boson state ( @xmath419 ) . let us see how the above discussion is changed in this case . the doublet charged higgs bosons with the mass of @xmath420 gev mainly decay into the @xmath421 and @xmath422 channels . thus , the branching ratio of the decay process @xmath423 , where @xmath389 represents @xmath391 and @xmath392 , is expressed in a non - zero @xmath94 case as @xmath424 where @xmath425 ( @xmath426 ) is the total width of @xmath365 at @xmath330 with the same mass as the decaying @xmath334 on the left - hand side of the above equation . the formula of @xmath427 is given in eq . ( [ s2width_0 ] ) , while @xmath428 , which is the same as the total decay width of the charged higgs boson in the thdm is given by @xmath429 where @xmath430 are fermion pairs which are kinematically allowed . in the type - ii yukawa couplings , we have @xmath431 in the thdm , the total decay width of the charged higgs boson ( @xmath419 ) for @xmath432 gev is about 470 kev . hence , if the mixing angle @xmath94 is not so small , the decay pattern of @xmath334 is dominated by that of the thdm charged higgs boson @xmath419 . in fig . [ fig : nonmix ] , we plot the branching ratio @xmath433 as a function of @xmath434 at @xmath302 gev for several values of @xmath343 . we only show the case with @xmath435 , where the result is independent of the type of the yukawa interaction . the coupling constant @xmath343 is taken to be @xmath436 and @xmath437 ( @xmath438 ) , which satisfy the phenomenological constraints given in sec . as expected , the branching ratio decreases as @xmath94 increases . when @xmath439 , @xmath433 is smaller than 10% for @xmath440 . for the smaller @xmath343 values , the branching ratio reduces more quickly . the branching ratio is not sensitive to @xmath294 unless the mass exceeds the threshold of the decay into a @xmath441 or @xmath442 pair . above the threshold of the @xmath441 pair production , the decay rate of @xmath443 is large due to the large mass of the top quarks , so that @xmath433 is substantial only for very small values of @xmath94 . finally , while the decay branching ratio can change drastically depending on the mixing angle @xmath94 , the production cross section for @xmath414 remains unchanged . in conclusion , the process @xmath444 can also be useful for testing the zee - model in the non - zero @xmath94 case , provided @xmath434 is not too large . in this paper , the higgs sector of the zee - model has been investigated , in which neutrino masses are generated radiatively . this model contains extra weak - doublet higgs field and singlet charged higgs field . we have studied indirect effects of these extra higgs bosons on the theoretical mass bounds of the lightest cp - even higgs boson , which are obtained from the requirement that the running coupling constants neither blow up to a very large value nor fall down to a negative value , up to a high - energy cut - off scale @xmath3 . for @xmath4 gev , the upper bound of @xmath176 is found to be about @xmath214 gev , which is almost the same value as the sm prediction . in the decoupling regime ( @xmath445 ) , the lower bound is found to be about @xmath446 gev for @xmath4 gev , which is much smaller than the lower bound in the sm , and is almost the same as that in the thdm . for smaller @xmath447 values , the bounds are more relaxed , similar to that of the sm . we have also investigated the allowed range of the coupling constants relevant to the weak - singlet higgs field . the most striking feature of the zee - model higgs sector is the existence of the weak - singlet charged higgs boson . we have examined the possible impact of the singlet charged - higgs boson on the neutral higgs boson search through radiative corrections . we found that its one - loop contributions to the @xmath7 width can be sizable . in the allowed range of the coupling constants the deviation from the sm prediction for this decay width can be about @xmath448% or near @xmath449% for @xmath302 gev and @xmath8 gev , depending on the sign of the coupling constants @xmath203 . the magnitude of the deviation is larger for lower @xmath6 values or for smaller @xmath136 values . for example , a positive deviation over @xmath450-@xmath451% is possible for @xmath452-@xmath14 gev , @xmath453 gev , and @xmath454 gev . in the decoupling limit ( i.e. when @xmath455 , where @xmath456 and @xmath199 ) , we expect that the production cross sections for @xmath457 , @xmath458 and @xmath459 in the zee - model are the same as those in the sm . however , a sizable change in the decay branching ratio of @xmath159 can alter the production rate of @xmath257 at the lhc , where this production rate can be determined with a relative error of 10 - 15%@xcite . also , such a deviation in the branching ratio of @xmath159 directly affects the cross section of @xmath460 , which can be measured with an accuracy of 16 - 22% at the future @xmath15 lc ( with @xmath461 gev and the integrated luminosity of 1 ab@xmath462)@xcite . therefore , the zee - model with low cut - off scales can be tested through the @xmath463 process at the lhc and the @xmath15 lc s . at the future photon colliders , the enhancement ( or reduction ) of the @xmath463 partial decay rate will manifest itself in the different production rate of @xmath0 from the sm prediction . a few per cent of the deviation in @xmath464 can be detected at a photon collider@xcite , so that the effects of the singlet charged higgs boson can be tested even if the cutoff scale @xmath6 is at the planck scale . the collider phenomenology of the singlet charged higgs boson has turned out to be completely different from that of the thdm - like charged higgs boson . the singlet charged higgs boson mainly decays into @xmath465 ( with @xmath466 or @xmath18 ) , while the decay mode @xmath467 is almost negligible due to the relation @xmath468 . this hierarchy among the coupling constants @xmath134 results from demanding bi - maximal mixings in the neutrino mass matrix generated in the zee - model to be consistent with the neutrino oscillation data . on the other hand , the thdm - like charged higgs boson decays mainly into either the @xmath19 mode or the @xmath380 mode , through the usual yukawa - interactions . hence , to probe this singlet charged higgs boson using the lep - ii data , experimentalists should examine their data sample with @xmath469 , @xmath470 , @xmath471 or @xmath403 , while the experimental lower mass bound of the thdm - like charged higgs boson is obtained from examining the @xmath472 , @xmath473 and @xmath474 events . using the published lep - ii constraints on the mssm smuon production ( assuming the lightest neutralinos to be massless ) , we estimate the current lower mass bound of this singlet charged higgs boson to be about 80 - 85 gev . the tevatron run - ii , lhc and future lc s can further test this model . finally , we comment on a case in which the singlet charged higgs boson ( @xmath332 for @xmath330 ) is the lightest of all the higgs bosons . for @xmath475 , the higgs sector of the zee - model can be further tested by measuring the production rate of @xmath476 . the branching ratio for @xmath329 can be large . for instance , for @xmath477 gev and @xmath306 gev , this branching ratio is about 12% for each @xmath478 , @xmath479 , @xmath480 or @xmath401 . the branching ratio decreases for larger masses of @xmath0 . moreover , the total decay width of @xmath0 can be largely modified when the decay channel @xmath481 is open . in this case , the decay branching ratios of @xmath482 , @xmath483 are also different from the sm predictions . in conclusion , the distinguishable features of the zee - model from the sm and the thdm can be tested by the data from lep - ii , the tevatron run - ii and future experiments at lhc and lc s . we are grateful to the warm hospitality of the center for theoretical sciences in taiwan where part of this work was completed . cpy would like to thank h .- j . he , j. ng and w. repko for stimulating discussions . sk was supported , in part , by the alexander von humboldt foundation . gll and jjt were supported , in part , by the national science council of r.o.c . under the grant no nsc-89 - 2112-m-009 - 035 ; yo was supported by the grant - in - aid of the ministry of education , science , sports and culture , government of japan ( no . 09640381 ) , priority area `` supersymmetry and unified theory of elementary particles '' ( no . 707 ) , and `` physics of cp violation '' ( no . 09246105 ) ; cpy is supported by the national science foundation in the usa under the grant phy-9802564 . here , we summarize the relevant rge s to our study . for the gauge coupling constants , we have [ sec : rgelist ] @xmath484 the rge s for the higgs - self - coupling constants of the doublets are calculated at the one - loop level as @xmath485 and those with respect to the additional singlet charged higgs are given by @xmath486 finally , the rge s for the yukawa - type coupling constants are obtained at one - loop level as @xmath487 where @xmath488 s.t . petcov , ; j. liu , ; b.k . pal , ; w. grimus , and g. nardulli , ; a.y . smirnov and z .- j . tao , ; p.h . frampton and s.l . glashow , ; g.c . mclaughlin and j.n . ng , ; a.s . jaurabh and s.d . rindani , ; k. cheung , o.c.w . kong , ; k. cheung , o.c.w . kong , hep - 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( ny ) _ * 148 * ( 1983 ) 295 ; m. sher , _ phys . _ * 179 * ( 1989 ) 273 ; d. kominis and r.s . chivukula , ; s. kanemura , t. kubota and e. takasugi , ; s. nie and m. sher , ; a. akeroyd , a. arhrib , e. naimi , . e. boos , j .- c . brient , d.w . reid , h.j . schreiber , and r. schanidze , hep - ph/0011366 ; rick van kooten , talk at the berkeley linear collider meeting in march 2000 ( http://needmore.physics.indiana.edu/~rickv/nlc/talks/berkeley_2000/berkeley_2000.html ) . m. ciuchini , g. degrassi , p. gambini and g.f . giudice , _ nucl . _ b * 527 * ( 1998 ) 21 ; p. ciafaloni , a. romanino and a. strumia , _ nucl . _ b * 524 * ( 1998 ) 361 ; f. borzumati and g. greub , _ phys . _ d * 58 * ( 1998 ) 074004 ; t.m . aliev and e.o . iltan , _ phys . _ d * 58 * ( 1998 ) 095014 . v. drollinger and a. sopczak , hep - ph/0102342 . s. kanemura , t. kasai , g .- lin , y. okada , j.j . tseng , and c .- p . yuan , hep - ph/0010233 . m. dima , talk given at lcws 2000 ( fermilab , october 2000 ) + http://www-lc.fnal.gov/lcws2000/ ; + aleph collaboration , . aleph collaboration , .

while the earth s magnetism has been studied for centuries starting with the first scientific monograph of gilbert ( 1600 ) , the question of the magnetism of other planets had received scant attention until recently because of the lack of relevant observations . only in 1955 clear evidence for the existence of a planetary magnetic field other than the geomagnetic one was obtained through the observation of the jovian decametric radio waves ( burke and franklin , 1955 ) . since it had been more or less accepted until the end of the 19-th century that geomagnetism arises from the remnant magnetization of the earth similar properties may have been assumed for the other terrestrial planets and the moon this view lost its appeal , however , when it became evident that the curie - temperature is exceeded in the earth below a depth of about 30 km . ferromagnetic materials in the earth s crust could thus explain only magnetic fields with a relatively short wavelength . the current period of intense research on the magnetism of planets other than that of the earth started with the first detailed measurement of jupiter s magnetic field by the pioneer 10 space probe in 1973 and the discovery of mercury s magnetism by mariner 10 in 1974 . in the early seventies also the development of the theory of magnetohydrodynamic dynamos had started in which the reaction of the lorentz force of the generated magnetic field is taken into account in physically realistic configurations ( childress and soward , 1972 ; busse , 1973 ; soward , 1974 ) . until that time dynamo theoreticians had focused their attention on the kinematic problem in which the possibility of growing magnetic fields driven by somewhat arbitrarily chosen velocity fields is considered . it must be remembered that only a few years earlier it had been demonstrated by backus ( 1958 ) and herzenberg ( 1958 ) in a mathematically convincing way that the homogeneous dynamo process of the generation of magnetic fields in a singly connected domain of an electrically conducting fluid is indeed possible . doubts about the feasibility of this process which had first been proposed by larmor ( 1919 ) had persisted after cowling ( 1934 ) had proved that purely axisymmetric fields could not be generated in this way . the complexity of the magnetohydrodynamic dynamo problem described by the nonlinearly coupled navier - stokes equations and the equation of magnetic induction had prevented progress in understanding planetary dynamos through analytical solutions . only the advent of powerful enough computers in the 1990-ies has allowed to solve numerically the coupled three - dimensional partial differential equations through forward integration in time . even today and for the foreseeable future the limits of computer capacity will permit the exploration of only a small fraction of the parameter space of interest for the understanding of planetary dynamos . in view of the difficulties of a rigorous theory of planetary dynamos , many attempts have been made to obtain simple similarity relationships which would fit the observed planetary magnetic moments as function of certain properties of the planets . some early proponents have gone as far as claiming the existence of a `` magnetic bode s law '' corresponding to a relationship between the magnetic moment and size or angular momentum of a planet in analogy to the titius - bode law for the radii of the orbits of the planets . just as in the latter case , however , attempts to derive a magnetic bode s law from basic physical principles have failed . other proposals have taken into account physical forces . since a common ingredient of planetary dynamos is the existence of a fluid part of the core with a sufficiently high electrical conductivity the latter parameter together with the core radius and the angular velocity of the planetary rotation usually enter the similarity relationships such as those proposed by hide ( 1974 ) , busse ( 1976 ) , jacobs ( 1979 ) and dolginov ( 1977 ) . malkus ( 1968 , 1996 ) has argued for the precession of the earth as the cause of geomagnetism and he and vanyo ( 1984 ) have demonstrated through laboratory experiments that precession and tides may cause turbulent motions in fluid planetary cores . dolginov ( 1977 ) proposed a scaling law for the precessional origin of all planetary magnetic fields . while a dynamo driven by turbulent flows caused by precession and tides can not be easily excluded in the case of the earth ( tilgner , 2005 ) , it is much less likely in the case of other planets such as uranus for which precessional torques are rather minute . just as a common precessional origin of planetary magnetism is not regarded as feasible , so have all other proposed similarity relationships lost in appeal and are no longer seriously considered . we shall return , however , to scaling relationships based more directly on the basic equations in section 6 . since the proposal of the geodynamo as the cause of the earth s magnetism had been in doubt for a long time before 1958 , numerous alternative proposals had been made in the literature . among these only the possibility that thermoelectric currents may generate a planetary magnetic field is still discussed in the case of mercury ( stevenson , 1987 ; giampieri and balogh , 2002 ) . for a discussion of the failings of the various proposals for non - dynamo origins of planetary magnetic fields we refer to the papers of bullard ( 1949 ) and stevenson ( 1983 ) . although the dynamo hypothesis of the origin of planetary magnetism is not without difficulties , it is the only one considered seriously at the present time with the possible exception of the just mentioned case of mercury . dynamos generally convert mechanical energy into magnetic one . in contrast to the technical dynamo which is characterized by a multiply connected region of high electrical conductivity , i.e. it depends on an appropriate wiring , planetary dynamos are referred to as homogeneous dynamos since they operate in a singly connected domain of high electrical conductivity . since flows in planetary cores with active dynamos are usually turbulent the small scale structure of the magnetic field is correspondingly chaotic . the large scale structure , however , can be quite regular . one distinguishes `` steady '' and oscillatory dynamos . the most famous example of the latter kind is the solar dynamo which exhibits a well defined period of about 22 years . the geodynamo , on the other hand , is a `` steady '' dynamo , even though it varies in its amplitude by a factor of two or more on the magnetic diffusion time scale and reverses its polarity on a longer time scale . a measure of the magnetic diffusion time is given by the decay time , @xmath0 , of the magnetic field in the absence of fluid motions . here @xmath1 and @xmath2 refer to the electrical conductivity and the magnetic permeability of the planetary core of radius @xmath3 . in the case of the earth the decay time is of the order of 20 000 years , but it may vary between a few hundred and a million years for other examples of planetary dynamos . the theory of homogeneous dynamos is based on maxwell s equations for the magnetic flux density @xmath4 , the electric current density @xmath5 and the electric field @xmath6 in the magnetohydrodynamic approximation in which the displacement current is neglected . this approximation is highly accurate as long as the fluid velocity is small compared to the velocity of light which is certainly the case for all planetary applications . these equations together with ohm s law for a moving conductor are given by @xmath7 where @xmath2 is the magnetic permeability of the fluid and @xmath1 is its electrical conductivity . these `` pre - maxwell '' equations have the property that they are invariant with respect to a galilei transformation , i. e. the equations remain unchanged in a new frame of reference moving with the constant velocity vector @xmath8 relative to the original frame of reference . indicating the variables of the new frame by a prime we find @xmath9 this invariance is the basis for the combination in mhd of equations ( 1 ) with the equations of hydrodynamics in their usual non - relativistic form . it is remarkable that this invariance does not only hold with respect to a galilei transformation , but with respect to a transformation to a rotating frame of reference as well . in that case @xmath8 is replaced by @xmath10 in equations ( 2 ) , but when @xmath11 is operating on any vector @xmath12 the term @xmath13 must be added on the right hand side , since even a constant vector field becomes time dependent when seen from a rotating frame unless it is parallel to @xmath14 . elimination of @xmath6 and @xmath5 from equation ( 1 ) yields the equation of magnetic induction @xmath15 which for a solenoidal velocity field @xmath16 and a constant magnetic diffusivity @xmath17 can be further simplified , @xmath18 this equation has the form of a heat equation with the magnetic field line stretching term on the right hand side acting as a heat source . this interpretation is especially useful for the dynamo problem . in order that a magnetic field @xmath4 may grow , the term on the right hand side of ( 4 ) must overcome the effect of the magnetic diffusion term on the left hand side . using a typical velocity @xmath19 and a typical length scale @xmath20 , the ratio of the two terms can be estimated by the magnetic reynolds number @xmath21 , @xmath22 only when @xmath21 is of the order one or larger may growing magnetic fields become possible . in the following we shall first consider the mathematical formulation of the problem of convection driven dynamos in rotating spherical shells in a simple form in which only the physically most relevant parameters are taken into account . before discussing dynamo solutions in section 5 we shall briefly outline in section 4 properties of convection it the absence of a magnetic field . applications to various planets and moons will be considered in section 6 of this article and some concluding remarks are given in section 7 . a sketch of the geometrical configuration that will be considered is shown in figure 1 . for the equations describing convection driven dynamos in the frame rotating with the angular velocity @xmath23 we use a standard formulation which has also been used for a dynamo benchmark ( christensen _ et al . but we assume different scales and assume that a more general static state exists with the temperature distribution @xmath24 where @xmath25 denotes the ratio of inner to outer radius of the spherical shell and @xmath20 is its thickness . @xmath26 is the temperature difference between the boundaries in the special case @xmath27 . in the case @xmath28 the static temperature distribution @xmath29 corresponds to that of a homogeneously heated sphere with the heat source density proportional to the parameter @xmath30 . the gravity field is given by @xmath31 where @xmath32 is the position vector with respect to the center of the sphere and @xmath33 is its length measured in units of @xmath20 . in addition to @xmath20 , the time @xmath34 , the temperature @xmath35 and the magnetic flux density @xmath36 are used as scales for the dimensionless description of the problem where @xmath37 denotes the kinematic viscosity of the fluid , @xmath38 its thermal diffusivity and @xmath39 its density . the boussinesq approximation is used in that @xmath39 is assumed to be constant except in the gravity term where its temperature dependence given by @xmath40 const . is taken into account . the dimensionless equations of motion , the heat equation for the deviation @xmath41 of the temperature field from the static distribution and the equation of magnetic induction thus assume the form @xmath42{\mbox{\boldmath $ r$ } } \cdot { \mbox{\boldmath $ v$ } } \nonumber \\ & \hspace*{1cm}= p ( \partial_t + { \mbox{\boldmath $ v$ } } \cdot \nabla ) \theta\\ & p_m\left(\frac{\partial}{\partial t } + { \mbox{\boldmath $ v$ } } \cdot \nabla\right){\mbox{\boldmath $ b$ } } - \nabla^2{\mbox{\boldmath $ b$ } } = p_m{\mbox{\boldmath $ b$ } } \cdot\nabla { \mbox{\boldmath $ v$}}\end{aligned}\ ] ] where @xmath43 is the unit vector in the direction of the axis of rotation and where @xmath44 includes all terms that can be written as gradients . the rayleigh numbers @xmath45 and @xmath46 , the coriolis parameter @xmath47 , the prandtl number @xmath48 and the magnetic prandtl number @xmath49 are defined by @xmath50 for simplicity @xmath51 will be assumed unless indicated otherwise . the notation @xmath52 will thus be used in the following . the prandtl number @xmath48 has been added as an important parameter of the problem even though @xmath53 is often assumed with the argument that all effective diffusivities are equal in turbulent media . the effective diffusivities of scalar and vector quantities in turbulent fluid flow differ in general , however , and large differences in the corresponding molecular diffusivities will not be erased entirely in the turbulent case . since the velocity field @xmath16 as well as the magnetic flux density @xmath4 are solenoidal vector fields , the general representation in terms of poloidal and toroidal components can be used , @xmath54 by multiplying the ( curl)@xmath55 and the curl of equation ( 6a ) by @xmath32 we obtain two equations for @xmath56 and @xmath57 @xmath58 \nabla^2 \phi + \tau q \psi - l_2 \theta \nonumber \\ & \hspace*{1cm}= - { \mbox{\boldmath $ r$ } } \cdot \nabla \times [ \nabla \times ( { \mbox{\boldmath $ v$ } } \cdot \nabla { \mbox{\boldmath $ v$ } } - { \mbox{\boldmath $ b$ } } \cdot \nabla { \mbox{\boldmath $ b$}})]\\ & [ ( \nabla^2 - \partial_t ) l_2 + \tau \partial_{\phi } ] \psi - \tau q\phi \nonumber \\ & \hspace*{1cm}= { \mbox{\boldmath $ r$ } } \cdot \nabla \times ( { \mbox{\boldmath $ v$ } } \cdot \nabla { \mbox{\boldmath $ v$ } } - { \mbox{\boldmath $ b$ } } \cdot \nabla { \mbox{\boldmath $ b$}})\end{aligned}\ ] ] where @xmath59 and @xmath60 denote the partial derivatives with respect to time @xmath61 and with respect to the angle @xmath62 of a spherical system of coordinates @xmath63 and where the operators @xmath64 and @xmath65 are defined by @xmath66 @xmath67 the equations for @xmath68 and @xmath69 are obtained through the multiplication of equation ( 6d ) and of its curl by @xmath32 , @xmath70\\ \nabla^2 l_2 g & = p_m [ \partial_t l_2 g - { \mbox{\boldmath $ r$ } } \cdot \nabla \times ( \nabla \times ( { \mbox{\boldmath $ v$ } } \times { \mbox{\boldmath $ b$ } } ) ) ] \end{aligned}\ ] ] either rigid boundaries with fixed temperatures as in the benchmark case ( christensen _ et al . _ 2001 ) , @xmath71 @xmath72 or stress - free boundaries with fixed temperatures , @xmath73 @xmath74 are often used . the latter boundary conditions are assumed in the following since they allow to cover numerically a larger region of the parameter space . the case @xmath75 will be considered unless indicated otherwise . it provides a good compromise for the study of both , the regions inside and outside the tangent cylinder . the latter is the cylindrical surface touching the inner spherical boundary at its equator . for the magnetic field electrically insulating boundaries are used such that the poloidal function @xmath68 must be matched to the function @xmath76 which describes the potential fields outside the fluid shell @xmath77 but computations for the case of an inner boundary with no - slip conditions and an electrical conductivity equal to that of the fluid have also been done . the numerical integration of equations ( 2 ) together with boundary conditions ( 4 ) proceeds with the pseudo - spectral method as described by tilgner and busse ( 1997 ) and tilgner ( 1999 ) which is based on an expansion of all dependent variables in spherical harmonics for the @xmath78-dependences , i.e. @xmath79 and analogous expressions for the other variables , @xmath80 and @xmath69 . @xmath81 denotes the associated legendre functions . for the @xmath33-dependence expansions in chebychev polynomials are used . for the computations to be reported in the following a minimum of 33 collocation points in the radial direction and spherical harmonics up to the order 64 have been used . but in many cases the resolution was increased to 49 collocation points and spherical harmonics up to the order 96 or 128 . it should be emphasized that the static state @xmath82 represents a solution of equations ( 6 ) for all values of the rayleigh numbers @xmath45 and @xmath46 , but this solution is unstable except for low or negative values of the latter parameters . similarly , there exist solutions with @xmath83 , but @xmath84 , for sufficiently large values of either @xmath45 or @xmath46 or both , but , again , these solutions are unstable for sufficiently large values of @xmath49 with respect to disturbances with @xmath85 . dynamo solutions as all solutions with @xmath85 are called are thus removed by at least two bifurcations from the basic static solution of the problem . finally we present in table 1 a list of the most important parameters used in the following sections . .important dimensionless parameters . [ cols= " < , < , < " , ] + for an introduction to the problem of convection in spherical shells we refer to the review of busse ( 2002a ) . convection tends to set in first outside the tangent cylinder in the form of thermal rossby waves for which the coriolis force is balanced almost entirely by the pressure gradient . the model of the rotating cylindrical annulus has been especially useful for the understanding of this type of convection . a rough idea of the dependence of the critical rayleigh number @xmath86 for the onset of convection on the parameters of the problem can be gained from the expressions derived from the annulus model ( busse , 1970 ) @xmath87 where @xmath88 refers to the mean radius of the fluid shell , @xmath89 , and @xmath90 to the corresponding colatitude , @xmath91 arcsin @xmath92 . the azimuthal wavenumber of the preferred mode is denoted by @xmath93 and the corresponding angular velocity of the drift of the convection columns in the prograde direction is given by @xmath94 . in figure 2 expressions ( 15a , c ) are compared with accurate numerical values which indicate that the general trend is well represented by expressions ( 15a , c ) . the same property holds for @xmath93 . for a rigorous asymptotic analysis in the case @xmath95 including the radial dependence we refer to jones _ et al . _ ( 2000 ) . it is evident from figure 2 that the agreement between expressions ( 15 ) and the numerical values deteriorates as low values of @xmath48 are approached . this behavior is caused by the fact that instead of the thermal rossby wave mode the inertial mode of convection becomes preferred at onset for sufficiently low prandtl numbers . it is characterized by convection cells attached to the equatorial region of the outer boundary not unlike the `` banana cells '' seen in the narrow gap experiment of figure 3 . the equatorially attached convection does indeed represent an inertial wave modified by the effects of viscous dissipation and thermal buoyancy . an analytical description of this type of convection can thus be attained through the introduction of viscous friction and buoyancy as perturbations as has been done by zhang(1994 ) and by busse and simitev ( 2004 ) for stress - free and by zhang ( 1995 ) for no - slip boundaries . according to ardes _ ( 1997 ) equatorially attached convection is preferred at onset for @xmath96 where @xmath97 increases in proportion to @xmath98 . a third form of convection is realized in the polar regions of the shell which comprise the two fluid domains inside the tangent cylinder . since gravity and rotation vectors are nearly parallel in these regions ( unless values of @xmath25 close to unity are used ) convection resembles the kind realized in a horizontal layer heated from below and rotating about a vertical axis . because the coriolis force can not be balanced by the pressure gradient in this case , the onset of convection is delayed to higher values of @xmath99 where convection outside the tangent cylinder has reached already high amplitudes . in the case of @xmath75 the onset of convection in the polar regions typically occurs at rayleigh numbers which exceed the critical values @xmath100 for onset of convection outside the tangent cylinder by a factor of the order @xmath101 . except for the case of very low prandtl numbers the retrograde differential rotation in the polar regions generated by convection outside the tangent cylinder tends to facilitate polar convection by reducing the rotational constraint . a tendency towards an alignment of convection rolls with the north - south direction ( busse and cuong 1977 ) can be noticed , but this property is superseded by instabilities of the kppers - lortz type ( for an experimental demonstration see busse and heikes ( 1980 ) ) and by interactions with turbulent convection outside the tangent cylinder . typical features of low and high prandtl number convection are illustrated in figure 4 . the columnar nature of convection does not vary much with @xmath48 as is evident from the top two plots of the figure . at prandtl numbers of the order unity or less , - but not in the case of inertial convection- , the convection columns tend to spiral away from the axis and thereby create a reynolds stress which drives a strong geostrophic differential rotation as shown in the bottom left plot of the figure . this differential rotation in turn promotes the tilt and a feedback loop is thus created . at high values of @xmath48 the reynolds stress becomes negligible and no significant tilt of the convection columns is apparent . in this case the differential rotation is generated in the form of a thermal wind caused by the latitudinal gradient of the axisymmetric component of @xmath41 . among the properties of convection at finite amplitudes the heat transport is the most important one . customarily its efficiency is measured by the nusselt number which is defined as the heat transport in the presence of convection divided by the heat transport in the absence of motion . in the case of the spherical fluid shell two nusselt numbers can be defined measuring the efficiency of convection at the inner and the outer boundary , @xmath102 where the double bar indicates the average over the spherical surface . in addition local nusselt numbers @xmath103 are of interest where only the azimuthal average is applied , as indicated by the single bar , instead of the average over the entire spherical surface . examples of such measures of the dependence of the heat transport on latitude are shown in figure 5 . this figure demonstrates that at low supercritical rayleigh numbers the heat transport occurs primarily across the equatorial region , but as @xmath99 increases the heat transport in the polar regions takes off and soon exceeds that at low latitudes . this process is especially evident at the outer boundary . in the polar regions convection is better adjusted for carrying heat from the lower boundary to the upper one , and it is known from computations of the convective heat transport in horizontal layers rotating about a vertical axis that the value of @xmath104 may exceed the value in a non - rotating layer at a given value of @xmath99 in spite of the higher critical rayleigh number in the former case ( somerville and lipps , 1973 ) . outside the tangent cylinder the convective heat transport encounters unfavorable conditions in that the cylindrical form of the convection eddies is not well adjusted to the spherical boundaries . this handicap is partly overcome through the onset of time dependence in the form of vacillations in which the convection columns expand and contract in the radial direction or vary in amplitude . at prandtl numbers of the order unity and less another effect restricts the heat transport . the shear of the geostrophic differential rotation created by the reynolds stresses of the convection columns severely inhibits the heat transport . to illustrate this effect we have plotted in figure 6 in addition to the nusselt numbers the averages of the kinetic energy densities of the various components of the convection flow which are defined by @xmath105 where the angular brackets indicate the average over the fluid shell and @xmath106 refers to the azimuthally averaged component of @xmath56 , while @xmath107 is defined by @xmath108 . analogous definitions hold for the energy densities of the magnetic field , @xmath109 the total magnetic energy density @xmath110 averaged over the fluid shell is thus given by @xmath111 . figure 6 is instructive in that it demonstrates both , convection in the presence and in the absence of its dynamo generated magnetic field . as is evident from the right part of figure 6 with @xmath112 relaxation oscillations occur in which convection sets in nearly periodically for short episodes once the energy @xmath113 of the differential rotation has decayed to a sufficiently low amplitude . but as soon as convection grows in amplitude , the differential rotation grows as well and shears off the convection columns . after convection has stopped the differential rotation decays on the viscous time scale until the process repeats itself . as long as the magnetic field is present @xmath113 is suppressed owing to the action of the lorentz force and high nusselt numbers are obtained . the dynamo generated magnetic field thus acts in a fashion quite different from that of a homogeneous field which typically counteracts the effects of rotation and tends to minimize the critical value of the rayleigh number when the elsasser number , @xmath114 assumes the value @xmath115 in the case of a plane layer ( chandrasekhar 1961 ) or values of the same order in the case of a sphere ( fearn 1979 ) or in the related annulus problem ( busse 1983 ) . [ tl ] [ tl ] [ tl ] it appears that dynamos are generated by convection in rotating spherical shells for all parameter values as long as the magnetic reynolds number is of the order @xmath116 or higher and the fluid is not too turbulent . since @xmath21 can be defined by @xmath117 where the kinetic energy density , @xmath118 , increases with @xmath99 , increasing values of the rayleigh number are required for dynamos as @xmath49 decreases . in planetary cores the latter parameter may assume values as low as @xmath119 or @xmath120 , but numerical simulation have achieved so far only values somewhat below @xmath121 . the trend of increasing @xmath99 with decreasing @xmath49 is evident in figure 7 where results are shown for two different values of @xmath47 and @xmath48 . it is also evident from the results in the upper left corner of the figure that an increasing @xmath99 may be detrimental for dynamo action . this is a typical property of marginal dynamos at the boundary of dynamos in the parameter space ( christensen _ et al . _ 1999 , simitev and busse 2005 ) . since convection at the relevant values of @xmath99 is chaotic this property also holds for the generated magnetic field . the magnetic energy must be finite near the boundary of dynamos in the parameter space because a sufficient amplitude of the fluctuating components of the convection flow can be obtained only when the magnetic field is strong enough to suppress the differential rotation . this property appears to hold even at high prandtl numbers where differential rotation occurs only in the form of a relatively weak thermal wind . even in its chaotic state convection at high values of @xmath47 exhibits a strong symmetry with respect to the equatorial plane at high values of @xmath47 , at least as long as convection in the polar regions is still weak . because of this symmetry magnetic fields can be generated either with the same symmetry as the convection flow , i.e. @xmath68 and @xmath69 are antisymmetric with respect to equatorial plane in which case one speaks of a quadrupolar dynamo , or the magnetic field exhibits the opposite symmetry with symmetric functions @xmath68 and @xmath69 in which case one speaks of a dipolar dynamo . of special interest are hemispherical dynamos ( grote and busse 2001 ) in which case the fields of dipolar and quadrupolar symmetry have nearly the same amplitude such that they cancel each other either in the northern or the southern hemisphere . examples of typical dynamos with different symmetries are shown in figure 8 . here it is also evident that low @xmath48 and high @xmath48 dynamos differ in the structure of their mean toroidal fields . while their mean poloidal dipolar fields exhibit hardly any difference , the high @xmath48 dynamo is characterized by strong polar azimuthal flux tubes which are missing in the low @xmath48 case . the reason for this difference is that the radial gradient of the differential rotation in the polar regions is much stronger for high @xmath48 than for low @xmath48 . as the rayleigh number increases and polar convection becomes stronger the convection flow looses some of its equatorial symmetry and the magnetic field can no longer easily be classified . usually the dipolar component becomes more dominant in cases of dynamos which started as either quadrupolar or hemispherical dynamos at lower values of @xmath99 . it should be emphasized that quadrupolar and hemispherical dynamos are usually oscillatory in that new azimuthal flux tubes of alternating sign emerge in the equatorial plane , move towards higher latitudes , while old flux tubes are dissipated in the polar region . there also exist dipolar dynamos which exhibit the same type of oscillations . they are typically found in the parameter space near the region where hemispherical dynamos occur . to illustrate the dynamo oscillations additional plots separated at equal distances in time before the corresponding plot in the lower three rows of figure 8 are shown in figure 9 . the four plots in each case cover approximately half a period of oscillation . it must be realized , of course , that the oscillations are not strictly periodic since they occur in a turbulent system . in this respect they resemble the solar cycle with its 22 year period . because the solar dynamo is believed to operate at the bottom of the solar convection the propagation of the dynamo wave is towards low latitudes on the sun . a remarkable feature of the dipole oscillation in figure 9 is that the polar flux tubes hardly change in time . the oscillation appears to be confined to the region outside the tangent cylinder . at a lower value of @xmath49 the strong polar flux tubes even inhibit the oscillation of the mean poloidal field as shown in figure 10 . this case has been called the `` invisibly '' oscillating dynamo since at some distance from the boundary of the spherical fluid shell the oscillation of the dynamo can hardly be noticed . + the relatively simple magnetic fields displayed in figures 8 , 9 , and 10 should not be regarded as representative for high values of the rayleigh number when @xmath99 exceeds its critical value @xmath100 by an order of magnitude or more . in that case the equatorial symmetry has nearly disappeared owing to the dominant convection in the polar regions and non - axisymmetric components of the field tend to exceed axisymmetric ones as shown , for example , in figure 11 . the strong growth with increasing @xmath21 of the non - axisymmetric magnetic flux is also evident in figure 12 when energies corresponding to different values of @xmath49 are compared . an important question is the average strength of the magnetic field generated by the dynamo process in dependence on the parameters . two concepts are often used in attempts to answer this question . in astrophysical situations the equipartition between magnetic and kinetic energy is used as a guide . while such a balance may hold locally as , for example , in the neighborhood of sunspots , it is not likely to be applicable to global planetary magnetic fields . in the earth s core , for example , the magnetic energy density exceeds the kinetic one by a factor of the order of @xmath122 . the second concept is based on the property that an elsasser number @xmath123 of the order unity appears to correspond to optimal conditions for convection in rotating systems in the presence of an applied nearly uniform magnetic field . while this idea may be useful as a first rough estimate , the results of numerical dynamo simulations do not support this concept very well . the examples shown in figure 12 exhibit @xmath123-values differing by an order of magnitude and even larger variations have been reported by simitev and busse ( 2005 ) and by christensen and aubert ( 2005 ) . another important feature demonstrated in figure 12 is the change in the structure of the magnetic field with increasing prandtl number . while for @xmath124 and lower values the mean poloidal field is small in comparison with the fluctuating components , this situation changes dramatically at about @xmath125 for @xmath126 such that for higher @xmath48 the mean poloidal field becomes dominant . usually this field is dipolar . this change is associated with the transition from the geostrophic differential rotation generated by reynolds stresses to the thermal wind type differential rotation caused by a latitudinal temperature gradient . while the magnetic energy @xmath110 may exceed the total kinetic energy @xmath127 by orders of magnitude in particular for high prandtl numbers , ohmic dissipation is usually found to be at most comparable to viscous dissipation in numerical simulations . but this may be due to the limited numerically accessible parameter space . the brief introduction of this section to convection driven dynamos in rotating spherical fluid shells can only give an vague impression of the potential of numerical simulations for the understanding of planetary magnetism . many more examples of such simulations can be found in the literature . usually they have been motivated by applications to the geodynamo and @xmath53 is assumed in most cases for simplicity . for some recent systematic investigations we refer to christensen et al . ( 1999 ) , grote et al . ( 2000 ) , jones ( 2000 ) , grote and busse ( 2001 ) , kutzner and christensen ( 2000 , 2002 ) , busse et al . ( 2003 ) , simitev and busse ( 2005 ) and other papers referred to therein . in many respects it is too early to model the dynamo process in particular planets . the numerical simulations of the kind discussed in the preceding sections are still rather removed from the parameter regime relevant to planetary interiors . only most recently attempts have been made to extrapolate results to high rayleigh numbers and high values of @xmath47 ( christensen and aubert , 2006 ) . moreover , only the most important physical parameters have been taken into account and relevant properties such as compressibility and other deviations from the boussinesq approximation have been considered only in special models applied to the earth s core ( glatzmaier and roberts 1996 ) or to the sun ( brun _ et al . _ 2004 ) . on the other hand , many essential parameter values of planetary cores are not sufficiently well known to provide a basis for the development of specific dynamo models . much of the future progress of the field will thus depend on the mutual constraints derived from observational evidence and from theoretical conclusions in order to arrive at a better understanding of the workings of planetary dynamos . l@c@c@c@c@c ' '' '' planet & planetary radius & dipole moment & core radius & angular rate of & magn . + ( satellite ) & @xmath128 ( @xmath129 m ) & ( @xmath130t@xmath131 ) & ( @xmath129 m ) & rot . @xmath23 ( @xmath119s@xmath132 ) & ( m@xmath55/s ) + ' '' '' mercury & 2.439 & 0.0025 & 1.9 & 0.124 & @xmath133 2 + earth & 6.371 & 0.31 & 3.48 & 7.29 & 2 + mars & 3.389 & @xmath134 & @xmath133 1.5 & 7.08 & @xmath133 2 + jupiter & 69.95 & 4.3 & @xmath135 & 17.6 & @xmath136 + saturn & 58.30 & 0.21 & @xmath137 & 16.2 & @xmath138 + uranus & 25.36 & 0.23 & @xmath139 & 10.1 & @xmath140 + neptune & 24.62 & 0.14 & @xmath141 & 10.8 & @xmath140 + ganymede & 2.63 & 0.0072 & 0.7 & 1.02 & 4 + + @xmath142 since the giant planets do not possess a well - defined boundary of a highly conducting core , + values of the most likely region of dynamo activity are given . + @xmath143 remnant magnetism of the martian crust appears to require an ancient dynamo + with a field strength of at least 10 times the earth s magnetic field . in table 2 some properties related to planetary magnetism have been listed . the dipole moments of the planets have been given as multiples of @xmath144 where @xmath128 denotes the mean radius of the planet or satellite . thus the numerical value indicates the field strength in @xmath145 in the equatorial plane of the dipole at the distance @xmath128 from its center . we have included mars in the table , although it does have an active dynamo at the present time . the strong magnetization of the martian crust indicates , however , that a strong field dynamo must have operated in the early history of the planet . the most important question of the theory of planetary magnetism is the dependence of the observed field strength on the properties of the planet . we have already discussed in the historical introduction various proposals for such dependences based on _ ad hoc _ assumptions , but these have been abandoned by and large . only the concept of an elsasser number @xmath123 of the order unity is still frequently used . stevenson ( 2003 ) shows that a value @xmath146 fits most of the planets with a global magnetic field quite well , but in cases such as mercury and saturn only a value of the order @xmath147 can be estimated for @xmath123 . in order to save the concept different dynamo regimes must be assumed . equilibration balances for weaker magnetic fields have been proposed as , for instance , in the case when the strength of convection is not sufficient to attain the @xmath148 balance . for details see stevenson ( 1984 ) . an important balance often invoked in discussions of planetary dynamos ( see , for example , stevenson ( 1979 ) and jones ( 2003 ) ) is the mac - balance ( braginsky 1967 ) where it is assumed that lorentz force , buoyancy force and pressure gradient are all of the same order as the coriolis force , while viscous friction and the momentum advection term are regarded as negligible in the equations of motion . the neglection of the latter term is well justified for high prandtl numbers , but it is doubtful for values of @xmath48 of the order unity or less . inspite of their smallness in comparison with the coriolis force , the divergence of the reynolds stress can generate the most easily excitable mode of a rotating fluid , namely the geostrophic differential rotation which can not be driven by the coriolis force , the pressure gradient or buoyancy . of course , an excitation by the lorentz force is possible in principle . the latter force , however , usually inhibits the geostrophic differential rotation . since the mean azimuthal component of the magnetic field is typically created through the shear of the differential rotation , the lorentz force opposes the latter according to lenz rule . christensen and aubert ( 2006 ) have recently introduced a concept in which the equilibrium strength of planetary magnetism does no longer depend on the rate of rotation @xmath23 , but instead depends only on the available power for driving the dynamo . they find that scaling laws can be obtained once rayleigh number , nusselt number and dimensionless buoyancy flux @xmath149 have been defined through quantities that no longer involve molecular diffusivities . this is a surprising result since it requires the presence of rotation and can not be achieved in a non - rotating system . the final estimate obtained for the strength of the magnetic field in the dynamo region fits the cases of the earth and jupiter quite well , but yields the result that mercury s field could not be generated by a buoyancy driven dynamo since the magnetic reynolds number would be too small . in the cases of the outer planets relevant parameters are not sufficiently well known to draw definitive conclusions . finally we like to draw attention to what has been called stevenson s paradox , namely that a too high electrical conductivity is detrimental for dynamo action in planetary cores . according to the wiedemann - franz law of condensed matter physics the thermal conductivity @xmath150 of a metal is proportional to its electrical conductivity times the temperature , @xmath151 , where l has approximately the same value for all metals . this is due to the fact that in metals electrons dominate both , the transports of heat and of charge . as a consequence there is an upper bound on the electrical conductivity for which a dynamo driven by thermal convection is possible because a high value of @xmath1 implies a low temperature change with increasing depth needed to carry the heat from the interior of the planetary core . but as soon as the temperature change with pressure falls below its isentropic value thermal convection disappears and a stably stratified environment is obtained which makes it difficult to drive a dynamo by other types of motions such as compositional convection . ignoring the latter possibility stevenson ( 2003 ) has sketched the diagram shown in figure 13 where the marginal nature of dynamos in terrestrial as well as in the `` icy '' planets is indicated . in the following brief characterizations of the magnetisms of various planets and moons are given . in terrestrial planets and satellites dynamos are possible in the liquid parts of their iron cores . such cores always include light elements which depress the temperature of freezing and can give rise to compositional buoyancy in the presence of a solidifying inner part of the core . in jupiter and especially in saturn convection may also be driven in part by compositional buoyancy since hydrogen and helium are immiscible in certain regions and helium drops may rain out . in numerical dynamo models the different sources of buoyancy are usually not distinguished , however . it has already become apparent that the origin of mercury s magnetism is especially enigmatic . besides the hypothesis of an active dynamo , thermoelectric currents and remnant magnetism of the crust compete as explanations . the latter possibility had been discarded for a long time since runcorn s theorem states that a homogeneous spherical shell can not be magnetized by an interior magnetic field in such a way that a dipolar field can be observed from the outside after the interior source has been removed ( runcorn , 1975 ) . aharonson et al . ( 2004 ) have shown , however , that plausible inhomogeneities of the crust could possibly explain a remnant magnetic field created by an ancient dynamo . the idea that a convection driven dynamo operates in a fluid outer core of mercury has received some support from observations of mercury s librations ( margot et al . , 2004 ) and from numerical simulations ( stanley et al . , 2005 ) . the amplitudes of the small periodic variations ( librations ) of mercury s rotation caused by the gravitational pull of the sun on the non - axisymmetric mass distribution of mercury appear to be too large for a fully solidified planet . this suggests that the solid inner core is separated from the mantle by a fluid outer core in which convection flows may occur . the numerical simulations of dynamos in the thin shell of the outer core have demonstrated that the dipole strength measured from the outside may not be representative for the strength of the magnetic field at the core - mantle boundary and the criteria mentioned above may be satisfied after all . the question , however , whether there is a sufficiently strong source of buoyancy to sustain convection and the dynamo over the age of the planet has not yet been answered satisfactorily . space probes have clearly shown that there is no dynamo operating presently in venus . this result is surprising since venus is very similar to the earth in many other aspects . that the period of rotation is only @xmath152 of a day should not matter much since the coriolis parameter @xmath47 is still huge . more important is the fact that the cooling of the planet is less efficient because of the apparent absence of mantle convection and plate tectonics . a solid inner core may not have yet started to grow in venus and vigorous convection does not occur in the core , at least not at the present time . neither moon nor mars have an active dynamo , but the magnetized rocks in their crusts suggest that in their early history about 4 billion years ago dynamos may have been operating in the iron cores of these bodies . the small lunar core with a radius of the order of 350 km and the age of the magnetized lunar rocks put severe constraints on a possible dynamo origin of lunar magnetism as discussed by stegman et al . in their model these authors try to accommodate in particular the apparent sudden onset of a lunar dynamo at a time @xmath153 ago . because of its larger iron core the possibility of a temporarily operating martian dynamo is more likely , but the strong magnetization of parts of the martian crust requires a field of at least ten times the strength of the present geomagnetic field . for a recent review of the implications of martian magnetism for the evolution of the planet we refer to stevenson ( 2001 ) . the jovian magnetic field is rather similar to the geomagnetic one sharing with it a dipole axis that is inclined by an angle of the order of 10 degrees with respect to the axis of rotation . that the higher harmonics of the field are relatively stronger than those in the case of the earth suggest that the jovian dynamo is driven at a more shallow layer than the earth s outer core . this assumption is in accordance with the transition to metallic hydrogen which is expected to occur at a pressure of @xmath154 corresponding to a region with a radius of 0.9 of the jovian radius ( nellis et al . , 1996 ) . the transition is not a sharp one as suggested by earlier models of jupiter s interior , but a gradual one leading to a decrease of the magnetic diffusivity down to about @xmath155 . since this value yields a rather high value for the elsasser number @xmath123 stevenson ( 2003 ) has argued that the dynamo is most active above the region of highest conductivity . jones ( 2003 ) , on the other hand , accepts the value @xmath156 for @xmath123 , but must admit a high magnetic reynolds number of the order @xmath157 at which an effective dynamo may no longer be possible . a detailed model of a convection driven jovian dynamo should include effects of compressibility and depth dependent electrical conductivity , but none has yet been published . saturn s magnetic field with a strength of @xmath158 near the poles is much weaker than that of jupiter which must be attributed primarily to its deeper origin in the planet . the transition pressure of @xmath159 is reached only at about half the planetary radius . a property that has received much attention is that the saturnian magnetic field is nearly axisymmetric with respect to the axis of rotation . this property can not be interpreted as a contradiction to cowling s ( 1934 ) theorem since it has long been shown that fields with arbitrarily small deviations from axisymmetry can be generated by the dynamo process ( braginsky 1976 ) . stevenson ( 1982 ) gave a reasonable explanation for the almost axisymmetric saturnian field by demonstrating that the differential rotation in the stably stratified , but still electrically conducting gas shells above the dynamo region will tend to shear off all non - axisymmetric components of the field . the problem may be a bit more intricate as has been pointed out by love ( 2000 ) who showed that sometimes fields with solely non - axisymmetric components are found in such situations . a metallic liquid probably does not exist in uranus and neptune whose interiors consist mostly of ice and rocks . because of the pressure dissociation of water deeper regions of these planets are characterized by an ionic conductivity which is lower than typical metallic conductivities by between one and two orders of magnitude . the mixture of water , methane , ammonia and other ices is sufficiently fluid that convection can occur and that a dynamo is thus possible . the fact that the observed magnetic field do not show an alignment with the axis planetary rotation and that quadrupolar and octupolar components are comparable to the dipolar components of the fields has been attributed to a dynamo operating in a thin shell ( ruzmaikin and starchenko , 1991 ; stanley and bloxham , 2004 ) . this seems be hardly necessary , however , since even convection driven dynamos in thick shells often exhibit such fields when small scale components dominate as in the case of high rayleigh numbers and low prandtl numbers . see , for example , figure 11 which shows a magnetic field rather similar to that of uranus as displayed in the paper of connerney ( 1993 ) . a special difficulty for a convection driven dynamo in the interior of uranus is caused by its low emission of heat . holme and bloxham ( 1996 ) suggest that a typical dynamo would involve more ohmic dissipation than corresponds to the heat flux emitted from the interior of the planet . it came as a great surprise when the measurements of the galileo spacecraft indicated that jupiter s moon ganymede possesses a global magnetic field for which an active dynamo inside the satellite seems to be the only realistic explanation ( kivelson et al . , 1996 ) , although the possibility of a remnant magnetism can not easily be excluded ( crary and bagenal 1998 ) . it must be kept in mind that ganymede is the largest satellite in the solar system which exceeds even mercury in size . nevertheless the estimated radius of its iron core is only about 660 km and it is hard believe that it could still be partly molten unless it contain a lot of radioactive potassium 40 or ganymede was captured into a resonance in its more recent history ( showman _ et al . the fact that ganymede s magnetic moment is nearly aligned with the ambient jovian magnetic has led to the suggestion that the ambient field could aid significantly ganymede s dynamo . sarson et al . ( 1997 ) have investigated this question with the result that the ambient field is too weak to exert much influence . on the other hand , in the case of io which is much closer to jupiter the interaction between the ambient field and the liquid iron core can explain the observed structure of the magnetic field without the assumption of an active dynamo . there are no other satellites in the solar system where an active dynamo must be suspected . the substantial magnetization of many iron meteorites suggest , however , that several differentiated proto - planets have had dynamos in the early days of the solar system . it is apparent from the above discussions that dynamo theory does not yet have much specific information to contribute to the interpretation of the observed magnetic properties of planets and satellites in the solar system . even possible interactions between thermal and compositional buoyancies have not yet been taken fully into account ( glatzmaier and roberts 1996 , busse 2002b ) . some typical properties are already apparent , however , as for instance : * dynamos that are dominated by a nearly axial dipole and exhibit a magnetic energy that exceeds the kinetic one by orders of magnitude as in the case of the earth are typical for high effective prandtl numbers as must be expected when convection is primarily driven by compositional buoyancy . the rather earth like appearance of the dynamo of glatzmaier and roberts ( 1995 ) is in part due to the high value of @xmath48 used in their computational model . * dynamos exhibiting strong higher harmonics are more likely driven by thermal convection corresponding to prandtl numbers of the order unity or less . * dynamo oscillations are a likely phenomenon in the presence of a sufficiently strong differential rotation . they may not always be apparent in the poloidal magnetic field seen at a distance from the dynamo region . * considerations based on the effects on convection of an imposed homogeneous magnetic field can not directly be applied to the case of convection driven dynamos . more details on the parameter dependence of dynamos will certainly emerge in the future as the increasingly available computer capacity will allow extensions of the parameter space accessible to computer simulations . space probes such as messenger in the case of mercury will provide much needed detailed information on planetary magnetic fields . eventually we may learn more about their variation in time which is one of the most interesting properties of planetary magnetic fields . as in the case of stellar magnetism where the study of star spots and stellar magnetic cycles is contributing to the understanding of solar magnetism it may eventually become possible to learn about the magnetism of extrasolar planets and apply this knowledge for an improved understanding of solar system dynamos . undoubtedly the field of planetary magnetism will continue to be an exciting one ! : : aharonson o , zuber m t , solomon s c 2004 crustal remanence in an internally magnetized non - uniform shell : a possible source for mercury s magnetic field ? 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in the past decades , raman spectroscopy@xcite techniques were successfully applied to carbon compounds , such as graphite ( see ref . and references therein ) and carbon nanotubes.@xcite upon the discovery of graphene,@xcite raman spectroscopy has proven to be a powerful tool to identify the number of layers , structure , doping , disorder , and to characterize the phonons and electron - phonon coupling.@xcite so far , most of the attention was focused on the position and width of the raman peaks . here we present a detailed calculation of the _ intensities _ of the multiphonon raman peaks in graphene . raman scattering involves an electron - hole pair as an intermediate state ; we show that the multiphonon raman peaks are strongly sensitive to the dynamics of this electron - hole pair . thus , raman scattering can be used as a tool to probe this dynamics . writing the low - energy hamiltonian of the interaction of electrons with the crystal vibrations and the electromagnetic field from pure symmetry considerations , we describe the system in terms of just a few independent coupling constants , considered to be parameters of the theory . the electron scattering rate is introduced phenomenologically as another parameter . the results of the present calculation are used to extract information about these parameters from the raman peak intensities , measured experimentally . as shown below , the raman intensities strongly depend on the electron scattering rate ; moreover , the electron - phonon and electron - electron contributions to this rate can be separated . this is especially important as there are very few techniques giving experimental access to electron scattering rates , which , in turn , determine the transport properties of graphene samples . besides , the quasiclassical character of the process imposes a severe restriction on the electron and hole trajectories which can contribute to the two - phonon raman scattering : upon the phonon emission the electron and the hole must be scattered backwards . this restriction results in a significant polarization memory : it is almost three times more probable for the scattered photon to have the same polarization as the incidend photon than to have the orthogonal polarization . also , the raman intensities depend on electron - phonon coupling constants ; to reproduce the experimental results , one has to take into account renormalization of these coupling constants by electron - electron interaction . this renormalization is missed by local or semi - local approximations to the density - functional theory , typically used for the _ ab initio _ calculation of the coupling constants . since graphene is a non - polar crystal , raman scattering involves electronic excitations as intermediate states : the electromagnetic field of the incident laser beam interacts primarily with the electronic subsystem , and emission of phonons occurs due to electron - phonon interaction . the matrix element of the process can be schematically represented as @xmath0 here @xmath1 is the initial state of the process ( the incident photon with a given frequency and polarization , and no excitations in the crystal ) , @xmath2 is the final state ( the emitted photon and @xmath3 phonons left in the crystal ) , while @xmath4 , @xmath5 , label the intermediate states where no photons are present , but an electron - hole pair is created in the crystal and @xmath6 phonons have been emitted . @xmath7 and @xmath8 , @xmath5 are the energies of these states , and @xmath9 is the inverse lifetime of the electron ( hole ) due to collisions . @xmath10 and @xmath11 stand for the terms in the system hamiltonian describing interaction of electrons with the electromagnetic field and with phonons , respectively . in the calculations we do not include the phonon broadening , assuming the phonon states to have zero width . first , this approximation is consistent with the available experimental information : the phonon width is about @xmath12 at most,@xcite while the electronic broadening is at least an order of magnitude higher ( see the discussion below , sec . [ sec:2intensities ] ) . second , this approximation is irrelevant provided that we calculate the integrated intensities of the raman peaks , since they are determined by the total spectral weight of the phonon state which does not depend on the phonon broadening . ( color online ) . schematic representation of the role of electron dispersion ( dirac cones , shown by solid lines ) in the one - phonon ( a , b ) and two - phonon raman scattering ( c , d ) . vertical solid arrows represent interband electronic transitions accompanied by photon absorption or emission ( photon wave vector is neglected ) , dashed arrows represent phonon emission , the horisontal dotted arrow represents the impurity scattering.,width=264 ] the photon wave vector is negligible , so momentum conservation requires that the sum of the wave vectors of emitted phonons must vanish ( provided that the impurity scattering is neglected ) . for the same reason raman scattering on one intervalley phonon must be impurity - assisted [ process ( b ) in fig . [ fig : res ] , giving rise to the so - called @xmath13 raman peak ] . @xmath13 peak is absent in the experimental raman spectrum of graphene,@xcite showing that impurity scattering is indeed negligible in these samples . looking at the intermediate electronic states involved in the raman scattering ( fig . [ fig : res ] ) , we notice that for one - phonon scattering [ processes ( a ) , ( b ) ] at least one intermediate state must be virtual , since energy and momentum conservation can not be satisfied simultaneously in all processes . thus , at least one of the factors in the denominator of eq . ( [ ramanmatrixelement= ] ) must be of the order of the phonon frequency @xmath14 [ for the impurity assisted scattering one of the electron - phonon matrix elements in the numerator of eq . ( [ ramanmatrixelement= ] ) should be replaced by the electron - impurity matrix element ] . for the two - phonon scattering [ process ( c ) ] all intermediate states can be real , so that all energy mismatches in the denominator of eq . ( [ ramanmatrixelement= ] ) can be nullified simultaneously and the result is determined by the electron scattering rate @xmath9 . we emphasize the qualitative difference between the _ fully resonant _ process ( c ) and the double - resonant@xcite process ( b ) , where one intermediate state is still virtual . we also note the analogous difference between the two - phonon processes ( c ) and ( d ) in fig . [ fig : res ] : only the process ( c ) is fully resonant , the other one involves an energy mismatch of @xmath15 . as a result , its amplitude will be smaller by a factor @xmath16 . obviously , these arguments can be extended to all multi - phonon processes with odd and even number of phonons involved : in order to annihilate radiatively , the electron and the hole must have opposite momenta ; if the total number of emitted phonons is odd , the electron and the hole must emit a different number of phonons , which is incompatible with energy conservation in all processes . our main focus will thus be on even - phonon processes , as their intensities are determined by the electronic scattering . the full resonance picture presented above assumes the mirror symmetry between the electron and the hole spectra . the electron - hole asymmetry can be included as a correction to the dirac spectrum : the electron and the hole energies can be written as @xmath17 , where @xmath18 is the momentum counted from the dirac point , @xmath19 is the dirac velocity , and @xmath20 is the asymmetry parameter . the energy scale @xmath21 , quantifying the role of the asymmetry in the raman scattering is defined as @xmath22 , where @xmath23 and @xmath24 are the frequencies of the incident and the scattered photon ( the details are given in secs . [ sec : warping ] , [ sec : intdev ] ) . namely , the arguments of the previous paragraph hold if @xmath25 . in the opposite case , it is @xmath21 that determines the smallest value of the denominators in eq . ( [ ramanmatrixelement= ] ) . we will always assume that both @xmath26 . in the real space , the typical size of the region of space probed by the electron - hole pair in the fully resonant two - phonon raman scattering , is @xmath27 . for the doubly resonant defect - induced one - phonon scattering , the inverse energy mismatch @xmath28 determines the time duration of the process by virtue of the uncertainty principle , so the length scale of the process in the real space is @xmath29 . although this length scale is much shorter than @xmath30 , it is still much greater than the lattice constant or the electron wavelength @xmath31 . the fully resonant raman scattering , where the energy is conserved in each of the elementary scattering processes , admits a simple quasiclassical description , described qualitatively in this subsection , and justified rigorously in secs . [ sec:2raman],[sec:4raman ] . let us denote by @xmath32 the energy of the electron and the hole in the photoexcited pair . initially , it is given by the half of the excitation frequency @xmath23 , @xmath33 . after the emission of @xmath3 phonons it is decreased by @xmath34 ; assuming @xmath35 , we neglect this decrease in the qualitative considerations . thus , during all the time taken by the raman scattering , electron and hole can be viewed as wave packets of the size @xmath36 , propagating across the crystal along classical trajectories . the electron and the hole are created in the same region of space of the size @xmath36 around some point @xmath37 at the moment of the arrival of the excitation photon . at this initial moment they have opposite momenta @xmath38 , and opposite velocities @xmath39 ( if the electron - hole asymmetry is taken into account , the two velocities will have slightly different magnitude ) , so they move along the straight lines , their positions being @xmath40 , @xmath41 . after a typical time @xmath42 they undergo some scattering processes ( e. g. , phonon emission ) , where their momenta and ( generally speaking ) energies are changed . each such elementary scattering process occurs during a short time @xmath43 . thus , the trajectories of the electron and the hole after their creation are represented by broken lines , with the typical segment length @xmath44 ( the electron mean free path ) . the crucial point is that in order to recombine radiatively and contribute to raman signal , the electron and the hole should meet again within a spatial region of the size @xmath36 , and have opposite momenta . the latter condition automatically implies that the number of the phonons emitted by the electron and the hole is the same . these considerations are illustrated by fig . [ fig : trajectories ] . in the presence of a significant electron - hole asymmetry , @xmath45 , the described picture is modified . namely , one of the segments of either the electron or the hole trajectory has the length @xmath46 instead of @xmath30 , the corresponding time travelling being restricted by the phase mismatch rather than by collisions . ( color online ) . ( a ) an example of a quasiclassical electron - hole trajectory contributing to the four - phonon raman scattering . ( b , c ) trajectories with emission of two phonons , not contributing ( b ) and contributing ( c ) to the two - phonon raman scattering . in all pictures the lightning represents the incident photon which creates the pair . the solid lines denote the free propagation of the electron and the hole . the flash represents the radiative recombination of the electron - hole pair . the dashed lines denote the emitted phonons.,title="fig:",width=302 ] ( color online ) . ( a ) an example of a quasiclassical electron - hole trajectory contributing to the four - phonon raman scattering . ( b , c ) trajectories with emission of two phonons , not contributing ( b ) and contributing ( c ) to the two - phonon raman scattering . in all pictures the lightning represents the incident photon which creates the pair . the solid lines denote the free propagation of the electron and the hole . the flash represents the radiative recombination of the electron - hole pair . the dashed lines denote the emitted phonons.,title="fig:",width=302 ] ( color online ) . ( a ) an example of a quasiclassical electron - hole trajectory contributing to the four - phonon raman scattering . ( b , c ) trajectories with emission of two phonons , not contributing ( b ) and contributing ( c ) to the two - phonon raman scattering . in all pictures the lightning represents the incident photon which creates the pair . the solid lines denote the free propagation of the electron and the hole . the flash represents the radiative recombination of the electron - hole pair . the dashed lines denote the emitted phonons.,title="fig:",width=302 ] for the single - phonon raman peaks the commonly accepted notations are `` @xmath47 '' for the peak at @xmath48 corresponding to emission of an optical phonon with zero wave vector , and `` @xmath13 '' for the defect - induced peak at @xmath49 corresponding to emission of an optical phonon with the wave vector near @xmath50 or @xmath51 points of the brillouin zone.@xcite sometimes one also distinguishes the so - called @xmath52-peak at @xmath53 . this peak is also defect - induced , and corresponds to emission of an optical phonon with a small wave vector @xmath54 . as mentioned above , in the present work we study only the clean graphene , hence @xmath13 and @xmath52 peaks are of no interest to us . unfortunately , there is no single commonly accepted system for labelling of the multiphonon raman peaks . the strong peak at @xmath55 , corresponding to emission of two phonons with the opposite wave vectors near the @xmath50 and @xmath51 points , was historically called @xmath56 ( as it is not defect - induced ) ; sometimes it is denoted by @xmath57 or by @xmath58 to stress that it is the second overtone of the @xmath13 peak . the peak at @xmath59 corresponding to emission of two phonons with two opposite wave vectors near the @xmath60 point is sometimes called @xmath61 , @xmath62 or @xmath63 . the latter notation reflects the fact that the frequency of this peak is not exactly the double of that of the @xmath47 peak , but rather the double of the defect - induced @xmath52 peak . in the following we use the notation @xmath64 to denote the peak corresponding to emission of @xmath3 phonons with wave vectors within @xmath65 from the @xmath60 point and of @xmath66 phonons with wave vectors within @xmath65 from the @xmath50 or @xmath51 points . for multiphonon peaks this nomenclature is unambiguous . thus , the peaks at @xmath55 and at @xmath59 will be called @xmath67 and @xmath68 , respectively . in the clean graphene the only one - phonon raman process , allowed by the momentum conservation corresponds to the emission of the @xmath69 optical phonon with zero wave vector and frequency @xmath48 . for this process the situation turns out to be drastically different from that described by the qualitative considerations of sec . [ sec : fullresonance ] . as shown in sec . [ sec:1raman ] , if one approximates the electron spectrum by the dirac cones , the numerator of eq . ( [ ramanmatrixelement= ] ) vanishes due to high symmetry of the low - energy electronic dirac hamiltonian ( as compared to the microscopic symmetry of the crystal ) . thus , the main contribution to the raman amplitude comes from the regions of the electronic brillouin zone far from the dirac points . as a consequence , the typical energy mismatch in the denominator of eq . ( [ ramanmatrixelement= ] ) is of the order of the whole electronic bandwidth . thus , the raman process , responsible for the @xmath48 peak , is completely off - resonant and the picture shown in fig . [ fig : res ] for the process ( a ) is wrong . as a result , the intensity of the peak is expected to be insensitive to most external parameters : polarization , electron concentration , degree of disorder , etc . to characterize this intensity , one has to introduce an additional parameter into the theory which has no simple relation to the parameters of the low - energy effective hailtonian . the resulting intensity of the peak is given by eq . ( [ ig= ] ) ; it is proportional to the fourth power of the excitation frequency which is the standard result for raman scattering when the difference between the frequencies of the incident and scattered photons is small . this dependence also agrees with the experimental results of ref . . note that the results described above do not hold for the defect - induced peak at @xmath49 . for this peak the double resonance picture,@xcite shown in fig . [ fig : res ] , process ( b ) , is fully adequate . as the phonons are emitted by electrons with momentum @xmath70 , the largest possible phonon momentum is @xmath71 , corresponding to the electron and hole backscattering ( for the @xmath67 peak at 2700 @xmath72 we count the phonon momenta from the @xmath50 and @xmath51 points ) . it would be natural to expect that any pair of phonons with opposite momenta @xmath73 and @xmath74 can be emitted , the only exception being @xmath75 which is prohibited by symmetry@xcite and the nearby ones which are suppressed due to the smallness of the matrix elements . these arguments would predict the width of the peak to be of the order of @xmath76 , where @xmath77 is the @xmath50 phonon group velocity ; besides , the shape of the peak would be strongly asymmetric : a sharp cutoff on the high - energy side at the frequency @xmath78 due to the resonance restriction , and a smooth drop - off towards zero at @xmath79 due to the matrix element suppression . the phonon dispersion can be deduced from the dependence of the frequency of the impurity - assisted one - phonon @xmath13 peak in graphite on the excitation energy @xmath23 : @xmath80.@xcite thus , for @xmath81 ev these arguments give the width of the @xmath67 peak to be about @xmath82 @xmath72 . however , the experimentally observed width is only about 30 @xmath72 at @xmath83 ev , and its shape is quite symmetric.@xcite the observed small width of the peak is explained by the quasiclassical picture , presented in sec . [ sec : realspace ] . if upon the emission of phonons the electron and the hole are scattered by an arbitrary angle , as shown in fig . [ fig : trajectories](b ) , they will not be able to meet at the same spatial point in order to recombine radiatively and contribute to the two - phonon raman peak . only if the scattering is backwards , this event is possible , as illustrated by fig . [ fig : trajectories](c ) . this condition fixes the wave vectors of the emitted phonons to be @xmath84 . the small deviations of the scattering angle from @xmath85 are restricted by the quantum diffraction , and the width of the two - phonon raman peaks , instead of being @xmath86 , is determined by a much smaller energy scale ( see the discussion below ) . the dominance of the electron and hole backscattering manifests itself in the polariazion memory of the raman signal . if the incident light is linearly polarized , the probability of excitation of the electron - hole pair with a given direction of momenta is proportional to @xmath87 , where @xmath88 is the angle between the electric field vector of the light and the momenta . thus , upon backscattering and radiative recombination , the probability to detect a photon of the same polarization as the original one is @xmath89 , and that of the orthogonal polarization is @xmath90 . averaging over @xmath88 , we obtain the ratio of intensities for the detection of polarization parallel and perpendicular to that of the incident light to be @xmath91 . this ratio may be slightly decreased due to a finite aperture ( see the discussion in sec . [ sec : ramanprobab ] ) . the calculation of the intensities of the two - phonon raman peaks is performed in sec . [ sec:2raman ] . the explicit expressions for the intensities of the @xmath67 and @xmath68 peaks , obtained under the assumption of dirac spectrum for the electrons , are represented by eqs . ( [ i2kend= ] ) and ( [ i2gend= ] ) . both are proportional to @xmath92 , where @xmath9 is the electron ( hole ) inelastic scattering rate . if the latter is smaller that the electron - hole asymmetry @xmath21 , then , according to the arguments of sec . [ sec : fullresonance ] , it is @xmath21 that restricts the energy denominators from below . formally , this results in the replacement ( [ i2ktrig= ] ) in both eqs . ( [ i2kend= ] ) , ( [ i2gend= ] ) . numerically , @xmath93 ( see , e. g. , ref . ) , so the relative correction to eq . ( [ i2kend= ] ) for small @xmath21 can be estimated as @xmath94 ^ 2/2\sim-{10}^{-4}(\omega_{in}/2\gamma)^2 $ ] . the total electronic broadening @xmath9 was measured by time - resolved photoemission spectroscopy to be 20 mev in ref . and @xmath95 mev in ref . ( all values taken for @xmath96 ev ) . a recent angle - resolved photoemission spectroscopy ( arpes ) measurement gives a significantly larger value for @xmath97 ( ref . ) . thus , the case @xmath98 seems to be more relevant for the description of experiments , than the opposite one . the raman matrix element corresponding to emission of two phonons with given wave vectors @xmath99 and @xmath100 is given by eq . ( [ mq2ph= ] ) for @xmath98 . from this dependence one can deduce the lineshape of the two - phonon peaks @xmath101 : @xmath102^{3/2}},\ ] ] where @xmath103 , @xmath104 is the central frequency for each peak , and @xmath105 is the group velocity of the corresponding phonon . thus , the full width at half maximum ( fwhm ) of each peak is given by @xmath106 for @xmath107 the dependence of the raman matrix element on @xmath108 is described by eq . ( [ warpedbackscattering= ] ) . the lineshape corresponds to _ two _ peaks separated by @xmath109 . experimentally , one sees just one @xmath67 peak with the fwhm about 30 @xmath72 at the excitation frequency @xmath110.@xcite this corresponds to an unrealistically large value of @xmath111 . most likely , this indicates that two - phonon peaks are broadened by other mechanisms , not taken into account in the present work . in particular , eq . ( [ di2dw= ] ) neglects ( i ) the broadening of the phonon states , and ( ii ) the anisotropy of the phonon dispersion ( trigonal warping of the phonon spectrum ) . a detailed study of these effects would require introduction of additional parameters into the theory , so we prefer to postpone such study for the future work . it is worth emphasizing again that the _ integrated _ intensity of the peaks , which is the main focus of the present study , does not depend on these details . in view of the results of the present paper it is worth mentioning the experimental measurements of the intensity @xmath112 as a function of doping . while in ref . no significant dependence was observed , ref . , where higher doping levels were reached , shows quite a strong dependence of @xmath113 on doping . the intensity @xmath114 of the off - resonant single - phonon @xmath48 peak should not depend on doping ( although the phonon width does exhibit such a dependence , the total spectral weight of the phonon state , determining the integrated intensity of the peak , must be preserved ) . at the same time , the intensity @xmath112 , if determined by the electron inelastic lifetime , should be sensitive to the concentration of carriers . indeed , in the intrinsic graphene at low temperatures the photoexcited carriers do not participate in electron - electron collisions , as the phase space volume is restricted.@xcite as the carriers are added to the system , the electron - electron collisions become possible , thus the total @xmath115 increases , and the intensity @xmath112 is decreased , in qualitative agreement with the observation of ref . . the motivation to study the four - phonon raman process comes from the following picture for the fully resonant processes . the incident photon creates an electron and a hole real quasiparticles which can participate in various scattering processes . if the electron emits a phonon with a momentum @xmath99 , the hole emits a phonon with the momentum @xmath100 , and after that the electron and the hole recombine radiatively , the resulting photon will contribute to the two - phonon raman peak . if they do not recombine at this stage , but each of them emits one more phonon , and they recombine afterwards , the resulting photon will contribute to the four - phonon peak , _ etc . _ three - phonon processes , not being fully resonant , are not interesting in this context . besides phonon emission and radiative recombination , electron and hole are subject to other inelastic scattering processes , which can also be viewed as emission of some excitations of the system . in principle , raman spectrum should also contain the contrubution from these excitations , which are left in the system after the radiative recombination of the electron and the hole . the key point is that for real quasiparticles , the probability to undergo a scattering process @xmath116 is determined by the ratio of corresponding scattering rate @xmath117 to the total scattering rate @xmath118 , not by the history . this probability determines the relative _ frequency - integrated _ intensity of the corresponding feature in the raman spectrum . thus , the ratio of integrated intensity @xmath119 of the raman peak corresponding to @xmath120 @xmath50 phonons to that for @xmath121 @xmath50 phonons ( @xmath122 ) must be proportional to @xmath123 , where @xmath124 is the rate of emission of each of the two @xmath50 phonons , and the square comes from the phonon emission by the electron and the hole . this conclusion depends weakly on the relation between @xmath115 and @xmath21 , only through a logarithmic factor . in the doped graphene , the most obvious competitor of the phonon emission is the electron - electron scattering : the optically excited electron can kick out another one from the fermi sea , i. e. , to emit another electron - hole pair . thus , raman spectrum should contain contribution from electron - hole pairs ; however , their spectrum extends all the way to the energy of the photo - excited electron ( optical energy ) in a completely featureless way . thus , it can not be distinguished from the parasitic background which is always subtracted in the analysis of raman spectra , and can not be seen in the raman spectrum directly . however , assuming @xmath125 , where @xmath126 is the rate of emission of phonons from the vicinity of the @xmath60 point of the first brillouin zone , and @xmath127 is the electron - electron collision rate , one can extract the value of @xmath128 , relative to phonon emission rates from the experimental data . more precisely , in this way one obtains the rate of all inelastic scattering processes where the electron loses energy far exceeding the phonon energy . note that arguments leading to @xmath129 are not specific for graphene ; in fact , this is nothing but breit - wigner formula , applied once for the electron and once for the hole . multi - phonon raman scattering has been studied in wide - gap semiconductors both experimentally @xcite ( up to ten phonons were seen in the raman spectra of cds ) , and theoretically @xcite . in a wide - gap semiconductor an optically excited electron does not have a sufficient energy to excite another electron across the gap , so the electron - electron channel is absent . in addition , interaction with only one phonon mode is dominant , so the ratios of subsequent peaks are represented by a sequence of fixed numbers . the simple band structure ( one valley for cds in contrast to two valleys for graphene ) allowed a calculation of the whole sequence . a more complicated electronic band structure in graphene makes it problematic to calculate the whole sequence , so we restrict ourselves to the calculation of @xmath130 for the most intense four - phonon peak . this calculation is performed in sec . [ sec:4raman ] . its result depends , besides the relation between @xmath115 and @xmath21 , also on their relation to the energy scale @xmath131 , characterizing the phonon dispersion . in sec . [ sec:2intensities ] we have already discussed this energy scale ; for @xmath132 we have @xmath133 . the meaning of this energy scale is the difference between the energies of the electron and the hole after each of them has emitted two phonons with almost arbitrary momenta ( the only restriction is that the sum of all four phonon momenta must vanish ) . if @xmath134 , which seems to be the case ( see the discussion in the previous subsection ) then this difference can be neglected , and the intensity of the @xmath135 peak is given by eq . ( [ i4gammadispless= ] ) for @xmath98 [ which is likely to be the case relevant for most experiments , and which was reported in the short paper by the author ( ref . ) ] , and by eq . ( [ i4dehdispless= ] ) for @xmath107 ; the polarization memory is lost in both these cases , @xmath136 . in the case @xmath137 the intensity @xmath130 is given by eq . ( [ i4kdisp= ] ) , and a significant polarization memory is expected , up to @xmath138 . a thorough experimental study of the intensity @xmath130 ( in particular , its dependence on doping ) is still lacking . our prediction for the case @xmath139 , which we believe to be the experimentally relevant one , is@xcite @xmath140 in the calculations of the raman intensities , described above , electron - phonon coupling constants entered as parameters of the theory , without any assumptions about their values , except for the relations fixed by the symmetry of the crystal . in particular , the two - phonon peak intensities @xmath112 and @xmath141 are determined by two independent dimensionless coupling constants which we denote @xmath142 and @xmath143 [ see eq . ( [ lambdaphonon= ] ) for the definition ] . a simple estimate of the coupling constants can be obtained from the tight - binding nearest - neighbor model of the graphene crystal . in this model the only parameter characterizing the electron spectrum is the nearest - neighbor electronic matrix element @xmath144 , and the electron - phonon interaction is characterized by its change with the bond length , @xmath145 . in this model we obtain @xmath146 to be given by the inverse ratio of the corresponding phonon frequencies , about 1.2 ; the same result up to a few percent is obtained from the density - functional theory ( dft ) calculations of ref . . at the same time , by comparing the experimentally measured intensities of the different two - phonon peaks , and using the result of our calculation performed in sec . [ sec:2raman ] , we can independently extract the ratio of the coupling constants . according to the data of ref . , @xmath147 , which gives @xmath148 . to explain this discrepancy we first analyzed the effect of the electronic trigonal band warping , which affects @xmath112 and @xmath141 differently . the corresponding calculation is done in sec . [ sec : warping ] . for @xmath81 ev , we estimate the relative contribution of the warping term as @xmath149 for @xmath112 and @xmath150 for @xmath141 , which is far too little to account for the observed ratio @xmath151 . we are thus led to the conclusion that the observed ratio @xmath151 must be due to the difference of the coupling constants , not accounted for by the dft calculation . a similar conclusion about the insufficiency of the dft calculation of the electron - phonon coupling constants has been drawn in ref . , where an attempt was made to explain the experimental data obtained by arpes@xcite using the results of the dft calculation . at the same time , we should note that the dimensionless coupling constant @xmath143 for the phonons near the @xmath60 point , as calculated by dft@xcite ( @xmath152 ) , agrees reasonably well with the measured one : the measurements of the linear in the wave vector @xmath108 term in the phonon dispersion ( kohn anomaly due to electron - phonon interaction ) , @xmath153 , give @xmath154 ( see ref . ) ; the measurements of the dependence of the phonon frequency @xmath14 on the electron fermi energy @xmath155 : @xmath156 give @xmath157 ( ref . ) @xmath158 ( ref . ) . we show that the difference between the ratio @xmath148 extracted from the raman peak intensities , and @xmath159 obtained by the dft calculation,@xcite is due to the part of coulomb interaction between electrons , not picked up by the dft when local approximations are used for the exchange - correlation functional , such as the local density approximation ( lda ) or the generalized gradient approximation ( gga ) , namely , logarithmic renormalizations.@xcite coulomb interaction has been known to be a source of logarithmic renormalizations for dirac fermions.@xcite coulomb renormalizations in graphene subject to a magnetic field have been considered in ref . , coulomb effect on static disorder has been studied in refs . . essentially , the idea of the renormalization of the coupling constants is that the matrix element of the electron - phonon interaction should be taken not between the non - interacting electronic states , but between the states dressed by the coulomb interaction . if the typical electronic energy in the problem is @xmath32 ( @xmath160 in the case of raman scattering ) , the renormalization is determined by the coulomb interaction at all length scales from the shortest ones ( lattice constant ) to the electron wavelength @xmath31 . it is this long - range part of the exchange and correlation that is missed by the local approximations in the dft calculation , which take into account correctly only the short - range correlations ( at the distances of the order of the lattice constant ) . in sec . [ sec : coulombrg ] we calculate the renormalization of the dimensionless electron - phonon coupling constants ( a preliminary account of this work was given in the short paper@xcite ) , and show that the coupling constant @xmath143 for the phonons near the @xmath60 point is not renormalized ( hence the agreement between the value of @xmath143 calculated by the dft and measured in the experiments , as mentioned above ) , while the coupling constant @xmath142 for the phonons near the @xmath161 point , which is responsible for the @xmath67 raman peak , is enhanced by the coulomb interaction . this enhancement depends on the electronic energy , as shown in fig . [ fig : rgflow ] . for the electronic energy @xmath162 this enhancement is in quantitative agreement with the measured ratio @xmath151 , provided that the screening of the coulomb interaction by the substrate is weak . the dependence of the enhancement on the electronic energy translates into the dependence of @xmath151 on the excitation frequency , which can be checked experimentally . similarly , as the coulomb interaction is screened by the substrate with a dielectric constant @xmath163 ( its high - frequency value ) , the dependence of @xmath151 on @xmath163 can also serve as an experimental check of the theory . we also show in sec . [ sec : phononrg ] that the electron - phonon coupling itself is a source of logarithmic renormalizations . however , due to the smallness of the coupling constants this effect is much weaker than the effect of the coulomb interaction . in sec . [ sec : hamiltonian ] the low - energy hamiltonian of the interaction of electrons with the crystal vibrations and the electromagnetic field is written from pure symmetry considerations . in sec . [ sec : symmetry ] the symmetry of the graphene crystal is reviewed . in sec . [ sec : electrons ] the symmetry considerations are used to write the electronic part of the hamiltonian . [ sec : adddirac ] is dedicated to the symmetry analysis of the dirac part of the electron hamiltonian , whose symmetry is significantly higher than the symmetry of the crystal . [ sec : parphonons ] is dedicated to the symmetry analysis of the in - plane crystal vibrations . in secs . [ sec : eopt ] and [ sec : eac ] we write the hamiltonian of interaction of electrons with the optical and acoustical vibrations , respectively . [ sec : perpphonons ] is dedicated to the symmetry analysis of the out - of - plane vibrations of the graphene crystal . in sec . [ sec : emfield ] the hamiltonian of the interaction of electrons with the electromagnetic field is written . sec . [ sec : ramangeneral ] describes the general scheme of the calculation of raman peak intensities using the standard perturbation theory . in sec . [ sec : propagators ] the green s functions are introduced , and in sec . [ sec : ramanprobab ] the general expression for the raman scattering probability is derived . in sec . [ sec : inelastic ] we discuss the electron inelastic scattering , and calculate the electronic self - energy due to the electron - phonon coupling . in sec . [ sec:1raman ] one - phonon raman scattering is discussed , and it is shown that the calculation of the one - phonon peak intensity can not be performed within the low - energy theory . in sec . [ sec:2raman ] the two - phonon raman peak intensities are calculated , first , under the assumption of the dirac electron spectrum ( sec . [ sec:2ramandirac ] ) , and then taking into account the trigonal band warping and electron - hole asymmetry ( sec . [ sec : warping ] ) . in sec . [ sec:4raman ] the intensity @xmath130 of the most intense four - phonon peak is calculated . [ sec : coulombrg ] , [ sec : phononrg ] are dedicated to the renormalization of the electron - phonon coupling constants due to coulomb interaction and due to the electron - phonon interaction , respectively . since the typical energy of the incident photon ( about @xmath164 ev ) is much smaller than the @xmath85-electron bandwidth ( @xmath165 ev ) , one can expect the low - energy excitations to play the dominant role . in this section we employ standard symmetry analysis@xcite to fix the form of the low - energy hamiltonian . we prefer not to choose any specific basis and use algebraic properties . the carbon atoms of graphene form a honeycomb lattice with two atoms per unit cell , labeled @xmath166 and @xmath167 ( fig . [ fig : lattice ] ) , the distance between nearest neighbors being @xmath168 . three out of four electrons of the outer shell of each carbon atom form strong @xmath169 bonds with its three nearest neighbors , and represent no interest to us . the remaining @xmath85 orbitals ( one per each carbon atom ) give rise to the half - filled @xmath85 band . the honeycomb lattice with two atoms ( @xmath166 and @xmath167 ) per unit cell . tripled unit cells are shown by dashed hexagons.,width=302 ] .irreducible representations of the groups @xmath170 and @xmath171 and their characters.[tab : c6vc3v ] [ cols="^,^,^,^,^,^,^",options="header " , ] polarization operator with arbitrary matrix vertices @xmath172 , @xmath173 , @xmath174 ( we denoted @xmath175 ) , can be conveniently calculated in the coordinate representation:@xcite @xmath176 if @xmath177 matrices are also present in the vertices , tracing of them is trivial : the matrices in the two vertices must coincide for the trace not to vanish , then their product is equal to the unit matrix . it is convenient to switch to the imaginary time @xmath178 , @xmath179 . then the green s function in the coordinate representation can be found by using the analogy with the 3d coulomb problem . namely , we introduce the third dimension @xmath180 , and notice that @xmath181 since the inverse of the 3d laplacian is the coulomb potential , we obtain @xmath182 using auxiliary relations ( here @xmath183 ) , 99 l. i. mandelshtam and g. s. landsberg , z. phys . * 50 * , 169 ( 1928 ) ; 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superconducting films are candidate substances for the improvement of electronics technology in a myriad of applications . while the low resistance is very attractive in this regard , it has proved difficult to control the nonlinear behaviour of such materials in response to electromagnetic field@xcite . when a magnetic field is strong enough to penetrate into a superconductor in the form of quantised magnetic flux tubes , the vortex state obtains as a mixed state of superconducting phase punctuated by the vortices themselves . vortices are surrounded by a supercurrent and can be forced into motion by the current resulting from an applied field . as a topological defect , a vortex is not only stable under perturbations@xcite but can not decay . the collection of vortices in a type - ii superconductor forms what is called vortex matter , and it is this which determines the physical properties of the system rather then the underlying material properties , in particular driving phase transitions@xcite . in the mixed state , a superconductor is not perfect ; it exhibits neither perfect diamagnetism nor zero electrical resistance . the transport current @xmath0 generates a lorentz force @xmath1 on the vortex and forces it into motion , dissipating energy . in reaching thermal equilibrium , energy is transferred via interactions between phonons and quasiparticle excitations . small - scale imperfections such as defects scatter the quasiparticles , affecting their dynamics . in _ superconductors , impurities are plentiful and vortices experience a large friction . this implies a fast momentum - relaxation process . in contrast is the _ clean _ limit , where impurities are rare and no such relaxation process is available . it is in this situation of slow relaxation that the hall effect appears . generally , the @xmath2-@xmath3 phase diagram@xcite of the vortex matter has two phases . in the _ pinned phase _ vortices are trapped by an attractive potential due to the presence of large - scale defects , thus resistivity vanishes . this phase contains what are known as glass states . there is then the _ unpinned phase _ in which vortices can move when forced and so a finite resistivity appears . this phase is also known as the flux - flow region and can be of two types . one type is a liquid state where vortices can move independently ; the other type is a solid state in which vortices form a periodic abrikosov lattice@xcite resulting from their long - range interacton . one model for the transition between the pinned and unpinned phases appears in @xcite . in the unpinned phase , the system is driven from equilibrium and experiences a relaxation process . there are several ways to describe such a system . a microscopic description@xcite invoking interactions between a vortex and quasi - particle excitations at the vortex core provides a good understanding of friction and sports good agreement with experiments in the sparse - vortex region @xmath4 . there is also a macroscopic description , the london approach , where vortices are treated either as interacting point - like particles or an elastic manifold subject to a pinning potential , driving force and friction@xcite . in the small - field region , vortices behave as an array of elastic strings . in the dense - vortex region @xmath5 , where the magnetic field is nearly homogeneous due to overlap between vortices , ginzburg - landau ( gl ) theory , which describes the system as a field , provides a more reasonable model . in dynamical cases , time - dependent gl ( tdgl ) theory is appropriate@xcite ; in gl - type models , additional simplification can come from the lowest landau level ( lll ) approximation which has proven to be successful in the vicinity of the superconducting - normal ( s - n ) phase transition line @xmath6 . this has been pursued in the static case@xcite ( without driving force ) and in the dynamic case with a time - independent transport current@xcite . it may be noted that in the glass state , zero resistance within the lll approximation can not be attained@xcite . based on tdgl theory , we will study the dynamical response of a dense vortex lattice forced into motion by an alternating current induced by an external electromagnetic field . vortices are considered which are free from being pinned and thermally excited , which in addition to thermal noise would produce entanglement and bending . we assume the vortices can transfer work done by an external field to a heat bath . experimentally , a low - temperature superconductor far away from the clean limit is the best candidate for attaining these conditions . we do not consider thermal fluctuation effects specific to high - temperature superconductors . in a dissipative system driven by a single - harmonic electric field @xmath7 , long after its saturation time we can expect the system to have settled into steady - state behaviour , where the vortices are vibrating periodically with some phase . the tdgl model in the presence of external electromagnetic field is analysed and solved in . the dynamical s - n phase transition surface @xmath8 is located in @xmath9-space . this surface coincides with the mean - field upper - critical field @xmath10 in the absence of the applied field , and with the phase - transition surface in the presence of the constant driving field considered by hu and thompson@xcite . we will provide an analytical formalism for perturbative expansion in the distance to @xmath11 , valid in the flux - flow region . the response of vortex matter forced into motion by the transport current is studied in . the current - density distribution and the motion of vortices are treated in . in analysing the vortex lattice configuration in , a method is utilised whereby the heat - generation rate is maximised . next are discussed power dissipation , generation of higher harmonics , and the hall effect . an experimental comparison is made in with far - infrared ( fir ) measurement on nbn . finally , some conclusions are made in . let us consider a dense vortex system prepared by exposing a type - ii superconducting material to a constant external magnetic field @xmath12 with magnitude @xmath13 . we also select the @xmath14 axis of the superconductor to be in the @xmath15 direction . let the superconductor carry an alternating electric current along the @xmath16 direction , generated by an electric field @xmath17 as shown in . such a system when disturbed from its equilibrium state will undergo a relaxation process . for our system , the tdgl equation@xcite is a useful extension of the equilibrium gl theory . in the dense - vortex region of the @xmath2-@xmath3 phase diagram , vortices overlap and a homogeneous magnetic field obtains . describing the response of such a system by a field , the order parameter @xmath18 in the gl approach , is more suitable than describing vortices as particle - like flux tubes , as is done in the london approach@xcite . a strongly type - ii superconductor is characterised by its large penetration depth @xmath19 and small coherence length @xmath20 , @xmath21 . the difference between induced magnetic field and external magnetic field is @xmath22 . in the vicinity of the phase - transition line @xmath10 vortices overlap significantly , and @xmath23 making @xmath24 small . in this case , the magnetic field may be treated as homogeneous within the sample . we will have in mind an experimental arrangement using a planar sample very thin compared with its lateral dimensions . since the characteristic length for inhomogeneity of electric field@xcite @xmath25 is then typically large compared with sample thickness , this implies that the electric field may also be treated as homogeneous throughout@xcite , eliminating the need to consider maxwell s equations explicitly . in equilibrium , the gibbs free energy of the system is given by@xcite @xmath26&=&\int\td { \bf r } \bigg\ { \frac{\hbar^2}{2m_{ab}}|{\bf d}\psi|^2 + \frac{\hbar^2}{2 m_c}|\partial_z \psi|^2 \nonumber\\ & & { \ \ \ \ } -\alpha ( \tnst - t)|\psi|^2+\frac{\beta}2 |\psi|^4 \bigg\}\end{aligned}\ ] ] where @xmath27 is the critical temperature at zero field . covariant derivatives employed here preserve local gauge symmetry and are two - dimensional ; @xmath28 and @xmath29 . governing the dynamics of the field @xmath18 is the tdgl equation @xmath30 this determines the characteristic relaxation time of the order parameter . microscopic derivation of tdgl can be found in @xcite in which the values of @xmath31 , @xmath32 and @xmath33 are studied . in the macroscopic case , these are viewed simply as parameters of the model . at microscopic scale , disorder is accounted for by @xmath33 , the inverse of the diffusion constant ; the relation of @xmath33 to normal - state conductivity is discussed in . in standard fashion , @xmath34 while @xmath35 . our set of equations is completed@xcite by including ampre s law , writing for the total current density @xmath36 as we shortly make a rescaling of quantities , we have written @xmath37 subscripts here for clarity . the first term is the normal - state conductivity . the second term can be written using a maxwell - type equation relating the vector - potential with the supercurrent , @xmath38 .\end{aligned}\ ] ] this is a gauge - invariant model ; we fix the gauge by considering the explicit vector potential @xmath39 and @xmath40 , corresponding to an alternating transport current . each vortex lattice cell has exactly one fluxon . we do not assume the electric field and the motion of vortices are in any particular direction relative to the vortex lattice , by way of rendering visible any anisotropy . for convenience , we define some rescaled quantities . the rescaled temperature and magnetic field are @xmath41 and @xmath42 . @xmath43 denotes the mean field upper - critical field , extrapolated from the @xmath27 region down to zero temperature . in the @xmath44-@xmath45 plane of the crystal we make use of _ magnetic length _ @xmath46 . we define @xmath47 where @xmath48 . the scale on the @xmath14-axis is @xmath49 with @xmath50 . the coordinate anisotropy in @xmath15 is absorbed into this choice of normalisation , as can be seen in . the order parameter @xmath18 is scaled by @xmath51 . the time scale is normalised as @xmath52 . therefore , frequency is @xmath53 . note that @xmath54 is then inversely proportional to @xmath45 . the amplitude of the external electric field is normalised with @xmath55 so that @xmath56 . after our rescaling the tdgl equation takes the simple form @xmath57 where the operator @xmath58 is defined as @xmath59 with our specified vector potential , covariant derivatives are @xmath60 , @xmath61 and @xmath62 . we define @xmath63 for convenience . the tdgl equation is invariant under translation in @xmath15 , thus the dependence of the solution in the @xmath15 direction can be decoupled . @xmath58 is not hermitian ; @xmath64 where the conjugation is with respect to the usual inner product , defined below . we will make extensive use of the eigenfunctions of @xmath58 and @xmath65 in what follows . the eigenvalue equation @xmath66 defines the set of eigenfunctions of @xmath58 appropriate for our analysis ; this can be seen in . the convention is that @xmath67 when and only when @xmath68 . taking corresponding in @xmath58 and switches it in the resulting @xmath69 to get the ` corresponding ' eigenfunction @xmath70 for @xmath65 . ] eigenfunctions of @xmath65 to be @xmath71 , the orthonormality @xmath72 may be chosen , so long as @xmath73 . shown in , crystal structure determines linear combinations of these basis elements with respect to @xmath74 ; the resulting @xmath75 functions are then useful for expansion purposes below . the inner product is @xmath76 where the brackets @xmath77 denote an integral over space and time . to define averages over only time or space alone , we write @xmath78 or @xmath79 respectively . ; ( b ) typical small angle , plotted for @xmath80 . in the static case , both @xmath81 and @xmath80 are two particular angles will correspond to the same energy . the applied constant magnetic field @xmath82 is along the @xmath83 direction and time - dependent oscillating electric field @xmath84 is in the @xmath16 direction . @xmath85 is the the apex angle of the two defining lattice vectors . the two vectors for the rhombic vortex lattice are @xmath86 and @xmath87 where @xmath88 . the motion of vibrating vortices , indicated by the red arrow , is disccused in . , title="fig:",width=151 ] ; ( b ) typical small angle , plotted for @xmath80 . in the static case , both @xmath81 and @xmath80 are two particular angles will correspond to the same energy . the applied constant magnetic field @xmath82 is along the @xmath83 direction and time - dependent oscillating electric field @xmath84 is in the @xmath16 direction . @xmath85 is the the apex angle of the two defining lattice vectors . the two vectors for the rhombic vortex lattice are @xmath86 and @xmath87 where @xmath88 . the motion of vibrating vortices , indicated by the red arrow , is disccused in . , title="fig:",width=151 ] states of the system can be parametrised by @xmath89 . by changing temperature @xmath90 , a system with some fixed @xmath91 may experience a normal - superconducting phase transition as temperature passes below a critical value @xmath92 . such a point of transition is also known as a _ bifurcation point_. the material is said to be in the normal phase when @xmath18 vanishes everywhere ; otherwise the superconducting phase obtains , with @xmath18 describing the vortex matter . because of the vortices , the resistivity in the superconducting phase need not be zero . the s - n phase - transition boundary @xmath92 separates the two phases . to study the condensate , we will use a bifurcation expansion to solve . we expand @xmath18 in powers of distance from the phase transition boundary @xmath93 . as in the static case , we can locate the dynamical phase - transition boundary by means of the linearised tdgl equation@xcite . this is because the order parameter vanishes at the phase transition , and we do not need to consider the nonlinear term . the linearised tdgl equation is written @xmath94 of the eigenvalues of @xmath58 , only the smallest one @xmath95 , corresponding to the highest superconducting temperature @xmath93 , has physical meaning . the s - n phase transition occurs when the trajectory in parameter space intersects with the surface @xmath96 where the lowest eigenvalue is calculated in ; @xmath97 utilising a @xmath45-independent frequency @xmath98 and amplitude @xmath99 of input signal , we write @xmath100 in the absence of external driving field , @xmath101 , the phase - transition surface coincides with the well - known static - phase transition line @xmath102 in the mean - field approach . with time - independent electric field at @xmath103 , where the vortex lattice is driven by a fixed direction of current flow , the dynamical phase - transition surface coincides with that proposed in @xcite , but with a factor of @xmath104 . this amplitude difference is familiar from elementary comparisons of dc and ac circuits . in the above equation , we can see that in the static case @xmath101 , the superconducting region is @xmath105 . in addition when @xmath106 , the superconducting region in the @xmath45-@xmath90 plane is smaller than the corresponding region in the static case , as can be seen in ( a ) . finally , increasing frequency will increase the size of the superconducting region , as in ( b ) ; in the high - frequency limit , the area will reach its maximum , which is the superconducing area from the static case . as with any damped system , response is diminished at higher frequencies . the superconducting state does not survive at small magnetic field ; for example at @xmath107 in ( a ) , the material is in the normal state over most of the @xmath2-@xmath3 phase diagram . later in this paper we will consider interpretation of this phenomenon . in particular , when discussing energy dissipation in , we will see that the main contribution to the dissipation is via the centre of the vortex core . at small magnetic field , since there are fewer cores to dissipate the work done by the electric field , the superconducting state is destroyed and the order parameter vanishes . as a function of @xmath45 for various @xmath99 at @xmath108 and ( b ) @xmath93 as a function of @xmath99 for various @xmath109 at @xmath110 . the straight line in ( a ) is the @xmath101 curve and corresponds to the mean - field phase - transition line @xmath102 . states above each line are normal phase while the region below each line is superconducting . @xmath99 suppresses the superconducting phase as shown in ( a ) , while @xmath109 removes this suppression effect , as shown in ( b ) . , title="fig:",width=158 ] as a function of @xmath45 for various @xmath99 at @xmath108 and ( b ) @xmath93 as a function of @xmath99 for various @xmath109 at @xmath110 . the straight line in ( a ) is the @xmath101 curve and corresponds to the mean - field phase - transition line @xmath102 . states above each line are normal phase while the region below each line is superconducting . @xmath99 suppresses the superconducting phase as shown in ( a ) , while @xmath109 removes this suppression effect , as shown in ( b ) . , title="fig:",width=158 ] that the vortex matter dominates the physical properties of the system is especially pronounced in the pinning - free flux - flow region . here we solve by a bifurcation expansion@xcite . since the amplitude of the solution grows when the system departs from the phase transition surface where @xmath111 , we can define a distance from this surface as @xmath112 and expand @xmath18 in @xmath113 . the tdgl in terms of @xmath113 is @xmath114 where @xmath115 is the operator @xmath58 shifted by its smallest eigenvalue . @xmath18 is then written @xmath116 and it is convenient to expand @xmath117 in terms of our eigenfunctions of @xmath118 @xmath119 in principle , all coefficients @xmath120 in can be obtained by using the orthogonal properties of the basis , which are explained in . inserting @xmath18 from equation into tdgl equation , and collecting terms with the same order of @xmath113 , we find that for @xmath121 @xmath122 and for @xmath123 @xmath124 for @xmath125 @xmath126 and so on . observing , the solution for the equation is @xmath127 where @xmath128 is a particular linear combination of all eigenfunctions with the smallest eigenvalue . the coefficient of @xmath129 can be obtained by calculating the inner product of @xmath130 with , @xmath131 in the same way , the coefficient of the next order @xmath132 , can be obtained by finding the inner product of @xmath133 with the @xmath125 equation , @xmath134 the inner product of @xmath135 on gives the coefficient for @xmath136 @xmath137 and @xmath138 the solution of tdgl is then @xmath139 in this paper we will restrict our discussion to the region near @xmath93 where the next - order correction can be disregarded ; @xmath140 we would like to emphasise that our discussion at this order is valid in the vicinity of the phase - transition boundary and in particular for a superconducting system without vortex pinning . in such a system , vortices move in a viscous way , resulting in flux - flow resistivity ; no divergence of conductivity is expected . our results based on were calculated at @xmath129 order , where only the lowest eigenvalue @xmath141 of the tdgl operator @xmath58 makes an appearance . the next - order correction is at order @xmath132 , and there is now a contribution from higher landau levels . from the symmetry argument in @xcite , as long as the hexagonal lattice remains the stable configuration for the system , the next - order contribution comes from the sixth landau level with a factor @xmath142 . even in the putative case of a lattice deformed slightly away from a hexagonal configuration , the next contributing term is @xmath143 , since in our system the lattice will remain rhombic . the vortex lattice has been experimentally observed since the 1960s and its long - range correlations have been clearly observed@xcite with dislocation fraction of the order @xmath144 . remarkably , the same techniques can be used to study the structure and orientation of moving vortex lattice with steady current@xcite , and with alternating current in the small - frequency regime@xcite . in this subsection , we will discuss the configuration of the vortex lattice in the presence of alternating transport current in the long - time limit . in the dynamical case , the presence of an electric field breaks the rotational symmetry of an effectively isotropic system to the discrete symmetry @xmath145 . in contrast , a rhombic lattice preserves at least a symmetry of this kind along two axes , and the special case of a hexagonal lattice preserves sixfold symmetry . the area of a vortex cell is determined by the quantised flux in the vortex , which is @xmath146 in terms of our rescaled variables . as shown in , we choose a unit cell @xmath147 defined by two elementary vectors @xmath148 and @xmath149 . we will first construct a solution for an arbitrary rhombic lattice parameterised by an apex angle @xmath85 . consideration of translational symmetry in the @xmath150 direction leads to the discrete parameter @xmath151 . in we show that in the long - time limit the lowest - eigenvalue steady - state eigenfunctions of @xmath58 must therefore combine to form @xmath152{2 \sigma}\sum_{l=-\infty}^{\infty } e^{i \frac{\pi}2 l(l-1 ) } e^{i k_l ( x- v \sin\omega \tau/\omega ) } u_{k_l}(y,\tau ) .\ ] ] here @xmath128 is normalised as @xmath153 the function @xmath154 is given by @xmath155 ^ 2 } , \end{aligned}\ ] ] with @xmath156 } .\end{aligned}\ ] ] in analogy with a forced vibrating system in mechanics , a phase @xmath157 and a reduced velocity @xmath158 have been introduced for convenience in . the zero electric - field limit , large - frequecy limit and zero frequency limit are consistent with previous studies concluded in . in our approximation , the @xmath159 in is a time - independent quantity from and @xmath160 where @xmath161 . in the small signal limit @xmath162 , @xmath159 reduces to the abrikosov constant . the abrikosov constant with either @xmath80 or @xmath163 minimises the gl free energy in the static state@xcite . to be more explicit , @xmath159 can be expanded in terms of the amplitude of input signal . in powers of @xmath164 , @xmath165 and we find it convenient to write in terms of @xmath166 . the first term in @xmath159 is the abrikosov constant @xmath167 for hexagonal lattices @xmath168 , whereas for a square lattice @xmath169 . the next term in @xmath159 is @xmath170 with a coefficient @xmath171 we see that at high frequency , the correction in higher order terms of @xmath166 can be disregarded . in this section we discuss the current distribution and motion of vortices , energy transformation of the work done on the system into heat , nonlinear response and finally the hall effect . in addition to the conventional conductivity attributable to the normal state , there is an overwhelming contribution due to the superconducting condensate in the flux - flow regime , tempered only by the dissipative properties of the vortex matter . in this section we will examine the supercurrent density to investigate the motion of the vortex lattice . we consider a hexagonal lattice in a fully dissipative system ; the non - dissipative part known as the hall effect will be discussed in . the supercurrent density @xmath172 is obtained by substitution of the solution into . @xmath173 and @xmath174 where @xmath175 and @xmath154 is given in . observing , we conceptually split the current into two components . one part is the circulating current surrounding the moving vortex core as in the static case ; we refer to this component as the _ diamagnetic current_. the other part which we term the _ transport current _ is the component which forces vortices into motion . : ( a ) @xmath176 ; vortex cores move to the right . ( b ) @xmath177 ; vortex cores move to the left . vortices are drawn back and forth as the direction of the transport current density alternates . the magnitude of the current density has maximal regions which tend to circumscribe the cores ; the maxima in these regions move in the plane and their manner of motion can be described as leading the motion of the vortices by a small phase . the average current in a unit cell leads the motion of the vortex in time by a phase of @xmath178 . , title="fig:",width=158 ] : ( a ) @xmath176 ; vortex cores move to the right . ( b ) @xmath177 ; vortex cores move to the left . vortices are drawn back and forth as the direction of the transport current density alternates . the magnitude of the current density has maximal regions which tend to circumscribe the cores ; the maxima in these regions move in the plane and their manner of motion can be described as leading the motion of the vortices by a small phase . the average current in a unit cell leads the motion of the vortex in time by a phase of @xmath178 . , title="fig:",width=158 ] the diamagnetic current may be excised from our consideration by integrating the current density over the unit cell @xmath147 ; that is , we consider @xmath179 . we have @xmath180 and @xmath181 with our conventions , the transport current is along the @xmath16-direction . considering the lorentz force between the magnetic flux in the vortices and the transport current , we expect the force on the vortex lattice to be perpendicular . we identify the locations of vortex cores to be where @xmath182 . the velocity of the vortex cores turns out to be @xmath183 along the @xmath150-direction . note that vortex lattice moves coherently . the vortex motion the electric field with a phase @xmath184 which increases with frequency and reaches @xmath178 asymptotically . the maximum velocity of vortex motion @xmath185 decreases with increasing frequency . in we show the current distribution and the resultant oscillation of vortices . as anticipated , the transport current and the motion of vortices are perpendicular as the vortices follow the input signal . the current density diminishes near the core ; it is small there compared to its average value . in steady - state motion , since the vortices move coherently in our approximation , the interaction force between vortices is balanced as in the static case . since the system is entirely dissipative , the motion that the vortices collectively undergo is viscous flow . the vortex lattice responds to the lorentz driving force as a damped oscillator , and this is the origin of the frequency - dependent response . in static case the system is described by the gl equation . solving this equation , which is but with zero on the left - hand side , will select some lattice configuration . the global minima of the free energy correspond to a hexagonal lattice , while there may be other configurations producing local minima . in the static case the lattice configuration can be determined in practice by building an ansatz from the linearized gl solution@xcite and then using a variational procedure to minimise the full free energy . in the dynamic case , there is no free energy to minimise ; we must embrace another method of making a physical prediction regarding the vortex lattice configuration . let us follow @xcite and take as the preferred structure the one with highest heat - generation rate . though we have at present no precise derivation , our physical justification of this prescription is that the system driven out of equilibrium can reach steady - state and stay in condensate only if the system can efficiently dissipate the work done by the driving force . therefore , whatever the cause , the lattice structure most conducive to the maintenance of the superconducting state will correspond to the maximal heat generation rate . the heat - generating rate@xcite is @xmath186 \big\ } .\end{aligned}\ ] ] @xmath159 is given explictly in and is the only parameter involving the apex angle @xmath85 of the moving vortex lattice . here @xmath159 plays the same rle as the abrikosov constant @xmath187 in the static case . corresponding to maximising the heat - generating rate , the preferred structure can be obtained by simply minimising @xmath159 with respect to @xmath85 . this shows from the current viewpoint of maximal heat - generation rate that vortices are again expected to move coherently . in or it is seen that the moving lattice is distorted by the external electric field but this influence subsides at high frequency . numerical solution for minimising @xmath85 shows that while near the high - frequency limit there remain two local minima for @xmath159 with respect to @xmath85 , the solution near @xmath81 is favoured slightly over that at @xmath188 as the global minimum . this is as presented in ( a ) . the two minima tend to approach each other slightly as the frequency begins to decrease further . in an experimental setting , this provides an avenue for testing the empirical validity of the maximal heat generation prescription , in particular in terms of the direction of lattice movement@xcite . we put forth the physical interpretation that at high frequency the friction force becomes less important , and the distortion is lessened . since interactions dominate the lattice structure the system at high frequency will have many similarities with the static case . energy supplied by the applied alternating current is absorbed and dissipated by the vortex matter , and the heat generation does not necessarily occur when and where the energy is first supplied . in , we show an example of this transportation of energy by the condensate . on the left is shown a contour plot of the work @xmath189 done by the input signal ; points along a given contour are of equal power absorption . on the right of is shown the heat - generating rate@xcite , @xmath190 . the periodic maximal regions are near the vortex cores in both patterns . and @xmath191 : ( a ) work @xmath192 and ( b ) heat generating rate @xmath193 . the vortex cores are denoted by ` @xmath194 ' in both figures , shown in the @xmath150-@xmath16 plane . the maximum displacements of vortex cores are shown by the arrow . the maximal region around the core in ( a ) is elongated by the current . the similar horizontal broadening around the core in ( b ) is caused by the vortex motion . energy is transported ; maxima in ( a ) and ( b ) do not coincide . , title="fig:",width=158 ] and @xmath191 : ( a ) work @xmath192 and ( b ) heat generating rate @xmath193 . the vortex cores are denoted by ` @xmath194 ' in both figures , shown in the @xmath150-@xmath16 plane . the maximum displacements of vortex cores are shown by the arrow . the maximal region around the core in ( a ) is elongated by the current . the similar horizontal broadening around the core in ( b ) is caused by the vortex motion . energy is transported ; maxima in ( a ) and ( b ) do not coincide . , title="fig:",width=158 ] in ( b ) , one can see that the system dissipates energy via vortex cores . from a microscopic point of view , cooper pairs break into quasiparticles inside the core ; these couple to the crystal lattice through phonons and impurities to transfer heat . the interaction between vortices and excitation of vortex cores manifests as friction@xcite . the power loss of the system averaged over time and space is @xmath195 . @xmath196 \end{aligned}\ ] ] where @xmath197 is a bessel function of the first kind . ( upper panel ) and expansion parameter @xmath113 ( lower panel ) as a function of frequency @xmath109.,width=340 ] in is shown the power loss and also @xmath113 as a function of frequency . @xmath113 is proportional the density of cooper pairs , and can be thought of as an indication of how robust is the superconductivity . as frequency increases , while @xmath113 tends to an asymptotic value , @xmath198 achieves a maximum and then decreases ; this maximum is due to fluctuations of order parameter caused by the input signal . in a fully dissipative system as considered here , the maximum of each curve is not a resonance phenomenon but is instead caused by fluctuation of the order parameter resulting from the influence of the applied field . a parallel may be drawn between what we have observed in this section and the suppression of the superconductivity by macroscopic thermal fluctuations commonly observed in high - temperature superconductors . in our case , the vortices in a high-@xmath199 superconductor undergo oscillation due to the driving force of the external field . we may think of this as being analogous to the fluctuations of vortices due to thermal effects alone in a low-@xmath199 superconductor . although the method of excitation is different , the external electromagnetic perturbation in the present case essentially plays the same rle as the thermal fluctuations in low-@xmath199 situation . finally , we point out that @xmath113 seems to be an appropriate parameter for determining the amount of power loss . generically , it seems that for points deeped inside the superconducting region , that is at large @xmath113 compared with its saturation value at high @xmath109 , the power loss due to the dissipative effects of the vortex matter becomes suppressed . we suggest the possibility that this effect , which is navely intuitive , is in fact physical and more widely applicable than merely the present model . the practical application of superconducting materials is dependent on how well one can control the inherent nonlinear behaviour . in this section we will focus on the generation of higher harmonics in the mixed state , in response to a single - frequency input signal . the periodic transport current @xmath200 is an odd function of input signal , and it turns out that the response motion also contains only odd harmonics . from we can calculate the fourier expansion for transport current . @xmath201 , \ ] ] where the fourier coefficient @xmath202 is @xmath203 e^{-i(2n+1)\theta } .\ ] ] we see the response goes beyond simple ohmic behaviour and the coefficients are proportional to @xmath113 . experimentally , one way of measuring these coefficients is a lock - in technique@xcite which is adept at extracting a signal with a known wave from even an extremely noisy environment . to make contact with more standard parameters and satisfy our intuition , we expand the first two harmonics in terms of @xmath164 . the fundamental harmonic , @xmath204 expanded in powers of @xmath205 is @xmath206 where @xmath207 . the first term is the ohmic conductivity denoted as @xmath208 , and is reminiscent of drude conductivity for free charged particles . this is not an unexpected parallel , since the cooper pairs in a superconducting system can be imagined to behave like a free - particle gas . taking this viewpoint , in the small - signal limit , the ratio @xmath209 gives the relaxation time of the charged particles . subsequent higher - order corrections all contain @xmath54 in such a way that their contributions are suppressed at large @xmath54 . the coefficient of the @xmath210 harmonic expanded in powers of @xmath205 is @xmath211 which decreases quickly with increasing @xmath54 . in , we show the generation of higher harmonics for three different states in the dynamical phase diagram . for each harmonic labeled by @xmath212 , @xmath213 as a function of @xmath109 has the same onset as @xmath113 . we can see that @xmath213 reaches a maximum and then starts to decay while @xmath113 saturates . the coefficients of harmonics with @xmath214 decay to zero in the high @xmath109 limit , where the state is well inside the superconducting region . we pointed out in and reaffirm here that @xmath113 plays a significant rle in determining the extent of nonlinearity in the system . in turn , the parameter which controls this is @xmath54 . when @xmath54 is large , @xmath113 is brought closer to its saturation value @xmath215 , causing the higher harmonics to be suppressed , and also lessening distortion of the vortex lattice . finally , for a given harmonic , @xmath213 is generally smaller when @xmath215 is smaller ; this can be seen by comparing ( a ) and ( c ) of . one might point out that the nonlinear behaviour is decreased at , for example , large @xmath54 . nevertheless , we view the parameter @xmath216 as more intrinsic to the system , rather than simply characterising the input signal . a limited parallel can be drawn between the effect of thermal noise in high-@xmath199 superconducting systems , and the effect of the electromagnetic perturbation in our present case . it seems that in either case the fluctuation influence can be reduced by moving the state deeper inside the superconducting region . [ r][r]@xmath217 [ r][r]@xmath218 [ r][r]@xmath219 [ r][r]@xmath220 in contrast to the fully dissipative system we have considered , in this section we will discuss an effect caused by the non - dissipative component , namely the hall effect . in a clean system , vortices move without dissipation ; a transverse electric field with respect to current appears . the non - dissipative part is subject to a gross - pitaevskii description , using a type of nonlinear schrdinger equation@xcite , with a non - dissipative part to the relaxation @xmath33 from . the fully dissipative operator @xmath58 in our previous discussion can be generalised by using a complex relaxation coefficient @xmath221 . we thus define @xmath222 the ratio @xmath223 is typically on the order of @xmath224 for a conventional superconductor , and @xmath225 for a high-@xmath199 superconductor@xcite . the hall effect is small here . in normal metals , the non - dissipative part gives the cyclotron frequency . if @xmath226 is the relaxation time of a free electron in a dirty metal , then for typical values of @xmath227 the hall effect becomes negligible . because the supply of conducting electrons is limited , the transverse component increases at the expense of the longitudinal component as the mean free path of excitations grows it is equivalent to an increase in the imaginary part of the relaxation constant at the expense of the real part . the eigenvalues and eigenfunctions of @xmath228 can be obtained easily by replacing the @xmath164 in previous results with @xmath229 , @xmath54 with @xmath230 and @xmath231 with @xmath232 . the transport current along the @xmath150-direction is no longer zero in the presence of the non - dissipative component ; it is propotional to @xmath233 . the frequency - dependent hall conductivity can be obtained from the first - order expension in @xmath164 , @xmath234 while the hall contribution in the @xmath16 direction is expected to be negligible , as it is of the order of @xmath235 . in principle , the crossover between non - dissipative systems and dissipative systems can be tuned using the ratio @xmath233 . in a non - dissipative system , which is the clean limit , the hall effect is important and taking account of the imaginary part of tdgl is necessary . on the contrary , in a strong dissipative system where excitations are in thermal equilibrium via scattering , the tdgl equation gives satisfactory agreement . far - infrared spectroscopy can be performed using monochromatic radiation which is pulsed at a high rate , known as fast far - infrared spectroscopy . this technique sports the advantage of avoiding overheating in the system , making it a very effective tool in observing the dynamical response of vortices . in particular , one can study the imaginary part of conductivity contributed mainly from superconducting component . in is shown a comparison with an nbn experiment measuring the imaginary part of conductivity . the sample has the gap energy @xmath236 mev . the resulting value of @xmath237 is larger than the value expected from bcs theory@xcite . we consider frequency - dependent conductivity in the case of linearly polarised incident light with a uniform magnetic field along the @xmath15 axis . the theoretical conductivity contains both a superconducting and a normal contribution . the total conductivity is obtained from the total current as in where the normal - part conductivity in the condenstate is the conductivity appearing in the drude model . according to our previous discussion , the nonlinear effect of the input signal on nbn is unimportant in the thz region , which corresponds to @xmath238 . an approximation where the flux - flow conductivity includes only the term @xmath239 from , and hall coefficient @xmath240 from is shown in and the agreement with experiment is good . the nave way in which we have treated the normal - part contribution is essentially inapplicable to the real - part conductivity . this is because the real - part conductivity contains information about interactions with the quasi - particles inside the core , making further consideration necessary@xcite . [ r][r]@xmath241 k and @xmath242 t calculated using of @xcite . normal - state conductivity is @xmath243 @xmath244 and relaxation time of an electron @xmath245 fs are taken from @xcite . the theoretical curve has one fitting parameter @xmath246.,title="fig:",width=226 ] the time - dependent ginzburg - landau equation has been solved analytically to study the dynamical response of the free vortex lattice . based on the bifurcation method , which involves an expansion in the distance to the phase transition boundary , we obtained a perturbative solution to all orders . we studied the response of the vortex lattice in the flux - flow region just below the phase transition , at first order in this expansion . we have seen that there are certain parameters which can be tuned using the applied field and temperature , providing a feasible superconducting system where one can study precise control of nonlinear phenomena in vortex matter . under a perturbation by electromagnetic waves , the steady - state solution shows that there is a diamagnetic current circulating the vortex core , and a transport current parallel to the external electric field with a frequency - dependent phase shift and amplitude . vortices move perpendicularly to the transport current and coherently . using a technique of maximising the heat - generation rate , we showed that the preferred structure based on energy dissipation is a hexagonal lattice , with a certain level of distortion appearing as the signal is increased or the frequency is lowered . energy flowing into the system via the applied field is dissipated through the vortex cores . we showed that the superconducting part may be thought of as having inductance in space and time . we have written transport current beyond a simple linear expression . a comparison between different harmonics of three different states in our four - dimensional parameter space indicated that the nonlinearity becomes unimportant at high frequency and small amplitude , and the influence of the input signal is decreased when the system moves deeper inside the superconducting region , away from the phase - transition boundary . to observe the configuration of moving vortices , techniques such as muon - spin rotation@xcite , sans@xcite , stm@xcite and others@xcite seem to be promising options . to provide the kind of input signal considered here , methods such as short - pulse fir spectroscopy as used in @xcite might be applied . the coefficient we defined in corresponds to conductivity . we have also seen that a simple parametrisation by complex quantities like conductivity and surface impedence is insufficient to capture the detailed behaviour of the system ; in performing experiments , it should be kept in mind that the nonlinearity can be measured in terms of more appropriate variables as we have shown . we have viewed the forcing of the system by the applied field to be somewhat analogous to thermal fluctuations , in the sense that they both result in vibration of the vortex lattice . hence , the influence of the electromagnetic fluctuation is stronger at the nucleation region of superconductivity than deep inside the superconducting phase . besides , since at high frequency the motion of vortices is limited , the influence from electric field is suppressed , as is the hall effect . fruitful discussions with b. rosenstein , v. zhuravlev , and j. kolek are greatly appreciated . the authors kindly thank g. bel and p. lipavsk for critical reading of the manuscript and many useful comments . the authors also have benefitted from comments of a. gurevich . we thank j. r. clem for pointing out to us reference @xcite . ` nsc99 - 2911-i-216 - 001 ` we follow @xcite to estimate the coefficient @xmath33 which characterises the relaxation process of the order parameter . @xmath33 is the inverse of the diffusion coefficient for electrons in the normal state . for a strongly - scattering system , as in the dirty limit@xcite , the ratio between the relaxation times of order parameter @xmath247 and the vector potential ( or current ) @xmath248 is @xmath249 by definition of the thermal critical field @xmath250 and @xmath251 , as in @xcite , we know the ratio of the two parameters is @xmath252 the coherence length at zero temperature can be written in terms of @xmath43 as @xmath253 and in terms of effective mass as @xmath254 . as a result , @xmath255 with @xmath33 , we can retrieve the experimental quantities from the rescaled ones used in calculation . using the @xmath37-subscripted original variables , we write the electric field @xmath256 , and the frequency @xmath257 . the current density @xmath258 . in the case of linear response , we have @xmath259 where @xmath260 . we consider the linearised time - dependent ginzburg - landau equation , which has been written in our chosen gauge . we wish to find the set of eigenfunctions of @xmath58 corresponding to the lowest eigenvalue . based on knowledge of the solution in the static case , we solve the now time - dependent problem by making the following ansatz@xcite . the electric field along the @xmath16-direction breaks rotational symmetry in the @xmath150-@xmath16 plane , so we write @xmath261 after substitution of @xmath262 for @xmath263 in , comparison of coefficients of powers of @xmath16 gives the following differential equations in @xmath231 . @xmath264 the solutions are @xmath265 .\ ] ] for a steady - state solution , @xmath266 , we have @xmath267 . @xmath268 as with @xmath269 , we have here @xmath270 . @xmath271 where @xmath272 is a normalisation constant . the resulting eigenvalue is @xmath273 now , although in a more realistic treatment of the system , one may introduce some boundary condition restricting @xmath274 to a certain set of values , here we simply select the smallest eigenvalue @xmath275 available to us by setting @xmath274 to zero . thus equipped with the set of eigenfunctions corresponding to our lowest eigenvalue , we deem them to be the first elements of our basis , labeled by @xmath141 and @xmath74 . these eigenfunctions of @xmath58 are @xmath276 with @xmath277 ^ 2 } , \end{aligned}\ ] ] and @xmath278 } .\end{aligned}\ ] ] in the same way , the corresponding eigenfunctions of @xmath279 can be obtained . @xmath280 with @xmath281 where @xmath282 } \end{aligned}\ ] ] according to our normalisation condition . the lowest eigenvalue of @xmath279 is @xmath283 . in the @xmath284 limit , the system reduces to the case of constant electric field . the eigenfunctions and eigenvalues are then consistent with those obtained by hu and thomson@xcite . in the limit of zero electric field , the eigenfunctions and eigenvalues reduce to those of the lowest landau level static - state solution@xcite . this is also the same in the @xmath285 limit . 99 d. e. oates , _ overview of nonlinearity in hts : what we have learned and prospects for improvement _ , j. supercond . novel magn . * 20 * , 3 ( 2007 ) ; 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over the last decades , numerous data on hot giant electric dipole resonance ( gdr ) parameters have been accumulated . since the field seems to have matured , it might be useful at this point to gather all present data in one comprehensive compilation in a uniform format . the introduction is organized in the following way : first , we give some theoretical motivation why gdr parameters might be dependent on temperature , hence what question the present compilation tries to address . then , in subsection [ sect : exp ] , we give an overview over typical experimental techniques used for measuring hot gdr parameters ; in subsection [ sect : smc ] we describe different ways of how hot gdr parameters are extracted from such experiments by means of statistical - model calculations . although we have not attempted to extract any gdr parameters from such calculations ourselves , we believe it is helpful for the reader to get a general impression of the experimental and data - analysis part of the compiled works . in subsection [ sect : data ] we explain how different sets of gdr parameterizations from the original articles were brought onto a common footing . finally , we end the introduction with a statement regarding our policy and a note on references . a first good understanding of statistical @xmath0 emission was gained from the works of brink @xcite and axel @xcite who realized that average electric - dipole ( @xmath1 ) transition strengths in different energy regimes can be described in a unified fashion by assuming that the gdr can be built on any excited state , and that the gdr properties do not depend on the temperature of the excited state in question . this so - called brink - axel hypothesis has been refined in the past to allow for temperature- and spin - dependent widths . a model to motivate such a modification takes into account shape fluctuations of the nucleus . since the ground - state gdr splits into two components for a nucleus with static deformation , and the splitting depends on the degree of deformation @xcite , it is reasonable to assume that at finite temperatures , when the nucleus can explore a large volume in deformation space , the gdr response will be an average over different deformations and hence , different splittings . the result within this adiabatic damping model will be a more diffuse and certainly wider gdr than the ground - state gdr . quantitatively , assuming a fermi - gas level density , one can write the nuclear entropy in the microcanonical ensemble as @xmath2 , where @xmath3 is some potential energy proportional to the square of the deformation @xmath4 . the entropy is trivially maximized for @xmath5 ; expanding @xmath6 for small @xmath7 yields @xmath8 . with @xmath9 , the probability distribution to find a nucleus with energy @xmath10 and deformation @xmath11 becomes @xmath12 where the second factor represents a gaussian distribution of deformations around @xmath5 and a width of @xmath13 . assuming that the splitting of the gdr into two components is roughly proportional to the nuclear deformation , the shape - fluctuation model predicts an increase in width of the gdr roughly proportional to @xmath14 . moreover , there is also a potential spin dependence of the gdr width which stems from the possibility of spin - induced deformation . finally , orientation fluctuations of the nucleus and nuclear - structure effects such as pairing can influence the temperature dependence of the gdr in different energy regimes . several groups have calculated temperature - dependent gdr widths along these lines @xcite and a simple scaling law has emerged @xcite . investigations of the low - energy tail of the gdr have also yielded indications for a temperature - dependent gdr width . it was , e.g. , noted by popov @xcite in @xmath15 experiments on sm nuclei that the @xmath0 strength function tends to approach a finite value for @xmath16 for @xmath0 transitions in the quasicontinuum ( below the neutron separation energy ) . this experimental observation led kadmenski , markushev , and furman ( kmf ) to propose a @xmath0 strength - function model for spherical nuclei with a temperature - dependent width @xcite based on the effect of in - medium nucleon - nucleon collisions . the proposed temperature dependence was derived within migdal s theory of fermi liquids and has the form @xmath17 . the model was later improved by sirotkin @xcite who included the pauli exclusion principle , and it was extended to deformed nuclei within the framework of the generalized lorentzian model by kopecky and uhl @xcite and by inclusion of a coupling term between the @xmath1 operator and the quadrupole deformation according to mughabghab and dunford @xcite . the kmf model ( taken at constant temperatures ) and its extensions have been successfully applied to improve @xmath0 and isomeric production cross sections @xcite and they have been used for direct fits of measured low - energy @xmath0 strength functions @xcite . the connection of collisional - damping models with hot gdr parameters has been made in @xcite . unlike the measurement of the gdr by ground - state photo - absorption cross - section measurements @xcite , measurements of hot gdr parameters can be performed in many different ways . one of the simplest ways is by fusion - evaporation reactions where only @xmath0 rays are detected @xcite . such measurements are the most inclusive reactions , since the high - energy @xmath0 yield which competes with particle and especially neutron evaporation is representative for a range of different product nuclei , excitation energies , and spins . moreover , it is not necessarily guaranteed that all detected @xmath0 rays stem from fusion - evaporation reactions . other reactions such as inelastic or deep inelastic scattering can compete and yield @xmath0 rays from target or projectile - like fragments . to improve the sensitivity of such experiments to the fusion - evaporation reaction channel , typical gates such as , e.g. , @xmath0-multiplicity filters @xcite , detection of heavy evaporation residues @xcite , and detection of evaporated , light charged particles such as protons or @xmath18 particles @xcite can be performed . the resulting @xmath0-ray spectra are more exclusive , not only in terms of the product nuclei from which high - energy @xmath0 rays are emitted , but also in terms of the spin and the excitation - energy range investigated . for example , a gate on different @xmath0 folds translates rather directly into certain spin regions of the investigated product nucleus @xcite . a gate on evaporated light charged particles will not only reduce the average charge and mass of the product nucleus , but it will also reduce its average excitation energy , since the evaporated particles will carry away some part of the initial excitation energy of the compound nucleus @xcite ; hence applying such a gate will test the gdr at somewhat lower temperatures than the fully inclusive experiment . in the same way , gating on the @xmath0 sum energy @xcite or on specific product nuclei by means of detecting in coincidence discrete , known low - energy @xmath0 transitions @xcite will also influence the average spin and excitation - energy region from which the high - energy @xmath0 rays are emitted , since one effectively biases the competition between high - energy @xmath0 decay and neutron evaporation in one or the other direction . other , more rarely used gating conditions are , e.g. , the isomeric @xmath0 decay by discrete transitions @xcite or the @xmath18 decay of a product nucleus @xcite , both of which have similar implications for the average spin and excitation - energy range from which prompt high - energy @xmath0 rays are observed . for heavy nuclei , an added difficulty is the possibility of fission of the compound nucleus . typically , for low spins , the production of an evaporation residue dominates while for high spins fission will become the dominant exit channel @xcite . hence , by gating on evaporation residues or fission fragments , one effectively selects a spin region from which high - energy @xmath0 emission is observed @xcite . in the case of the fission exit channel , one also observes high - energy @xmath0 emission from the fission fragments themselves @xcite , though typically at significantly higher @xmath0 energies owing to the much lower mass of the fission fragments . another complication is the fact that high - energy @xmath0 emission can occur from the compound nucleus ( which is desired ) , or during the saddle - to - scission motion after the nucleus has passed the fission barrier @xcite . also , for the heaviest nuclei investigated , it is not clear whether a true compound nucleus forms which is confined by some fission barrier or whether one observes direct fission or just the formation of a mononucleus . where we suspect the latter as in , e.g. , @xcite , the extracted data do not enter the present compilation . in some cases , one employs inelastic ( see , e.g. , @xcite ) or deep inelastic scattering @xcite to excite the target or projectile nucleus . by measuring the kinetic energy of at least one of the products , one can reconstruct the reaction kinematics event by event and it is possible to obtain initial excitation - energy indexed coincident high - energy @xmath0 spectra with only one beam energy . otherwise , the excited nucleus can be treated in the same way as a compound nucleus which is formed in a fusion - evaporation reaction . high - energy @xmath0 spectra are typically analyzed using a statistical - model calculation . in the first step , total fusion cross sections and maximum @xmath19 or average @xmath20 angular momenta are determined . typically , total fusion cross sections can be verified by experiment ; maximum angular momenta are calculated by the theory of either winther @xcite or swiatecki @xcite . average angular momenta can then be determined by @xmath21 . in the next step , the decay of the highly excited compound nucleus is modeled . in many cases , this simply involves a hauser - feshbach - type theory @xcite into which particle and @xmath0 transmission coefficients ( sometimes including higher - than-@xmath1 multipolarities ) as well as nuclear level densities enter . typically , particle transmission coefficients are not discussed in great detail in the compiled works . the level - density models are either the phlhofer model @xcite ( the default in the statistical - model code cascade @xcite ) or the reisdorf model @xcite . the phlhofer model relies on the local dilg _ parameterization @xcite for excitation energies up to and slightly above the nucleon separation energy , while it interpolates then to a regime where the level - density parameter @xmath22 becomes proportional to the nuclear mass number @xmath23 . the reisdorf approach builds on the generalized superfluid model by ignatyuk _ @xcite , but it uses a global parameterization for the asymptotic level density parameter @xmath22 . in one case @xcite , the level - density model by fineman _ @xcite is used in the data analysis . _ _ _ statistical - model calculations are often adapted to different experimental situations . for light compound nuclei near the @xmath24 line , an isospin - dependent formalism is often used @xcite . also , the wigner energy @xcite is sometimes included in the level - density parameterization @xcite . for large excitation energies , pre - equilibrium emission due to direct and semi - direct reaction mechanisms are often taken into account ( see , e.g. , @xcite ) . especially the peqag2 code @xcite has been developed for this purpose . in the case of fissile compound nuclei , the fission channel and the decay of excited fission fragments are modeled as well ( see , e.g. , @xcite ) . when gating conditions were applied in the experiment , they are usually reflected in the statistical - model calculation as well , which often implies the need to use a monte - carlo simulation tool such as in @xcite . in some cases , also asymmetries @xmath25 of @xmath0 emission are calculated ( see , e.g. , @xcite ) , however , experimental asymmetries do not enter into the present work ( with the exception of their influence on the sign of the quoted deformation as in , e.g. , @xcite ) . typically , calculated high - energy @xmath0 spectra are compared to their experimental counterparts , and the @xmath0 transmission coefficients ( parameterized by one- or two - component lorentzians multiplied by a factor @xmath26 ) are varied until the best fit is obtained . if absolute values in the high - energy region are compared , the fit is often normalized to the data in an energy region of 37 mev ( see , e.g. , @xcite ) , far below the peak of the gdr . statistical uncertainties are usually determined in the normal fashion by varying gdr parameters until the quality of the fit deteriorates . systematic uncertainties are more difficult to estimate . a good way is , e.g. , to perform several fits to the experimental data with different level - density parameterizations as in @xcite or differing sets of other input parameters into the statistical - model calculation ( say , e.g. , those which describe the dynamic of the fission process such as the nuclear viscosity which governs the timescale of the saddle - to - scission motion as in @xcite ) . the range of resulting gdr parameters might give a good indication of the size of the systematic error . other sources of systematic errors concern the experimental conditions . some of the most important problems there involve inefficient neutron-@xmath0 discrimination ( often done by time - of - flight techniques as in @xcite , or by simply considering @xmath0 rays only at backward angles as in @xcite ) , contamination of high - energy @xmath0 spectra by cosmic rays ( which can be greatly reduced by coincidence measurements as in @xcite ) , target impurities , pile - up ( see , e.g. , @xcite for a thorough investigation of these two effects ) , and add - back issues . add back is a technique often used for an array of small detectors where for high - energy @xmath0 rays one observes a significant amount of ( i ) compton scattering from one detector into a neighboring one , and ( ii ) pair production with subsequent annihilation @xmath0 rays being detected in neighboring detectors . the add - back technique remedies this situation by adding the deposited energies in neighboring detectors and ( rightly ) consider such events as stemming from one single @xmath0 ray ( see , e.g. , @xcite ) . pile up is a problem especially for large detectors , where two or more coincident @xmath0 rays hit the same detector and their energies add up and are falsely registered as one high - energy @xmath0 ray ( see , e.g. , the discussion in @xcite ) . if one applies the add - back technique , however , pile up can also occur when two coincident @xmath0 rays hit two neighboring detectors . obviously , for any given detector array , there is an optimal balance between the benefit of applying the add - back technique and the possible distortions of the high - energy @xmath0 spectrum due to pile up . where we found that this balance was not met @xcite , we have rejected the data for the present compilation . some other physical background involves nuclear bremsstrahlung which is emitted in the first moments of the fusion process where individual nucleons of the projectile are greatly de - accelerated in the proximity of target nucleons and emit high - energy @xmath0 rays ( see , e.g. , the discussion in @xcite ) . these @xmath0 rays can either be modeled ( and hence subtracted from the high - energy @xmath0 spectrum which is to be fitted by the statistical - model calculation as in @xcite ) , or they are simply considered as a source of systematic error as in @xcite . another possible source of high - energy @xmath0 rays is the pre - equilibrium @xmath0 emission during the formation of the compound nucleus which has been investigated by measuring high - energy @xmath0 spectra for ( isospin ) symmetric and asymmetric reactions as in , e.g. , @xcite . since in the extreme , such reactions essentially probe the di - nuclear system and not a compound nucleus ( see , e.g. , @xcite ) , the resulting data are not entered into the present compilation . in general , in our compilations we focus more on low - energy data to avoid complications due to non - compound sources of high - energy @xmath0 rays , hence , data concerning the saturation or increase of the gdr width at very high excitation energies such as in @xcite are typically omitted . in our data treatment , the first step was to determine the target isotope from the context ( for those few articles where it was not stated explicitly ) . ranges of laboratory energy @xmath27 , initial excitation energy @xmath28 , and initial spin @xmath29 were replaced by their central values . average initial spins @xmath30 were determined from maximum spins @xmath31 by means of @xmath21 for the case of fusion - evaporation reactions but irrespective of gating conditions . ranges in final spin @xmath32 after gdr @xmath0 emission were replaced by central values as well ; the widths of the @xmath32 ranges were converted into @xmath33 values which were preferred in this case over regular uncertainties due to the often non - gaussian distribution of final spins . hot gdr parameterizations can take many different forms . the most common is probably the parameterization in terms of a centroid @xmath10 , width @xmath34 , and maximum @xmath35 of an equivalent lorentzian photon - absorption cross section . the maximum @xmath35 is often formulated as a fraction @xmath6 of the thomas - reiche - kuhn ( trk ) sum rule which describes the integral @xmath36 mev mb of the lorentzian in terms of neutron @xmath37 , proton @xmath38 , and mass @xmath23 numbers @xcite . for two - component lorentzian parameterizations , often the total @xmath39 , and either the ration @xmath40 or the relative fraction @xmath41 are given . in either case , the given parameters were converted into @xmath42 and @xmath43 values . sometimes , the total is assumed to fulfill the trk while only the ratio or the relative fraction @xmath44 is determined by the fit . in these cases , the errors are marked by an asterisk to indicate correlations . in other cases , no information is given on the total . in those cases , the fraction @xmath43 is given in terms of the fraction @xmath42 in the table . in case of gdr widths , some authors reduce their number of fit parameters by introducing a phenomenological relation between the widths and the centroids of a two - component lorentzian according to @xmath45 , where @xmath46 becomes the fit parameter @xcite . in such cases , we have calculated the widths @xmath34 including their errors . however , the errors are again correlated and marked by an asterisk . in the case of two gdr centroids , some authors give the average @xmath47 and the ratio @xmath48 . sometimes this ratio is replaced by an average deformation @xmath11 which can be related to @xmath48 by either @xmath49 ( see , e.g. , @xcite ) or @xmath50 ( see , e.g. , @xcite ) , where @xmath51 and @xmath52 denote the centroids of the gdr components due to oscillations perpendicular and parallel to the symmetry axis . in the case where the original article did not mention which of the two formulas applies @xcite , the two formulas resulted in values for @xmath53 and @xmath54 within 0.2 mev of each other , a difference far less than the quoted statistical error . in general , however , we did not concern ourselves with deformations . only when oblate deformation was established from , e.g. , the asymmetry @xmath25 , a - sign was added to the deformation parameter ( in cases where the original article only provided the absolute value , see , e.g. , @xcite ) . in our treatment of errors , the first step was to add quadratically statistical and systematical errors ( when quoted separately ) . where a range of systematic uncertainties is given , we adopted the center value of that range as a representative systematic uncertainty . when uncertainties are given in terms of a @xmath33 , it was converted by @xmath55 . in general , rigorous error propagation was performed . however , no original work published the full covariance matrix for the fitted gdr parameters . therefore , the derived errors are only representative of the true errors under the assumption that the originally fitted parameters are fully uncorrelated . in cases where we determined more parameters than were originally fitted , an asterisk denotes the correlations which were introduced to the errors . in cases where gdr parameters were held fixed during the fit , a little f was added in the table instead of an error . some works do not cite errors at all . in such cases we have made no attempt to estimate the errors . in a few cases where @xmath53 and @xmath54 were given without errors , while @xmath56 was given with error , we could not find a good method to translate this error into errors of the individual values , and hence @xmath53 and @xmath54 remain without errors ( see , e.g. , @xcite ) . although we have tried to avoid a true evaluation of the original articles , we have excluded some of them from the present compilation . typically , where high - 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equilibrium particle emission , but always before particle evaporation + l.d.&level density parameterization : r = reisdorf , p = phlhofer , i = ignatyuk , d = dilg , f = fineman ; in some cases , the wigner energy is included in the parameterization + @xmath59&level density parameter ( mev ) , often used only for conversion of excitation energy to temperature of the compound nucleus ; the default value for the phlhofer l.d . parameterization is @xmath60 mev which should be assumed if no value is given + @xmath61&final average angular momentum of the compound nucleus ( @xmath57 ) after high - energy @xmath0 emission and influenced by gating conditions + @xmath33&full width at half maximum of the final angular momentum distribution ( @xmath57 ) + @xmath62&final average excitation energy of the compound nucleus ( mev ) after high - energy @xmath0 emission and influenced by gating conditions + @xmath63&final temperature of the compound nucleus ( mev ) after high - energy @xmath0 emission and influenced by gating conditions + code&statistical model computer code used for fitting the experimental spectra : c = cascade ( many different modifications exist ) , p = pace , p2=peqag2 ; if no code is given , most likely some version of the code cascade was used + gate&gating condition : @xmath18d=@xmath18 decay , @xmath0c=@xmath0 coincidences , d@xmath0t = discrete @xmath0 transitions , did = discrete isomeric decay , dis = deep inelastic scattering , er = evaporation residues , f=@xmath0-fold , ff = fission fragments , ipc = internal pair conversion , is = inelastic scattering , lcp = light charged particle , @xmath0m=@xmath0 multiplicity , @xmath64=@xmath0 sum energy , st = subtraction technique + @xmath42&strength of the first component of the gdr ( fraction of the trk sum rule ) + @xmath53&centroid of the first component of the gdr ( mev ) + @xmath65&width of the first component of the gdr ( mev ) + @xmath42&strength of the second component of the gdr ( fraction of the trk sum rule ) + @xmath53&centroid of the second component of the gdr ( mev ) + @xmath65&width of the second component of the gdr ( mev ) + @xmath33&full width at half maximum of the gdr cross - 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the dmrg algorithm was introduced by steven white @xcite , as an algorithm for calculating ground state properties of principally one - dimensional strongly correlated systems in condensed matter physics . the connection between dmrg and matrix product states @xcite ( also known as finitely correlated states ) was first made by rommer and stlund @xcite , who identified the thermodynamic limit of dmrg with a position - independent matrix product wavefunction . although dmrg had already proven itself to be useful empirically , this was an important step in rigorously establishing the physical basis of the algorithm due to the concrete and easy to manipulate form of matrix product states . work on the spectra of density matrices @xcite , later formulated as scaling of the von neumann entropy @xcite has placed the algorithm on a firm footing , showing that the required computational effort ( realized via the basis dimension @xmath0 ) is essentially a function of the entanglement of the wavefunction @xcite , which for one - dimensional ground - states scales at worst logarithmically with the system size @xcite . computationally , mps algorithms came to the fore with the assistance of a quantum information perspective , leading to algorithms for periodic boundary conditions @xcite , and finite temperature algorithms based on density operators @xcite . at around the same time , methods for simulation of real time evolution were developed in dmrg @xcite , which can also benefit from mps formulations @xcite . the common theme of mps approaches is to allow algorithms that operate on multiple , distinct wavefunctions at the same time . this is possible in the original formulation of dmrg only by constructing a mixed effective hilbert space that is weighted appropriately to represent all of the relevant states simultaneously . this is inefficient , as the algorithms typically scale as @xmath1 ( or up to @xmath2 for periodic boundary conditions @xcite ) in the number of basis states @xmath0 , so increasing @xmath0 so as to represent multiple states in the same basis is typically much slower than performing separate operations on each basis . in addition , the mixed basis approach lacks flexibility . while traditional dmrg programs calculate the wavefunction and a few ( often predetermined ) expectation values or correlation functions , if instead the wavefunction is calculated in the mps representation of eq . ( [ eq : mpwavefunction ] ) it can be saved for later use as an _ input _ for many purposes . perhaps the simplest such operation beyond the scope of traditional dmrg is to calculate the _ fidelity _ , or _ overlap _ between the wavefunctions obtained from separate calculations . in the mps formulation , this calculation is straightforward . nevertheless the determination of the scaling function for the fidelity of finite - size wavefunctions for different interaction strengths , provides a new tool for investigating phase transitions and crossover phenomena @xcite . indeed , due to the simplicity of the calculation the fidelity is likely in the coming years to be the first choice for quantitatively determining critical points . similar measures of entanglement , such as the concurrence and single- and two - site entropy @xcite , are also straightforward to calculate , hence the mps formalism allows us to apply directly the emerging tools of quantum information to the study of realistic systems in condensed matter physics . an alternative measure , the loschmidt echo @xcite is important because , unlike many of the quantum information theoretic measures , this is directly accessible in experiments while showing the rich behavior of the simpler measures . the loschmidt echo is more time - consuming to measure numerically as it requires a full time evolution simulation rather than a direct measurement , nevertheless it is well within the current state of the art @xcite . in this paper , we focus on the case of open boundary condition matrix product states . this does not preclude calculation of periodic systems , however the entanglement of such periodic states is increased such that in the large @xmath3 limit ( where @xmath3 is the lattice size ) , the number of states kept tends to the square of that required for open boundary conditions @xcite . algorithms exist for periodic boundary conditions @xcite and infinite systems @xcite ( not to be confused with the ` infinite - size ' dmrg algorithm ) , and the basic formulas introduced here carry over to these cases , but we do not describe the specific algorithms here . in sec . [ sec : mps ] , we introduce the basic formulation of matrix product states , and formulas for the fidelity . [ sec : operators ] is devoted to a new approach , whereby we construct the hamiltonian operator itself as an mps , with many advantages . we cover some remaining details of the dmrg algorithm in sec . [ sec : dmrg ] , before discussing in detail the use of abelian and non - abelian quantum numbers in sec . [ sec : quantumnumbers ] . we finish with a few concluding remarks in sec . [ sec : conclusions ] , including some observations on finite temperature states . we denote an mps on an @xmath3-site lattice by the form @xmath4 the local index @xmath5 represents an element of a local hilbert space at site @xmath6 . the two important cases we cover here are when @xmath5 runs over a @xmath7-dimensional local basis for a wavefunction @xmath8 , in which case we refer to this as a matrix product wavefunction ( mpw ) , or @xmath5 is a @xmath9-dimensional local basis for all operators acting on a local site , which we refer to as a matrix product operator ( mpo ) . in this paper , we use mps for a generic state irrespective of the form of the local space , and use mpw or mpo as necessary when the distinction between wavefunctions and operators is important . in general , the matrix product form can also represent periodic @xcite and infinite ( non - periodic ) states @xcite , but here we use only the open - boundary form equivalent to the wavefunction obtained by dmrg @xcite . to enforce this boundary condition , we require the left - most matrix @xmath10 to be @xmath11 dimensional , and the right - most matrix @xmath12 to be @xmath13 . here we have introduced @xmath0 as the basis size , or dimension of the _ matrix basis _ of the @xmath14-matrices . this quantity is often denoted @xmath15 , or sometimes @xmath16 , in the quantum information literature , but we emphasize it is exactly the same quantity in all cases . in general @xmath0 is position dependent , as we do not require the @xmath14-matrices to be square even away from the boundary . because of the 1-dimensional basis at the boundaries we can regard the mps wavefunction to be a sequence of operators attached to left and right ( or outgoing and incoming ) vacuum states . this makes the operator product in eq . ( [ eq : mpwavefunction ] ) an ordinary number , so the trace operation can be dropped . in practice , a mps state with no particular constraints on the form of the @xmath14-matrices is numerically difficult to handle . we are always free to insert some product of a non - singular @xmath17 operator @xmath18 and its inverse @xmath19 in the middle of our mps , thus we can apply an arbitrary transformation to the matrix basis of an @xmath14-matrix , as long as we make the corresponding transformation to its neighbor . using this freedom , we can transform the @xmath14-matrices into a form where they are orthonormalized , that is , we prefer that they satisfy one of two possible constraints , @xmath20 states satisfying these conditions are orthonormalized in the sense that if all @xmath14-matrices to the left of some matrix @xmath21 are orthogonalized in the left - handed sense , then the basis on the left - hand side of @xmath21 is orthonormal ( _ ie _ the identity operator in the effective hilbert space is trivial ) . conversely , if all @xmath14-matrices to the right of @xmath21 are orthogonalized in the right - hand sense , then the basis on the right - hand side of @xmath21 is orthogonal . usually , we want both these conditions to be true simultaneously . note that it is not , in general , possible for _ all _ of the @xmath14-matrices ( including @xmath21 itself ) to be in orthonormal form at the same time . there are several ways of transforming an arbitrary mps into this normalized form . two ways that we consider here are the singular value decomposition ( svd ) , and the related reduced density matrix , as used in dmrg @xcite . the simplest , and in principle the fastest , is the svd , well - known from linear algebra @xcite . for example , for the left - handed orthogonality constraint on @xmath22 , where we have re - inserted the matrix indices @xmath23 , we consider @xmath24 to be a single index of dimension @xmath25 , giving an ordinary @xmath26 dimensional matrix , and carry out the singular value decomposition , @xmath27 where @xmath28 is column - orthogonal , @xmath29 , and @xmath30 is row - orthogonal , @xmath31 . @xmath15 is a non - negative diagonal matrix containing the singular values . this form coincides with the schmidt decomposition , where @xmath15 gives the coefficients of the wavefunction in the schmidt basis @xcite . the matrix @xmath28 therefore satisfies the left - handed orthogonality constraint , so we use this as the updated @xmath14-matrix , and multiply the @xmath14-matrix on the right by @xmath32 . this implies that the @xmath14-matrix on the right is no longer orthonormalized ( even if it was originally ) , but we can apply this procedure iteratively , to shift the non - orthogonal @xmath14-matrix to the boundary or even beyond it at which point the @xmath33 @xmath14-matrix coincides with the norm of the wavefunction . an important point here is that we can choose to discard some of the states , typically those that have the smallest singular value . this reduces the matrix dimension @xmath0 , at the expense of introducing an approximation to our wavefunction , such that the squared norm of the difference of our approximate and exact wavefunctions is equal to the sum of the squares of the discarded singular values . note however that the singular values only correspond to the coefficients of the schmidt decomposition if all of the remaining @xmath14-matrices are orthogonalized according to eq . ( [ eq : normalizationconstraint ] ) . if this is not the case , the singular values are not useful for determining which states can be safely discarded . alternatively , we can construct the reduced density matrix , obtained by tracing over half of the system . this is achieved by @xmath34 which is a @xmath35 matrix , with @xmath0 eigenvalues coinciding with the values on the diagonal of @xmath36 , and the remaining eigenvalues are zero . again , the eigenvalues are only meaningful if the remaining @xmath14-matrices are appropriately orthogonalized . the utility of the density matrix approach over the svd , is that we can introduce mixing terms into the density matrix which can have the effect of stabilizing the algorithm and accelerating the convergence , which is further discussed in sec . [ sec : dmrg ] . the overlap of two mps is an operation that appears in many contexts . for wavefunctions this gives the fidelity of the two states , and for operators this is equivalent to the operator inner product @xmath37 which induces the frobenius norm . direct expansion of the mps form yields , @xmath38 due to the open boundary conditions , the direct product @xmath39 reduces to an ordinary @xmath17 matrix , after which can construct successive terms recursively , via @xmath40 with an analogous formula if we wish to start at the right hand side of the system and iterate towards the left boundary , @xmath41 for the purposes of numerical stability , it is advisable for the @xmath14- and @xmath42-matrices to be orthogonalized in the same pattern , that is , @xmath43-matrices are associated exclusively with the left - hand orthogonality constraint and @xmath44-matrices are associated with the right - hand orthogonality constraint . if we iterate all the way to the boundary , the @xmath43- ( or @xmath44- ) matrix ends up as a @xmath33 matrix that contains the final value of the fidelity . alternatively , we can iterate from both ends towards the middle and calculate the fidelity as @xmath45 . the key notion of the matrix product scheme is that of _ local updates _ ; that is , we modify , typically though some iterative optimization scheme , one ( or perhaps a few ) @xmath14-matrix while keeping the remainder fixed . a useful and flexible alternative is the _ center matrix formulation _ , where , instead of modifying an @xmath14-matrix directly , we introduce an additional matrix @xmath46 into the mps , @xmath47 this allows us to preserve orthogonality of the matrices at all times ; matrices @xmath48 for @xmath49 are normalized always according to the left - handed constraint , and matrices for @xmath50 are normalized according to the right - handed constraint . we directly modify only the matrix @xmath46 which simplifies the local optimization problem as @xmath46 is just an ordinary matrix . to introduce the local degrees of freedom , say for the @xmath51 states , we _ expand _ the basis for @xmath21 . that is , we replace the @xmath17 dimensional matrices @xmath21 with @xmath52 matrices @xmath53 , given by @xmath54 and introduce the @xmath26 dimensional center matrix @xmath55 with @xmath56 running over @xmath25 states . this does nt change the physical wavefunction , as @xmath57 . similarly , we can expand the basis for the @xmath14-matrix on the right side of @xmath46 , to give the effect of modifying either a single @xmath14-matrix , or two ( or more ) at once . in the center matrix formulation , the singular value decomposition required for truncating the basis is simply the ordinary svd on the matrix @xmath58 , and we multiply ( for a right - moving iteration ) @xmath59 , which preserves the left - handed orthogonality constraint , and @xmath60 which is not orthogonal , but becomes so when we again expand the basis to construct the new @xmath46 matrix . the density matrix in the center matrix formulation is simply @xmath61 or @xmath62 for left and right moving iterations respectively . for readers already familiar with dmrg , the center matrix corresponds exactly with the matrix form of the _ superblock wavefunction _ @xcite . the utility of the mps approach is realized immediately upon attempting manipulations on the wavefunction eq . ( [ eq : mpwavefunction ] ) . suppose we have two distinct mps , defined over the same local hilbert spaces , @xmath63 the superposition @xmath64 is formed by taking the sum of the matrix products , @xmath65 , which can be factorized into a new mps , @xmath66 , with @xmath67 to preserve the one - dimensional basis at the boundaries , the direct sum is replaced by a concatenation of columns or rows , for the left and right boundary respectively . this procedure results in an mps of increased dimension , @xmath68 . thus , after constructing these matrices we need to re - orthogonalize the state , and then we can , if necessary , truncate the basis size to a given truncation error , which is well defined here and measures the exact difference between the original and truncated wavefunctions . alternatively , the normalized and truncated mps @xmath69 can be constructed directly , by calculating the overlap matrices @xmath43 between @xmath70 and @xmath71 . from the @xmath43-matrices introduced in eq . ( [ eq : ematrix ] ) , we can construct directly the orthogonalized reduced density matrices of @xmath69 and truncate the basis as required , in a single step . this approach has better computational scaling than the two - step procedure of first orthogonalizing and then truncating , especially when the number of mps in the superposition is large . but in general , iterative optimization approaches , where we use a dmrg - like algorithm to optimize the overlap @xmath72 , have even better performance scaling with large @xmath0 or more states in the superposition . a useful generalization of the mps structure eq . ( [ eq : mpwavefunction ] ) is to use it to represent an operator ( an mpo ) instead of a wavefunction . this has been used before for calculating finite - temperature density matrices @xcite , but here we instead want to use this structure to represent the hamiltonian operator itself . all hamiltonian operators with finite - range interactions have an _ exact _ mps representation with a relatively small matrix dimension @xmath73 . for example , the ising model in a transverse field has a dimension @xmath74 , and the fermionic hubbard model has dimension @xmath75 . we use the capital letter to distinguish from the dimension of the wavefunction , @xmath0 . similarly , the local dimension of the upper index is denoted here by @xmath15 , which is usually just equal to @xmath9 , but slightly complicated in the case of non - abelian quantum numbers ( see sec . [ sec : nonabelian ] ) . we denote an mpo by the form @xmath76 where again we require that the first and last dimensions are @xmath77 , for open boundary conditions . the orthogonality constraint used previously for the mps , eq . ( [ eq : normalizationconstraint ] ) , is not appropriate for hamiltonian operators . when applied to an operator , the usual orthogonality constraints utilize the ( frobenius ) operator norm , which scales exponentially with the dimension of the hilbert space . with this normalization , components of an mpo hamiltonian , such as the identity operator or some interaction term , tend to differ in magnitude by some factor that increases exponentially with the lattice size . arithmetic manipulations on such quantities is a recipe for catastrophic cancellation @xcite resulting in loss of precision . mixing operators with a unitary transformation ( for example @xmath78 ) , will lead to a disaster if @xmath79 and @xmath80 differ by a sufficiently large order of magnitude , @xmath81 for typical double - precision floating point arithmetic . but such rotations are inevitable in the orthogonalization procedure because in general the operator inner product @xmath82 will not be zero . instead we completely avoid mixing different rows / columns of the operator @xmath73-matrices , only collapsing a row or column if it is exactly parallel with another row or column . in this case , the actual norm of each component of the @xmath14-matrices is irrelevant , as they are never mixed with each other ( but see also the discussion of the single - site algorithm in sec . [ sec : dmrg ] ) . for physical hamiltonian operators this remains essentially optimal , with the minimum possible matrix dimension @xmath73 . the only operators for which this orthogonalization scheme does not produce an optimal representation are operators that have a form analogous to an aklt @xcite state where the local basis states of the @xmath83 chain are replaced by local operators . the resulting operator contains an exponentially large number of @xmath84-body interactions for all @xmath85 . we know of no physical hamiltonians of this form . given a hamiltonian as a sum of finite - range interactions , it is possible to construct the operator @xmath73-matrices such that they are entirely lower ( or upper ) triangular matrices , thus in principle we can ` normalize ' the matrices via some kind of generalized @xmath86 or @xmath87 decomposition . in practice we do nt need to do this , as the hamiltonian can easily be constructed in lower - triangular form from the beginning . imposing , again without loss of generality , that the top - left and bottom - right components of the operator @xmath73-matrices are equal to the identity operator @xmath88 , we can construct the sum of @xmath89-site local terms @xmath90 as a position - independent mps , @xmath91 which we regard as a @xmath92 matrix , the elements of which are @xmath93 dimensional local operators . for nearest - neighbor terms , @xmath94 , we have @xmath95 with the obvious generalization to @xmath84-body terms . the direct sum and direct product of lower triangular matrices is itself lower triangular , thus this form can be preserved throughout all operations . for open boundary conditions , the left ( right ) boundary @xmath96 ( or @xmath97 ) matrices are initialized to @xmath98 and @xmath99 respectively . the principal advantage of formulating the hamiltonian operator ( and indeed , _ all _ operators needed in the calculation ) in this way that it can be manipulated extremely easily , amounting to a symbolic computation on the operators . this is in contrast to the ad hoc approach used in past dmrg and mps approaches where the block transformations required for each operator are encoded manually , with limited scope for arithmetic operations . in particular , the sum of operators is achieved via eq . ( [ eq : mpssum ] ) . products of operators are achieved by matrix direct product ; given mpo s @xmath14 and @xmath42 , the product @xmath100 is given by the matrices @xmath101 an implication of this is that the square of an mpo has a matrix dimension of at most @xmath102 , which , since @xmath73 is usually rather small , means that it is quite practical to calculate expectation values for higher - order moments , for example of the _ variance _ @xmath103 which has been mentioned previously @xcite as it gives a rigorous _ lower bound _ on the energy ( although with no guarantee that this corresponds to the ground - state ) . in practice this lower bound is too wide to be useful in all but the simplest cases , but of more interest is the property that the variance is , to first order , proportional to the squared norm of the difference between the exact and numerical wavefunctions , and therefore also proportional to the truncation error @xcite ( see sec . [ sec : dmrg ] ) . thus , this quantity gives a quantitative estimate of the goodness of the wavefunction even for situations where the truncation error is not available . for our numerical mps algorithms the variance takes the role of the precision @xmath104 in numerical analysis @xcite via @xmath105 . of a similar form to the product of two operators , the action of an operator @xmath73 on a wavefunction @xmath106 gives a wavefunction @xmath107 with matrix elements , @xmath108 the mpo formulation also gives a natural form for the evaluation of expectation values , similarly to the fidelity of eq . ( [ eq : overlap ] ) , @xmath109 where the @xmath43- and @xmath44-matrices now have an additional index @xmath110 that represents the terms in the mpo @xmath73 , with a recursive definition @xmath111 where again we can either iterate all the way to a boundary , at which point the @xmath110 index collapses down to one - dimensional and the @xmath112 or @xmath113 are @xmath33 dimensional matrices containing the expectation value , or we can iterate from both boundaries and meet at the middle , where our expectation value is given by the scalar product eq . ( [ eq : expectation ] ) . incidentally , given that the identity operators occur in a fixed location in the operator @xmath73-matrix ( ie . at the top - left and bottom - right of the @xmath73-matrix ) this fixes the index @xmath110 of the reduced hamiltonian and identity operators for the left and right partitions of the system . that is , in the application of the hamiltonian @xmath114 matrices to the wavefunction we are guaranteed that the @xmath115 component of @xmath116 corresponds precisely to @xmath117 , and the @xmath118 component corresponds to @xmath119 . thus , even after an arbitrary series of mpo computations we can still identify exactly which component of the @xmath120 matrices corresponds to the block hamiltonian . this is useful for eigensolver preconditioning schemes @xcite , for example to change to a basis where the block hamiltonian is diagonal . . the @xmath121 symbols denote the impurity magnetization calculated via adaptive time dmrg , the dashed line is a guide to the eye . parameters of the calculation were ( in units of bandwidth ) , @xmath122 , on a log - discretized wilson chain with @xmath123 . at time @xmath124 , the hamiltonian was switched to @xmath125 and @xmath126 . , scaledwidth=80.0% ] as an example of the utility of this approach , fig . [ fig : timeexample ] shows the time evolution of the magnetization of the impurity spin in the single impurity anderson model ( siam ) , where the ground - state is obtained with a small magnetic field which is then turned off at time @xmath124 . the mps operator approach readily allows the evaluation of the commutators required for a small @xmath127 expansion of the expectation value of an observable in the heisenberg picture , @xmath128 - \frac{t^2}{2!\hbar^2}[h,[h , a ] ] - \frac{it^3}{3!\hbar^3}[h,[h,[h , a ] ] ] + \cdots \ ; .\ ] ] since the number of terms in the repeated commutator will , in general , increase exponentially the accessible time - scales from such an expansion are clearly limited . nevertheless this is a very fast and accurate way to obtain short - time - scale dynamics , and in this example @xmath129 order is easily enough to determine the @xmath130 relaxation rate . for this calculation , the terms up to @xmath131 took a few seconds to calculate , while the @xmath132 term took 6 minutes and the @xmath133 term took just over an hour , on a single processor 2ghz athlon64 . this time was divided between calculating the mpo matrix elements ( the dimension of which was just over 2500 at the impurity site ) , and calculating the expectation value itself . we now have all of the ingredients necessary to construct the dmrg algorithm for determining the ground - state . indeed , given the previous formulations , the dmrg itself is rather simple ; using the center matrix formulation , we iteratively improve the wavefunction locally by using the center matrix @xmath46 as input to an eigensolver for the ground - state of the hamiltonian . the details of this calculation are precisely as for dmrg , already covered elsewhere @xcite . an important component of dmrg , which has been neglected in some matrix product approaches , is the truncation error . if only a single site is modified at a time , the maximum number of non - zero singular values is bounded by the matrix dimension @xmath0 , thus the matrix dimension can not be incrementally increased as the calculation progresses . some way of avoiding this limitation is practically essential for a robust algorithm . the original dmrg formulation @xcite solved this problem by modifying two a - matrices simultaneously , equivalent to expanding the matrix dimension for both the left and right @xmath14-matrices in eq . ( [ eq : centermatrixmps ] ) so that the center matrix has dimension @xmath134 . a scheme for single - site algorithms that avoids the limit on the singular values was introduced by steven white @xcite , which uses a mixed density matrix constructed by a weighted sum of the usual reduced density matrix and a perturbed density matrix formed by applying the @xmath112-matrices ( on a right - moving sweep ) or @xmath113-matrices ( on a left - moving sweep ) of the hamiltonian , @xmath135 where @xmath136 is some small factor that fixes the magnitude of the fluctuations . this solves nicely the problem of the bound on the number of singular values and introduces fluctuations into the density matrix that give the algorithm good convergence properties , often better than the two - site algorithm . a minor problem is that the scaling of the @xmath112 matrices is not well defined , in that we can scale the @xmath112-matrices by an arbitrary @xmath137 non - singular matrix @xmath138 , while at the same time scaling the @xmath44-matrices by @xmath19 . for one- and two - site terms , there is an ` obvious ' scaling factor to use , whereby the scaling factors are chosen such that the ( frobenius ) operator norm of the @xmath43 and @xmath44-matrices are identical , but i do nt know how this would apply more generally . an alternative that appears interesting is to apply the full hamiltonian to a density operator for the full system , @xmath139 , constructed from the left ( right ) reduced density matrix and the right ( left ) identity operator , followed by a trace over the right ( left ) partition . in mps form , this operation is @xmath140 where @xmath141 is an @xmath137 coefficient matrix . however , this scheme often fails ; incorporating the @xmath142 matrix reduces the fluctuations such that @xmath143 differs little from @xmath144 itself and the algorithm typically fails to reach the ground - state . the single - site algorithm @xcite corresponds to choosing @xmath145 . a useful compromise appears to be using only the _ diagonal _ elements , such that @xmath146 , but this is surely not the last word on this approach . both the two - site and mixed single - site algorithm inevitably result in a reduction in the norm of the wavefunction by truncating the smallest non - zero eigenvalues of the density matrix . the sum of the discarded eigenvalues , summed over all iterations in one sweep , is equal to the truncation error @xmath147 , familiar from dmrg @xcite ( but note that it is common in the literature to quote an average or maximum truncation error _ per site _ ) . this quantity is useful in giving an estimate of the error in the wavefunction in this is , for a properly converged wavefunction , proportional to the norm of the difference between the exact ground - state and the numerical approximation . the presence of the truncation error explains why the bare single - site algorithm , despite having slightly better variational wavefunction than the two - site or mixed single - site algorithms @xcite , converges much slower ; the single site algorithm is a highly constrained optimization within an @xmath0-dimensional basis , whereas the two - site and mixed single - site algorithms are selecting the optimal @xmath0 basis states out of a pool of a much larger set of states , namely the discarded states at each iteration ( total @xmath148 states ) . while the notion of truncation error remains useful in mps algorithms , for the purposes of error analysis we much prefer the variance eq . ( [ eq : variance ] ) as being a direct measure of the accuracy of the wavefunction , independent of the details of a particular algorithm @xcite . low - lying excited states can be constructed using this algorithm . this has been done in the past in dmrg by targeting multiple eigenstates in the density matrix , but the mps formulation allows a substantial improvement . namely , it is easy to incorporate into the eigensolver a constraint that the obtained wavefunction is orthogonal to an arbitrary set of predetermined mps s . that is , after constructing the mps approximation to the ground - state , we can , as a separate calculation , determine the first excited state by running the algorithm again with the constraint that our obtained wavefunction is orthogonal to the ground - state . this is achieved by constructing the @xmath43-matrices that project the set of states to orthogonalize against onto the local hilbert space . these matrices are precisely those used in constructing the fidelity , eq . ( [ eq : overlap ] ) , thus given the center matrix of some state @xmath149 , we project this onto the current hilbert space @xmath150 , and as part of the eigensolver , orthogonalize our center matrix against this state . this is a very fast operation , much faster than even a single hamiltonian - wavefunction multiplication . so it is quite practical to orthogonalize against a rather large number of states , the practical limit is rather on numerical limitations in orthogonalizing the krylov subspace in the eigensolver . if this is combined with an eigensolver capable of converging to eigenvalues in the middle of the spectrum ( say , the lowest eigenvalue larger than some bound @xmath151 ) , then we need only a small number of states to orthogonalize against , say half a dozen states immediately above @xmath151 in energy . in our numerical tests it seems to be rather common to skip eigenvalues , which is why we can not simply orthogonalize against a single state . with a larger number of states to orthogonalize against , skipping eigenvalues is less of a problem as we are likely to recover the missing eigenstate on a later calculation . using this approach , quantities such as the level spacing statistics can be determined for system sizes far beyond exact diagonalization @xcite . an important feature of matrix product states is that they can easily be constrained by quantum numbers representing the global symmetries of the hamiltonian , as long as the symmetry is not broken by the spatial geometry of the mps . for example , internal rotational symmetries such as @xmath152 and @xmath153 @xcite can be maintained exactly , but for a real - space lattice we can not utilize the momentum in the same way because the representation itself violates this symmetry . to achieve this , we impose a symmetry constraint on the form of the @xmath14-matrices , so that they are _ irreducible tensor operators_. that is , under a symmetry rotation the matrix @xmath154 for each local degree of freedom @xmath155 transforms according to an irreducible representation @xmath156 of the global symmetry group . this is a very general procedure , that is applicable to essentially all mps approaches and generalizations thereof . for abelian symmetries , the representations are one - dimensional therefore the set of quantum numbers labeling the irreducible representations also forms a group , which we can write as @xmath157 for two representations @xmath158 and @xmath159 , where @xmath160 denotes the group operation . thus to incorporate abelian symmetries into the algorithm we simply attach a quantum number to all of the labels appearing in the mps , with the constraint that each @xmath14-matrix transforms irreducibly , so that the only non - zero matrix elements are @xmath161 where @xmath162 are the quantum numbers attached to the local basis state and left and right matrix basis states respectively . we have suppressed here indices not associated with a quantum number , a convention which will be followed for the remainder of the paper . by convention , for our open boundary condition mps we choose the right hand vacuum state to have quantum number zero . the symmetry constraint eq . ( [ eq : quantumnumbers ] ) then implies that the quantum number at the left hand vacuum will denote how the state as a whole transforms ( the _ target state _ , in dmrg terminology ) . this is the only real difference between dmrg and mps algorithms , in that the dmrg convention is to construct both blocks starting from a scalar ( quantum number zero ) vacuum state , so that the superblock wavefunction is a tensor product of two ket states , @xmath163 whereas for the mps formulation the superblock wavefunction is represented by a scalar operator with a tensor product basis @xmath164 with quantum numbers @xmath165 . this means that , in contrast to the usual formulation of dmrg , the target quantum number is not encoded in the superblock but rather in the left vacuum state . a consequence of this is that dmrg is capable of representing simultaneously states with different quantum numbers , but an mps is not . this is an important detail , for example in the calculation of dynamical correlations , as both the correction vector @xcite and the similar ddmrg @xcite algorithm require a basis optimized for both the ground - state @xmath166 and the so - called lanczos - vector @xmath167 , where @xmath14 is some operator that may not be scalar . however , the mps formulation allows significant optimizations to these algorithms whereby the the calculation of the ground - state is decoupled from that of the lanczos vector @xcite and the two need never appear in the same basis . if the symmetry group is large enough that some elements do not commute with each other , then it is no longer possible to construct a basis that simultaneously diagonalizes all of the generators hence the approach of the previous section needs some modification . instead , we label the basis states by quantum numbers that denote the representation , which is no longer simply related to the group elements themselves as the representations are in general no longer ordinary numbers , but instead are matrices of dimension @xmath168 , and eq . ( [ eq : abeliangrouprep ] ) no longer applies . for @xmath153 , we choose to label the representations by the total spin @xmath155 , being related to the eigenvalue of of the spin operator , @xmath169 . assuming that all of the required operations can be formulated in terms of manipulations of these representations , we have a formulation that is _ manifestly _ @xmath153 invariant ; the rotational invariance is preserved at all steps and at no time in the calculation is it necessary to choose an axis of quantization @xcite . this supersedes the earlier approach based on the clebsch - gordan coefficients @xcite . the non - abelian formulation is an important optimization , because it increases the performance of the algorithm by an order of magnitude or more @xcite compared with the abelian case , while enabling more accurate and detailed information about the ground - state magnetization . the basic ingredient that enables this rotationally invariant construction is the wigner - eckart theorem @xcite , which we can state as : when written in an angular momentum basis , each matrix element of an irreducible tensor operator is a product of two factors , a purely angular momentum dependent factor ( the `` clebsch - gordan coefficient '' ) and a factor that is independent of the projection quantum numbers ( the `` reduced matrix element '' ) . we formulate the algorithm in such a way that we store and manipulate only the reduced matrix elements , factorizing out completely the clebsch - gordan coefficients . the efficiency improvement resulting from the non - abelian formulation is that the matrix dimensions @xmath0 and @xmath73 now refer to the number of irreducible representations in the basis , which is typically much smaller than the total degree of the representation . for a scalar state , this equivalence is precise : a single representation of degree @xmath84 in the non - abelian approach results in @xmath84 degenerate eigenstates when the symmetry is not used , with a corresponding improvement in efficiency . we do not give here a full introduction to the theory of quantum angular momentum , rather we present , in the style of a reference , the important formulas required to manipulate mps wavefunctions and operators . for a comprehensive introduction see for example references @xcite . using the normalization convention of biedenharn @xcite , we define the matrix elements of a tensor operator @xmath170}$ ] transforming as a rank @xmath171 tensor under @xmath153 rotations , as @xmath172}_m \ , | \ , jm \ , \rangle } = { \langle \ , j ' \ , \| \ , { \mbox{\boldmath $ t$}}^{[k ] } \ , \| \ , j \ , \rangle } \ ; { \mbox{$c { } ^{j}_{m } { } ^{k}_{m } { } ^{j'}_{m'}$ } } \ ; , \ ] ] where @xmath173 is the clebsch - gordan ( cg ) coefficient , @xmath174 label the representation of @xmath153 , and @xmath175 and @xmath176 label the projections of the total spin onto the @xmath177-axis . using the orthogonality of the clebsch - gordan coefficients , this defines the reduced matrix elements , @xmath178 } \ , \| \ , j \ , \rangle } = \sum_{mm } { \mbox{$c { } ^{j}_{m } { } ^{k}_{m } { } ^{j'}_{m'}$ } } { \langle \ , j'm ' \ , | \ , t^{[k]}_m \ , | \ , jm \ , \rangle } \ ; , \ ] ] where @xmath179 is arbitrary . note that this normalization is _ not _ the same as that used by varshalovich _ et . al _ @xcite , whom instead use an additional factor @xmath180 in the reduced matrix elements . this is a tradeoff ; some formulas simplify slightly with this normalization , but the normalization used here has the advantage that the reduced matrix elements of scalar operators ( with @xmath181 ) coincide with the actual matrix elements as all of the relevant clebsch - gordan coefficients are equal to unity . given the definition of the reduced matrix elements , we formulate the remaining formulas without further reference to the axis of quantization , except as an intermediate step to relate the reduced matrix elements prior to factorizing out the clebsch - gordan coefficients . the coupling of two operators is just as for the coupling of ordinary spins ; @xmath182}$ } } \times { \hbox{${\mbox{\boldmath $ t$}}^{[k_2]}$ } } \right]^{[k ] } \ ; , \ ] ] which denotes the set of operators with components @xmath182}$ } } \times { \hbox{${\mbox{\boldmath $ t$}}^{[k_2]}$ } } \right]^{[k]}_{\mu } = \sum_{\mu_1 \mu_2 } { \mbox{$c { } ^{k_1}_{\mu_1 } { } ^{k_2}_{\mu_2 } { } ^{k}_{\mu}$ } } { \hbox{${s}^{[k_1]}_{\mu_1}$ } } { \hbox{${t}^{[k_2]}_{\mu_2}$ } } \ ; .\ ] ] applying the wigner - eckart theorem gives , after a few lines of algebra , @xmath183}$ } } \times { \hbox{${\mbox{\boldmath $ t$}}^{[k_2]}$ } } \right]^{[k ] } \ , \| \ , j \ , \rangle } \\ = ( -1)^{j+j'+k } \sum_{j '' } \sqrt{(2j''+1)(2k+1 ) } { \mbox{$\left\ { \begin{array}{ccc } \!{j'}\ ! & \!{k_1}\ ! & \!{j''}\ ! \\ \!{k_2}\ ! & \!{j}\ ! & \!{k}\ ! \end{array } \right\}$ } } \\ \times { \langle \ , j ' \ , \| \ , { \hbox{${\mbox{\boldmath $ s$}}^{[k_1]}$ } } \ , \| \ , j '' \ , \rangle } { \langle \ , j '' \ , \| \ , { \hbox{${\mbox{\boldmath $ t$}}^{[k_2]}$ } } \ , \| \ , j \ , \rangle } \ ; , \end{array } \label{eq : irredproduct}\ ] ] where @xmath184 denotes the @xmath185 coefficient @xcite . a special case of the coupling law eq . ( [ eq : irredproduct ] ) that we will need is when the operators act on different spaces , such that they have a tensor product form @xmath186}_{\mu_1}$ } } & = & { \hbox{${t}^{[k_1]}_{\mu_1}$}}(1 ) \otimes i(2 ) \ ; , \\ { \hbox{${t}^{[k_2]}_{\mu_2}$ } } & = & i(1 ) \otimes { \hbox{${t}^{[k_2]}_{\mu_2}$}}(2 ) \ ; . \end{array}\ ] ] here @xmath187 denotes the identity operator and @xmath188}$}}(i)$ ] is an irreducible tensor operator with respect to the angular momentum @xmath189 of part @xmath6 of a two - part physical system ( @xmath190 ) . the total angular momentum of the system is @xmath191 . in this case , we write the coupling as @xmath192}$ } } \mathbf{\times } { \hbox{${\mbox{\boldmath $ t$}}^{[k_2]}$ } } { \big]}$}^{[k]}}$ ] @xmath193 @xmath194}$}}(1 ) \mathbf{\otimes } { \hbox{${\mbox{\boldmath $ t$}}^{[k_2]}$}}(2 ) { \big]}$}^{[k]}}$ ] . repeated application of the wigner - eckart theorem to these tensor operators gives , after some algebra , @xmath195}$}}(1 ) \mathbf{\otimes } { \hbox{${\mbox{\boldmath $ t$}}^{[k_2]}$}}(2 ) { \big]}$}^{[k ] } } \ , \| \ , j \ , ( j_1j_2\alpha_1\alpha_2 ) \ , \rangle } \vspace{0.25 cm } \\ = { \mbox{$\left [ \begin{array}{ccc } \!{j_1}\ ! & \!{j_2}\ ! & \!{j}\ ! \\ \!{k_1}\ ! & \!{k_2}\ ! & \!{k}\ ! \\ \!{j'_1}\ ! & \!{j'_2}\ ! & \!{j'}\ ! \end{array } \right]$ } } { \langle \ , j'_1 \ , ( \alpha'_1 ) \ , \| \ , { \hbox{${\mbox{\boldmath $ t$}}^{[k_1]}$}}(1 ) \ , \| \ , j_1 \ , ( \alpha_1 ) \ , \rangle } { \langle \ , j'_2 \ , ( \alpha'_2 ) \ , \| \ , { \hbox{${\mbox{\boldmath $ t$}}^{[k_2]}$}}(2 ) \ , \| \ , j_2 \ , ( \alpha_2 ) \ , \rangle } \ ; , \end{array } \label{eq : tensorproductcoupling}\ ] ] where @xmath196 $ } } \equiv [ ( 2j'_1 + 1)(2j'_2 + 1)(2j+1)(2k+1)]^{\frac{1}{2 } } { \mbox{$\left\ { \begin{array}{ccc } \!{j_1}\ ! & \!{j_2}\ ! & \!{j}\ ! \\ \!{k_1}\ ! & \!{k_2}\ ! & \!{k}\ ! \\ \!{j'_1}\ ! & \!{j'_2}\ ! & \!{j'}\ ! \end{array } \right\}$ } } \ ; , \ ] ] and the term in curly brackets is the wigner @xmath197 coefficient , which can be defined as a summation over @xmath185 coefficients @xcite , @xmath198 we can define an operator norm , corresponding to the usual frobenius norm , such that @xmath199}$}}||^2_{\mbox{\tiny frob } } = \tr { \hbox{${\mbox{\boldmath $ x$}}^{[k]}$ } } \cdot { \hbox{${\mbox{\boldmath $ x$}}^{\dagger[k]}$ } } = \tr { \hbox{${\mbox{\boldmath $ x$}}^{\dagger[k]}$ } } \cdot { \hbox{${\mbox{\boldmath $ x$}}^{[k]}$ } } \ ; .\ ] ] after some arithmetic , we see that @xmath199}$}}||^2_{\mbox{\tiny frob } } = \sum_{j'j } ( 2j'+1 ) |{\langle \ , j ' \ , \| \ , { \hbox{${\mbox{\boldmath $ t$}}^{[k]}$ } } \ , \| \ , j \ , \rangle}|^2\ ] ] for the center - matrix formalism , we need the transformation @xmath200}$}}_{ij } \rightarrow \sum_k \ ; c_{ik } \ ; { \hbox{${\mbox{\boldmath $ a'$}}^{[s]}$}}_{kj}\ ] ] where @xmath171 is a @xmath201 dimensional index that encapsulates both a @xmath202 and a @xmath203 index runs over the clebsch - gordan expansion of @xmath204 . ] : @xmath205 . requiring @xmath206}$}}_{kj}$ ] to satisfy the right orthogonality constraint , @xmath207 , this requires @xmath208}$}}_{kj } = \delta_{j'j}\delta_{s 's } \quad \left[\mbox{with } k \simeq ( s',j')\right]\ ] ] with @xmath209}$}}_{ij'}\ ] ] in the other direction , we need @xmath200}$}}_{ij } \rightarrow \sum_k \ ; { \hbox{${\mbox{\boldmath $ a'$}}^{[s]}$}}_{ik } \ ; c_{kj}\ ] ] where @xmath210 . requiring @xmath206}$}}_{ik}$ ] to satisfy the left orthogonality constraint , @xmath211 , this requires @xmath208}$}}_{ik } = \delta_{s 's } \delta_{i'i } \sqrt{\frac{2k+1}{2i+1 } } \quad \left[\mbox{with } k \simeq ( s',i')\right]\ ] ] and @xmath212}$}}_{j'j } \sqrt{\frac{2i+1}{2k+1}}\ ] ] the most natural definition for a matrix product operator has two lower indices and three upper , @xmath213}_{s'i ' } { } ^{si}\ ] ] which transforms as the product of two operators of rank @xmath214 $ ] , with matrix elements @xmath215}_{r}$ } } \ , | \ , sq ; jm \ , \rangle } = { \langle \ , s';j ' \ , \| \ , { \hbox{${\mbox{\boldmath $ m$}}^{[k]}$ } } \ , \| \ , s;j \ , \rangle } \ ; { \mbox{$c { } ^{s}_{q } { } ^{k}_{r } { } ^{s'}_{q'}$ } } \ ; { \mbox{$c { } ^{j}_{m } { } ^{k}_{r } { } ^{j'}_{m'}$ } } \ ; .\ ] ] note that the product of an operator and a state requires a contraction of the index @xmath155 , which has the symmetry of over two _ lower _ indices , and then shifting the result index @xmath202 from upper to lower . for @xmath153 , the required phase factor is @xmath216 , giving the rule @xmath217}$ } } = { \hbox{${\mbox{\boldmath $ ( ma)$}}^{[s']}$ } } = \sum_s ( -1)^{s+k - s ' } { \hbox{${\mbox{\boldmath $ m$}}^{[k]}$}}^{s 's } \otimes { \hbox{${\mbox{\boldmath $ a$}}^{[s]}$ } } \ ; . \label{eq : operatorstateproduct}\ ] ] the action of a matrix - product operator on another matrix product operator is @xmath218}$ } } = { \hbox{${\mbox{\boldmath $ m$}}^{[m]}$ } } { \hbox{${\mbox{\boldmath $ n$}}^{[n]}$ } } \ ; , \ ] ] which corresponds to the ordinary ( contraction ) product in the local basis and the tensor product in the matrix basis , and therefore results in the product of a @xmath185 and a @xmath197 coefficient from equations eq . ( [ eq : irredproduct ] ) and eq . ( [ eq : tensorproductcoupling ] ) respectively . for the evaluation of matrix elements , we need the operation @xmath219 on expanding out the reduced matrix elements , we see immediately that the coupling coefficient is @xmath220}$}}_{i'j ' } = \sum_{a , i , j , k , s , s ' } { \mbox{$\left [ \begin{array}{ccc } \!{j}\ ! & \!{s}\ ! & \!{j'}\ ! \\ \!{a}\ ! & \!{k}\ ! & \!{a'}\ ! \\ \!{i}\ ! & \!{s'}\ ! & \!{i'}\ ! \end{array } \right]$ } } { \hbox{${\mbox{\boldmath $ m$}}^{[k]}$}}^{s's}_{a'a } { \hbox{${\mbox{\boldmath $ a$}}^{[s']}$}}^*_{i'i } { \hbox{${\mbox{\boldmath $ b$}}^{[s]}$}}_{j'j } { \hbox{${\mbox{\boldmath $ f$}}^{[a]}$}}_{ij}\ ] ] conversely , from the left hand side , @xmath221 is @xmath222}$}}_{ij } = \sum_{a',i',j',k , s',s } \frac{2i'+1}{2i+1 } { \mbox{$\left [ \begin{array}{ccc } \!{j}\ ! & \!{s}\ ! & \!{j'}\ ! \\ \!{a}\ ! & \!{k}\ ! & \!{a'}\ ! \\ \!{i}\ ! & \!{s'}\ ! & \!{i'}\ ! \end{array } \right]$ } } { \hbox{${\mbox{\boldmath $ e$}}^{[a']}$}}_{i'j ' } { \hbox{${\mbox{\boldmath $ m$}}^{[k]}$}}^{s's}_{a'a } { \hbox{${\mbox{\boldmath $ a$}}^{[s']}$}}^*_{i'i } { \hbox{${\mbox{\boldmath $ b$}}^{[s]}$}}_{j'j}\ ] ] on interchanging @xmath223 , @xmath224 , this becomes the equation for a direct operator - matrix - product multiply . but using the center - matrix formalism , we want instead the operation @xmath225 where @xmath46 and @xmath226 transform as scalars , the quantum numbers impose @xmath227 , @xmath228 . this is essentially a scalar product @xmath229 , and the coupling coefficients drop out . in this paper , we have presented an introduction to the mps formulation of the dmrg algorithm for the calculation of ground- and excited states of one - dimensional lattice hamiltonians . the mps formulation is extremely flexible , allowing the possibility for algorithms that act on several distinct wavefunctions at once . the simplest such algorithms are for the fidelity and expectation values involving unrelated wavefunctions , @xmath230 and @xmath231 , which are difficult to extract from conventional dmrg . this gives access to new tools for the analysis of quantum phase transitions , by measuring the scaling function and exponents for the fidelity between ground - states as a function of the interaction strength . in addition , the mps formulation allow optimized versions of algorithms for dynamical correlations @xcite and time evolution @xcite , which remains a fertile area for continued algorithmic improvements . finally , we note that in the simulation of finite temperature states via a density operator or purification @xcite in the absence of dissipative terms that mix the particle numbers between the real and auxiliary systems , the symmetries of the system are _ doubled _ , such that the symmetries of the hamiltonian are preserved by the real and auxiliary parts independently . for simulations in a canonical ensemble , this leads to a significant efficiency improvement that , as far as we know , has not yet been taken into consideration . thanks to ulrich schollwck and thomas barthel for many stimulating conversations . while preparing this manuscript , we learned that a rotationally invariant formulation using the clebsch - gordan coefficients @xcite has been applied to the tebd algorithm for infinite systems @xcite .

quenched disorder induces an extrinsic inhomogeneity in otherwise translationally invariant pure systems . as a result , the equilibrium and out - of - equilibrium physics of disordered systems may be influenced by rare collective events , such as `` avalanches '' , statistically unlikely regions , such as `` droplets '' or `` griffiths regions '' , and by the proliferation of `` metastable states '' . our objective is to develop a theory describing the long - distance physics of such systems through a nonperturbative functional renormalization group ( np - frg ) method and our first focus is the equilibrium behavior of the random - field model . we showed in previous papers , which we refer to as i@xcite and ii@xcite , that the effect of avalanches and droplets can be captured by an approach based on the rg flow of the cumulants of the renormalized disorder , provided that the full functional dependence of the latter is accounted for.@xcite a functional rg is therefore required to let the singular behavior due to rare events or regions emerge along the flow , as also shown in the case of the equilibrium and forced behavior of manifolds in a random environment.@xcite the issue of metastable states , which , at zero temperature where the concept is well defined , refers to the presence of many minima of the microscopic hamiltonian ( bare action ) not simply related by symmetry transformations , is a recurring conundrum in theories of disordered systems . in the case of the random - field ising model ( rfim ) under study , it has a clear manifestation . the critical behavior of the model being controlled by a zero - temperature fixed point,@xcite the long - distance physics can be described through the properties of the ground state which , due to the presence of the random field , is obtained as the solution of a stochastic field equation . by standard field - theoretic manipulations , this leads to a theory expressed in terms of superfields . parisi and sourlas@xcite showed that a supersymmetry of the theory , more specifically the invariance under rotations of the underlying superspace , implies the property of dimensional reduction , according to which the critical behavior of the rfim in @xmath4 dimensions is identical to that of the pure ising model in @xmath5 dimensions . the property however has been proven to be wrong in low enough dimension.@xcite at the same time , it has been understood that the superfield construction breaks down because of the presence of many solutions of the stochastic field equation.@xcite however , in the parisi - sourlas formalism , it is not possible to disentangle breaking of superrotational invariance and collapse of the formalism due to the appearance of multiple solutions . in the companion paper , henceforth referred to as paper iii,@xcite we have shown how to resolve the above conundrum and to combine an extended superfield approach with the np - frg formalism . in particular , ground - state selection can be achieved by adding a weighting factor involving an auxiliary temperature and letting at the end of the manipulations the auxiliary temperature go to zero in the exact np - frg equations for the cumulants of the renormalized disorder . the resulting property of `` grassmannian ultralocality '' then allows one to specifically investigate supersymmetry ( superrotational invariance ) and its spontaneous breaking along the rg flow . such an investigation is the purpose of the present paper . the outline of the article is as follows . in sec . [ sec : model ] , we briefly summarize the main steps of the np - frg in a superfield formalism that we have developed in paper iii . this allows us to recall definitions , notations , and rg equations that will be used in the present article . in particular , we stress two important , and distinct , formal properties of the superfield theory : grassmannian ultralocality " and superrotational invariance " . we also consider the rg flow of the ward - takahashi identities associated with the latter and the consequences for the choice of the infrared regulator . in sec . [ sec : surot invariance ] , we show through our np - frg formalism that the superrotational invariance nonperturbatively leads to dimensional reduction . we conclude the section by building a scenario for a spontaneous breaking of the superrotational invariance along the rg flow that is based on our previous results on the appearance of a cusp " in the functional dependence of the cumulants of the renormalized random field,@xcite and we propose a continuation of the np - frg flow equations when spontaneous breaking has taken place . [ sec : approximation scheme ] is devoted to the development of a supersymmetry - compatible nonperturbative approximation scheme for the exact np - frg equations . it relies on combined truncations of the cumulant expansion and the expansion in spatial derivatives of the field . we also provide details on the numerical resolution . in sec . [ sec : results ] we present the results . we show in particular that breakdown of the dimensional reduction predictions for the critical exponents of the rfim occurs below a critical dimension @xmath0 . we compute the critical exponents as a function of dimension down to @xmath2 and we find good agreement with the best available estimates in @xmath2 and @xmath3 . finally , we show that scaling is described by three independent exponents , contrary to a proposed conjecture.@xcite a short account of this work has appeared in ref . we start by briefly recalling the main features of the formalism presented in the preceding article , with the associated definitions and notations . in paper iii,@xcite we have developed a np - frg theory for describing the equilibrium long - distance physics of the rfim that is based on an extension of the parisi - sourlas@xcite supersymmetric formalism . the latter relies on the fact that the critical behavior of the model is dominated by disorder - induced fluctuations , thermal fluctuations being subdominant , and can therefore be studied by looking directly at zero temperature . the equilibrium properties are then described by the ground state which is solution of the following stochastic field equation @xmath6}{\delta \varphi(x ) } = j(x),\ ] ] where we have added an external source ( a magnetic field ) @xmath7 conjugate to the @xmath8 field and the action @xmath9 $ ] is given by @xmath10 where @xmath11 , @xmath12 , with @xmath13 a random ( magnetic ) field sampled from a gaussian distribution of zero mean and variance @xmath14 . two key ingredients of our extension of the parisi - sourlas formalism are : \(1 ) the need to consider multiple copies ( or replicas ) of the original system with the same disorder , each copy being coupled to a different applied source , in order to generate cumulants of the renormalized disorder with their full functional dependence , thereby allowing for the emergence of a nonanalytic behavior in the field arguments , \(2 ) the introduction of a weighting factor involving an auxiliary temperature @xmath15 to the solutions of the stochastic field equation , so that when @xmath15 appraches 0 only the ground state contributes to the generating functional . by using standard field - theoretical techniques,@xcite one then ends up with a superfield theory for multiple copies in a curved superspace , with @xmath16=\exp ( \mathcal w^{(\beta)}[\{\mathcal j_a\}])=\overline{\exp(\sum_{a=1}^n\mathcal w_h^{(\beta ) } [ \mathcal j_a ] ) } \\&= \int \prod_{a=1}^{n}\mathcal q\phi_a \exp \bigg(-s^{(\beta)}[\{\phi_a\ } ] + \sum_{a=1}^{n } \int_{\underline x } \mathcal j_a(\underline x ) \phi_a(\underline x)\bigg ) , \end{aligned}\ ] ] where the multicopy action is given by @xmath17 = \sum_{a=1}^{n } \int_{\underline{x } } \left [ \frac 12 ( \partial_{\mu}\phi_a(\underline{x}))^2+u_{b}(\phi_a(\underline{x}))\right ] \\&- \frac{\delta_b}{2}\sum_{a_1=1}^{n}\sum_{a_2=1}^{n } \int_{x}\int_{\underline{\theta}_1\underline{\theta}_2 } \phi_{a_1}(x,\underline{\theta}_1)\phi_{a_2}(x,\underline{\theta}_2 ) . \end{aligned}\ ] ] in the above equations , we have introduced a superspace ( coordinates @xmath18 ) comprising the @xmath4-dimensional euclidean space ( coordinates @xmath19 ) and a @xmath20-dimensional grassmannian space ( anticommuting coordinates @xmath21 ) ; the metric is flat in the euclidean sector and curved in the grassmannian one with the curvature proportional to @xmath22 . ( for instance , the integral over superspace is defined as @xmath23 . ) we have also defined superfields @xmath24 , with one auxiliary bosonic ( response " ) field @xmath25 and two auxiliary fermionic ( ghost " ) fields @xmath26 and @xmath27 , and associated supersources @xmath28 . the action in eq . ( [ eq_superaction_multicopy ] ) is invariant under a large group of ( bosonic and fermionic ) symmetries . when expanded in increasing number of sums over copies , the generating functional of the connected green s functions , @xmath29 $ ] , gives access to the cumulants of the random generating functional @xmath30 $ ] . we have next applied the np - frg formalism to this superfield theory . this proceeds by first adding an infrared ( ir ) regulator that enforces a progressive account of the fluctuations to the bare action , @xmath31 where the two cutoff functions @xmath32 and @xmath33 are related through @xmath34 with @xmath35 the euclidean momentum , @xmath36 the strength of the renormalized random field , and @xmath37 the field renormalization constant . this ensures that all symmetries of the theory are satisfied . this includes the superrotational invariance found when the theory is restricted to a single copy and to @xmath38 : it then corresponds to @xmath39 , a property that is valid at the microscopic ( uv ) scale @xmath40 ( see also below ) . one next introduces the effective average action@xcite @xmath41=-\mathcal w_k^{(\beta)}[\{\mathcal j_a\}]+\sum_a\int_{\underline x } \phi_a(\underline x ) \mathcal j_a(\underline x ) -\delta s_k^{(\beta)}[\{\phi_a\}]$ ] , whose dependence on the ir cutoff @xmath42 is governed by an exact renormalization - group equation ( erge ) . the effective average action can also be expanded in increasing number of sums over copies , each term of the expansion being then related through the legendre transform to the cumulants of @xmath30 $ ] . we refer to the @xmath43th order term of the expansion of @xmath41 $ ] as the @xmath43th `` cumulant of the renormalized disorder '' . these expansions in increasing number of sums over copies lead to systematic algebraic manipulations that allow one to derive from the erge for @xmath41 $ ] a hierarchy of coupled erge s for the cumulants of the renormalized disorder . we have unveiled an important property of the random generating functional @xmath30 $ ] : the latter satisfies a specific dependence on the grassmann coordinates , which we have called grassmannian ultralocality " , if and only if a unique solution of the stochastic field equation is included in its computation . this translates into an ultralocal " property of the cumulants of the renormalized disorder . grassmannian ultralocality `` becomes a property of the superfield theory when @xmath44 ; it is also asymptotically found for finite @xmath22 when @xmath45 ( after going to dimensionless quantities ) . it then reflects the desired ground - state dominance . when this property is satisfied ( _ e.g. _ by setting @xmath46 ) , the erge s for the cumulants simplify and only involve ultralocal '' quantities that can be evaluated for physical fields @xmath47 , _ i.e. _ for superfields that are uniform in the grassmann subspace . for illustration , we give below the erge s for the first two cumulants under the property of grassmannian ultralocality " , which will be needed in the following : @xmath48= \\&- \dfrac{1}{2 } \tilde{\partial}_t \int_{x_2x_3}\widehat{p}_{k;x_2x_3}[\phi_1 ] \big(\gamma_{k2;x_2,x_3}^{(11)}\left[\phi_1,\phi_1\right ] - \widetilde{r}_{k;x_2x_3}\big ) \end{split}\ ] ] and @xmath49=\\ & \dfrac{1}{2 } \tilde{\partial}_t \int_{x_3x_4}\big \{- \widehat{p}_{k;x_3x_4}\left[\phi_1\right ] \gamma_{k3;x_3,.,x_4}^{(101)}\left[\phi_1,\phi_2,\phi_1\right ] + \\ & \widetilde{p}_{k;x_3x_4}\left[\phi_1,\phi_1\right ] \gamma_{k2;x_3x_4,.}^{(20)}\left[\phi_1,\phi_2\right ] + \frac{1}{2}\widetilde{p}_{k;x_3x_4}\left[\phi_1,\phi_2\right ] \\ & \times \left ( \gamma_{k2;x_3,x_4}^{(11)}\left[\phi_1,\phi_2\right ] - \widetilde{r}_{k;x_3x_4}\right ) + perm(12)\big \ } , \end{split}\ ] ] where @xmath50 denotes the expression obtained by permuting @xmath51 and @xmath52 , @xmath53 , @xmath54 is a short - hand notation to indicate a derivative acting only on the cutoff functions ( _ i.e. _ , @xmath55 ) , and the propagators @xmath56 and @xmath57 ( denoted by @xmath58}$ ] and @xmath59}$ ] in paper iii@xcite ) are defined as @xmath60=\left(\gamma _ { k,1}^{(2 ) } [ \phi ] + \widehat r_k\right ) ^{-1}\ ] ] and @xmath61= \widehat { p}_{k } [ \phi_1 ] ( \gamma _ { k,2}^{(11)}[\phi_1 , \phi_2 ] -\widetilde r_k ) \widehat { p}_{k } [ \phi_2 ] , \ ] ] and superscripts indicate functional differentiation with respect to the field arguments . generically , the flow of @xmath62 $ ] involves three types of quantities : the propagators @xmath56 and @xmath57 , second functional derivatives of @xmath63 in which all the arguments are different , and second functional derivatives of @xmath64 with two of their arguments equal to each other . a graphical representation of the hierarchy of erge s is provided is appendix c of the companion paper iii . in the present paper we will focus on these flow equations , which correspond to the @xmath44 limit and describe the equilibrium properties associated with the ground state ( recall that the `` bath '' temperature is also at @xmath65 ) . we will briefly comment on the rg flow of corrections to grassmannian ultralocality " when @xmath22 is large but finite . finally , it is worth stressing that to obtain the flow equation for @xmath66 $ ] with its full functional dependence on the @xmath43 field arguments , one needs to consider at least @xmath43 copies . if one considers less than @xmath43 copies , the equation necessarily involves @xmath63 and its derivatives in which several of the arguments are equal . as we will show later , breaking of supersymmetry and breakdown of dimensional reduction are precisely related to the analyticity properties of the functionals when taking the limit of equal arguments . formally , the whole hierarchy of flow equations for the cumulants can thus be obtained by considering an arbitrary large number of copies . when grassmannian ultralocality " is satisfied , the curvature @xmath22 disappears from the flow equations [ see eqs . ( [ eq_flow_gamma1_ulapp],[eq_flow_gamma2_ulapp ] ) ] . formally , the latter are then the same as those obtained from considering a flat grassmann subspace . we have shown in paper iii that in this case , and provided one restricts the supersources such that one effectively recovers a one - copy theory , the theory is invariant under superrotations " that mix the euclidean and grassmannian sectors of the superspace . the associated generators are @xmath67 and @xmath68 . as any linearly realized continuous symmetry , superrotational invariance leads to a set of ward - takahashi ( wt ) identities for the one - particle irreducible ( 1pi ) generating functional , _ i.e. _ at scale @xmath42 the effective average action . for a flat superspace and a restriction to one copy , the wt identity for the effective average action reads @xmath69=0\ ] ] and similarly with @xmath70 . one can now check that the above wt identities are _ a priori _ stable under the rg flow , _ i.e. _ @xmath71=0,\ ] ] where @xmath72 generically indicates a component @xmath73 and , since there is no curvature , @xmath74 . to prove eq . ( [ eq_flow_ward ] ) , we rewrite the erge for the effective average action when the superfield theory is restricted to one copy and is considered in a flat superspace ( @xmath38 ) : @xmath75=\frac 12 \int_{\underline{x}_1\,\underline{x}_2 } \left ( \partial_t \mathcal r_{k;\underline x_1,\underline x_2}\right ) \mathcal p_{k;\underline x_1,\underline x_2}[\left\lbrace \phi_a \right\rbrace ] , \ ] ] where the regulator function @xmath76 with @xmath77 and the modified ( full ) propagator @xmath78 is the inverse in the sense of operators of @xmath79 . ( inserting the `` multilocal '' expansion in the grassmann coordinates and using the assumption of grassmannian ultralocality " then leads to erge s for the cumulants which are just eqs . ( [ eq_flow_gamma1_ulapp],[eq_flow_gamma2_ulapp ] ) and their extension to higher orders with all copy superfields taken as equal in both sides of the equations . ) the flow of the left - hand side of eq . ( [ eq_flow_ward ] ) can then be expressed as @xmath80\\ & = \frac{1}{2}\tilde{\partial}_t \int_{\underline{x}_1\,\underline{x}_2\ , \underline{x}_3}\phi(\underline{x}_1)\mathcal q_{\underline{x}_1 } \left ( \mathcal p_{k;\underline{x}_2\underline{x}_3}[\phi]\gamma_{k;\underline{x}_3\underline{x}_2\underline{x}_1}^{(3)}[\phi]\right ) \\ & = \frac{1}{2}\tilde{\partial}_t \int_{\underline{x}_2\,\underline{x}_3}\mathcal p_{k;\underline{x}_2\underline{x}_3}[\phi ] \int_{\underline{x}_1}\phi(\underline{x}_1)\mathcal q_{\underline{x}_1 } \gamma_{k;\underline{x}_3\underline{x}_2\underline{x}_1}^{(3)}[\phi ] , \end{aligned}\ ] ] where we have used that @xmath81 does not act on @xmath82 . assuming that the wt identity , eq . ( [ eq_ward_k ] ) , is satisfied down to the scale @xmath42 ( we study its further evolution ) , one has @xmath83= - \left ( \mathcal q_{\underline{x}_2 } + \mathcal q_{\underline{x}_3}\right ) \gamma_{k;\underline{x}_3\underline{x}_2}^{(2)}[\phi ] , \end{aligned}\ ] ] which is obtained by differentiating twice eq . ( [ eq_ward_k ] ) . after an integration by parts and a relabel of the dummy variables @xmath84 , one then finds @xmath85\\&= -\frac{1}{2}\tilde{\partial}_t \int_{\underline{x}_2\,\underline{x}_3}\gamma_{k;\underline{x}_3\underline{x}_2}^{(2)}[\phi]\ ; \left(\mathcal q_{\underline{x}_2 } + \mathcal q_{\underline{x}_3}\right ) \mathcal p_{k;\underline{x}_2\underline{x}_3}[\phi ] . \end{aligned}\ ] ] provided that one chooses the infrared cutoff function such that @xmath86 one can replace @xmath87 $ ] by @xmath87 + \mathcal r_{k;\underline{x}_3\underline{x}_2}$ ] in eq . ( [ eq_dem_susy_dr ] ) . by using the fact that the latter is equal to the inverse modified propagator @xmath88)_{\underline{x}_3\underline{x}_2}$ ] , one can rewrite the right - hand side of eq . ( [ eq_dem_susy_dr ] ) ( up to the trivial factor of @xmath89 ) as @xmath90)_{\underline{x}_3\underline{x}_2}\ ; \mathcal q_{\underline{x}_2 } \mathcal p_{k;\underline{x}_2\underline{x}_3}[\phi]\\&=\tilde{\partial}_t \int_{\underline{x}_2\,\underline{x}_4}\delta_{\underline{x}_2\underline{x}_4}\int_{\underline{x}_3}(\mathcal p_{k}^{-1}[\phi])_{\underline{x}_3\underline{x}_4}\ ; \mathcal q_{\underline{x}_2 } \mathcal p_{k;\underline{x}_2\underline{x}_3}[\phi]\\&=\tilde{\partial}_t \int_{\underline{x}_2\,\underline{x}_4}\delta_{\underline{x}_2\underline{x}_4 } \mathcal q_{\underline{x}_2}\ ; \int_{\underline{x}_3}(\mathcal p_{k}^{-1}[\phi])_{\underline{x}_3\underline{x}_4 } \mathcal p_{k;\underline{x}_2\underline{x}_3}[\phi]\\&=\tilde{\partial}_t \int_{\underline{x}_2\,\underline{x}_4}\delta_{\underline{x}_2\underline{x}_4 } \mathcal q_{\underline{x}_2}\ ; \delta_{\underline{x}_2\underline{x}_4}. \end{aligned}\ ] ] this last expression is easily shown to be identically zero , so that one finally obtains that eq . ( [ eq_flow_ward ] ) is satisfied at scale @xmath42 . the same reasoning can be repeated with the identity associated with @xmath91 . therefore , if one starts with an initial condition that is superrotationally invariant , which can be enforced by suppressing fluctuations ( see section ii - a and paper iii ) , the above derivation proves that the wt identities associated with the superrotational invariance are _ a priori _ preserved along the rg flow if the regulator satisfies eq . ( [ eq_condition_cutoff ] ) . the latter condition can be reexpressed as @xmath92=0 , \end{aligned}\ ] ] where we have used the fact that @xmath93 and @xmath94 are translationally invariant functions of the euclidean coordinates , with therefore @xmath95 and similarly for @xmath94 . going to fourier space , this implies that @xmath93 and @xmath94 are related through @xmath96 which , as already stated , corresponds to eq . ( [ eq_cutoffs_relation ] ) with @xmath39 , a condition that we choose to enforce at the beginning of the flow when @xmath97 . such a choice of cutoff then avoids an _ explicit _ breaking of superrotational invariance . an important property of the theory is that the superrotational invariance ( which for simplicity will often be denoted by the acronym susy , for supersymmetry , in the following ) leads to dimensional reduction . we consider here the case where grassmannian ultralocality " is satisfied so that the curvature @xmath22 drops out of the flow equations as in eqs . ( [ eq_flow_gamma1_ulapp],[eq_flow_gamma2_ulapp ] ) , and we assume that susy is indeed obeyed when the theory is restricted to one copy ( see above ) . we showed in paper iii@xcite that susy implies nontrivial wt identities relating cumulants of different orders . more specifically , the following wt identity at the scale @xmath42 will be needed below : @xmath98 & - \frac{\delta_b}{2 } ( x_1^\mu - x_2^\mu)\gamma_{k1;x_1x_2}^{(2)}[\phi ] = \\&- \int_{x_3}\phi(x_3 ) \partial_{3\mu}\gamma_{k2;x_1x_3,x_2}^{(21)}[\phi,\phi ] . \end{aligned}\ ] ] to prove the dimensional reduction property , it is then useful to single out a @xmath20-dimensional subspace of the @xmath4-dimensional euclidean space . we define @xmath99 with @xmath100 and @xmath101 and we consider superfields that are uniform in the 2-dimensional grassmannian and the 2-dimensional euclidean subspaces , _ i.e. _ @xmath102 . for such fields , the wt identity in eq . ( [ eq_wt_susy ] ) implies that @xmath103 = \delta_b\ , \partial_{p^2}\gamma_{k1;y_1y_2}^{(2)}[p^2;\phi],\ ] ] where @xmath43 is the @xmath20-dimensional momentum obtained from a spatial fourier transform over @xmath104 [ by using translational invariance we have as usual defined @xmath105 and similarly for @xmath106 . the same type of relation holds between @xmath107 and @xmath108 . with the help of the above relations , the erge for the first cumulant @xmath109 $ ] , eq . ( [ eq_flow_gamma1_ulapp ] ) , can be reexpressed as @xmath110&=- \big(\frac{\delta_b}{2 } \big ) \tilde \partial_t \int d^{d-2}y_1 \int d^{d-2}y_2 \int \frac{d^{2}p}{(2\pi)^2}\\&\times \widehat{p}_{k;y_1y_2}[p^2 , \phi]\ ; \partial_{p^2}\left ( \gamma_{k1;y_1y_2}^{(2)}[p^2,\phi ] + \widehat{r}_k(p^2)\right ) \\&= - \big(\frac{\delta_b}{2 } \big ) \tilde \partial_t \int d^{d-2}y_1 \int \frac{d(p^2)}{4\pi}\\&\times \partial_{p^2}\left [ \log \left ( \gamma_{k1}^{(2)}[p^2,\phi ] + \widehat{r}_k(p^2)\right)\right ] _ { y_1y_1 } , \end{aligned}\ ] ] where , in the last expression , @xmath111 and @xmath32 are functions of @xmath112 but operators in the @xmath113-dimensional euclidean space spanned by the coordinate @xmath114 ( since @xmath115 is _ not _ uniform ) . the integral over @xmath112 is easily performed . choosing an infrared cutoff function that becomes independent of @xmath42 when its argument @xmath112 is at the uv scale , it only remains : @xmath116=&\big(\frac{\delta_b}{4\pi } \big ) \frac 12 \tilde{\partial}_t \int d^{d-2}y \\&\times \big [ \log\left ( \gamma_{k1}^{(2)}[p^2=0,\phi ] + \widehat{r}_k(p^2=0)\right)\big ] _ { yy}. \end{aligned}\ ] ] up to the trivial factor @xmath117 and with the identifications @xmath118\equiv \gamma_{k}[\phi ] , \gamma_{k1}^{(2)}[p^2=0,\phi]\equiv \gamma_{k}^{(2)}[\phi ] , \widehat{r}_k(p^2=0)\equiv r_k,\ ] ] eq . ( [ eq_dim_red ] ) coincides with the erge for the effective average action of a standard scalar @xmath119 theory@xcite in dimension @xmath5 . this provides another nonperturbative demonstration@xcite of the property of dimensional reduction for supersymmetric scalar field theories , first put forward by parisi and sourlas.@xcite we have shown within a nonperturbative implementation of the rg that the supersymmetry of the theory , and more specifically the superrotational invariance for one copy ( susy ) , imply the property of dimensional reduction . as one knows that dimensional reduction does not hold in low enough dimension , what then goes wrong in the formalism ? the answer is that some ( super)symmetries or identities must be spontaneously broken along the rg flow . we make the assumption , which will be supported by actual calculations , that only the superrotational invariance ( for the theory restricted to a single copy ) may be broken along the rg flow , all other ( super)symmetries and properties , most significantly the `` grassmannian ultralocality '' encoding ground - state dominance ( when @xmath44 ) , remaining unaltered . from the above proof , it follows that failure of dimensional reduction implies that the wt identity between @xmath120 and @xmath111 breaks down : a singularity must occur along the flow , which invalidates the wt identity ; the equality in eq . ( [ eq_flow_ward ] ) and some step in its derivation must go wrong at some scale @xmath42 . from our previous work,@xcite we anticipate that this arises due to the appearance of a strong enough nonanalytic dependence of the second cumulant of the renormalized random field @xmath121 $ ] in its field arguments when the latter two , @xmath51 and @xmath52 , become equal . ( one should keep in mind that the cumulants are invariant under permutations of their arguments ; consequently , @xmath120 is an even function(al ) in @xmath122 . ) as a result , some higher - order derivative ( _ i.e. _ , a 1pi vertex deriving from @xmath120 ) blows up and the whole hierarchy of coupled erge s and wt identities for one copy ceases to be valid . in order to envisage the possible scenario for spontaneous susy breaking and breakdown of dimensional reduction , it is instructive to study the structure of the exact hierarchy of coupled rg flow equations , whose lowest - order examples are given in eqs . ( [ eq_flow_gamma1_ulapp],[eq_flow_gamma2_ulapp ] ) . the following exposition is not meant to be rigorous , but only to provide some heuristic arguments . consider first the hierarchy of erge s for the proper vertices evaluated for uniform fields ( in euclidean space ) that is obtained by repeated functional differentiation of the erge for the effective average action . for this part of the reasoning , the case of a simple scalar field theory is sufficiently illustrative . the erge for the effective average action then reads @xmath123=\frac{1}{2}\int_q \partial_t r_k(q^2 ) \left(\gamma_k^{(2)}[\phi]+r_k \right)_{q ,- q}^{-1}.\ ] ] after differentiating this equation @xmath124 times , it is easily realized that for @xmath125 the erge for any generic 1pi vertex @xmath126 depends on all lower - order proper vertices ( of order @xmath127 ) as well as on some @xmath128 s and @xmath129 s . two cases then arise : when @xmath130 , the flow equation is _ linear _ in @xmath126 itself , whereas in the other case when @xmath131 , the flow equation may be _ nonlinear _ in @xmath126 itself . the same structure applies to the present theory . a consequence is that , if no other 1pi vertices diverge first , @xmath126 can only diverge at the end of the flow ( @xmath45 ) when @xmath130 ( due to the linearity of the corresponding erge ) whereas a divergence may occur at a finite scale when @xmath132 . to get more insight , we have to look more thoroughly into the structure of the rg equations and consider the 1pi vertices associated with the cumulants separately [ see eqs . ( [ eq_flow_gamma1_ulapp],[eq_flow_gamma2_ulapp ] ) for illustration ] . we further make the following plausible assumptions ( in order to have a renormalizable theory ) : \(i ) the @xmath133-copy 1pi vertices @xmath134 always stay finite , as do the cumulants of the renormalized random field @xmath135 and their first - order derivatives . the latter may however be discontinuous ; the condition on the first derivatives amounts to assuming that no `` supercusp''@xcite stronger than linear appears in the cumulants when two arguments become equal . \(ii ) since we expect a nonanalytic behavior occuring as the copy fields become equal in the reduction to an effective @xmath133-copy system , we also anticipate that derivatives with respect to symmetric combinations of the field arguments are always bounded ( _ e.g. _ , @xmath136 \vert_{\phi } \vert < + \infty$ ] , where the derivative is evaluated for all replica fields equal to @xmath137 ) . a `` cusp''@xcite in @xmath135 with @xmath138 implies that one of its second derivatives , which means a proper vertex of order at least @xmath139 , blows up . similarly , for having a `` subcusp''@xcite in @xmath120 , at least one of its third derivatives , hence a proper vertex of order at least @xmath139 , should blow up . as we have seen above that the associated flow equations are linear , we conclude that the divergence of such proper vertices ( with @xmath130 ) can not be the triggering event for a singularity at a finite scale . on the other hand , due to the nonlinearity of the erge s for the second derivatives of @xmath120 , a `` cusp '' may appear in the latter at a finite scale . to summarize this discussion : one expects that breakdown of dimensional reduction requires the presence of a _ cusp _ in the field dependence of @xmath120 , cusp that should appear _ at a finite scale _ , which by analogy with what occurs for a manifold in a random environment@xcite we call the `` larkin '' scale , during the rg flow . by contrast , weaker nonanalyticities in @xmath120 as well as nonanalyticities in higher - order cumulants ( if no cusp has appeared in @xmath120 ) can only appear at the fixed point , in the limit @xmath140 , thereby preserving the dimensional - reduction property . indeed , the derivation leading to eq . ( [ eq_dim_red ] ) then remains valid . another nontrivial question is then raised : if superrotational invariance ( susy ) is spontaneously broken , _ how can one continue the rg flow for the effective average action ? _ the answer , which again should be verified in calculations , is as follows : \(i ) keeping in mind that superrotational invariance is explicitly broken in the presence of multiple copies ( see section [ sec : model ] ) and that it is only recovered in the process of reducing the whole problem to a single - copy system , spontaneous breaking may then be described by rejecting any implicit assumption of analyticity of the field dependences in the latter process . one should therefore restrict the hierarchy of erge s to those equations for cumulants that are considered for generic ( nonequal ) field arguments , so that a putative nonanalytic dependence in these arguments can freely emerge along the rg flow.@xcite \(ii ) one assumes that except for the superrotational invariance , all of the properties and symmetries of the effective average action @xmath141 remain valid . in particular , this applies to the property of grassmannian ultralocality " of the random generating functional and to its consequences for the cumulants of the renormalized disorder . as discussed in the preceding paper , this property is obeyed when taking the limit @xmath46 in the erge s for the cumulants , which leads to eqs . ( [ eq_flow_gamma1_ulapp],[eq_flow_gamma2_ulapp ] ) and their extensions to higher orders . it keeps track of the fact that only the ground state is considered for each copy . the validity of the property of grassmannian ultralocality " is therefore distinct from that of superrotational invariance , a distinction that can not be made in the original parisi - sourlas formalism . \(iii ) the cutoff functions satisfy the relation in eq . ( [ eq_cutoffs_relation ] ) which contains as a special case the explicit superrotational invariance when @xmath39 . in practice , the typical strength of the renormalized random field @xmath36 is defined from @xmath142 evaluated for some specific uniform field configuration and the field renormalization constant @xmath37 is obtained from @xmath143 also evaluated for some uniform field configuration . finally , the cusp in @xmath144 , which is associated with spontaneous susy breaking , must be stable upon further evolution with the infrared scale @xmath42 and must not generate `` supercusps '' . from the structure of the erge for @xmath144 , it is expected that a `` linear cusp''@xcite satisfies these requirements and provides a mechanism for dimensional reduction _ if _ it persists at the fixed point ( see appendix [ appendixa ] ) . up to this point , many of the previous considerations remain plausible conjectures . we now provide an implementation of the np - frg that allows us to check their validity . the hierachy of erge s of course can not be solved exactly and the next step is to provide a susy - compatible nonperturbative approximation scheme . from our previous work,@xcite we know that an efficient scheme relies on a joint truncation in the _ derivative expansion _ , which approximates the long - distance behavior of the 1pi vertices,@xcite and in the _ expansion in cumulants of the renormalized disorder_. the truncations however must be combined in a way that does not explicitly break the supersymmetry , more precisely the superrotational invariance ( susy ) . to implement this requirement , we use the wt identities . as is clear from the discussion in the preceding section , a minimal truncation must at least include the second renormalized cumulant . when susy is not broken , the wt identity [ see eq . ( [ eq_wt_susy_vertex ] ) ] imposes that for a uniform field @xmath145 is given by the derivative of @xmath146 with respect to @xmath147 . ( a similar relation is satisfied by the infrared cutoff functions . ) consider now the derivative expansion . in the lowest order ( lpa for `` local potential approximation '' ) , the derivative of @xmath111 is a field - independent constant , which would force @xmath120 to be also field independent : this is clearly too crude to describe the physics of the rfim . the first order of the derivative expansion leads to @xmath146 of the form @xmath148 , with @xmath149 the `` effective average potential '' ( `` gibbs free energy '' for a magnetic system ) and @xmath150 a field renormalization function . when the wt identity is satisfied , this in turn requires that @xmath151 which corresponds to a local approximation for the second cumulant . it is easy to generalize this reasoning by taking into account the whole set of wt identities for the proper vertices obtained from the cumulants : a susy - compatible approximation at order @xmath152 consists of taking @xmath153 at the order @xmath152 of the derivative expansion , @xmath154 at the order @xmath155 of the derivative expansion , ... , @xmath156 in the local approximation , and all higher - order cumulants equal to zero.@xcite this provides a scheme of successive approximations that in principle can be used to check the robustness of the results and if necessary improve them . the minimal truncation that can already describe the long - distance physics of the rfim and does not explicitly break susy is then the following : @xmath157=\int_{x}\left [ u_k(\phi(x))+\frac{1}{2}z_k(\phi(x))(\partial_{\mu}\phi(x))^2 \right ] , \end{aligned}\ ] ] @xmath158=\int_{x}v_k(\phi_1(x),\phi_2(x)),\ ] ] with the higher - order cumulants set to zero.@xcite inserted in the erge s for the cumulants , eqs . ( [ eq_flow_gamma1_ulapp ] ) and ( [ eq_flow_gamma2_ulapp ] ) , the above ansatz provides @xmath159 coupled flow equations for the @xmath133-copy effective average potential @xmath149 ( or its derivative ) that describes the thermodynamics of the system , the field renormalization function @xmath150 , and the @xmath20-copy effective average potential @xmath160 from which one obtains the second cumulant of the renormalized random field at zero momentum , @xmath161 . susy is obeyed when @xmath162 , which is easily satisfied at the uv scale @xmath97 . in order to search for the fixed point that controls the critical behavior , the flow equations must be recast in a scaled form . this can be done by introducing appropriate scaling dimensions ( see refs . [ ] and paper iii@xcite ) . near a zero - temperature fixed point,@xcite one has the following scaling dimensions : @xmath163 and a renormalized temperature is introduced as @xmath164 , with @xmath165 and @xmath166 related through @xmath167 . more precisely , we define running anomalous dimensions @xmath168 and @xmath169 as @xmath170 @xmath171 where @xmath172 and @xmath36 have been introduced in sec . one also has @xmath173 the dimensionless counterparts of @xmath174 will be denoted by lower - case letters , @xmath175 . the resulting equations are @xmath176 \big [ z_{k}(\varphi ) -\delta_{k,0}(\varphi)\big ] + l_1^{(d)}(\varphi)\big[-\frac{d}{2 } z_k'(\varphi ) + \delta_{k,0}'(\varphi)\big]\\&+\frac{1}{2}(\eta_k-\bar\eta_k)\big[z_k'(\varphi)j_2^{(1,d+2)}(\varphi)+u'''_k(\varphi)j_2^{(1,d)}(\varphi)\big ] \big\ } , \end{split}\ ] ] @xmath177 + 4 l_{3}^{(d)}(\varphi ) z_{k}'(\varphi ) u'''_k(\varphi)[z_{k}(\varphi ) \\&- \delta_{k,0}(\varphi ) ] - l_{2}^{(d)}(\varphi ) \big[\frac{2d+1}{2 } z_{k}'(\varphi)^2 + z_{k}''(\varphi)(z_{k}(\varphi ) - \delta_{k,0}(\varphi ) ) \\ & - 2 z_{k}'(\varphi ) \delta_{k,0}'(\varphi)\big ] + m_{4}^{(d-2)}(\varphi)u'''_k(\varphi)^2 + \frac{2}{d}m_{4}^{(d)}(\varphi ) u'''_k(\varphi)\times \\&[(d+2)z_{k}'(\varphi)- 2 \delta_{k,0}'(\varphi ) ] + \frac{1}{d}m_{4}^{(d+2)}(\varphi)z'_k(\varphi)[(d+4)z_{k}'(\varphi)- \\ & 4 \delta_{k,0}'(\varphi ) ] - \frac{8}{d}[m_{5}^{(d+4)}(\varphi ) z'_k(\varphi)^2 + 2 m_{5}^{(d+2)}(\varphi ) z'_k(\varphi)u'''_k(\varphi ) + \\ & m_{5}^{(d)}(\varphi ) u'''_k(\varphi)^2][z_{k}(\varphi ) - \delta_{k,0}(\varphi ) ] -\frac{1}{2}(\eta_k-\bar\eta_k)\big[z_k''(\varphi)\times\\&j_2^{(1,d)}(\varphi)-(4+\frac 2d)z_k'(\varphi)^2j_3^{(1,d+2)}(\varphi)-4z_k'(\varphi)u'''_k(\varphi)\times\\ & j_3^{(1,d)}(\varphi)+\frac 2dz_k'(\varphi)^2h_4^{(d+2)}(\varphi)+\frac 4dz_k'(\varphi)u'''_k(\varphi)h_4^{(d)}(\varphi)+\\ & \frac 2d u'''_k(\varphi)^2h_4^{(d-2)}(\varphi)\big ] \big\ } , \end{split}\ ] ] @xmath178 ^ 2 + l_{2}^{(d)}(\varphi_1)v_k^{(20)}(\varphi_1,\varphi_2)\\ & \times [ \delta_{k,0}(\varphi_1)-z_k(\varphi_1 ) ] + \frac{d-2}{2 } l_{1}^{(d-2)}(\varphi_1)v_k^{(20)}(\varphi_1,\varphi_2 ) + \\ & \frac{1}{2 } m_{1,1}^{(d-2)}(\varphi_1,\varphi_2 ) - n_{1,1}^{(d-2)}(\varphi_1,\varphi_2)[\delta_k(\varphi_1,\varphi_2)-z_k(\varphi_1 ) ] \\ & -\frac{1}{2}(\eta_k-\bar\eta_k ) \big[\delta_k(\varphi_1,\varphi_2)j_{1,1}^{(1,d)}(\varphi_1,\varphi_2)+j_{1,1}^{(2,d)}(\varphi_1,\varphi_2)\\&+v_k^{(20)}(\varphi_1,\varphi_2)j_{2}^{(1,d)}(\varphi_1)\big ] + perm(12 ) \big\ } , \end{split}\ ] ] where @xmath179 , a prime denotes a derivative with respect to the field ( when only one argument is present ) , @xmath180 , @xmath181 , and @xmath182 denotes the terms obtained by permuting @xmath183 and @xmath184 . the functions @xmath185 , @xmath186 , @xmath187 , @xmath188 , @xmath189 , @xmath190 , @xmath191 and @xmath192 appearing in the above flow equations are `` dimensionless threshold functions '' which are defined from the infrared cutoff functions that satisfy@xcite @xmath193 and @xmath194 with the prime again denoting a derivative . the dependence of the dimensionless threshold functions on the dimensionless fields comes through @xmath195 and @xmath196 . their definitions are given in appendix [ appendixb ] . the properties of these threshold functions have been extensively discussed.@xcite they decay rapidly when their arguments @xmath197 become large , which , since @xmath198 is the square of a renormalized mass , ensures that only modes with a mass smaller than @xmath42 contribute to the flow in eqs . ( [ eq_u_ising]-[eq_v_ising ] ) . these threshold functions essentially encode the nonperturbative effects beyond the standard one - loop approximation.@xcite note that @xmath199 and @xmath196 are even functions of @xmath8 and that , due to the @xmath200 symmetry and the permutational symmetry of the cumulants in their field arguments , @xmath201 satisfies @xmath202 , and similarly for @xmath203 . finally , the initial conditions for eqs . ( [ eq_eta_z],[eq_etabar_delta],[eq_u_ising],[eq_z_ising],[eq_v_ising ] ) at the uv scale @xmath40 ( @xmath204 ) are obtained from the bare action in eq . ( [ eq_superaction_multicopy ] ) for @xmath38 and @xmath205 , _ i.e. _ , @xmath206 the last equality being valid in the ( relevant ) region where the bare potential has a nontrivial minimum;@xcite after a trivial rescaling of the fields so that the explicit dependence on @xmath207 disappears , one also has @xmath208 which implies @xmath209 . this initial condition trivially satisfies @xmath210 which ensures that invariance under superrotations is obeyed at the beginning of the flow . an important property of the present theory occurs in the limit @xmath211 when @xmath203 is regular enough . for convenience , we introduce the variables @xmath212 and @xmath213 . the property can be stated as follows : if the behavior of @xmath214 is regular enough when @xmath215 , _ i.e. _ , is such that @xmath216 with @xmath217 when @xmath215 , then the flow of @xmath218 coincides with that of @xmath196 . this is precisely the wt relation in eq . ( [ eq_wt_susy_vertex ] ) when evaluated at zero momentum ; indeed , in dimensionless form , this equation is just @xmath219 . within the present ansatz for the effective average action , dimensional reduction then follows . to prove the equality of the two flow equations , we first derive eq . ( [ eq_v_ising ] ) with respect to @xmath183 and @xmath184 to obtain the rg equation for @xmath214 , change variables to @xmath220 and take the limit @xmath221 by assuming the above small @xmath114 behavior [ eq . ( [ eq_deltak_small_y ] ) ] . the derivation makes use of the properties of the dimensionless threhold functions that are summarized in appendix [ appendixb ] . this leads to @xmath222 \\ & - \frac{2d-3}{2 } l_{2}^{(d)}(\varphi ) \delta_{k,0}'(\varphi)^2 + m_{4}^{(d-2)}(\varphi)u'''_k(\varphi)^2 + 2 m_{4}^{(d)}(\varphi ) \times \\&u'''_k(\varphi)\delta_{k,0}'(\varphi ) + m_{4}^{(d+2)}(\varphi)\delta_{k,0}'(\varphi)^2 + l_k[\varphi;\delta_{k,0}-z_k]\big\ } , \end{split}\ ] ] where @xmath223 $ ] is a polynomial of degree two in @xmath224 $ ] and its derivatives , with coefficients that are function of @xmath8 and such that @xmath225\equiv 0 $ ] . subtracting from the above flow equation the one for @xmath226 in eq . ( [ eq_z_ising ] ) leads to an equation for the difference @xmath227 in the form @xmath228 = m_k[\varphi ; \delta_{k,0}-z_k ] - ( \bar\eta_k-\eta_k ) z_k(\varphi),\ ] ] where @xmath229 $ ] is another polynomial of degree two in @xmath224 $ ] and its derivatives such that @xmath230\equiv 0 $ ] . [ note that due to their definitions , @xmath231 and @xmath168 are equal as soon as @xmath219 . ] since the initial condition satisfies @xmath232 and @xmath233 , the only solution to eq . ( [ eq_diff_ising ] ) is @xmath234 at all scales , which provides the desired demonstration . by inserting this result in eq . ( [ eq_u_ising ] ) , one obtains @xmath235 which coincides with the flow equation for @xmath236 in the @xmath237 theory without random field in dimension @xmath5 ( use the relation @xmath238 and compare with eq . ( 4.33 ) of ref . [ ] ) . the same exercise can be repeated for the flow of @xmath196 , thereby completing the proof of dimensional reduction in this nonperturbative approximation . on the other hand , a spontaneous breaking of susy and of the associated wt identity occurs whenever @xmath239 diverges and @xmath214 has a cusp - like singularity in the form @xmath240 as @xmath241 . we expect that such a cusp appears at a finite `` larkin '' scale @xmath242 through the formation of a boundary layer as @xmath243 and @xmath215 in the form @xmath244^{2\varpi } + y^2},\ ] ] with @xmath245 an ( _ a priori _ unknown ) exponent , as can be checked by inserting in the flow equations . the boundary layer provides the appropriate initial conditions to further continue the flow below @xmath242 with a cuspy function . finally , it is worth comparing the above np - frg equations with those we have previously derived without the superfield formalism@xcite in a minimal approximation based on the same ansatz as in eqs . ( [ eq_ansatz1],[eq_ansatz2 ] ) . the two main differences are that : \(i ) the present equations are directly considered at zero temperature ( and zero auxiliary temperature ) and subdominant terms proportional to @xmath246 do not appear in the beta functionals . \(ii ) more importantly , since from the previous formalism we had no insight into the requirements for avoiding an _ explicit _ breaking of the underlying superrotational invariance at the single - copy level , we had for simplicity set the infrared cutoff function @xmath94 to zero . as a result , the wt identity associated with superrotational invariance was violated at _ all _ scales . the dimensional - reduction property could therefore not be exactly recovered when it was expected to be valid.@xcite we are interested in the long - distance physics of the rfim near its critical point and in the associated ( once unstable ) fixed point . such information can be obtained from the solution of the above coupled frg flow equations with appropriate boundary conditions . we have carried out the three following types of resolution . \(1 ) we have investigated the location of the breakdown of susy associated with the appearance of a cusp in @xmath203 . to this end , we have solved the set of coupled flow equations which is obtained when @xmath214 has a sufficiently regular ( no cusp ) behavior as @xmath247 , _ i.e. _ @xmath215 : see eq . ( [ eq_deltak_small_y ] ) . as already shown , the flow of @xmath218 then coincides with that of @xmath196 , and the np - frg equations for @xmath248 and @xmath196 form a closed set . in addition , we have solved the equation for the second derivative of @xmath249 with respect to @xmath250 , @xmath239 , which depends on @xmath248 and @xmath196 . a divergence in @xmath239 signals the appearance of a cusp in @xmath203 . the numerical task of solving this system of three coupled partial differential equation is reasonable . we discretize the problem both in the time and field directions . we use finite differences to evaluate the field derivatives that appear in the beta functions and use an explicit scheme to find the ( rg ) `` time '' evolution of the functions . \(2 ) when a cusp is present , solving in their full glory the set of coupled partial differential equations for functions of up to two field arguments is a hard task . we have discretized the field variables , using a 2-dimensional grid to solve the equation for @xmath203 . for convenience , we have switched variables from @xmath251 to @xmath252 and @xmath213 and restricted the grid to a trapezoidal region with @xmath253 and @xmath254 . observe that the flow of @xmath203 depends on @xmath255 and @xmath256 and their derivatives . in this sense , the flow equation for @xmath214 is nonlocal since its evolution depends not only on its property in a neighborhood of @xmath257 but also on its values at points faraway in the @xmath257 plane . it is therefore important to choose the trapezoidal grid such that , for all point in the grid , the information necessary for computing the evolution of the function at this point is inside the trapezoid . in the larger simulations we had 60 points in the @xmath258 direction and 50 in the @xmath114 direction . we have solved the set of coupled equations with two kinds of _ modus operandi _ : ( 2-i ) first , for selected values of the dimension @xmath4 we have searched the `` cuspy '' fixed point by dichotomy , starting from `` cuspy '' initial conditions ( @xmath259 ) . depending on the dimension and the regulator parameters , there appears from time to time numerical instabilities . such instabilities showed up for instance in the ( physically not very interesting ) large - field region , where the threshold functions are very small . to keep the latter under control , we have therefore used slightly modified flow equations : this is discussed in appendix [ appendix c ] . ( 2-ii ) secondly , once a proper `` cuspy '' fixed point has been identified in some given @xmath4 , we have followed this fixed point by continuously decreasing or increasing @xmath4 . this was done by looking directly for the roots of the beta functions by means of the hybrid algorithm , using as an initial guess the fixed point found at the previous iteration . we have studied the cuspy fixed point down to @xmath1 ; it is indeed difficult to go down to lower @xmath4 because the anomalous dimensions @xmath260 and @xmath166 become large in low dimensions : @xmath261 ( see also ref . \(3 ) when approaching the critical dimension @xmath262 for both susy and dimensional - reduction breaking from below , the procedures ( 2-i ) and ( 2-ii ) above become inefficient . we have used instead an expansion of the function @xmath263 in @xmath264 with , in practice , @xmath265 note that the expansion can not be applied in the close vicinity of @xmath262 . indeed , one then expects the approach @xmath266 to proceed nonuniformly in @xmath114 through a boundary layer in @xmath267 . it is easy to check that a similar boundary - layer phenomenon is found in the more easily handled 1-loop equation for the rf@xmath268 m near @xmath3,@xcite around the critical value @xmath269 , _ i.e. _ when @xmath270 . to compute the anomalous dimensions , we have defined the amplitudes @xmath37 and @xmath36 appearing in eqs . ( [ eq_eta_z],[eq_etabar_delta ] ) by choosing the specific configuration of the fields as equal to zero . this amounts to imposing @xmath271 @xmath272 which fixes @xmath168 and @xmath169 . in addition to the anomalous dimensions @xmath260 and @xmath166 ( obtained as the fixed - point values @xmath273 and @xmath274 ) , we have computed the critical exponent @xmath275 that controls the escape of the rg flow away from the fixed point along the unstable ( relevant ) direction . this was done by diagonalizing the flow equations around the fixed point and extracting the negative eigenvalue which identifies with @xmath276 . finally , we have chosen a dimensionless cutoff function @xmath277 [ see eqs . ( [ eq_scaled_hatr ] ) and ( [ eq_scaled_wider ] ) ] of the form @xmath278 where @xmath279 [ not to be confused with @xmath280 . this function satisfies the requirements @xmath281 and @xmath282 . it is a modification of the function used in refs . the parameters @xmath283 , @xmath284 and @xmath285 can be optimized via stability considerations ( see refs . [ , , ] ) . they are varied to find a region of values for which the computed critical exponents are stable under changes of the parameters . for more details , see appendix [ appendix c ] . our main result is a numerical confirmation of the scenario put forward above : the np - frg equations allowing us to continuously follow the critical behavior and the associated fixed point as a function of dimension @xmath4 , we find that there is a critical dimension @xmath262 separating a region @xmath286 where susy is valid all along the rg flow ( except possibly right at the fixed point ) and dimensional reduction applies from a region @xmath287 where susy is broken at a finite ir scale along the rg flow and a breakdown of dimensional reduction takes place . by using the procedure ( 1 ) detailed in the above subsection , we have numerically located this critical dimension as @xmath288 ( the precise value has a residual dependence on the chosen cutoff function ) . note that the value @xmath0 obtained here is consistent with the value found in our previous , and somewhat cruder , np - frg approach of the rf@xmath268 m when extrapolating the transition line @xmath289 down to @xmath290 ( see figure 4 of paper ii@xcite ) . for initial conditions of the rg flow at or near the critical point , the second derivative @xmath291 blows up at a finite rg `` time '' @xmath292 for @xmath293 , whereas it stays finite up to the fixed point for @xmath286.@xcite we illustrate the difference of behavior between these two cases in fig . [ fig_larkin ] for a field configuration @xmath294 . np - frg flow of @xmath295 in the regime where susy is valid . the initial conditions at @xmath97 ( _ i.e. _ , @xmath204 ) for @xmath296 and @xmath297 , with @xmath298 , are taken at the fixed - point solution , @xmath299 and @xmath300 [ @xmath301 , and those for @xmath302 and @xmath303 are chosen as @xmath304 and @xmath305 . the upper ( color online blue ) curve corresponds to @xmath306 and one observes that @xmath295 tends to a finite fixed - point value . the lower ( color online red ) curve corresponds to @xmath307 shows a divergence at a finite rg `` time '' @xmath308 . ] the divergence of the full function @xmath239 when @xmath287 is shown in fig . [ fig_larkin_3d ] . ( due to the @xmath200 symmetry , it is more convenient to represent the functions in terms of @xmath298 . ) np - frg flow of @xmath309 for @xmath307 . the initial conditions at @xmath97 ( _ i.e. _ , @xmath204 ) for @xmath296 and @xmath310 are taken at the fixed - point solution , @xmath299 and @xmath300 , and those for @xmath302 and @xmath303 are chosen as @xmath304 and @xmath305 . one observes that the divergence first takes place for small values of @xmath311 . ] we find , as can be anticipated from an analysis of the np - frg equations for @xmath312 , @xmath218 and @xmath239 , that the latter stays finite at @xmath313 and that its fixed - point value for @xmath314 behaves as @xmath315 , as seen in fig . [ fig_d_dr_dpf ] . fixed - point solution @xmath316 for dimensions @xmath317 . note the square - root behavior @xmath318 as a function of dimension . ] when considering dimensions smaller than @xmath262 , one must study the full dependence of the function @xmath319 . we show in fig . [ fig_larkin_y ] the evolution of this function for @xmath320 . starting from a constant function , one can clearly see that a linear cusp in @xmath114 appears in a finite rg time @xmath321 , close to 0.7 for the case shown . np - frg flow of @xmath322 for @xmath323 . the initial conditions at @xmath97 ( _ i.e. _ , @xmath204 ) for @xmath296 and @xmath310 are taken at the fixed - point solution , @xmath299 and @xmath300 , and that for @xmath249 is chosen as @xmath324 . one observes that a linear cusp appears at a finite rg time . by construction @xmath325 along the flow . ] the appearance of a cusp along the flow leads to a breakdown of the superrotational invariance and of the associated wt identities . this is illustrated for the @xmath2 fixed point in fig [ fig_d0z3d ] : there , @xmath326 , which implies a breaking of the wt identity in eq . ( [ eq_wt_approx ] ) . fixed - point solution in @xmath2 for @xmath300 ( solid line ) and @xmath327 ( dashed line ) . the two functions differ for a large enough field ( by construction , they coincide at @xmath320 ) . ] the different asymptotic behaviors at large @xmath311 are easily deduced from eqs . ( [ eq_z_ising],[eq_v_ising ] ) , from which we show that @xmath328 and @xmath329 . in @xmath2 , @xmath330 is very small , which implies that the function @xmath327 decreases slowly to @xmath331 . fixed - point solution @xmath332 for dimensions ranging from 3 ( steepest ) to 4.8 ( smoother curve ) . the variable @xmath311 has been rescaled by a factor @xmath333 so that all curves can be represented on the same plot . ] fixed - point solution @xmath300 for dimensions ranging from 3 ( steepest ) to 4.8 ( smoother curve ) . the variable @xmath311 has been rescaled by a factor @xmath333 so that all curves can be represented on the same plot . ] we also display in figs . [ fig_uppf ] and [ fig_zpf ] the fixed - point solutions @xmath332 and @xmath300 for different dimensions . one can see that the lower the dimension the steeper the curves . this means that for a numerical study of the critical properties in low dimensions , we need to discretize the field dependence in the np - frg equations with a small mesh , which entails a large number of points . the numerical integration is therefore more difficult and even becomes intractable in practice . finally , we show in fig . [ fig_deltapf ] the full dependence of the fixed - point solution @xmath334 . fixed - point solution @xmath334 in @xmath2 . note that the point @xmath335 corresponds to the lower right corner . the function is even in @xmath114 ( with a cusp around @xmath221 ) and the part corresponding to @xmath336 is not shown . ] we now turn to the results concerning the critical exponents . we begin with the anomalous dimensions @xmath260 and @xmath337 which are determined at the ( critical ) fixed point . ( we recall that @xmath260 characterizes the spatial decay of the `` connected '' pair correlation function and @xmath166 that of the `` disconnected '' pair correlation function at criticality . ) their dependence on the spatial dimension @xmath4 is shown in fig . [ fig_eta ] . anomalous dimensions @xmath260 and @xmath338 as functions of the spatial dimension @xmath4 . the dashed lines are lower bounds for the anomalous dimensions [ @xmath339 and @xmath340 for @xmath260 and @xmath166 respectively ] and the crosses correspond to predictions from computer simulations and ground - state determinations . ] above @xmath0 , we find that @xmath260 and @xmath166 rigorously coincide , @xmath341 , which is the signature of dimensional reduction . ( their value is moreover equal to that found in the same order of the derivative expansion for the pure ising model in dimension @xmath5 . ) below @xmath262 , the two anomalous dimensions bifurcate and @xmath342 . note that the predicted values satisfy the required bounds , _ i.e. _ , @xmath343 , @xmath344 , @xmath345 . we give in table [ tab_results_eta_etab ] our estimates for @xmath260 and @xmath166 in @xmath2 and @xmath3 , where a comparison is possible with numerical determinations from computer studies ( monte carlo calculations and @xmath65 ground - state determinations ) . .[tab_results_eta_etab ] comparison between the anomalous dimensions obtained in the present work and in computer studies ( a : [ ] ; b : [ ] ; c : [ ] ; d : [ ] ; e : [ ] ; f : [ ] ; g : [ ] ; h : [ ] ; i : [ ] ; j : [ ] ; k : [ ] ; l : [ ] ; m : [ ] ; n : [ ] ) . the error bars on the results from simulations and ground - state computations can be found in the original papers.the values of the temperature exponent @xmath346 that we obtain are @xmath347 in @xmath2 and @xmath348 in @xmath3 , in excellent agreement with the other studies . [ cols="<,<,<,<",options="header " , ] one can see that the agreement is good or very good . ( real - space rg studies in @xmath2@xcite provide results in the same range , with an anomalous dimension @xmath260 between @xmath349 to @xmath350 and an exponent @xmath165 between @xmath351 and @xmath352 . ) as is well known from the study of simpler models such as the pure @xmath268 model , the way to further improve the accuracy of the exponents would be to consider higher orders of the approximation scheme presented in section iv - a . we have not been able to perform our np - frg calculation down to the lower critical dimension @xmath353 , where the values @xmath354 and @xmath355 are exactly known . the numerical resolution of the flow equations become extremely arduous in low dimension where the anomalous dimensions become large and approach their lower bound , @xmath339 and @xmath340 respectively . in addition , one also encounters numerical difficulties as one approaches the critical dimension @xmath262 from below . indeed , as mentioned in section iv - d , it is anticipated that the limit @xmath356 , @xmath215 is nonuniform with a boundary layer in @xmath267 . more work will be needed to solve this boundary - layer problem numerically . [ fig_eta ] also provides evidence that the claim according to which the two exponents @xmath260 and @xmath166 are related by a fixed ratio , @xmath357,@xcite can not be right in general . it is true that the relation @xmath357 is exact in @xmath353 ( and according to a phenomenological rg,@xcite also at first order in @xmath358 ) and is very closely obeyed by the numerical estimates in @xmath2 and @xmath3 ( see table [ tab_results_eta_etab ] ) . however , the latter type of `` numerical evidence '' can always be challenged , and it is actually impossible to reach a definite conclusion on the sole basis of numerical results in selected dimensions ( @xmath359 ) . on the other hand , the overall and continuous dependence on spatial dimension that we provide through the np - frg leads to a different and firmer type of answer : since one goes from @xmath341 for @xmath360 to @xmath361 above , the relation @xmath357 can not be always valid . finally , we also display our results in @xmath2 and @xmath3 for the correlation length exponent @xmath275 , which is associated with the relevant direction around the critical fixed point , in table i. the agreement with the available data from computer studies is excellent in @xmath3 and fair in @xmath2 . ( note that in the real - space rg studies in @xmath2@xcite , the exponent @xmath275 is in general too large , with values ranging from @xmath362 to @xmath363 . ) all the other critical exponents are obtained in the np - frg through the expected relations : @xmath364 , @xmath365 , @xmath366 , etc.@xcite the validity of the theoretical description is clearly confirmed by both the overall consistency and the quantitative accuracy of the predictions in @xmath2 and @xmath3 . to assess the effect of a nonzero auxiliary temperature and of a deviation from `` grassmannian ultralocality '' , we have considered the simplest `` non - ultralocal '' contribution to the cumulants introduced in section vii - b of paper iii . combined with the above truncation of the `` ultralocal '' contributions of effective average action [ eqs . ( [ eq_ansatz1],[eq_ansatz2 ] ) ] , this amounts to expressing the first cumulant as ( see eq . ( 136 ) of paper iii ) @xmath367 = \int_{\underline{\theta}_1 } \bigg ( \gamma_{k1 } [ \phi_1(\underline{\theta}_1)]+\frac{1}{2\beta}(1 + \beta \bar \theta_1 \theta_1)\times \\ & \int_x y_{k}(\phi_1(x,\underline{\theta}_1))\partial _ { \theta_1 } \phi_1(x,\underline{\theta}_1)\partial_{\bar \theta_1 } \phi_1(x,\underline{\theta}_1 ) \bigg ) , \end{aligned}\ ] ] with @xmath109 $ ] given in eq . ( [ eq_ansatz1 ] ) , whereas the second - order cumulant is taken as purely `` ultralocal '' and given in eq . ( [ eq_ansatz2 ] ) . the frg flow equation for the function @xmath368 is directly obtained from eq . ( 139 ) of paper iii . in a dimensionless form , it reads @xmath369 l^{(d)}_2(\varphi)+(\bar \eta_k-\eta_k)j^{(1,d)}_2(\varphi)\big\}+ o(t_k ) , \end{split}\ ] ] where the threshold functions are defined in appendix [ appendixb ] . in deriving the above expression from eq . ( 139 ) of the preceding paper , we have used the dimensionless quantity @xmath370 defined through @xmath371 and the fact that as @xmath372 , @xmath373 the above equation is valid provided that @xmath374 which asymptotically goes as @xmath375 is subdominant with respect to @xmath376 . as discuss just below , this is indeed true , although @xmath370 goes to zero at the fixed point . the flow of the `` ultralocal '' quantities in the limit @xmath46 has already been solved and we can introduce the corresponding solution in eq . ( [ eq_flow_ydimensionless ] ) . we have solved the latter equation and found that @xmath377 goes to zero as one approaches the fixed point , in such a way that the dimensionful quantity @xmath378 goes to a constant @xmath379 : this is illustrated in fig . 10 for @xmath2 . of the dimensionless field with @xmath380 . after a transient regime , the function tends to a constant . ] in addition , we can investigate the effect of the `` non - ultralocal '' contribution to the flow of the `` ultralocal '' quantities when the auxiliary temperature @xmath15 is different from zero . this can be done by considering the erge s for the `` ultralocal '' components of the first and the second cumulants given in eqs . ( 134 ) and ( 135 ) of paper iii . from the above results ( in particular , the fact that @xmath376 goes to a finite constant when @xmath372 ) , we know that @xmath381 is asymptotically equal to @xmath382 : see eq . ( [ eq_hatqp_approx ] ) and discussion below . the flow of the dimensionless first cumulant therefore follows eq . ( 151 ) of paper iii and the contribution of the `` non - ultralocal '' piece goes as @xmath246 times a well - behaved function(al ) , with @xmath383 and @xmath384 . the flow of the second cumulant is also of the form of eq . ( 151 ) in the preceding paper , but the dimensionless beta function due to the `` non - ultralocal '' piece is now potentially singular when a cusp appears in the limit of zero ( auxiliary ) temperature of the second cumulant . more specifically , one finds that @xmath385+perm(12 ) \big\ } , \end{split}\ ] ] where @xmath386 is the zero ( auxiliary ) temperature beta functional obtained by taking derivatives with respect to @xmath183 and @xmath184 of the right - hand side of eq . ( [ eq_v_ising ] ) and we have omitted subdominant terms coming from the difference between @xmath387 and @xmath388 ( see above ) . we can now follow the derivation done for the effect of the ( bath ) temperature in the formalism without superfields ( see paper ii@xcite ) . after changing variables from @xmath389 to @xmath390 and @xmath391 , one obtains the solution for @xmath392 in the close vicinity of the fixed point , when both @xmath246 and @xmath114 go to zero , in the form of a boundary layer @xmath393 + o(t_k^2 ) , \end{split}\ ] ] where @xmath394 , @xmath395 , @xmath396 are well - behaved functions of @xmath19 and @xmath397 is equal to the absolute value of the coefficient of the cusp in the zero - temperature fixed - point function @xmath398 . the connection between this boundary - layer phenomenon and the presence of rare low - energy excitations above the ground - state known as `` droplets''@xcite is the same as that already discussed in our previous work@xcite ( see also refs . [ , ] ) and the discussion is not repeated here . in this paper and the preceding one @xcite we have extended our np - frg approach of disordered systems , which was initiated in the previous articles of this series.@xcite the objective was to discuss the property of dimensional reduction and its breakdown in the rfim from the parisi - sourlas@xcite perspective of an underlying supersymmetry and its breaking . we have reformulated the np - frg in a superfield formalism and , to do so , we have proposed a solution for properly selecting the ground state among the many metastable states that are present at zero temperature . through the introduction of an appropriate regulator and a supersymmetry - compatible nonperturbative approximation , we have been able to follow the supersymmetry ( more precisely , the superrotational invariance ) and its spontaneous breaking along the rg flow . despite the fact that the effective hamiltonian ( or microscopic action ) in the presence of a random field has numerous minima in the region of interest near the critical point , dimensional reduction need not be systematically broken . by implementing a np - frg flow for the cumulants of the renormalized disorder , our work shows that there is a finite range of dimension below the upper critical dimension , @xmath399 , for which this multiplicity of minima has no effect on the long - distance properties of the model . more precisely , the scaling behavior around the critical point conforms to the dimensional - reduction predictions . the associated fixed point is characterized by a nonanalytic dependence of the effective action in the dimensionless fields , at odds with a naive description based on perturbation theory , but the nonanalyticity is too weak to alter the spectrum of critical exponents . it is only by going below a critical dimension @xmath262 , which we numerically find around @xmath400 , that spontaneous breaking of the supersymmetry ( superrotational invariance ) takes place . the robustness of our theoretical description , which explains the pending puzzles concerning the critical behavior of the rfim , is supported by the good agreement obtained between the np - frg predictions for the critical exponents and the available values from computer studies . finally , one may wonder whether the formalism developed here can be useful in a different context . an obvious extension is a study of the long - distance physics of self - avoiding branched polymers and the associated property of reduction to the yang - lee edge singularity problem in two fewer dimensions.@xcite dimensional reduction is known to hold in this case,@xcite and this provides a benchmark model to test the ability of our nonperturbative approximation scheme to reproduce this property . another extension concerns the hysteresis behavior and out - of - equilibrium phase transitions of the rfim at zero temperature when driven by a slow change of the external magnetic field.@xcite work in those directions is under way . more challenging , and far more speculative too , is the question known as the `` gribov ambiguity '' in the nonperturbative quantization of nonabelian gauge field theories.@xcite the standard faddeev - popov gauge - fixing procedure aims at restricting the functional integral over the gauge field to nonequivalent gauge configurations ( _ i.e _ , configurations that are not obtainable one from another by a gauge transformation ) . unfortunately , the gauge - fixing conditions that respect lorentz invariance and internal ( color ) symmetry , such as the landau gauge condition , are inconsistent because they admit solutions that are equivalent up to a gauge transformation ( `` gribov copies '' ) . there has been a strong theoretical effort to overcome this problem . this `` gribov ambiguity '' does not affect the perturbative regime , so that calculations at high energy can be performed in the standard faddeev - popov approach . however , very little is known on its influence in the nonperturbative regime . the gribov - zwanziger model @xcite enables one to reduce the number of copies taken into account in the functional , but not to a single one , and , up to now , there is no unambiguous gauge - fixing procedure . the strong connections between the faddeev - popov and the parisi - sourlas formalisms , and their common failure when the relevant field equation has a multiplicity of solutions , have been underlined.@xcite the relevance of the tools and concepts developed in the present and the preceding papers to the problem of the gribov ambiguity has yet to be investigated . to illustrate that a linear cusp in @xmath144 does not generate a supercusp and may break dimensional reduction , we consider the erge for the fourier transform of the 2-point proper vertex @xmath401 ( for uniform fields ) in the limit @xmath402 . we switch to the variables @xmath403 and @xmath404 and we study a situation in which a cusp is present : @xmath405 where @xmath406 and @xmath407 ( recall that @xmath120 is even in @xmath137 and @xmath408 separately ) . with the assumption concerning the behavior of the proper vertices detailed in sec . iii - b , we find that the flow of @xmath120 , where for simplicity we take @xmath409 , can be written in the @xmath406 limit : @xmath410 , \end{aligned}\ ] ] where @xmath411 denotes the terms that could be obtained by directly considering the flow of @xmath412 and by assuming a regular behavior of its beta - functional when @xmath406 . the contribution due to the derivatives of the third cumulant @xmath413 is an even function of @xmath408 ; the @xmath414 term contributes to the regular component of the beta - functional and the next term is a @xmath415 ) with @xmath416 . the above equation indicates that a `` supercusp '' with @xmath417 leads to an ill - defined flow for @xmath412 . for @xmath418 ( cusp ) one should also study the erge for the third cumulant to determine the leading nonanalyticity ( _ i.e. _ , @xmath419 ) and therefore study the stability of the cusp under further rg flow . in any case , eq . ( [ eq_flow_cusp_stab ] ) shows that a cusp weaker than linear , with @xmath420 , has no effect on the flow of @xmath412 and , since @xmath421 is obtained from the latter , is not expected to modify dimensional reduction . a linear cusp ( @xmath422 ) on the other hand provides a possible mechanism for the failure of dimensional reduction while not _ a priori _ generating stronger nonanalyticities . in order to actually lead to breakdown of dimensional reduction , the linear cusp must of course remain in the renormalized second cumulant up to the appropriate fixed point . the various dimensionless threshold functions correspond to the various @xmath133-loop integrals involving the infrared cutoff functions and the propagators ( after account of the scaling dimensions ) . more specifically , they are defined as follows:@xcite @xmath423 @xmath424 @xmath425 with @xmath426 , @xmath279 , and @xmath427 acts only on the dimensionless function @xmath428 that is contained in the @xmath429s@xcite(by definition , @xmath430 $ ] ) . note that , in addition to the dependence on @xmath431 and @xmath432 , the threshold functions explicitly depend on the scale @xmath42 via the running anomalous dimension @xmath168 . ( note also that the functions @xmath433 and @xmath434 are not symmetric in the exchange of the indices @xmath133 and @xmath20 . ) the generic properties of these dimensionless threshold functions are discussed in detail in ref . [ ] . in the present study , it is also necessary to introduce additional threshold functions , which accounts for the fact that the cutoff function @xmath435 has an anomalous scaling compared to @xmath436 when dimensional reduction is broken . these threshold functions then always appear multiplied by @xmath437 in the flow equations . we define : @xmath438 and @xmath439 we summarize here a number of relations that are useful in the developments of sec . one has @xmath440 , @xmath441 , @xmath442 , @xmath443 , with @xmath444 . in addition , the functions @xmath445 s and @xmath446 s are related by @xmath447 from the above definitions , one also straightforwardly finds that @xmath448 @xmath449 and @xmath450 @xmath451 @xmath452 similar relations hold for the functions @xmath453 and @xmath454 . from these expressions one can obtain the derivatives with respect to the field arguments @xmath455 by using that @xmath431 and @xmath456 . dependence of the anomalous dimensions @xmath260 ( upper panel ) , @xmath166 ( middle panel ) and of the critical exponent @xmath275 ( lower panel ) on the parameter @xmath283 of the ir cutoff function in eq . ( [ eq_chosen_r ] ) . the `` principle of minimal sensitivity '' leads to a determination of the anomalous exponents @xmath457 , @xmath458 , @xmath459 ( at the minima ) . the variations of the anomalous dimensions are very small over a wide range of @xmath283 and enables us to evaluate the precision to @xmath460 and @xmath461 . @xmath275 is more sensitive to the regulator parameters and we estimate @xmath462,title="fig : " ] dependence of the anomalous dimensions @xmath260 ( upper panel ) , @xmath166 ( middle panel ) and of the critical exponent @xmath275 ( lower panel ) on the parameter @xmath283 of the ir cutoff function in eq . ( [ eq_chosen_r ] ) . the `` principle of minimal sensitivity '' leads to a determination of the anomalous exponents @xmath457 , @xmath458 , @xmath459 ( at the minima ) . the variations of the anomalous dimensions are very small over a wide range of @xmath283 and enables us to evaluate the precision to @xmath460 and @xmath461 . @xmath275 is more sensitive to the regulator parameters and we estimate @xmath462,title="fig : " ] dependence of the anomalous dimensions @xmath260 ( upper panel ) , @xmath166 ( middle panel ) and of the critical exponent @xmath275 ( lower panel ) on the parameter @xmath283 of the ir cutoff function in eq . ( [ eq_chosen_r ] ) . the `` principle of minimal sensitivity '' leads to a determination of the anomalous exponents @xmath457 , @xmath458 , @xmath459 ( at the minima ) . the variations of the anomalous dimensions are very small over a wide range of @xmath283 and enables us to evaluate the precision to @xmath460 and @xmath461 . @xmath275 is more sensitive to the regulator parameters and we estimate @xmath462,title="fig : " ] in the exact formulation of the np - frg , the results of the flow equations are independent of the ir regulator , which is only an intermediate means that does not affect the long - distance physics . once approximations are introduced , a residual dependence on the choice of regulator however remains . ( a similar situation occurs in perturbation theory where a residual dependence on the parameters of the borel resummation procedure is observed . ) one can in some sense optimize the choice of ir cutoff function by demanding that the output , say the critical exponents , satisfies a property of `` minimal sensitivity '' such that by varying the characteristics of the cutoff function around the optimum , a minimal variation of the exponents results . this guarantees the stability of the results . we apply this procedure to the type of function given in eq . ( [ eq_chosen_r ] ) where the parameters @xmath283 , @xmath284 , and @xmath285 can be varied . we illustrate in fig . [ fig_etapms ] the dependence of the anomalous dimensions in @xmath2 on the parameter @xmath283 when @xmath284 and @xmath285 are kept fixed at the values @xmath463 and @xmath464 respectively ( these latter values were determined by preliminary variational studies to find minimal sensitivity ) . one can see that there is a large domain of values in which @xmath260 and @xmath166 vary little ( say by less than @xmath465 ) . we use the `` principle of minimal sensitivity '' to determine the best estimate of the critical exponents [ note that the minima of @xmath466 and @xmath467 happen at very close values of @xmath283 ] . we also estimate the error bars from the observation that when @xmath283 varies in the range [ 1.5,3 ] , the maximum variation of @xmath260 is of 0.03 and that of @xmath338 is of 0.04 . these are the values that we have reported in sec . [ sec : results ] . during the numerical integration of the flow , we encountered numerical instabilities in physically unimportant regions ( typically , at large fields ) . the origin of these instabilities can be understood as follows . in the region of large fields , the threshold functions rapidly decrease to zero . in this limit , the flow of a function is given by the dimensional part , _ i.e. _ the flow obtained by setting @xmath469 [ _ e.g. _ @xmath470 . this flow no longer depends on the second derivative of the associated function with respect to the field . however , this diffusive - like term [ @xmath471 in our previous example ] is very important to stabilize the numerical integration . in practice , when necessary , we have therefore added by hand a small diffusive contribution to the flow equations and checked that the numerical results are stable under varying the strength of these extra terms . the physics associated with griffiths phases " is however unlikely to be captured by such an approach . in this case the rare regions that are involved are exponentially suppressed as their size increases and produce very weak ( essential ) singularities in the thermodynamics . such a behavior does not show up easily once the average over disorder has been performed , as done in the cumulants . droplets on the other hand are `` power - law rare '' regions in their size and have a clear signature in the functional dependence of the cumulants of the renormalized disorder ( see _ e.g. _ the detailed work by balents and ledoussal on the random manifold model.@xcite ) the same is true for avalanches " , which are collective discontinuous events occurring in _ typical _ samples at zero temperature for exceptional values of the applied source . we define `` supercusp '' , `` cusp '' , and `` subcusp '' in @xmath472 according to the order of the nonanalyticity in @xmath473 when @xmath402 : @xmath474 corresponds to a supercusp , @xmath475 to a `` cusp '' , and @xmath419 a noninteger @xmath476 to a subcusp . a `` linear cusp '' is associated with @xmath477 . this can be generalized to higher - order cumulants of the renormalized random field . note that there does not seem to be any alternative route , for instance by introducing new sources conjugate to operators that break susy explicitly.we have not found any useful additional wt identities when the invariance under superrotations is broken . to check this , we have followed zinn - justin s procedure,@xcite which amounts to performing transformations on the action and adding sources to the new generated independent operators . however , introducing new operators to compensate for the terms breaking susy breaks additional symmetries , including rotations in euclidean subspace , and an increasing number of independent operators are generated at each new iteration of the procedure . if one also wishes to describe microscopic details associated with a specific definition of the bare action and keep a finite uv cutoff @xmath40 , one has to generalize the function @xmath277 to @xmath478 , with @xmath479 when @xmath480 and @xmath481 ; a possible choice of @xmath482 is @xmath483 where @xmath484 is the heaviside step function . when @xmath485 for @xmath147 fixed , the infrared cutoff function @xmath486 diverges as @xmath487 whereas @xmath488 goes to @xmath489 , which fulfills the requirements described in paper iii.@xcite the constraint of no explicit susy breaking and the associated wt identities are not easily implemented in other approximation schemes . for instance one could envisage to replace the derivative expansion by a more powerful scheme to describe the full momentum dependence of the proper vertices , as proposed in ref . [ ] . however , it is not obvious then to combine this approximation with the expansion in cumulants . note that if the bare random field is not gaussian distributed , one can not bluntly neglect all higher - order cumulants . one should then check that , as expected , the non - gaussian behavior that is present at the bare level does not change the long - distance physics . to assess this point , it is necessary to include in the description at least the third cumulant and then consider the next order of the approximation scheme . when @xmath317 , one anticipates a subcusp in @xmath490 as @xmath491 , with no implication for the dimensional reduction . numerically however , it is very hard to find evidence for such subcusps as this would involve computing high orders in the derivatives with respect to the fields : for instance , one should be able to follow the fourth derivative of @xmath214 with respect to @xmath114 and check for its divergence at the fixed point above @xmath262 . one can also check that the `` magnetic - field eigenvalue '' @xmath492 corresponding to the relevant direction associated with odd eigenfunctions in the field arguments is analytically obtained in any dimension as @xmath493 , which as expected is the dimension of the fields ; the corresponding eigenfunctions @xmath494 , @xmath495 , and @xmath496 are expressed in terms of the fixed - point functions , namely : @xmath497 , @xmath498 , and @xmath499 .

the large magellanic cloud ( lmc ) has revealed a very complex structure both in the stellar and in the gaseous component . the elongation of the stellar disk in the direction of the galactic center , its substantial vertical thickness , the warp and the strong asymmetric bar are naturally predicted by numerical simulations as a result of the gravitational interaction between the lmc and the galaxy ( bekki & chiba 2005 , mastropietro et al . 2005 , hereafter m05 ) . the old stellar distribution appears to be quite smooth in the outer parts of the disk , with no signs of spiral structures out to a radius of 10 kpc @xcite . within the same radius the hi large scale structure reveals the presence of several asymmetric features that have no equivalent in the old stellar disk . the gaseous disk is characterized by the presence of an elongated region located at the south - east of the galaxy and aligned with the border of the optical disk , where the column density distribution shows a steep increase @xcite . since the lmc proper motion vector is directed to the east , it appears natural to associate this high density region with ram - pressure acting on the leading edge of the disk due to the orbital motion of the lmc and its consequent interaction with the diffuse hot gas in the halo of the milky way ( mw ) . the presence of an extended hot halo surrounding galaxies and in hydrostatic equilibrium within the dark matter potential is expected by current models of hierarchical structure formation . in the mw , x - ray absorption lines produced by hot ( @xmath1 k ) gas are detected in the spectra of several bright agn @xcite . some ionization features discovered in the magellanic stream and high velocity clouds indicate that this distribution of hot gas extends well beyond the galactic disk ( @xmath2 kpc ) . constraints from dynamical and thermal arguments fix its density in a range between @xmath3 and @xmath4 cm @xmath5 at the lmc distance from the galactic center ( but kaufmann et al . 2009 suggest a value ten times higher ) . @xcite have performed a detailed analysis of the lmc global star formation rate using asymptotic giant branch stars . they find an irregular and patchy distribution in age , with the youngest carbon - rich systems located at the south - east of the disk . the present star formation activity is rather clumpy and concentrated in stellar complexes characterized by intense hii emission and associated with bright h@xmath0 filamentary bubbles . most of these very young structures lie on the south - east of the disk , in the proximity of 30 doradus , the largest star forming region of the lmc , some are located in the bar and the remainder form an asymmetric pattern that covers the entire disk with no apparent relation to the global geometry of the satellite . it is not clear which is the overall physical mechanism responsible for triggering star formation with the observed asymmetric pattern and different models have been proposed in the past . the stochastic self - propagating star formation ( sspsf ) model predicts a clear age gradient in the lmc s stellar complexes , with the edges being younger with respect to the center , in contradiction with observations @xcite . @xcite proposed a scenario where the bow shock originated by the motion of the lmc through the hot galactic halo compresses the leading edge of the disk and induces star formation . the pressure at the south - eastern edge of the lmc is indeed 10 times higher than the average in the rest of the lmc @xcite . this model , which assumes the orbital motion vector lying in the plane of the disk , predicts increasing ages of the stellar complexes in the direction of the rotation , due to the fact that the material compressed at the front side of the disk moves , in time , away to the side . the youngest systems would indeed lie at the south - east border of the disk , where the relative velocity between the corotating interstellar medium and the external diffuse gas is maximum . several giant structures along the outer east and north edge of the lmc actually show a progression in age in a clockwise direction : moving from south - east to the north lmc 2 , 30 doradus and lmc 3 , lmc 4 , ngc1818 . in particular the difference in age between 30 doradus and lmc4 is exactly their distance along the border of the disk divided by the satellite s rotational velocity ( harris , private communication ) . studied the recent star formation history of the lmc using cepheids and other supergiant stars and found that although the majority of the star formation events in the last 30 myr are concentrated on the east border , others are distributed across the entire disk in partial contrast with the bow shock induced star formation model , that can not explain them . in this work we use high resolution sph simulations to study the effects of the interaction between the lmc interstellar medium and the diffuse hot halo of the mw . we want to investigate whether the ram - pressure acting on the leading edge of the lmc disk is responsible for the increase in density observed in the south - east and for triggering star formation . the analytic model of @xcite assumes a pure edge - on model , but according to @xcite the present angle between the lmc disk and the orbital motion is nearly @xmath6 . even in the absence of precession and nutation , this angle is subjected to large variations during the orbital period in such a way that compression produced by the external hot gas can affect in time both edge - on and face - on . moreover , the ram - pressure felt by the lmc is not constant and has a maximum when the satellite approaches the perigalacticon . the motion of the lmc through the hot halo of the mw during the last 1 gyr is modeled using `` test wind tunnel '' simulations with increasing ram - pressure values . the paper is structured as follows . section 2 describes the models and the star formation criteria adopted , section 3 illustrates the results of simulations without star formation , focussed on the investigation of pure effects of compression on the lmc interstellar medium while section 4 describes the runs where star formation is activated . several simulations have been performed , assuming different star formation models , disk inclinations and hot halo densities . the initial conditions of the simulations are constructed using the technique described by @xcite . our disk galaxy model is a multi - component system with a stellar and gaseous disk embedded in a spherical nfw @xcite dark matter halo . the density profile of the dark matter halo is adiabatically contracted in response to baryonic infall @xcite . the stellar disk follows an exponential surface density profile of the form : @xmath7 where @xmath8 and @xmath9 are the disk mass and radial scale length ( in cylindrical coordinates ) , respectively , while the thin vertical structure has a scale length @xmath10 : @xmath11 the gaseous disk is characterized by an exponential profile with the same radial and vertical scale length as the stellar component and by a constant density layer which extends up to 8@xmath9 . the structural parameters of the disk and the halo are chosen so that the resulting rotation curve resembles that of a typical bulgeless late - type ( sc / sd ) disk galaxy . they are similar to those adopted in m05 for the initial lmc model and reproduce quite well the peak of the rotation curve inferred by @xcite ( fig . as seen in m05 , the interaction with the mw does not affect significantly the stellar and dark matter mass in the inner @xmath12 kpc of the lmc and consequently the global rotation curve within this radial range . the choice of an extended gaseous component for the initial lmc model is motivated by the fact that spiral galaxies in the local universe are commonly observed to be embedded in extended disks of neutral hydrogen significantly larger than their stellar component . as seen in m05 , the combined effect of tidal interactions and ram - pressure stripping can remove a significant fraction of gas from a lmc disk orbiting within the hot halo of the mw , with a ram - pressure stripping radius which is a factor of three smaller than the initial radius of the gaseous disk . also in the case of a lmc with orbital velocities significantly higher @xcite hydrodynamic and gravitational forces together are effective in resizing and reshaping the extended gaseous disk of the satellite beyond 8 kpc . in the present work we neglect the presence of gravitational forces focusing on the effects of pure ram - pressure . therefore we do not expect to see a significant decrease in the radius of the gas distribution . however , in order to take in account the loss of cold gas from the disk of the satellite and the star formation events , we assumed an initial amount of gas in the disk which is about @xmath13 times larger than the hi mass in the lmc ( @xmath14 @xmath15 according to putman et al . 2003 ) . the mass within the virial radius is set equal to @xmath16 m@xmath17 and the fraction of mass in the disk is @xmath18 , equally distributed between the gaseous and stellar component . the contribution of the different components to the global rotation curve , assuming a disk scale length @xmath19 kpc and a dark halo concentration @xmath20 ( where @xmath21 is defined as @xmath22 , with @xmath23 and @xmath24 virial and scale radius of the nfw halo , respectively ) is plotted in fig . [ rotcurve ] . the halo spin parameter , which sets the disk scale length in our modeling , is @xmath25 , where @xmath26 relates the angular momentum @xmath27 and the total energy @xmath28 of a system with virial mass @xmath29 through the relation @xmath30 the initial stellar disk of the satellite galaxy has , within its scale radius @xmath9 , a central mass surface density of @xmath31 ( fig . [ diskprofile ] ) , that corresponds to a b - band surface brightness of @xmath32 , assuming a mass to light ratio @xmath33 . the central gas surface density is only @xmath34 since a significant fraction of gas is distributed in the external disk . assuming @xmath35 hi abundance this value corresponds to an hydrogen column density of @xmath36 within @xmath9 , comparable with the values observed by @xcite with the lmc parkes multibeam hi survey . in order to obtain a strongly stable disk against bar formation even in the presence of significant gas stripping and consequent perturbation of the satellite potential , the thickness of the stellar component is set such that the toomre s @xcite stability criterion is largely satisfied . in particular the toomre s parameter for the stellar disk : @xmath37 where @xmath38 is the radial velocity dispersion , @xmath39 is the local epicyclic frequency and @xmath40 the unperturbed stellar surface density , has a minimum at the disk scale length with @xmath41 . for a gaseous disk the stability of the disk is expressed in terms of the gas sound speed @xmath42 and surface density @xmath43 through the relation : @xmath44 the gaseous disk has initially a constant temperature of 10000 k , which implies @xmath45 and @xmath46 . according to and @xcite , the stability of a multicomponent disk is not guaranteed by the individual stability of its single constituents , due to the mutual gravitational interaction between gas and stars . stars are characterized by velocity dispersions 3 - 4 times larger than the typical sound velocities in the cold gaseous disk and even relatively small variations of the gaseous component parameters can significantly affect the stability of the whole disk . therefore we choose a large value of @xmath47 to contrast the effects of ram - pressure . in the case of a two components - gaseous and stellar - disk , the stability condition is expressed by @xmath48^{-1}>1 , \label{toomretotal}\ ] ] where @xmath49 and @xmath50 . [ qparameter ] illustrates the dependence of @xmath51 on the dimensionless wavenumber of the perturbation @xmath52 within three different regions of the disk : at the disk scale length @xmath9 , at @xmath53 kpc and in the external region ( @xmath54 kpc ) , where the gas component predominates . the criterium is always satisfied and the disk is stable against axisymmetric perturbations , independently of their wavelength . in order to check stability the disk was initially evolved in isolation for 1 gyr . = 8truecm = 8truecm = 8truecm all the simulations we now discuss were carried out using gasoline , a parallel tree - code with multi - stepping @xcite which is an extension of the pure n - body gravity code pkdgrav developed by @xcite . the code uses a spline kernel with compact support where the interaction distance for a particle @xmath55 is set equal to two times the smoothing length @xmath56 , defined as the @xmath57-th neighbour distance from the particle . in this paper @xmath58 . the internal energy of the gas is integrated using the antisymmetric formulation of @xcite that conserves entropy closely . dissipation in shocks is modeled using the quadratic term of the standard @xcite artificial viscosity . the balsara @xcite correction term is used to suppress the viscosity in non - shocking , shearing environments . the code includes radiative cooling for a primordial mixture of hydrogen and helium in collisional equilibrium . at temperatures below @xmath59 k the gas is entirely neutral and due to the lack of molecular cooling and metals , the efficiency of the cooling function drops rapidly to zero . we used a star formation recipe that includes density and temperature criteria , while converging flow criterium is not required in most of the simulations . gas particles are eligible to form stars only if the density of the star formation region has a minimum physical density corresponding to 0.1 hydrogen atoms per @xmath60 @xcite and an overdensity @xmath61 ( katz et al . 1996 ) , which basically restricts star formation to collapsed , virialized regions . the physical density threshold describes the steep drop in star formation rate observed in disk galaxies when the gas surface density is much lower than a critical value @xmath62 @xcite . the density threshold @xmath63 cm@xmath5 is compatible with observational results . according to @xcite , the star forming region has to be part of a converging flow that implies a local negative divergence of the sph velocity field . however , the converging flow criterium was introduced to describe star formation in cosmological simulations , where the geometry of the collapsing regions is approximatively spherical . in the case of star formation regions like 30 doradus , localized at the periphery of the lmc disk where gas particles relatively close in distance can have significantly different kinematics , this criterium leads to underestimate of the star formation rate . therefore the converging flow is not required in most of the simulations . a single run including the converging flow requirement has been performed for comparison . the star formation rate is assumed to be proportional to @xmath65 @xcite , where @xmath66 represents the volume density of the cold gas , and is given by the expression @xcite @xmath67 where the star formation timescale @xmath68 is the maximum between the local gas dynamical collapse time @xmath69 and the local cooling time . if the gas is already cool enough to form stars i.e @xmath70 , then @xmath71 is used . we assumed @xmath72k . the constant star formation rate parameter @xmath73 is chosen such that we reproduce the global lmc star formation rate @xcite . once a gas particle satisfies the above criteria , it spawns stars according to a probability distribution function . in particular , the probability @xmath74 that a gas particle forms stars in a time @xmath75 is modeled as @xmath76 a random number is then drawn to determine whether the gas particle forms stars during @xmath75 . for all the simulations in this paper @xmath77 myr . the newly created collisionless particle has the same position , velocity and softening length as the original gas particle while its mass is a fixed fraction @xmath78 of the parent gas particle , whose mass is reduced accordingly . following @xcite we assumed for our favorite models a star formation efficiency @xmath79 . up to six particles are then created for each gas particle in the disk . after its mass has decreased below @xmath80 of its initial value the gas particle is removed and its mass is re - allocated among the neighbouring particles . in order to study the influence of pure ram pressure on a galaxy model orbiting in a milky way halo , we performed `` wind tunnel '' simulations where the ram - pressure value varies with time . we represent the hot gas as a flux of particles moving along the major axis of an oblong of base equal to the diameter of the dark matter halo of the satellite and height @xmath81 , where @xmath82 is the velocity of the lmc at the perigalacticon and @xmath83 is the time scale of the simulation . the hot particles have an initial random distribution and a temperature @xmath84 . the box has periodic boundary conditions in order to restore the flow of hot gas that leaves the oblong . the galaxy model is at rest at the center of the oblong . kinematical data @xcite indicate that the lmc , presently located at @xmath85 kpc from the galactic center , is just past a perigalactic passage and has an orbital velocity of about 300 km s@xmath86 . recent proper motion measurements by @xcite and @xcite suggest that the velocity of the satellite is substantially higher ( almost 100 km s@xmath86 ) than previously estimated and consistent with the hypothesis of a first passage about the mw @xcite . in both scenarios the cloud is affected by the largest ram - pressure values during the last million years of its orbital evolution . indeed , while in the models proposed by @xcite the lmc does not enter the halo of the mw earlier than 1 gyr ago ( slightly different orbits are found in mastropietro 2008 ) , in m05 we have shown that the change in the orbital parameters due to dynamical friction strongly affects the ram - pressure stripping rate . even in the case of a `` low velocity '' model the largest ram - pressure on the satellite is expected during the last orbital semi - period ( about 1 gyr ) due to the increasing velocity and external gas density . we followed ram - pressure acting on the lmc s igm during the past 1 gyr . the density of the hot external gas increases with time , in such a way that in our low velocity model the external pressure experienced by the cold disk varies from @xmath87 to @xmath88 at the time corresponding to the pericentric passage . this is equivalent of assuming @xmath89 and @xmath90 , and a number density of the external gas that increases from @xmath91 to @xmath92 at @xmath85 kpc from the galactic center . these density values are comparable with those provided by m05 , who modeled the mw hot halo assuming a spherical distribution of gas that traces the dark matter profile , with a mean number density of @xmath93 within 150 kpc . we also consider the eventuality of a less dense galactic halo and performed runs where the gas density is a factor 10 lower . models with higher velocities @xcite and orbital parameters similar to those suggested by @xcite are also explored . in details , @xmath94 is the same as in the low velocity models since the higher orbital velocity at the beginning of the simulation ( about 250 km s@xmath86 ) is compensated by a lower external density ( @xmath95 , according to mastropietro 2008 . indeed 1 gyr ago the satellite has just passed through the virial radius of the mw ) . the maximum pressure felt by the disk is @xmath96 , that corresponds to a cloud moving with @xmath97 through an external hot medium of density @xmath92 . each galaxy model is simulated using 750000 particles , of which @xmath98 are in the dark halo and @xmath99 in the disk ( @xmath100 collisional and @xmath101 collisionless ) . the hot gas in the `` wind tunnel '' has @xmath102 particles , in such a way that the mass ratio @xmath103 between hot particles and particles in the disk is close to the unity even when the halo density is the largest . this choice permits to avoid the presence of scattering and numerical holes which artificially change the shape of the front edge and influence the morphology of the disk ( m05 ) . the gravitational spline softening is set equal to 0.5 kpc for the dark halo and the hot gas in the oblong , while it is 0.2 kpc for stars and gas in the disk . in order to study the effect of pure compression on the density distribution of cold gas in the lmc disk , we have run a first set of simulations where the gas cools radiatively but star formation was not activated . according to @xcite and @xcite , the present angle between the lmc s disk and its proper motion vector is roughly @xmath6 . even neglecting the effects of precession and nutation on the disk plane of the satellite @xcite this angle is expected to vary significantly during an orbital period , especially in the proximity of a pericentric passage due to rapid changes in the velocity vector . different relative orientations of the disk with respect to the orbital motion are therefore investigated . the inclination angle @xmath55 is defined as the angle between the angular momentum vector of the disk and the flux of hot particles in the wind tube , so that a galaxy moving edge - on through the external medium is characterized by @xmath104 , while the observed lmc disk would have @xmath105 . we explored cases with inclination angle @xmath55 of 90 , 45 and @xmath106 ( runs cool90 , cool45 and cool10 , respectively ) . with a hot halo temperature of @xmath107 k the relative velocity between the satellite and the external medium is supersonic ( sound speed @xmath108 km s@xmath86 and mach number @xmath109 and @xmath13 at the pericenter of the low and high velocity orbit , respectively ) and a bow shock forms in front of the disk ( fig . [ shockimage ] ) . since the cooling time of the post - shock gas is @xmath110 gyr , the shock can be considered adiabatic and hydrodynamical quantities at the two sides of the shock front are in first approximation related by the rankine - hugoniot equations for a stationary normal shock . = 8truecm for @xmath111 the jump conditions give @xmath112 and @xmath113 , where subscripts 1 and 2 denote upstream and downstream quantities . the ram - pressure @xmath114 actually felt by the galaxy behind the shock front is therefore smaller than that it would suffer due to the upstream flux of hot particles , but conservation of momentum flux across the shock discontinuity implies that the reduction in dynamical pressure has to be balanced by an increase in thermal pressure ( see also rasmussen et al . 2006 ) . fig . [ shock ] illustrates the behavior of hydrodynamical quantities across the shock discontinuity for a snapshot corresponding to the perigalacticon of a low velocity orbit . the disk inclination is @xmath115 . the horizontal axis is centered on the lmc stellar disk and oriented perpendicularly to the bow shock nose , with the shock located at @xmath116 kpc and the satellite moving towards increasing values of @xmath117 . the mach number derived by the temperature jump is @xmath118 , in good agreement with the theoretical value for a normal shock . only hot halo particles are considered in computations but , due to the sph nature of the simulations , close to the border of the disk we observe a further density increase and a sharp drop in temperature . the @xmath117-velocity profile is plotted in the system of reference where the pre - shock gas is at rest . = 8truecm pressure profiles across the shock are plotted in fig . [ shockpressure ] , where @xmath119 is defined as @xmath120 . the total pressure remains roughly constant until the edge of the disk . the steep increase at @xmath121 kpc is due the rapid growth in density at the border of the disk , which is not immediately followed by a decrease in temperature . for @xmath122 the shock wave is inclined with respect to the initial flow velocity and the rankine - hugoniot conditions apply to the normal components of the velocity across the shock discontinuity , while the component parallel to the shock front remain unchanged . the flow is therefore deflected toward an oblique shock wave and the jump at the shock discontinuity is smaller . an edge - on ( @xmath123 ) disk behaves like a wedge moving supersonically with the vertex facing upstream . if the wedge angle is smaller than or equal to the maximum flow deflection angle , the oblique shock becomes attached to the vertex of the wedge and the flow is deflected so that the streamlines are parallel to the surfaces of the wedge . the shock standoff distance thus depends on the external density profile of the collisional edge - on disk and on the mach number of the incident flow . in both low and high velocity edge - on models the shock results almost attached to the disk . figs . [ densitymaps][densitymaps10 ] illustrate for different values of the inclination angle @xmath55 changes in the disk gas density distribution as the satellite passes through increasing values of the external pressure , moving towards the perigalacticon . each couple of panels illustrates the state of the disk at increasing ( from the top to the bottom ) times along the orbit . panels on the left represent hi column density maps . the density contrast is chosen in order to highlight the density gradient in the external disk , since the gas distribution in the central regions of the lmc is dominated by the presence of the bar and a direct comparison with pure ram - pressure simulations is not possible . the color scale is logarithmic , with white corresponding to a density larger than @xmath124 @xmath125 and blue to values lower than @xmath126 @xmath125 . hot gas particles flow on to the disk from the left to the right of each plot , with increasing ram - pressure values from the first to the fourth image . the disk is seen face on and rotating clockwise . in the case of the edge - on run ( fig . [ densitymaps ] ) cold gas particles lying in the left - bottom quadrant of the disk feel the largest ram - pressure , due to the fact that their relative velocity with respect to the external medium is maximum . the rotational velocity of the external disk is @xmath127 km s@xmath86 , which implies a relative velocity at the pericenter of @xmath128 km s@xmath86 ( 450 km s@xmath86 in the case of a high velocity orbit ) . panels on the right represent the change in mean density and radius of the gaseous external disk as a function of the azimuthal angle @xmath129 . referring to the geometry of the hi density images on the left , @xmath130 corresponds to the bottom of the disk and increases clockwise in such a way that the disk moves in the direction of @xmath131 . the gas density and the mean radius are both calculated within sections of a three dimensional annulus with internal and external radius equal to 7 and 15 kpc , respectively . the initial azimuthal profiles ( not represented in the plots ) are flat since both these quantities have only radial dependence . as soon as the satellite starts moving through the surrounding medium , the external gas density develops a peak centered on @xmath132 : disk particles localized in regions of maximum ram - pressure get compressed and move on inner orbits , while their circular velocity increases consequently . after about a quarter of the orbital period the gas has reached its minimum radius and maximum local density . the gaseous disk becomes strongly asymmetric : compression at the front edge produces a density increase along the left border of the disk , evident in the hi maps even at early times . the high density region forms a thin ( @xmath133 kpc ) but continuous and well defined arc which has not an equivalent in the stellar distribution . at the perigalacticon this feature extends for almost @xmath134 with a density more than one order of magnitude higher than gas located at smaller radii . its average thickness ( @xmath135 1.5 kpc ) and velocity dispersion along the line of sight are larger than the average values in the rest of the disk . in the case of a satellite moving through the hot medium with an inclination angle different from @xmath136 the external pressure directed perpendicularly to the plane of the disk increases as cos@xmath55 while compression at the leading edge is much less pronounced . [ densitymaps45 ] and [ densitymaps10 ] refer to runs with inclination angles @xmath137 and @xmath115 . disks are shown face - on . the increment in density along the leading edge is smaller ( cool45 ) than in the edge - on model and almost absent for @xmath115 ( cool10 ) , while compression perpendicular to the plane of the disk produces local gravitational instabilities in the external gaseous disk ( also mayer , mastropietro & tran in preparation ) . this effect is more evident in the nearly face - on run cool10 where high density filaments delimitate regions where the local density is almost one order of magnitude lower . despite the absence of a peak in the azimuthal mean density profile , the integrated final density of cool10 is comparable to the other runs . [ dennosf ] represents the azimuthally averaged hi column density profile of the final disk configuration for the three simulations . as a result of the increase in density along the edge of the disk , the mean column density shows a secondary peak at large radius . a limb - brightened density profile has actually been observed by @xcite using the parkes multibeam hi survey of the lmc . the outer profiles of cool90 and cool45 are very similar , while in run cool10 the increment in density with respect to the original profile of the disk ( long - dashed red curve in the plot ) is located at larger radii ( @xmath138 kpc ) . indeed , due to compression of the leading edge the final gas distribution of cool90 and cool45 is asymmetric , the dense front edge being much closer to the center than the opposite border of the disk . therefore the hi peak in the external disk is located at relatively small radii , while the azimuthally averaged gas distribution is more extended than in the case of the run cool10 since ram pressure elongates the back side of the disk . again we stress the fact that our gaseous disk is more extended than it would be in a fully self consistent simulation including both gravitational and hydrodynamical forces . as seen in m05 and @xcite the combined effect of ram - pressure and tidal stripping is quite efficient in stripping gas from the outer satellite s disk , creating the tip of the magellanic stream already at large distances from the mw . the high density feature in cool90 would then form along the border of the disk at smaller radii and would not be easily subjected to further stripping due to the relatively ram - pressure values . [ cols="<,^,^,^,^,^,^,^",options="header " , ] [ sfruns ] the compressive increase in hi density is naturally associated with excess star formation . sph simulations can not follow the formation of molecular clouds but in first approximation the molecular gas fraction can be related to the density of atomic gas @xcite . the main parameters of star formation simulations are summarized in table [ sfruns ] . as we already mentioned in section 3 , our standard star formation model ( sf ) does not include the converging flow criterium and is characterized by an efficiency @xmath139 @xcite . the star formation rate parameter @xmath73 is initially set equal to @xmath140 . this model was adopted to run wind tube simulations with inclination angles @xmath141 ( sf90 , sf45 , sf10 ) ( table [ sfruns ] ) . we also investigated different star formation recipes requiring converging flows and assuming different values of @xmath73 and @xmath78 . we explored star formation rate parameter values in the range from 0.01 to 0.05 , that produce a star formation rate integrated over the entire disk comparable with the @xmath142 yr@xmath86 provided by @xcite . an efficiency @xmath143 implies that whenever a gas particle satisfies the star formation requirements , it is immediately turned into a single star particle of the same mass @xcite . in the last six simulations listed in table [ sfruns ] we used our standard star formation model sf to investigate the effects of different orbital parameters and gas halo densities runs sfv400 are characterized by a maximum ram pressure value corresponding to a perigalactic velocity of 400 km s@xmath86 @xcite . such high velocity disks when moving face - on through the external hot medium are strongly affected by local instabilities and star formation is consequently enhanced . the introduction of an artificial lower limit for the satellite gas temperature ( in runs sf10v400t12000 and sf10v400t15000 ) , higher than the cut - off in the cooling function , has the effect of reducing gravitational instabilities and fragmentation in the disk . this temperature threshold can be justified in order to crudely model the effect of the uv background and stellar feedback @xcite . finally , with simulations sf90ld and sf10ld in table [ sfruns ] we also consider the possibility of a galactic hot halo ten times less dense than our standard model . [ inclstarform ] illustrates the state of the newly formed stellar disk at increasing times along the satellite orbit . each couple of rows corresponds to one of the first three runs of table [ sfruns ] and is associated with a different inclination angle @xmath55 . the first row of the pair represents the face - on projection of the disk with the galaxy moving towards the left of the page and the same geometry as in fig . [ densitymaps ] . time increases from the left to the right . each small cross indicates a new star formation event at the time of the snapshot ( within a time interval of 40 myr ) while the circle delimitates the external disk ( @xmath144 kpc ) . stars form in the central regions as soon as the star formation algorithm is activated , but here the star formation activity of the inner disk is not represented . the second row represents the total mass @xmath145 of the newly formed stars in the external disk as a function of the disk azimuthal angle @xmath129 . in the case of a galaxy moving edge - on through the external medium ( sf90 ) , stars form at the leading edge of the disk when the ram - pressure becomes larger than @xmath146 dyn @xmath125 , at time @xmath147 gyr . the location of the star formation events initially corresponds to the hi column density peak observed in cool90 around @xmath148 ( fig . [ densitymaps ] ) . later on it expands along the entire front edge , creating a thin stellar arc well distinct from the star formation events that characterize the central disk . as soon as the satellite encounters ram pressure values comparable to those experienced by the lmc at the perigalacticon ( time @xmath149 gyr ) some episodes of star formation occur even on the back side of the disk ( last plot on the top right of fig . [ inclstarform ] ) , although they are not relevant in terms of new stellar mass formed . indeed @xmath145 shows a drastic drop at @xmath150 . runs with inclination @xmath151 are characterized by significant star formation only for values of the external pressure larger than @xmath152 dyn @xmath125 . in the case of the nearly face - on run sf10 , at @xmath153 gyr star formation occurs in the entire external disk . ram - pressure affects the plane of the disk almost perpendicularly and stars form along the delocalized and filamentary high density structures visible in fig . [ densitymaps10 ] ( mayer et al . in prep . ) . contrary to what has been found by @xcite who focused on higher ram - pressure values ( @xmath154 cm@xmath5 and @xmath155 km s@xmath86 ) typical of the outskirts of galaxy clusters , the newly formed stars are all located in the plane of the satellite s disk ( with the exception of the high velocity face - on run sf10v400 where about 10@xmath156 of the stars forms behind the disk ) . the star formation events appear to be distributed nearly homogeneously along the azimuthal profile of the external disk , although a small peak in @xmath145 is observable near @xmath123 . in fact , the orientation of the disk with respect to its orbital motion is not exactly face - on . the case of the intermediate run sf45 is more complex . in a first phase , for low ram pressure values , star formation is produced by compression at the leading edge and a thin star formation front although not so well defined as in the case of a pure edge - on model appears on the east side of the disk . as soon as the external pressure reaches a critical level compression directed perpendicularly to the disk becomes the dominant mechanism driving star formation . + converting sfrs to h@xmath0 luminosities according to @xcite : @xmath157 where sfr is the star formation rate averaged over the last 40 myr ( nearly two times the stellar age of 30 doradus ) , we obtain the h-@xmath0 maps illustrated in fig . [ lalpha ] . high emission regions are mainly concentrated in the external disk ( with the exception of run sfld90 where the ram - pressure exerted by the low density halo is not enough to induce star formation at the edge of the disk ) . the continuous stellar arc forming along the leading side of the disk in edge - on runs breaks up into several distinct and very luminous h-@xmath0 regions that more closely resemble the star - forming complexes observed on the eastern border of the lmc . the inclusion of stellar and supernovae heating which has been neglected in the present simulations could prevent further star - formation around highly emitting regions and consequently produce more compact and isolated star - forming complexes . nevertheless , modeling single star - formation complexes whose linear extension is smaller than our softening length is beyond the scope of this paper . the present inclination of the lmc s disk with respect to the orbital motion is about @xmath158 ( according to the convention adopted in this paper ) . since the satellite is currently near a perigalactic passage we expect the h-@xmath0 map at the leading border of the disk to be something in between pure edge - on runs and the run with inclination of @xmath159 . on the other side , it is very likely that the disk inclination during the phase of approach to the pericenter was different . indeed in @xcite we have simulated the lmc s orbit according to the new proper motion measurements of @xcite and found that the cloud enters the mw halo face - on and moves almost face - on during most of the last 1 gyr . it turns nearly edge - on only at the perigalacticon . this would have a remarkable effect on the star - formation history of the external disk during the last 1 gyr and some impact also on the h-@xmath0 maps . indeed , although the h-@xmath0 emission would be mostly concentrated on the eastern side of the disk due to the very recent edge - on motion , we expect to see some luminous clumps forming a patchy distribution on the entire disk , due to gravitational instabilities and subsequent star - formation induced by a nearly face - on compression of the gaseous disk before 30 myr ago . the high velocity edge - on run sf90v400 presents a more elongated and thinner stellar arc along the leading border , with a geometry similar to that obtained increasing the star formation rate parameter to 0.05 ( sf90c0.05 ) . the h-@xmath0 map of sf90v400 ( third panel of the third raw ) shows two distinct luminosity peaks . one is located at the south - east region of the disk , roughly corresponding to the position of 30doradus and the two compact emission regions n159 and n160 . = 9truecm fig . [ dmdt ] illustrates how the star formation rate of the external disk changes with time in the different models . the three black curves refer to the standard star formation runs sf , characterized by low orbital velocities . the edge - on disk starts forming stars earlier , but for large ram pressure values the star formation rate of sf10 and sf45 grows faster . at the perigalacticon passage sf45 has indeed a higher star formation rate than sf90 . in the case of an isolated lmc model star formation is almost absent for @xmath160 kpc . the remaining curves in fig . [ dmdt ] refer to different star formation recipes ( rows 4 - 9 of table [ sfruns ] ) . the location of the star formation events in the external regions of the disk does not change significantly choosing different parameters in the star formation algorithm . the star formation rate parameter @xmath73 determines the amount of new stars forming but does not affect the minimum threshold in ram - pressure neither the evolution of the star formation rate . in particular , in the case of edge - on runs after an initial steep increment the curve seems to converge to a constant value for increasing external pressures . the consequences of an increased star formation efficiency ( @xmath161 ) are almost negligible ( but not in h@xmath0 maps where only the very recent star formation rate is taken in account : compare the first panels of the first and second row ) while including the convergency requirement ( sfconv ) has nearly the same effect than reducing the star formation rate parameter of a factor two . = 9truecm for convenience , the star formation rate of the last six runs of table [ sfruns ] is plotted separately ( fig . [ dmdtalt ] ) . the high velocity edge - on model sf90v400 is characterized by a steeper increment in star formation at earlier times but later on the curve flattens and the star formation rate at the perigalacticon is similar to that of the low velocity case sf90 . differences in h-@xmath0 maps are produced by a difference of @xmath162 m@xmath163 myr@xmath86 about time = 1 gyr . on the contrary , the star formation generated by compression perpendicular to the disk increases with increasing ram - pressure values ( it shows a decrement only towards the end of the simulation ) and in the case of the high velocity run sf10v400 reaches a peak @xmath164 times higher than in sf10 . the star formation rate of sf10v400 is strongly affected by the introduction of an artificial temperature threshold . with a temperature floor of 15000 k we nearly suppress star formation in the external disk . however , a threshold of 12000 k is already very high for a lmc model and more typical of luminous disk galaxies like the mw . a temperature floor lower than 10000 k would not make sense since below this temperature the cooling function adopted in the present paper drops rapidly to zero . if , as pointed out by @xcite , the satellite is moving almost face - on until it gets very close to perigalacticon , we would not expect to see star formation before 0.6 gyr independently of the orbital velocity . differences between high and low velocity runs should be marginal also near the pericenter since star formation in edge - on runs assuming our standard prescriptions for star formation seems to saturate around @xmath165 m@xmath163 myr@xmath86 . finally , a hot halo ten times less dense than the one assumed in our favorite model would reduce drastically the star formation in a face - on lmc and suppress completely the star formation on the leading edge of the disk . = 9truecm the star formation rate of the entire disk is illustrated in fig . [ dmdttot ] . this plot is only indicative since we neglect the presence of the bar and its influence on the star formation history of the satellite . the star formation rate of a lmc model evolved in isolation is plotted for comparison . the largest contribution to sfr is given by star formation events in the central region of the disk . curves peak between 15 and 35 m@xmath163 myr@xmath86 , comparable with observations of the magellanic clouds photometric survey ( harris et al . in prep ) . the total star formation rate is clearly not affected by ram - pressure before 0.4 gyr . indeed , after an initial sharp increment curves are rather flat despite orientation and intensity of the external pressure . an initial burst in star formation is obtained only by increasing the star formation efficiency parameter to 0.05 . for time@xmath166 0.5 gyr the largest deviations from an average star formation rate of @xmath167 m@xmath163 myr@xmath86 are produced in the edge - on low density run sf90ld , where dm / dt drops to 10 m@xmath163 myr@xmath86 at time = 1 gyr , and in the high velocity face - on model sf10v400 whose star formation rate increases up to @xmath168 m@xmath163 myr@xmath86 toward the end of the simulation . differences among the other models are of the order of few m@xmath163 myr@xmath86 and vary with time so that it would be quite difficult to use the star formation history of the entire disk to test the cloud s orbital parameters and the hot halo density . [ agephi ] represents the mean stellar age of the external disk versus the azimuthal angle @xmath129 for the same selected runs of fig . [ dmdttot ] . the maximum increment in age in a clockwise direction is associated with edge - on runs , where stars forming at the leading edge move , in time , away to the side , due to the clockwise rotation of the disk . the youngest stars are located at @xmath169 . clearly the gradient in age is much weaker in models with @xmath151 . sf10v400 with a mean stellar age of @xmath170 myr forms stars earlier with respect to the other nearly face - on runs , while the difference between sf90v400 and the corresponding low velocity run sf90 is about 5 myr at the leading edge and not significant in the rest of the external disk . for the same runs we also plotted the final radial gas density profile ( fig . [ densf ] ) . in most of the cases a secondary peak is still present . = 8truecm = 8truecm we have performed high resolution `` wind tunnel '' simulations to study the effects of ram - pressure by a tenuous galactic hot halo on the hi morphology of the lmc s disk , its recent star formation history , and location of the youngest star forming regions . we did not focus on the mass loss produced by ram - pressure stripping since this would also be affected by tidal interactions . for the same reason our galaxies do not form any bar and we actually start with a galaxy model very stable against bar formation so that the pure effects of external pressure are more clearly visible . our lmc is a multi - component system with a spherical nfw halo , an exponential stellar disk and a gaseous disk that extends up to 8 times the stellar disk scale length . in each simulation the external flux of hot particles increases with time as the satellite approaches the perigalacticon in such a way that the pressure experienced by the disk is consistent with the lmc s orbital velocity and an average hot halo density of @xmath93 cm@xmath5 within 150 kpc from the galactic center . low velocity runs are characterized by a `` classic '' pericentric velocity of 300 km s@xmath86 @xcite while high velocity runs have velocities compatible with the new proper motion measurements of @xcite . we expect the angle between the lmc s disk and its proper motion to vary significantly during the last billion years of orbital interaction . we have defined the inclination angle @xmath55 as the angle between the angular momentum vector of the disk and the flux of hot particles in the wind tube , so that the observed lmc s disk would have @xmath171 . @xcite performed self consistent nbody / sph simulations of the interacting system mw / lmc adopting orbital constraints from the last lmc s proper motion measurements and found that the cloud enters the mw halo face - on ( @xmath172 ) , moving nearly face - on for most of the last billion years . it turns edge - on only about 30 myr ago . this means that the lmc is moving nearly edge - on close to the perigalactic passage , corresponding to the maximum ram - pressure values , consistently with the actual disk inclination measured by @xcite . we have performed several simulations varying the inclination angle of the disk , the star formation recipe and the intensity of the external pressure . we have shown that : * the compression of the leading border of an edge - on lmc disk can account for the high density hi region observed at the south east . in our simulations this high density feature is well defined ( with a mean density one order of magnitude higher than the surrounding gas ) and localized within 1.5 kpc from the border of the disk . its average thickness and velocity dispersion along the line of sight are larger than the average values in the rest of the disk . in cool90 it extends for almost @xmath134 and could also explain the origin of the spiral arm e described by @xcite , which does not have an equivalent in the stellar disk . * compression directed perpendicularly to the disk ( in runs with @xmath151 ) produces local instabilities in the gas distribution and a clumpy structure characterized by voids and high density filaments similar to those observed by the parkes multibeam hi survey ( see fig . 3 of staveley - smith et al . if the satellite was moving nearly face - on in the past and according to @xcite this is likely to happened during most of the lmc / mw orbital history ram - pressure could be responsible for the general mottled appearance of the hi disk . * as a result of the increase in density along the edge of the disk the mean hi column density shows a secondary peak at large radius , in agreement with observations . * the compression of the satellite s igm is naturally associated with induced star formation activity . we focussed on the external regions of the disk since the central parts of the real lmc would be dominated by the bar . edge - on disks start forming stars earlier , but for large ram - pressure values the star formation rate of runs with @xmath151 grows much faster . the high velocity edge - on model sf90v400 is characterized by a steeper increment in star formation at earlier times but later on the curve flattens and the star formation rate at the perigalacticon is similar to that of the low velocity case sf90 . on the other hand , the star formation generated by a compression perpendicular to the lmc s disk increases with increasing ram - pressure values and in the case of the high velocity run sf10v400 reaches a peak @xmath164 times higher than in sf10 . if the satellite is moving almost face - on until it gets very close to the perigalacticon we would expect not to see star formation before 0.6 gyr independently of the orbital velocity . differences between high and low velocity runs should be marginal also near the pericenter since star formation in edge - on runs assuming our standard prescriptions seems to saturate around @xmath165 m@xmath163 myr@xmath86 . * in edge - on models the star formation of the external disk is characterized by a thin stellar arc along the leading border , well distinct from the star formation events in the central disk . if sfr is converted in h@xmath0 luminosities , this arc breaks in several distinct and very luminous h@xmath0 regions that more closely resemble the star forming complexes observed on the eastern border of the disk . although the h@xmath0 emission is mostly concentrated on the eastern side as a consequence of the very recent edge - on motion , we expect to see some luminous clumps forming a patchy distribution on the entire disk , due to gravitational instabilities and subsequent star formation induced by a nearly face - on compression of the disk before 30 myr ago . as observed by @xcite stellar complexes on the leading edge show a progression in age in the clockwise direction , but a face - on compression in the recent past of the lmc would circumscribe this trend to the youngest stellar regions , with age @xmath173 30 - 40 myr . we would like to thank m - r . cioni , t. kaufmann and n. kallivayalil for useful discussions . the numerical simulations were performed on the zbox1 supercomputer at the university of zurich and on the local sgi - altix 3700 bx2 ( partly funded by the cluster of excellence `` origin and structure of the universe '' ) . this work was partly supported by the dfg sonderforschungsbereich 375 `` astro - teilchenphysik '' .

code coverage is a common metric in software and hardware testing that measures the degree to which an implementation has been tested with respect to some criterion . in its simplest form , one starts with a model of the program , and a partition of the behaviors of the model into _ coverage goals _ test _ is a sequence of inputs that determines a behavior of the program . the aim of testing is to explore as many coverage goals as possible , ideally as quickly as possible . in this paper , we give complexity results for several coverage problems . the problems are very basic in nature : they consist in deciding whether a certain level of coverage can be attained in a given system . it is thus somewhat surprising that the problems have not been considered previously in the literature . finite - state directed graphs have been used as program models for test generation of reactive systems for a long time ( see @xcite for surveys ) . a coverage goal is a partition of the states of the graph , and a test is a sequence of labels that determine a path in the graph . the maximal coverage test generation problem is to hit as many partitions as possible using a minimum number of tests . in the special case the partitions coincide with the states , the maximal coverage problem reduces to the chinese postman problem for which there are efficient ( polynomial time ) algorithms @xcite . in this paper , we show that the maximal coverage problem becomes np - complete for graphs with general partitions . we also distinguish between _ system complexity _ ( the complexity of the problem in terms of the size of the graph ) and the _ coverage complexity _ ( the complexity of the problem in terms of the number of coverage goals ) . then , the problem is nlogspace in the size of the graph ( but that algorithm uses space polynomial in the number of propositions ) . we consider the special case where the graph has a special `` reset '' action that takes it back to the initial state . this corresponds in a testing setting to the case where the system can be re - initialized before running a test . in this case , the maximal coverage problem remains polynomial , even with general partitions . directed graphs form a convenient representation for deterministic systems , in which all the choices are under the control of the tester . testing of non - deterministic systems in which certain actions are controllable ( under the control of the tester ) and other actions are uncontrollable lead to _ game graphs _ @xcite . a game graph is a directed labeled graph where the nodes are partitioned into tester - nodes and system - nodes , and while the tester can choose the next input at a tester node , the system non - deterministically chooses the next state at a system node . then , the test generation problem is to generate a test set that achieves maximal coverage no matter how the system moves . for general game graphs , we show the complexity of the maximal coverage problem is pspace - complete . however , there is an algorithm that runs in time linear in the size of the game graph but exponential in the number of coverage goals . again , the re - initializability assumption reduces the complexity of coverage : in case there is a re - initialization strategy of the tester from any system state , the maximal coverage problem for games is co - np - complete . dually , we show that the problem of whether it is possible to win a safety game while visiting fewer than a specified number of partitions is np - complete . finally , we consider the coverage problem in bounded time , consisting in checking whether a specified number of partitions can be visited in a pre - established number of steps . we show that the problem is np - complete for graphs , and is pspace - complete for game graphs . optimization problems arising out of test generation have been studied before in the context of both graphs and games @xcite . however , to the best of our knowledge , the complexities of the coverage problems studied here have escaped attention so far . while we develop our theory for the finite - state , discrete case , we can derive similar results for more general models , such as those incorporating incomplete information ( the tester can only observe part of the system state ) or timing . for timed systems modeled as timed automata , the maximal coverage problem is pspace - complete . for timed games as well as for ( finite state ) game graphs with incomplete information , the maximal coverage problem becomes exptime - complete . in this section we define _ labeled graphs _ and _ labeled games _ , and then define the two decision problems of coverage , namely , _ maximal coverage _ problem and _ coverage with bounded time _ problem . we start with definition of graphs and games . a _ labeled graph _ @xmath0 consists of the following component : 1 . a finite directed graph with vertex set @xmath1 and edge set @xmath2 ; 2 . the initial vertex @xmath3 a finite set of atomic propositions @xmath4 ; 4 . a labeling function @xmath5 that assigns to each vertex @xmath6 the set @xmath7 of atomic propositions true at @xmath8 . for technical convenience we will assume that for all vertices @xmath9 , there exists @xmath10 such that @xmath11 , i.e. , each vertex has at least one out - going edge . * paths in graphs and reachability . * given a labeled graph @xmath12 , a _ path _ @xmath13 in @xmath12 is a infinite sequence of vertices @xmath14 starting from the initial vertex @xmath3 ( i.e. , @xmath15 ) such that for all @xmath16 we have @xmath17 . a vertex @xmath18 is reachable from @xmath3 if there is a path @xmath19 in @xmath12 and @xmath20 such that the vertex @xmath21 in @xmath13 is the vertex @xmath18 . a _ labeled game graph _ @xmath22 consists of the components of a labeled graph along with a partition of the finite vertex set @xmath1 into @xmath23 . the vertices in @xmath24 are player 1 vertices where player 1 chooses outgoing edges , and analogously , the vertices in @xmath25 are player 2 vertices where player 2 chooses outgoing edges . again for technical convenience we will assume that for all vertices @xmath9 , there exists @xmath10 such that @xmath11 , i.e. , each vertex has at least one out - going edge . * plays and strategies in games . * a _ play _ in a game graph is a path in the underlying graph of the game . a strategy for a player in a game is a recipe to specify how to extend the prefix of a play . formally , a strategy @xmath26 for player 1 is a function @xmath27 that takes a finite sequence of vertices @xmath28 ending in a player 1 vertex @xmath6 , where @xmath29 and @xmath30 , representing the history of the play so far , and specifies the next vertex @xmath31 choosing an out - going edge ( i.e. , @xmath32 . a strategy @xmath33 is defined analogously . we denote by @xmath34 and @xmath35 the set of all strategies for player 1 and player 2 , respectively . given strategies @xmath36 and @xmath37 for player 1 and player 2 , there is a unique play ( or a path ) @xmath38 such that ( a ) @xmath15 ; ( b ) for all @xmath39 , if @xmath40 , then @xmath41 ; and if @xmath42 , then @xmath43 . * controllably recurrent graphs and games . * along with general labeled graphs and games , we will also consider graphs and games that are _ controllably recurrent_. a labeled graph @xmath12 is _ controllably recurrent _ if for every vertex @xmath18 that is reachable from @xmath3 , there is a path starting from @xmath18 that reaches @xmath3 . a labeled game graph @xmath12 is _ controllably recurrent _ if for every vertex @xmath18 that is reachable from @xmath3 in the underlying graph , there is a strategy @xmath26 for player 1 such that against all player 2 strategies @xmath37 , the path starting from @xmath18 given the strategies @xmath26 and @xmath37 reaches @xmath3 . controllable recurrence models the natural requirement that systems under test are _ re - initializable _ , that is , from any reachable state of the system , there is always a way to bring the system back to its initial state no matter how the system behaves . * the maximal coverage problem . * the _ maximal coverage problem _ asks whether at least @xmath44 different propositions can be visited . we now define the problem formally for graphs and games . given a path @xmath45 , let @xmath46 be the set of propositions that appear in @xmath13 . given a labeled graph @xmath12 and @xmath47 , the maximal coverage problem asks whether there is path @xmath13 such that @xmath48 . given a labeled game graph @xmath12 and @xmath47 , the maximal coverage problem asks whether player 1 can ensure that at least @xmath44 propositions are visited , i.e. , whether @xmath49 such that for all player 2 strategies @xmath50 we have @xmath51 . the maximal _ state coverage _ problem is the special case of the maximal coverage problem where @xmath52 and for each @xmath53 we have @xmath54 . that is , each state has its own label , and there are @xmath55 singleton partitions . * the coverage with bounded time problem . * the _ coverage with bounded time problem _ asks whether at least @xmath44 different propositions can be visited within @xmath56-steps . we now define the problem formally for graphs and games . given a path @xmath57 and @xmath58 , we denote by @xmath59 the prefix of the path of length @xmath60 , i.e. , @xmath61 . given a path @xmath57 and @xmath58 , we denote by @xmath62 . given a labeled graph @xmath12 and @xmath47 and @xmath63 , the coverage with bounded time problem asks whether there is path @xmath13 such that @xmath64 . given a labeled game graph @xmath12 and @xmath47 , the maximal coverage problem asks whether player 1 can ensure that at least @xmath44 propositions are visited within @xmath56-steps , i.e. , whether @xmath49 such that for all player 2 strategies @xmath50 we have @xmath65 . * system - tester game . * a _ system _ @xmath66 consists of the following components : * a finite set @xmath67 of states with the starting state @xmath68 . * a finite alphabet @xmath69 of input letters . * a transition relation @xmath70 . * a finite set of atomic propositions @xmath4 and a labeling function @xmath5 that assigns to each state @xmath71 the set of atomic propositions true at @xmath71 . we consider _ total _ systems such that for all @xmath72 and @xmath73 , there exists @xmath74 such that @xmath75 . a system is _ deterministic _ if for all @xmath72 and @xmath76 , there exists exactly one @xmath77 such that @xmath75 . the tester selects an input letter at every stage and the system resolves the non - determinism in transition to choose the successor state . the goal of the tester is to visit as many different propositions as possible . the interaction between the system and the tester can be reduced to a labeled game graph @xmath78 as follows : * _ vertices and partition . _ @xmath79 ; @xmath80 and @xmath81 ; and @xmath82 . * _ edges . _ @xmath83 . * _ labeling . _ @xmath84 and @xmath85 . the coverage question for game between tester and system can be answered by answering the question in the game graph . also observe that if the system is deterministic , then for all player 2 vertices in the game graph , there is exactly one out - going edge , and hence the game can be reduced to a labeled graph . in this paper we will present all the results for the labeled graph and game model . all the upper bounds we provide follow also for the game between tester and system . all the lower bounds we present can also be easily adapted to the model of the game between system and tester . in this section we study the complexity of the maximal coverage problem . in subsection [ subsec : maxexpl : graphs ] we study the complexity for graphs , and in subsection [ subsec : maxexpl : games ] we study the complexity for game graphs . we first show that the maximal coverage problem for labeled graphs is np - complete . [ thrm : maxexpl - graphs ] the maximal coverage problem for labeled graphs is np - complete . the proof consists of two parts . we present them below . 1 . _ in np . _ the maximal coverage problem is in np can be proved as follows . given a labeled game graph @xmath12 , let @xmath86 . we show first that if there is a path @xmath13 in @xmath12 such that @xmath48 , then there is a path @xmath87 in @xmath12 such that @xmath88 . if @xmath13 visits at least @xmath44 propositions , and there is a cycle in @xmath13 that does not visit a new proposition that is already visited in the prefix , then the cycle segment can be removed from @xmath13 and still the resulting path visits @xmath44 propositions . hence if the answer to the maximal coverage problem is `` yes '' , then there is a path @xmath87 of length at most @xmath89 that is a witness to the `` yes '' answer . since @xmath90 , it follows that the problem is in np . _ np - hardness . _ now we show that the maximal coverage problem is np - hard , and we present a reduction from the sat - problem . consider a sat formula @xmath91 , and let @xmath92 be the set of variables and @xmath93 be the set of clauses . for a variable @xmath94 , let 1 . @xmath95 be the set of indices of the set of clauses @xmath96 that is satisfied if @xmath97 is set to be true ; and 2 . @xmath98 be the set of indices of the set of clauses @xmath96 that is satisfied if @xmath97 is set to be false . + without loss of generality , we assume that @xmath99 and @xmath100 are non - empty for all @xmath101 ( this is because , for example , if @xmath102 , then we can set @xmath97 to be true and reduce the problem where the variable @xmath97 is not present ) . for a finite set @xmath103 of natural numbers , let @xmath104 and @xmath105 denote the maximum and minimum number of @xmath106 , respectively , for an element @xmath107 that is not the maximal element let @xmath108 denote the next highest element to @xmath109 that belongs to @xmath106 ; i.e. , ( a ) @xmath110 ; ( b ) @xmath111 ; and ( c ) if @xmath112 and @xmath113 , then @xmath114 . we construct a labeled game graph @xmath115 as follows . we first present an intuitive description : there are states labeled @xmath116 , and all of them are labeled by a single proposition . the state @xmath117 is an absorbing state ( state with a self - loop only ) , and all other @xmath118 state has two successors . the starting is @xmath119 . in every state @xmath118 given the right choice we visit in a line a set of states that are labeled by clauses that are true if @xmath118 is true ; and given the left choice we visit in a line a set of states that are labeled by clauses that are true if @xmath118 is false ; and then we move to state @xmath120 . we now formally describe every component of the labeled graph @xmath121 . the set of vertices is @xmath122 there is a vertex for every variable , and a vertex @xmath117 . there is a vertex @xmath123 iff @xmath124 , and there is a vertex @xmath125 iff @xmath126 , 2 . the set of edges is @xmath127 we now explain the role if each set of edges . the first edge is the self - loop at @xmath117 . the second set of edges specifies that from @xmath128 the next vertex is @xmath129 and similarly , from @xmath130 the next vertex is again @xmath129 . the third set of edges specifies that from @xmath97 there are two successors that are @xmath123 and @xmath131 where @xmath132 and @xmath133 . the final sets of edges specifies ( a ) to move in a line from @xmath134 to visit the clauses that are satisfied by setting @xmath97 as true , and ( b ) to move in a line from @xmath135 to visit the clauses that are satisfied by setting @xmath97 as false . fig [ fig : np - hardness ] gives a pictorial view of the reduction . 3 . the initial vertex is @xmath136 . @xmath137 , i.e. , there is a proposition @xmath138 for each clause @xmath138 and there is a proposition @xmath139 for all variables ; 5 . @xmath140 ; i.e. , every variable state is labeled by the proposition @xmath139 ; and we have @xmath141 and @xmath142 , i.e. , each state @xmath123 and @xmath125 is labeled by the corresponding clause that is indexes . + the number of states in @xmath115 is @xmath143 , and the reduction is polynomial in @xmath91 . in this graph the maximal number of propositions visited is exactly equal to the maximal number of satisfiable clauses plus 1 ( since along with the propositions for clauses the proposition @xmath139 for all variables is always visited ) . the proof of the above claim is as follows . given a path @xmath13 in @xmath144 we construct an assignment @xmath145 for the variables as follows : if the choice at a vertex @xmath97 is @xmath134 , then we set @xmath97 as true in @xmath145 , else we set @xmath97 as false . hence if a path in @xmath115 visits a set @xmath146 of @xmath147 propositions , then the assignment @xmath145 satisfies @xmath148 clauses ( namely , @xmath149 ) conversely , given an assignment @xmath145 of the variables , we construct a path @xmath150 in @xmath115 as follows : if @xmath97 is true in the assignment @xmath145 , then the path @xmath150 chooses @xmath134 at @xmath97 , otherwise , it chooses @xmath151 at @xmath97 . if @xmath145 satisfies a set @xmath67 of @xmath148 clauses , then @xmath150 visits @xmath152 propositions ( namely , the set @xmath153 of propositions ) . hence @xmath91 is satisfiable iff the answer to the maximal coverage problem with input @xmath115 and @xmath154 is true . the desired result follows . * hardness of approximation . * we note that from the proof theorem [ thrm : maxexpl - graphs ] it follows that the max - sat problem ( i.e. , computing the maximal number of clauses satisfiable for a sat formula ) can be reduced to the problem of computing the exact number for the maximal coverage problem . from hardness of approximation of the max - sat problem @xcite , it follows that the maximal coverage problem for labeled graphs is hard to approximate . the maximal coverage problem for labeled graphs that are controllably recurrent can be decided in ptime . to solve the maximal coverage problem for labeled graphs that are controllably recurrent , we compute the maximal strongly connected component @xmath155 that @xmath3 belongs to . since the graph is controllably recurrent , all states that are reachable from @xmath3 belong to @xmath155 . hence the answer to the maximal coverage problem is `` yes '' iff @xmath156 . the result follows . [ thrm : maxexpl - games ] the maximal coverage problem for labeled game graphs is pspace - complete . the proof consists of two parts . we present them below . 1 . _ in pspace . _ we argue that the maximal coverage problem for labeled game graph can be reduced to the coverage in bounded time problem . the reason is as follows : in a labeled game graph with @xmath157 vertices , if player 1 can visit @xmath44 propositions , then player 1 can visit @xmath44 propositions within at most @xmath158 steps ; because player 1 can always play a strategy from the current position that visits a new proposition that is not visited and never needs to go through a cycle without visiting a new proposition unless the maximal coverage is achieved . hence it follows that the maximal coverage problems for games reduces to the coverage in bounded time problem . the pspace inclusion will follow from the result of theorem [ thrm : pspace - bouexpl ] where we show that the coverage in bounded time problem is in pspace . 2 . _ pspace - hardness . _ the maximal coverage problem for game graphs is pspace - complete , even if the underlying graph is strongly connected . the proof is a reduction from qbf ( truth of quantified boolean formulas ) that is known to be pspace - complete @xcite , and it is a modification of the reduction of theorem [ thrm : maxexpl - graphs ] . consider a qbf formula @xmath159 defined on the set @xmath92 of variables , and @xmath93 are the clauses of the formula . we apply the reduction of theorem [ thrm : maxexpl - graphs ] with the following modification to obtain the labeled game graph @xmath115 : the partition @xmath160 of @xmath161 is as follows . for a variable @xmath97 if the quantifier before @xmath97 is existential , then @xmath162 ( i.e. , for existentially quantified variable , player 1 chooses the out - going edges denoting whether to set the variable true or false ) ; and for a variable @xmath97 if the quantifier before @xmath97 is universal , then @xmath163 ( i.e. , for universally quantified variable , the opposing player 2 chooses the out - going edges denoting whether to set the variable true or false ) . the state @xmath117 is a player 2 vertex , and all other vertex has an single out - going edges and can be player 1 state . given this game graph we have @xmath91 is true iff player 1 can ensure that all the propositions can be visited in @xmath164 . formally , let @xmath165 and @xmath166 denote the set of all strategies for player 1 and player 2 , respectively , in @xmath115 . then @xmath91 is true iff latexmath:[$\sup_{{\pi}_1 \in { \pi}_1^\phi } \inf_{{\pi}_2 \in { \pi}_2^\phi } observe that since @xmath117 is a player 2 state if we add an edge from @xmath117 to @xmath119 , player 2 will never choose the edge @xmath117 to @xmath119 ( since the objective for player 2 is to minimize the coverage ) . however , adding the edge from @xmath117 to @xmath119 makes the underlying graph strongly connected ( i.e. , the underlying graph of the game graph becomes controllably recurrent ; but player 1 does not have a strategy to ensure that @xmath119 is reached , so the game is not controllably recurrent ) . the desired result follows . * complexity of maximal coverage in controllably recurrent games . * we will now consider maximal coverage in controllably recurrent games . our analysis will use fixing memoryless _ randomized _ strategy for player 1 , and fixing a memoryless randomized strategy in labeled game graph we get a labeled markov decision process ( mdp ) . a labeled mdp consists of the same components as a labeled game graph , and for vertices in @xmath24 ( which are randomized vertices in the mdp ) the successors are chosen uniformly at random ( i.e. , player 1 does not have a proper choice of the successor but chooses all of them uniformly at random ) . given a labeled game graph @xmath168 we denote by @xmath169 the mdp interpretation of @xmath12 where player 1 vertices chooses all successors uniformly at random . an _ end component _ in @xmath170 is a set @xmath171 of vertices such that ( i ) @xmath171 is strongly connected and ( ii ) @xmath171 is player 1 _ closed _ , i.e. , for all @xmath172 , for all @xmath173 such that @xmath174 we have @xmath175 ( in other words , for all player 1 vertices , all the out - going edges are contained in @xmath171 ) . [ lemm - contr - mdp ] let @xmath12 be a labeled game graph and let @xmath169 be the mdp interpretation of @xmath12 . then the following assertions hold . 1 . let @xmath171 be an end - component in @xmath169 with @xmath176 . then @xmath177 . there exists an end - component @xmath178 with @xmath176 such that @xmath179 . we prove both the claims below . 1 . if @xmath171 is an end - component in @xmath170 , then consider a memoryless strategy @xmath50 for player 2 , that for all vertices @xmath180 , chooses a successor @xmath175 ( such a successor exists since @xmath171 is strongly connected ) . since @xmath171 is player 1 closed ( i.e. , for all player 1 out - going edges from @xmath171 , the end - point is in @xmath171 ) , it follows that for all strategies of player 1 , given the strategy @xmath50 for player 2 , the vertices visited in a play is contained in @xmath171 . the desired result follows . an optimal strategy @xmath181 for player 1 in @xmath12 is as follows : 1 . let @xmath182 and @xmath183 ; 2 . at iteration @xmath184 , let @xmath185 represents the set of propositions already visited . at iteration @xmath184 , player 1 plays a strategy to reach a state in @xmath186 ( if such a strategy exists ) , and then reaches back @xmath3 ( a strategy to reach back @xmath3 always exists since the game is controllably recurrent ) . 3 . if a new proposition @xmath187 is visited at iteration @xmath184 , then let @xmath188 . goto step ( b ) for @xmath189 iteration with @xmath190 . if no state in @xmath186 can be reached , then stop . + the strategy @xmath181 is optimal , and let the above iteration stop with @xmath191 . let @xmath192 , and let @xmath193 be the set of vertices such that player 1 can reach @xmath139 . let @xmath194 . then @xmath195 and player 2 can ensure that from @xmath3 the game can be confined to @xmath196 . hence the following conditions must hold : ( a ) for all @xmath197 , there exists @xmath198 such that @xmath174 ; and ( b ) for all @xmath199 , for all @xmath200 such that @xmath201 we have @xmath202 . consider the sub - graph @xmath203 where player 2 restricts itself to edges only in @xmath196 . a bottom maximal strongly connected component @xmath204 in the sub - graph is an end - component in @xmath169 , and we have @xmath205 it follows that @xmath171 is a witness end - component to prove the result . the desired result follows . [ thrm : maxexpl - contrgames ] the maximal coverage problem for labeled game graphs that are controallably recurrent is conp - complete . we prove the following two claims to establish the result . 1 . _ in conp . _ the fact that the problem is in conp can be proved using lemma [ lemm - contr - mdp ] . given a labeled game graph @xmath12 , if the answer to the maximal coverage problem ( i.e. , whether @xmath206 ) is no , then by lemma [ lemm - contr - mdp ] , there exists an end - component @xmath171 in @xmath169 such that @xmath207 . the witness end - component @xmath171 is a polynomial witness and it can be guessed and verified in polynomial time . the verification that @xmath171 is the correct witness is as follows : we check ( a ) @xmath171 is strongly connected ; ( b ) for all @xmath172 and for all @xmath200 such that @xmath174 we have @xmath208 ; and ( c ) @xmath209 . hence the result follows . 2 . _ conp hardness . _ we prove hardness using a reduction from the complement of the _ vertex cover _ problem . given a graph @xmath210 , a set @xmath211 is a _ vertex cover _ if for all edges @xmath212 we have either @xmath213 or @xmath214 . given a graph @xmath215 whether there is a vertex cover @xmath171 of size at most @xmath44 ( i.e. , @xmath216 ) is np - complete @xcite . we now present a reduction of the complement of the vertex cover problem to the maximal coverage problem in controallably recurrent games . given a graph @xmath210 we construct a labeled game graph @xmath217 as follows . let the set @xmath2 of edges be enumerated as @xmath218 , i.e. , there are @xmath219 edges . the labeled game graph @xmath220 is as follows . 1 . _ vertex set and partition . _ the vertex set @xmath221 is as follows : @xmath222 all states in @xmath2 are player 2 states , and the other states are player 1 states , i.e. , @xmath223 , and @xmath224 . the set @xmath225 of edges are as follows : @xmath226 intuitively , the edges in the game graph are as follows : from the initial vertex @xmath3 , player 1 can choose any of the edges @xmath227 . for a vertex @xmath228 in @xmath221 , player 2 can choose between two vertices @xmath229 and @xmath230 ( which will eventually represent the two end - points of the edge @xmath228 ) . from vertices of the form @xmath229 and @xmath230 , for @xmath231 , the next vertex is the initial vertex @xmath3 . it follows that from all vertex the game always comes back to @xmath3 and hence we have controllably recurrent game . 3 . _ propositions and labelling . _ @xmath232 , i.e. , there is a proposition for every vertex in @xmath1 and a special proposition @xmath233 . the vertex @xmath3 and vertices in @xmath2 are labeled by the special proposition @xmath233 , i.e. , @xmath234 ; and for all @xmath227 we have @xmath235 . for a vertex @xmath236 , let @xmath237 , where @xmath238 are vertices in @xmath1 , then @xmath239 and @xmath240 . note that the above proposition assignment ensures that at every vertex that represents an edge , player 2 has the choices of vertices that form the end - points of the edge . + the following case analysis completes the proof . * given a vertex cover @xmath171 , consider a player 2 strategy , that at a vertex @xmath241 , choose a successor @xmath236 such that @xmath242 . the strategy for player 2 ensures that player 1 visits only propositions in @xmath243 , i.e. , at most @xmath244 propositions . * consider a strategy for player 1 that from @xmath3 visits all states @xmath245 in order . consider any counter - strategy for player 2 and let @xmath211 be the set of propositions other than @xmath233 visited . since all the edges are chosen , it follows that @xmath171 is a vertex cover . hence if all vertex cover in @xmath215 is of size at least @xmath44 , then player 1 can visit at least @xmath154 propositions . + hence there is a vertex cover in @xmath215 of size at most @xmath44 if and only if the answer to the maximal coverage problem in @xmath12 with @xmath154 is no . it follows that the maximal coverage problem in controllably recurrent games is conp - hard . the desired result follows . * complexity of minimal safety games . * as a corollary of the proof of theorem [ thrm : maxexpl - contrgames ] we obtain a complexity result about _ minimal safety games_. given a labeled game graph @xmath12 and @xmath44 , the minimal safety game problem asks , whether there exists a set @xmath171 such that a player can confine the game in @xmath171 and @xmath171 contains at most @xmath44 propositions . an easy consequence of the hardness proof of theorem [ thrm : maxexpl - contrgames ] is minimal safety games are np - hard , and also it is easy to argue that minimal safety games are in np . hence we obtain that the minimal safety game problem is np - complete . in this section we study the complexity of the coverage in bounded time problem . in subsection [ subsec : bouexpl : graphs ] we study the complexity for graphs , and in subsection [ subsec : bouexpl : games ] we study the complexity for game graphs . [ thrm : bouexpl - graphs ] the coverage in bounded time problem for both labeled graphs and controllably recurrent labeled graphs is np - complete . we prove the completeness result in two parts below . 1 . _ given a labeled graph with @xmath157 vertices , if there a path @xmath13 such that @xmath64 , then there is path @xmath87 such that @xmath246 . the above claim follows since any cycle that does not visit any new proposition can be omitted . hence a path of length @xmath247 can be guessed and it can be then checked in polynomial time if the path of length @xmath248 visits at least @xmath44 propositions . 2 . _ in np - hard . _ we reduce the _ hamiltonian - path ( ham - path ) _ @xcite problem to the coverage in bounded time problem for labeled graphs . given a directed graph @xmath210 and an initial vertex @xmath6 , we consider the labeled graph @xmath12 with the directed graph @xmath215 , with @xmath6 as the initial state and @xmath249 and @xmath250 for all @xmath251 , i.e. , each vertex is labeled with an unique proposition . the answer to the coverage is bounded time with @xmath252 and @xmath253 , for @xmath86 is `` yes '' iff there is a ham - path in @xmath215 starting from @xmath6 . the desired result follows . * complexity in size of the graph . * we now argue that the maximal coverage and the coverage in bounded time problem on labeled graphs can be solved in non - deterministic log - space in the size of the graph , and polynomial space in the size of the atomic propositions . given a labeled graph @xmath12 , with @xmath157 vertices , we argued in theorem [ thrm : maxexpl - graphs ] that if @xmath44 propositions can be visited , then there is a path of length at most @xmath158 , that visits @xmath44 propositions . the path of length @xmath158 , can be visited , storing the current vertex , and guessing the next vertex , can checking the set of propositions already visited . hence this can be achieved in non - deterministic log - space in the size of the graph , and polynomial space in the size of the proposition set . a similar argument holds for the coverage in bounded time problem . this gives us the following result . [ thrm : nlogmaxexpl ] given a labeled graph @xmath254 , the maximal coverage problem and the coverage in bounded time problem can be decided in nlogspace in @xmath255 , and in pspace in @xmath256 . [ thrm : pspace - bouexpl ] the coverage in bounded time problem for labeled game graphs is pspace - complete . we prove the following two cases to prove the result . 1 . _ pspace - hardness_. it follows from the proof of theorem [ thrm : maxexpl - games ] that the maximal coverage problem for labeled game graphs reduces to the coverage in bounded time problem for labeled game graphs . since the maximal coverage problem for labeled game graphs is pspace - hard ( theorem [ thrm : maxexpl - games ] ) , the result follows . 2 . _ in pspace . _ we say that an _ exploration game tree _ for a labeled game graph is a rooted , labeled tree which represents an unfolding of the graph . every node @xmath257 of the tree is labeled with a pair @xmath258 , where @xmath6 is a node of the game graph , and @xmath259 is the set of propositions that have been visited in a branch leading from the root of the tree to @xmath257 . the root of the tree is labeled with @xmath260 . a tree with label @xmath258 has one descendant for each @xmath261 with @xmath262 ; the label of the descendant is @xmath263 . + in order to check if @xmath44 different propositions can be visited within @xmath56-steps , the pspace algorithm traverses the game tree in depth first order . each branch is explored up to one of the two following conditions is met : ( i ) depth @xmath56 is reached , or ( ii ) a node is reached , which has the same label as an ancestor in the tree . the bottom nodes , where conditions ( i ) or ( ii ) are met , are thus the leaves of the tree . in the course of the traversal , the algorithm computes in bottom - up fashion the _ value _ of the tree nodes . the value of a leaf node labeled @xmath258 is @xmath264 . for player-1 nodes , the value is the maximum of the values of the successors ; for player-2 nodes , the value is the minimum of the value of the successors . thus , the value of a tree node @xmath257 represents the minimum number of propositions that player 1 can ensure are visited , in the course of a play of the game that has followed a path from the root of the tree to @xmath257 , and that can last at most @xmath56 steps . the algorithm returns yes if the value at the root is at least @xmath44 , and no otherwise . + to obtain the pspace bound , notice that if a node with label @xmath265 is an ancestor of a node with label @xmath266 in the tree , we have @xmath267 : thus , along a branch , the set of propositions appearing in the labels increases monotonically . between two increases , there can be at most @xmath268 nodes , due to the termination condition ( ii ) . thus , each branch needs to be traversed at most to depth @xmath269 , and the process requires only polynomial space . the result follows . [ thrm : bouexpl - contrgames ] the coverage in bounded time problem for labeled game graphs that are controllably recurrent is both np - hard and conp - hard , and can be decided in pspace . it follows from the ( pspace - inclusion ) argument of theorem [ thrm : maxexpl - games ] that the maximal coverage problem for labeled game graphs that are controllably recurrent can be reduced to the coverage in bounded time problem for labeled game graphs that are controllably recurrent . hence the conp - hardness follows from theorem [ thrm : maxexpl - contrgames ] , and the np - hardness follows from hardness in labeled graphs that are controllably recurrent ( theorem [ thrm : bouexpl - graphs ] ) . the pspace - inclusion follows from the general case of labeled game graphs ( theorem [ thrm : pspace - bouexpl ] ) . theorem [ thrm : bouexpl - contrgames ] shows that for controllably recurrent game graphs , the coverage in bounded time problem is both np - hard and conp - hard , and can be decided in pspace . a tight complexity bound remains an open problem . * complexity in the size of the game . * the maximal coverage problem can alternately be solved in time linear in the size of the game graph and exponential in the number of propositions . given a game graph @xmath270 , construct the game graph @xmath271 where @xmath272 , @xmath273 iff @xmath274 and @xmath275 , @xmath276 for @xmath277 , @xmath278 , and @xmath279 . clearly , the size of the game graph @xmath203 is linear in @xmath215 and exponential in @xmath4 . now consider a reachability game on @xmath203 with the goal @xmath280 . player-1 wins this game iff the maximal coverage problem is true for @xmath215 and @xmath44 propositions . since a reachability game can be solved in time linear in the game , the result follows . a similar construction , where we additionally track the length of the game so far , shows that the maximal coverage problem with bounded time can be solved in time linear in the size of the game graph and exponential in the number of propositions . [ thrm : lintimemaxexpl ] given a labeled game graph @xmath168 the maximal coverage and the coverage in bounded time problem can be solved in linear - time in @xmath281 and in exponential time in @xmath256 . somewhat surprisingly , despite the central importance of graph coverage in system verification , several basic complexity questions have remained open . the basic setting of this paper on graphs and games can be extended in various directions , enabling the modeling of other system features . we mention two such directions . so far , we have assumed that at each step , the tester has complete information about the state of the system under test . in practice , this may not be true , and the tester might be able to observe only a part of the state . this leads to graphs and games of _ imperfect information _ @xcite . the maximal coverage and the coverage in bounded time problem for games of imperfect information can be solved in exptime . the algorithm first constructs a perfect - information game graph by subset construction @xcite , and then run the algorithm of theorem [ thrm : lintimemaxexpl ] , that is linear in the size game graph and exponential in the number of propositions , on the perfect - information game graph . thus , the complexity of this algorithm is exptime . the reachability problem for imperfect - information games is already exptime - hard @xcite , hence we obtain an optimal exptime - complete complexity . second , while we have studied the problem in the discrete , finite - state setting , similar questions can be studied for timed systems modeled as timed automata @xcite or timed game graphs @xcite . such problems would arise in the testing of real - time systems . we omit the standard definitions of timed automata and timed games . the maximal coverage problem for timed automata ( respectively , timed games ) takes as input a timed automaton @xmath282 ( respectively , a timed game @xmath282 ) , with the locations labeled by a set @xmath4 of propositions , and a number @xmath44 , and asks whether @xmath44 different propositions can be visited . an algorithm for the maximal coverage problem for timed automata constructs the region graph of the automaton @xcite and runs the algorithm of theorem [ thrm : nlogmaxexpl ] on the labeled region graph . this gives us a pspace algorithm . since the reachability problem for timed automata is pspace - hard , we obtain a pspace - complete complexity . similar result holds for the coverage in bounded time problem for timed automata . similarly , the maximal coverage and coverage in bounded time problem for timed games can be solved in exponential time by running the algorithm of theorem [ thrm : lintimemaxexpl ] on the region game graph . this gives an exponential time algorithm . again , since game reachability on timed games is exptime - hard @xcite , we obtain that maximal coverage and coverage in bounded time in timed games is exptime - complete . e. brinksma and j. tretmans . testing transition systems : an annotated bibliography . in _ movep 00 : modeling and verification of parallel processes _ , volume 2067 of _ lecture notes in computer science _ , pages 187195 . springer , 2000 . o. maler , a. pnueli , and j. sifakis . on the synthesis of discrete controllers for timed systems . in _ proc . of 12th annual symp . on theor . asp . of comp . _ , volume 900 of _ lect . notes in comp . _ springer - verlag , 1995 .

continuous measurement of quantum systems @xcite , the study of quantum systems states under the influence of observation prolonged in time , has been a topic of considerable activity in recent years . particularly , for the measurement of an individual microscopic system that is weakly coupled to measurement apparatus , the system s state and its conditioned evolution in time , the so - called diffusive - type quantum state trajectory @xcite , have been intensively explored for applications in quantum information and quantum control . some of the active fields are quantum state estimation ( i.e. , estimating the pre - measurement state of an ensemble of identically prepared systems @xcite ) , conditional reversal of measurement @xcite , and the preparation of entangled states @xcite . the ability to continuously measure quantum systems also opens the possibility of feedback control @xcite , which has also been investigated for topics such as the stabilization of coherent oscillations @xcite and rapid state purification @xcite . this growing interest in the quantum systems under weak continuous measurement has motivated a thorough analysis of quantum trajectory statistics . of notable importance are advances in experiments , such as the measurement of superconducting qubits @xcite , which has allowed the tracking of the trajectories of the quantum state with high fidelity in a single measurement run , allowing the statistics of selected subensembles of trajectories to be explored . the authors recently developed an action principle @xcite over a doubled quantum state space , based on a path integral representation of probability distributions of quantum trajectories . the action principle , implemented by extremizing of the stochastic path integral s action , was used to investigate the optimal behaviour of the trajectories with arbitrary constraints , such as fixing the final boundary condition @xcite . the stochastic path integral and the optimum likelihood approach provide a convenient way to investigate statistical distributions and globally optimal dynamics of quantum state evolution , in addition to the stochastic master equations describing the quantum trajectories and the lindblad master equations describing their average evolution @xcite . in this paper , we continue the development of the stochastic path integral formalism @xcite , to further explore advantages of having the full joint probability distribution of quantum trajectories . this includes computing statistical averages or expectation values with the ability to condition on definite quantum states at particular times . we present several examples of the formalism including a qubit system under the influence of measurement alone , measurement with concurrent unitary dynamics , and qubit measurement with feedback control . in these examples , the statistics of qubit trajectories are computed using developed techniques for the path integral , such as multi - dimensional gaussian integrals and diagrammatic expansion theory . moreover , in an example of qubit measurement with linear feedback , we utilize a phase portrait analysis to investigate the most likely behaviour of the system dynamics , revealing a simple and practical approach for qubit state stabilization . there have been past works on continuous quantum measurements with path integrals , so we wish to discuss how our approach bears both similarities and differences to them . an early approach suggested by feynman @xcite and later independently developed by mensky @xcite is a restricted path integral : a modified version of the feynman path integral to only sum over paths that contribute to a measurement record . caves , and also barchielli @xcite , constructed similar path integrals by adding coarse - graining ( resolution ) functions describing the effect of the measurement . in these path integral approaches , one considers the distribution of the measurement records ( it can be derived from the probability amplitude , see appendix [ app - otherpathint ] ) , whereas in our approach , we formulate a path integral in quantum state space to represent a joint probability distribution of the measurement readouts as well as quantum state trajectories . wei and nazarov discussed a different approach to continuous measurements using the keldysh path integral technique @xcite . in breuer and petruccione s path integral @xcite , the notion of the sum over pure state paths in hilbert space is applied , resulting in a different type of doubled state space that does not yield an action functional . the stochastic path integral formalism and the analysis of its action are also applied in classical stochastic processes . for instance , the formalism is used in studying the dynamics and distribution of transmitted electronic charge @xcite , the neural network @xcite , and the large deviations from typical behaviours ( rare events ) @xcite . our approach is similar to the martin - siggia - rose formalism @xcite , which involves adding conjugate fields through the fourier integral form of delta functionals enforcing diffusive dynamics . notably , one can think of the quantum trajectories on a finite dimensional state space as analogous to classical random processes in configuration space of that dimension . this paper is organized as follows . in section [ sec - mainspi ] , we review the stochastic path integral formalism and its extremized action equations , for a general finite - dimensional system with a markovian setup for weak continuous measurement . in section [ sec - qubit ] , a specific measurement setup for a qubit is presented , which is used throughout this paper . in section [ sec - qnd ] , we show that , in the case without qubit hamiltonian , we can perform the full path integration directly to get the multi - time correlation functions for the preselected and postselected qubit state . in section [ sec - purbexpand ] , the diagrammatic expansion theory is presented as an alternative approximation method for computing multi - time correlation functions , with examples for qubit measurement with rabi oscillation . in section [ sec - feedback ] , qubit measurement with feedback control and its optimal dynamics are investigated using the path integral and the action principle approaches . the conclusion can be found in section [ sec - conclusion ] . a series of supplementary discussions and some detailed calculations that are not included in the main text are presented in the appendixes . we consider the distribution of quantum state trajectories for continuous quantum measurement . a quantum state trajectory , or simply a quantum trajectory , is an evolution of a quantum state in time , conditioned on a detector readout realization . this conditional state trajectory is also known as a solution of stochastic master equations , unravelling master equations in lindblad form . let us discretize the measurement readout into @xmath0 time points and denote @xmath1 as a measurement realization . each @xmath2 is a readout obtained between time @xmath3 and @xmath4 , and is assumed dependent only on a quantum state right before its measurement ( markov assumption ) . we define a series of quantum states @xmath5 , written as a @xmath6-dimensional parametrized vector @xmath7 , where the components are the expansion coefficients of the density operator @xmath8 written in some orthogonal operator basis , such as the @xmath9 generalized gell - mann matrices @xmath10 of a @xmath11-state system @xcite . for a two - state system , the matrices @xmath10 for @xmath12 are the pauli matrices , and @xmath13 is a vector in bloch sphere coordinates . the quantum state trajectory can be computed with an update equation of the form , @xmath14 $ ] , taking into account the measurement back - action from the measurement readout @xmath2 , and also considering the unitary evolution from the measured system s hamiltonian . since the distribution of the measurement readout only depends on the quantum state right before the measurement in this markovian approach , we can then write the joint probability density function ( pdf ) of all measurement outcomes and state trajectories @xmath15 given an initial state @xmath16 and a set of other constraints @xmath17 as , @xmath18 this is a time step product of @xmath19 , the conditional probability distribution for the measurement outcome @xmath2 given the system state before the measurement @xmath20 , and @xmath21)$ ] , the ( deterministic ) conditional probability distribution for the quantum state after the measurement , given the state at the previous time step and the measurement readout . the prefactor @xmath22 $ ] in eq . is a function of quantum states and readouts at any times , accounting for constraints used in selecting a sub - ensemble of the quantum trajectories , such as initial - state and final - state conditions . the benefit of having the joint pdf eq . is that it contains all the statistical information about the system s evolution under measurement , and it allows us to selectively work with sub - ensembles of quantum trajectories simply by adding constraints ( or conditions ) to the joint distribution . statistical moments can be computed from this joint pdf by integrating over its variables . for example , an expected value of an arbitrary functional @xmath23 $ ] is given by @xmath24_1^n { \mathrm{d}}[r_k]_0^{n-1 } { \cal p}_{\zeta } { \cal a}$ ] , where we define a notation for the multiple integral , @xmath25_{1}^{n } \equiv \int\!\ ! { \mathrm{d}}{{\bm q}}_{1 } \cdots { \mathrm{d}}{{\bm q}}_{n}$ ] . direct integration of these quantities using the joint pdf as in eq . , however , can be a challenging task even for a simple qubit measurement problem . as such , we are motivated to write the joint pdf in a path integral form with an action ( or exponent ) , so we can perform the integration using techniques developed in quantum theory such as a diagrammatic perturbation theory . the path integral representation of the joint pdf in eq . can be attained by writing the delta functions for the state update @xmath26)$ ] for @xmath27 to @xmath28 in the fourier integral form , i.e. , @xmath29 for each @xmath30 and each component of the vector @xmath7 , and then rewrite other terms in exponential forms . the conjugate variables for those delta functions are denoted by @xmath31 for @xmath27 to @xmath28 . we refer to ref . @xcite and its appendixes for a thorough discussion about this transformation and the construction of the path integral . as a result , the joint pdf is then given in a path integral form , @xmath32_{0}^{n-1 } \exp({\cal s}),\end{aligned}\ ] ] where the integrals are over all possible paths of @xmath33 and the action @xmath34 is defined as , @xmath35 ) + \ln p(r_k | \bm{q}_k ) \bigg\}.\end{aligned}\ ] ] we note that @xmath36 is an additional term determined by the formation of @xmath37 in eq . , and @xmath38 is a prefactor absorbing normalization constants . for an example , we consider a sub - ensemble of trajectories that obey conditions on the initial and final states . the theoretical analysis of this kind of constraint is presented in ref . the initial state is fixed at @xmath39 and the final state is at @xmath40 . this leads to the constraint term @xmath41 in eq . and the preselected and postselected joint pdf , @xmath42 , which is written in a path integral form as , @xmath43_{-1}^{n } \exp({\cal s}_{{{\bm q}}_i , { { \bm q}}_f}),\end{aligned}\ ] ] where the path integral s action is , @xmath44 ) + \ln p(r_k | \bm{q}_k ) \bigg\}.\end{aligned}\ ] ] we note that , in eq . and , two additional conjugate variables , @xmath45 and @xmath46 , are introduced , because we have written the delta functions in @xmath47 , one for the initial state and another for the final state , in the fourier integral form . we can investigate the path integral s largest contribution by solving for its action s extrema . taking the variation of the action eq . over all the variables and setting it to zero , we obtain a set of difference equations , [ eq - diffeqs ] @xmath48&=0,\\ \label{eq - backwardp } -\bm{p}_{k-1 } + \frac{\partial}{\partial \bm{q}_k}\big\{\bm{p}_k\cdot \bm{\mathcal{e}}[\bm{q}_k , r_k ] + \ln p(r_k | \bm{q}_k)\big\}&=0 , \\ \frac{\partial}{\partial r_k}\big\ { \bm{p}_k \cdot \bm{\mathcal{e}}[\bm{q}_k , r_k ] + \ln p(r_k | \bm{q}_k)\big\}&=0.\end{aligned}\ ] ] the first , second and third equations are from taking derivatives over the conjugate variables @xmath31 from @xmath49 to @xmath28 , over the state variables @xmath20 from @xmath50 to @xmath28 , and over the measurement readout variables @xmath2 from @xmath27 to @xmath28 , respectively . the derivative over the final state variable @xmath51 gives a trivial equation @xmath52 , whereas the derivatives of the action over @xmath45 and @xmath46 force the boundary conditions @xmath39 and @xmath40 on the solutions of eqs . . we note that , in the case of no final boundary condition on the quantum state ( i.e. , @xmath53 or @xmath54 ) , the derivative of the action over @xmath51 instead gives @xmath55 , a final value of the conjugate variable . interestingly , this extremization of the action in eqs . reproduces the lagrange multiplier method , an optimization strategy that accommodates constraints . in our case , the optimized function is the last term of eq . , @xmath56 , subject to the constraints @xmath57 $ ] for @xmath58 ( enforcing the deterministic state update ) with the lagrange multipliers @xmath59 , and the boundary constraints @xmath39 and @xmath40 with the lagrange multipliers @xmath45 and @xmath46 therefore , a solution of the difference equations eqs . is a path that optimizes the log - likelihood of a quantum trajectory , @xmath56 , subject to the indicated constraints . the optimal path can be a local maximum , a local minimum , or a saddle point in the constrained probability space . for the optimal path that represents the local maximum , we call it the most likely path or the most probable path . the optimal path , a solution of the difference equations eq . and their boundary conditions , can be approximated by taking a time - continuous limit @xmath60 , changing the difference equations to a set of ordinary differential equations , which can then be solved analytically or numerically . this approximation is applicable because its solutions , the optimal readouts and the optimal quantum paths , are smooth functions of time . in the case of the optimal paths that maximize the log - likelihood of preselected and postselected quantum trajectories ( the most likely paths between two quantum states ) , these solutions have been experimentally verified with a superconducting transmon qubit @xcite . so far , we have presented the joint pdf and the path integrals in the time - discrete form . for a more compact representation , we introduce a time - continuous version of the discrete stochastic path integral , assuming that the system is not intrinsically discrete and a well - defined diffusive limit exists . the joint pdf , for example in eq . , is written as , @xmath61 \ , \exp({\cal s } ) \overset{\delta t \rightarrow 0}{= } { \cal n}\!\ ! \int \!\!\ ! { \cal d}{{\bm p}}\ , \exp({\cal s}),\end{aligned}\ ] ] where we have defined a notation for functional integrals , e.g. , @xmath62 $ ] . the action in eq . is given by , @xmath63 ) + { \cal f}[{{\bm q}},r]\big\},\end{aligned}\ ] ] where we have introduced @xmath64{\mathrm{d}}t$ ] as the time - continuous version of the state update equation @xmath65 $ ] , and defined @xmath66 $ ] as the linear - order expansion in time of the log - probability , i.e. , @xmath67+o(\delta t^2)\}$ ] . a proportionality factor of the latter is absorbed into the normalization factor @xmath38 . the time dependence of the variables in eq . and is suppressed for simplicity ( e.g. , @xmath68 ) . in the continuous version of the action , the state update equation @xmath69 $ ] can be derived from the first order expansion in @xmath70 of its discrete form @xmath65 $ ] using the exact readout probability distribution @xmath19 . as an alternative , one can consider using a stochastic master equation of the quantum state , where an ideal white noise @xmath71 is introduced with its single gaussian probability distribution . the latter substitution is valid in an idealized noise limit , which we discuss in more detail in section [ subsec - nonqnd ] ( where it stochastic equations are chosen in the diagrammatic perturbation approach ) and in appendix [ app - whitenoise ] . the choice of the stochastic state equations needs to be consistent with the readout ( or noise ) probability distribution used in obtaining the functional @xmath66 $ ] in eq . . a careful analysis is needed when considering the extremization of the action as described in eqs . , which , in the continuum limit , changes to a set of ordinary differential equations . we note that one can derive this same set of differential equations directly from extremizing the continuous - version action in eq . , if the state update equation @xmath69 $ ] is obtained from the first order expansion in @xmath70 of its discrete form ( this approach to the derivation is presented in ref . @xcite ) . for the action constructed using the it stochastic equations , its extremization does not give a correct set of differential equations describing the optimal paths . however , a stochastic path integral formulated using the it equations is more advantageous when computing functional integrals , which is presented in section [ sec - purbexpand ] . so far we have presented the formalism in the general context , the quantum measurement with measurement readouts @xmath72 and corresponding quantum states @xmath73 . in order to show examples and demonstrate how to compute interesting quantities using the path integral formalism , we choose a particular theoretical model , a qubit continuously measured by an apparatus assuming a weakly responding ( or coupled ) detector . this theoretical setup has long been a subject of interest in theories and experiments . some of the physical realizations of this system available with current technology are as follows : a single electron in a double quantum dot capacitively coupled to a quantum point contact detector @xcite , a superconducting qubit dispersively coupled to a microwave waveguide cavity @xcite , and quantum optics experiments such as a two - level atom monitored with homodyne optical measurement @xcite . in most of these experimental setups , one can characterize the qubit evolution as governed by the measurement back - action , the qubit unitary evolution , and extra dephasing due to loss of information or added noise . we write a time - discrete change for a qubit density matrix @xmath8 as @xmath74 $ ] , where we define a measurement operation @xmath75 = { { \hat { m}}}_{r_k } \rho { { \hat { m}}}_{r_k}^{\dagger}/{\text{tr}({{\hat { m}}}_{r_k } \rho { { \hat { m}}}_{r_k}^{\dagger})}$ ] , a unitary operation @xmath76 = e^{-i { { \hat { h } } } \delta t } \rho e^{i { { \hat { h } } } \delta t}$ ] , and an extra dephasing operation @xmath77 $ ] ( we set @xmath78 ) . the measurement back - action on a qubit state , considering the diffusive measurement as in refs . @xcite , is described by a measurement operator @xmath79 $ ] , where @xmath80 is a characteristic measurement time taken to separate the two gaussian distributions by two standard deviations . the measurement readout distribution is computed from the trace of the measurement operator and the qubit state , @xmath81 with the bloch sphere coordinates of the qubit state given by @xmath82 . we denote a qubit hamiltonian by @xmath83 characterizing the unitary evolution during the measurement , and we define the dephasing operation @xmath77 $ ] accounting for additional dephasing on the system , such as detection inefficiency and dephasing due to the environment . we model the dephasing operation as an extra dephasing rate @xmath84 on the off - diagonal elements of the qubit density matrix @xmath8 . by expanding the state update equation @xmath74 $ ] to first order in @xmath70 , and taking a continuum limit @xmath60 , we obtain a set of differential equations , [ eq - contstate ] @xmath85 which turns out to be analogous to the stochastic master equation in stratonovich interpretation as mentioned in ref . @xcite . using the state update equations in eqs . and the readout distribution in eq . , we have the qubit action in continuous form ( as in eq . ) , @xmath86 where we have approximated the logarithm of the readout distribution as @xmath87 , omitting the second term and its higher order expansion because they can be absorbed into the prefactor @xmath38 . we note that the extremization of this action and its most likely paths with fixed initial and final states are presented in more detail in ref . @xcite . in the remainder of this paper , we will focus on applying the path integral formalism to this particular qubit measurement system , though in different regimes of the hamiltonian parameters . for example , in the next section , we study the `` plain '' qubit measurement where we assume that there is no qubit hamiltonian , i.e. , @xmath88 , and the qubit s evolution comes only from the measurement and the dephasing mechanism . in this section , we present examples as to how we can use the path integral and its optimal path to compute statistical moments of the measured qubit trajectories . as it turns out , analytic solutions are possible for the case of quantum non - demolition measurement ( @xmath89 , where the system hamiltonian commutes with total system - detector hamiltonian ) with a constraint on the initial and final states of the quantum trajectories . in this case , the path integral can be written only in the @xmath90-coordinate , i.e. , @xmath91 , because the evolution of @xmath90 is independent of the other two coordinates , @xmath92 and @xmath93 . the @xmath92 and @xmath93 degrees of freedom can be found directly from @xmath94 and @xmath90 , so they will be dropped from the discussion until section [ sec - purbexpand ] . in the following subsections , we consider the plain measurement case when @xmath88 for simplicity . we show how to compute the path integral using the preselected and postselected joint pdf , @xmath95 . we first integrate this joint pdf over all intermediate variables to get the probability density function @xmath96 of arriving at the final state @xmath97 after time @xmath98 , given the initial state @xmath99 . then , we show how to generalize the integration procedures to compute statistical quantities such as averages , variances , and correlation functions of the qubit trajectories in this simplified case . we note here that even though we present the derivation in the plain measurement case , the result should still be valid for the case when @xmath100 . in the latter case , the evolution of the @xmath92 and @xmath93 coordinates of the qubit will be deterministically oscillating with the frequency @xmath101 , without affecting the evolution of the @xmath90 coordinate . in order to compute the probability distribution of the final qubit state given the initial state separated by time @xmath98 , we start with the joint pdf @xmath102 of the qubit trajectories @xmath103 and measurement outcomes @xmath104 with fixed boundary states , and then integrate over all variables except the final state @xmath97 , @xmath105_0^{n}{\mathrm{d}}[r_k]_0^{n-1 } { \cal p}_{z_i , z_f},\end{aligned}\ ] ] where , as before , we have defined the multi - variable integral , e.g. , @xmath106_{0}^{n } = \int\!\ ! { \mathrm{d}}z_{0 } \cdots { \mathrm{d}}z_{n}$ ] . from the discussion in section [ sec - mainspi ] , the joint pdf is given by a product over the sliced time variables , @xmath107 . the term describing the state update equation is a delta function , @xmath108 with its argument derived from the @xmath90-coordinate of the density matrix equation @xmath109 $ ] , whereas , the probability distribution for the readout eq . is expressed in this form , @xmath110 neglecting higher orders in @xmath70 . substituting these two expressions , eqs . and , into the joint pdf , we get @xmath111 where we have used @xmath112 to represent the delta function in eq . . we then integrate out the measurement readouts by transforming the delta function of @xmath90 into a delta function in the readout variable , @xmath113 , using the transformation relation @xmath114 = \sum_i \delta[y - y_i]/|\partial_y f(y)|_{y_i = f^{-1}(x)}$ ] , where the sum extends over all solutions of @xmath115 . after integrating over the measurement readout and also over the boundary state @xmath116 , and @xmath117 , the joint pdf is transformed to a new joint pdf , a function of only the intermediate @xmath90-variables , @xmath118 where the action @xmath34 is , @xmath119 we note that we have simplified the above equation by defining a new variable @xmath120 to write the joint pdf in a compact form . this joint pdf of the @xmath90-variable is one of the main results shown in this paper . it is also important to point out that since we have already integrated over the boundary state variable @xmath116 and @xmath117 , the delta functions for the boundary states have applied to the states @xmath121 and @xmath122 in eqs . and . the next step is to perform the integration over the intermediate variables @xmath123 for @xmath124 . these integrals can be done easily by expanding @xmath123 around the optimal solution denoted by @xmath125 . we substitute @xmath126 for all @xmath30 s in the action , and perform integration by parts with the vanishing boundaries , @xmath127 . this substitution exactly simplifies to @xmath128 = { \cal s}[{\bar u } ] - \frac{\tau_m}{2 \delta t } \sum_{k=0}^{n-1}(\eta_{k+1}-\eta_k)^2 $ ] ( see appendix [ app - actionpure ] for more detail ) . we then change the integral measures to @xmath129 for @xmath130 and write the probability distribution in eq . in terms of @xmath131-variables , @xmath132)}{1-z_{f}^2 } \\ & \times \exp\left\{- \frac { \tau_m}{2 \delta t}\sum_{k=0}^{n-1}(\eta_{k+1}-\eta_k)^2\right\},\end{aligned}\ ] ] where the optimal path @xmath125 is a solution of the extremization of the action , which gives @xmath133 . the solution is @xmath134 with the two constants of integration @xmath135 and @xmath136 . the optimal solution ( the most likely path ) is given by , @xmath137 where we wrote @xmath138 . we note that in the time - continuum limit , this extremization is equivalent to the vanishing functional derivative of the action , @xmath139/\delta u(t ) = \tau_m \ddot{u } = 0 $ ] , which is similar to the euler - lagrange equation for the classical trajectory of a free particle in the @xmath140 coordinate transformation . the integrals in eq . , with the probability distribution in @xmath131-variable eq . , is now in this form , @xmath141_1^{n-1 } p ( \ { \eta_k \}_1^{n-1 } , z_f | z_i).\end{aligned}\ ] ] fortunately , they are gaussian integrals in multiple dimensions which can be calculated from the matrix integral , @xmath142 , where , in our case , @xmath143_1^{n-1}= \int \!\ ! { \mathrm{d}}\eta_1\cdots { \mathrm{d}}\eta_{n-1}$ ] , and the matrix @xmath144 and the vector @xmath145 are given by , @xmath146 noting that @xmath147 . after substituting the optimal solution @xmath125 into @xmath148 $ ] and performing the integrals over @xmath149 in eq . , we obtain the distribution of the final state fixing the initial state and the duration of time @xmath150 , @xmath151 where we have defined @xmath152 as the optimal measurement readout . this solution is exactly the same as what we would get from the change of variables of the probability distribution , @xmath153 , where @xmath154 is a time - average measurement readout . the derivation of the latter distribution is presented in the methods section of ref . @xcite . we have derived the probability distribution for the measured qubit given the fixed initial and final states , as shown in eq . and . in this subsection , we consider computing the qubit trajectory s statistical quantities that can be written in this expectation form , @xmath155_1^{n-1}{\cal a}\,\ , \frac{p ( \ { z_k \}_1^{n-1},z_f | z_i)}{p(z_f|z_i)},\\ = & \frac{\sqrt{\text{det}{\bm m}}}{(2 \pi)^{\frac{n-1}{2 } } } \int { \rm d}[\eta_k]_1^{n-1}{\cal a}\,\ , e^{-\frac{1}{2}{\bm \eta}^t\cdot { \bm m } \cdot { \bm \eta}},\end{aligned}\ ] ] where @xmath156 is an arbitrary functional of @xmath157 at any time @xmath158 for @xmath159 and the matrix @xmath144 is the same as defined in eq . . note that we have used the notation @xmath160 for a statistical average conditioned on fixed initial and final states , @xmath99 and @xmath97 . since the probability distribution in the @xmath131-variables is gaussian , it is preferable to write @xmath90 in terms of @xmath131 , replacing @xmath90 with @xmath161 , and then perform the integration . for example , a conditional average of @xmath157 at time @xmath158 is given by , @xmath162 expanding to second order in @xmath131 , where the primes indicate derivatives over the @xmath140-variable . in the second line , we still use the bracket @xmath160 for the conditional average of the variable @xmath131 , even if its boundary values are shifted to @xmath163 . the expectation values in terms of @xmath131 can be computed using the definition of moments in the second line of eq . . because it is a multi - dimensional gaussian integral , we find the result from wick s theorem , @xmath164 where the left hand side is an expectation value of even numbers of @xmath131 at times @xmath165 , and the right hand side is a product of @xmath166 elements of the inverse matrix @xmath167 , summing over all possible pairings between the @xmath131 s on the left side . any other statistical averages with odd numbers of @xmath131 will vanish . as an example , pairing four @xmath131-variables , @xmath168 , yields the sum , @xmath169 . standard deviation ) of the measured qubit trajectories . the preselected state ( initial state ) and the postselected state ( final state ) are @xmath170 and @xmath171 , respectively . the total time is @xmath172 . the numerical data , plotted as gray dots , are analyzed from @xmath173 trajectories computed with time steps of the size @xmath174 , zero qubit hamiltonian , and the final selection tolerance @xmath175 . the numerical data shows excellent agreement with the theoretical solutions . the three fluctuating curves are randomly chosen individual trajectories . the inset shows the average ( dashed green ) and the most likely path ( magenta ) on a bloch sphere plotted in the @xmath92-@xmath90 plane.,width=321 ] the elements of the inverse matrix @xmath167 are of a simple form , @xmath176 for @xmath177 ( the full matrix is presented in appendix [ app - invmatrix ] ) . knowing these matrix elements , we can then compute any expectation values of the type shown in eq . , such as a two - time correlation , @xmath178 for @xmath179 , and statistical moments of even orders of @xmath131 , @xmath180 where we have used the discrete time notation @xmath181 and @xmath182 , and the double factorial prefactor @xmath183 for a positive integer @xmath184 . this double factorial comes from the number of ways of pairing @xmath185 identical variables . substituting these quantities into the expectation value eq . , we then obtain the preselected and postselected average of @xmath157 , @xmath186 keeping terms up to first order of @xmath187 , which is equivalent to second order of @xmath131 because the conditional average of the second order of @xmath131 scales as @xmath187 , i.e. , @xmath188 . this solution eq . is justified in a weak - coupling limit ( @xmath189 ) , however , the full solution to all orders can be computed and is presented in appendix [ app - fullqnd ] . using the same approximation , we can also compute another interesting quantity , the variance , keeping terms up to first order of @xmath187 , @xmath190 where its full solution to all orders in @xmath187 is also presented in appendix [ app - fullqnd ] . finally , we compute a two - time correlation in @xmath90-coordinate , keeping up to first order in @xmath187 , @xmath191 for @xmath192 , which decreases linearly in @xmath3 for a fixed @xmath158 . we show in figure [ fig - qnd ] , the average @xmath193 , its variance , and the optimal path ( the most likely path ) @xmath194 , for the preselected and postselected trajectories , compared with data from a numerical simulation using the monte carlo method ( see appendix [ app - numer ] for more detail ) . we generate @xmath173 trajectories with a postselection time @xmath172 , showing an excellent agreement with the analytical solutions . we have shown in the previous section that the preselected and postselected moments ( or expected values ) for quantum trajectory variables , in the plain measurement case , can be computed by expanding the path integral around its optimal path . since the action is gaussian , however , the path integral approach can be useful in other cases as well , such as to compute the moments and expectation values of qubit trajectories when there is no final state condition ( state postselection ) , or with non - zero qubit hamiltonian . in this section , we present an alternative method using a diagrammatic perturbation expansion of the action , similar to how feynman diagrams are used in computing path integrals in quantum field theory . in the following , we start with a brief introduction to the perturbation method ( a standard method used in quantum field theory ) and show how one can compute expectation values of system variables from moment generating functionals . we will then go on to the examples , for the case of qubit trajectories with non - zero hamiltonian and no state postselection . we note that even though most of the derivations are shown in time - continuous form for simplicity , the most straightforward derivations are in time - discretized version , and some of these are shown in appendixes [ app - sourceterm ] and [ app - heaviside ] . let us suppose that a quantity of interest is an expectation value given in a path integral form , @xmath195 where the quantity @xmath156 is a functional of a system variable @xmath196 , the integral s action @xmath34 is a functional of the system variable @xmath196 and its conjugate variable @xmath197 , and @xmath38 is a constant normalized factor . although we are writing @xmath196 and @xmath197 as one - dimensional variables here , a generalization to arbitrary dimensional vectors is straightforward . to compute the integral above , we write the action as a sum of two parts , @xmath198 , one being terms in the action that have bilinear forms , ( e.g. , @xmath199 ) , which we call the _ free action _ , @xmath200 and call the rest of terms in the action the _ interaction action _ @xmath201 . we then define a free generating functional @xmath202 $ ] , a path integral of the free action adding extra source terms with functions @xmath203 and @xmath204 , @xmath205 = & \,\,{\cal n}\!\!\int\!\ ! \mathcal{d } { x}\mathcal{d } { \tilde{x}}e^{{\cal s}_f + \int\!\ ! { \mathrm{d}}t { \tilde{x}}(t ) { \tilde{j}}(t ) + \int \!\!{\mathrm{d}}t { j}(t ) { x}(t ) } \\ = & \,\,\exp\left\{\int \!\!{\mathrm{d}}t { \mathrm{d}}t ' { j}(t ) { g}(t , t'){\tilde{j}}(t ' ) \right\},\end{aligned}\ ] ] where @xmath206 is an inverse of @xmath207 in eq . satisfying @xmath208 , and we assume that the gaussian integrals in the first line produce a factor @xmath209 that leaves no prefactor in the second line . note that these integrals converge when @xmath197 is purely imaginary . moreover , if the bilinear terms are instead quadratic terms ( e.g. , @xmath210 ) , the free action will be of the form @xmath211 , and there will be only one source term @xmath212 leading to a different generating functional @xmath213 = \exp\left\{\frac{1}{2}\int\!\!{\mathrm{d}}t{\mathrm{d}}t ' { j}(t ) g(t , t ' ) { j}(t')\right\}$ ] . the generating functional eq . is then used to compute a _ free moment _ defined as @xmath214 . the free moment of the variable @xmath215 ( or @xmath216 ) at time @xmath217 is simply a functional derivative of the generating functional over the variable @xmath218 ( or @xmath219 ) , taking both variables @xmath220 and @xmath204 to be zero at the end . by looking at the second line of eq . ( [ eq - genfunz ] ) , one can see that the simplest non - vanishing free moment is a two - point correlation function @xmath221\big|_{{j}= { \tilde{j}}= 0 } = g(t , t')$ ] . generalizing this to moments of multiple time points , we get @xmath222\bigg|_{{j}={\tilde{j}}=0},\\ { \nonumber}=&\sum_{\substack{\text{all pairings between}\\ \text{variables $ { \tilde x}$ and $ x$ } } } g(t_{j_{k_1 } } , t_{j_{k_2}})\ , \cdots \,g(t_{j_{k_{2m-1 } } } , t_{j_{k_{2m}}}),\end{aligned}\ ] ] where the summation is for all possible pairings between the variables @xmath196 s and their conjugate @xmath197 s . the number of propagators in the summation on the right is equal to the number of pairs presented in the expectation bracket on the left . from this , one can see that if there is at least one unpaired variable , the moment will vanish . using these definitions of the free moments , one can then compute a path integral as in eq . where @xmath198 and the arbitrary functional @xmath156 is a function of the physical variable @xmath196 , @xmath223 \rangle \equiv & \,\,{\cal n}\!\ ! \int \!\!{\cal d}{x}{\cal d}{\tilde{x}}e^{{\cal s}_f+{\cal s}_i[{x},{\tilde{x}}]}{\cal a}[{x}],\\ { \nonumber}=&{\cal a}\left[\frac{\hat \delta}{\delta { j}}\right]e^{{\cal s}_i[\frac{\hat\delta}{\delta { j}},\frac{\hat\delta}{\delta { \tilde{j}}}]}{\cal z}_f[{j},{\tilde{j}}]\bigg|_{{j}={\tilde{j}}=0},\\ = & \langle { \cal a}[{x}]e^{{\cal s}_i[{x},{\tilde{x}}]}\rangle_f , \label{eq - expa}\end{aligned}\ ] ] where in the second line we wrote the interaction action @xmath201 and the arbitrary functional @xmath156 as operators acting on the free generating functional @xmath224 $ ] . the perturbation expansion refers to the series expansion of the term @xmath225\}$ ] in eq . . in some cases , infinite series can be summed over , giving an exact analytic result , although in many others , an approximation is needed in order to truncate the series . an approximation can be made when there is a small parameter appearing in any ( or all ) of the terms in the interaction action @xmath201 , where the expansion is straightforward , keeping terms up to any desired order of the small parameter . another type of approximation , which we present here in more detail , is similar to a semiclassical expansion in quantum mechanics , keeping terms up to any order of a small parameter @xmath226 that appears as an inverse in front of the action , for example , @xmath227 \right\},\end{aligned}\ ] ] where we explicitly write the the free action in a bilinear form . to compute an expectation value of a functional @xmath156 using this action , one needs to expand the exponential of the interaction action and then evaluate the free moments in terms of the propagators . the order of expansion is controlled by the parameter @xmath226 . there is a factor of @xmath228 for every single term in the expansion of @xmath201 , and a factor of @xmath226 for every propagator that emerges . this is the basic idea behind the loop expansion in quantum theory , where @xmath226 plays the role of the @xmath229 in the feynman path integral . we will discuss more about the loop expansion in section [ sec - smallnoiseapprox ] . this loop expansion based on the small parameter @xmath226 , in our quantum measurement case , can be considered as a small noise expansion around a saddle point solution ( a solution that extremizes the action ) . moreover , as in the feynman path integral , under the approximation of the small parameter @xmath226 , a saddle point approximation can also be applied to estimate a path integral , using an expansion of the action around the saddle point solution . for our stochastic path integral , the saddle point approximation gives an estimation of total probability density for the joint probability distribution that the path integral represents . now we can apply the perturbation approach to our quantum measurement problem . we use the joint probability density function introduced in section [ sec - mainspi ] in its generalized form @xmath230 , a joint pdf of the measurement outcomes and the quantum states given an arbitrary set of constraints @xmath17 . statistical averages or expectation values using the joint pdf are then given in this form , @xmath231{\mathrm{d}}[r_k ] { \cal p}_{\zeta}\ , { \cal a},\\ \overset{\delta t \rightarrow 0}{\approx}&{\cal n}\!\!\!\int\!\!\!{\cal d}{\bm q}{\cal d } r { \cal d}{\bm p } \,\exp({\cal s}){\cal a},\end{aligned}\ ] ] where , in the second line , we have taken the time - continuous limit and written the joint pdf in the path integral form as @xmath232 , noting that the time - continuous action is given by eq . . in the following examples , we present the perturbative expansion approach in computing the statistical moments of qubit trajectories . we focus on the case when the qubit hamiltonian does not commute with the measurement operators ( @xmath233 ) and without a final state constraint . the theoretical setup for the qubit with rabi oscillation is analogous to the one used in ref . @xcite , where the qubit hamiltonian is @xmath234 . here we can calculate important quantities such as average trajectories and correlation functions , for the case when there is only an initial state fixed and not the final state . for simplicity of the diagrammatic expansions , we make a white noise approximation and variable transformations to the qubit system . | cm | > m3.8cm| > m2.5 cm | type & labels of vertices & full forms & diagrams + type 1 ( initial ) & @xmath235 & @xmath236 & ( b2 ) ; ( bp ) ; ( b2 ) ( bp ) ; ( bp ) circle ( .06 cm ) ; + type 2 & @xmath237 , @xmath238 , @xmath239 , @xmath240 , @xmath241 , @xmath242 & @xmath243 & ( b2 ) ; ( bp ) ; ( bpp ) ; ( c0 ) ; ( c1 ) ; ( c2 ) ; ( b2 ) ( bp ) ; ( bp)(bpp ) ; ( bp ) circle ( .06 cm ) ; ( bp ) sin ( c1 ) ; ( bp ) sin ( c2 ) ; + type 3 & @xmath244 , @xmath245 , @xmath246 & @xmath247 & ( b2 ) ; ( bp ) ; ( bpp ) ; ( b2 ) ( bp ) ; ( bp)(bpp ) ; ( bp ) circle ( .06 cm ) ; + we make an idealized white noise limit on the qubit measurement . the variance of the measurement readout distribution @xmath248 is assumed to be very broad , which in this case means @xmath249 , justifying an approximation of the two gaussian distribution in eq . to a single gaussian distribution , @xmath250 ( see appendix [ app - whitenoise ] for more detail ) . the measurement readout is then approximated to be its mean plus a noise , @xmath251 , where @xmath71 is the gaussian noise with variance @xmath252 , independent of the qubit state @xmath90 . because the nature of this noise is highly fluctuating , in the derivation of the state update equations , we need to keep an expansion up to a second order in @xmath70 , replacing @xmath253 @xcite . this leads to the it stochastic differential equations @xcite , [ eq - itoform ] @xmath254 where @xmath255 are bloch sphere coordinates for the qubit and a dephasing rate @xmath256 is now a total dephasing rate @xmath257 . the white noise @xmath71 has a gaussian probability distribution @xmath258 . before substituting the state update above into the action of the path integral , we can make changes in the variables of the system in order to simplify the later perturbation process . this is to avoid infinite series related to the linear terms in eqs . . we define a new set of variables @xmath259 where @xmath260 and @xmath261 is a matrix that diagonalizes the linear terms of eqs . , @xmath262 the eigenvalues are @xmath263 , @xmath264 and @xmath265 , where we define @xmath266 . the diagonalizing matrix @xmath261 and the transformation of the system variables @xmath255 are described in the matrix equation , @xmath267 which gives the following transformation : @xmath268 , @xmath269 , and @xmath270 , and the inverse transformation : @xmath271 , @xmath272 , and @xmath273 . the stochastic differential equations with this new set of variables are , [ eq - transformsde ] @xmath274 where we have defined @xmath275 and @xmath276 , @xmath277 , @xmath278 . now we are ready to construct the action of the path integral , substituting the update equations and the log - likelihood function @xmath279 = - \frac{1}{2}\xi^2 $ ] into the action eq . . we write the action in two separated terms , @xmath198 , where [ eq - actions ] @xmath280 are the free and interaction actions , respectively . we note that the additional term @xmath281 is not merely a time - continuous form of the first term in eq . ( we only consider the initial condition ) . this is because the actual initial condition term @xmath282 can be removed simply by integrating over the initial variable @xmath16 , forcing the condition @xmath283 to the variables at time @xmath284 . in place of the initial term , when we write the free action eq . in this form , @xmath285 , there will be leftover terms which contribute to the term @xmath281 , resulting in , @xmath286 this can be easily shown with a discretized version of the path integral , which is presented in more detail in appendix [ app - sourceterm ] . we note also that , in the rest of the paper , the time integral is always from @xmath287 to @xmath288 , unless stated otherwise . the propagators ( or green s functions ) are computed from the inverse green s functions , which in this particular case are @xmath289 , where @xmath290 ( or @xmath291 ) for the first three terms in eq . , and @xmath292 for the noise term . with an identity relation , @xmath293 , the propagators ( green s functions , or two - point correlation functions ) are given by , [ eq - propagators ] @xmath294 where @xmath295 for the first three lines . it is important to note that @xmath296 is a left continuous heaviside step function , i.e. , @xmath297 and @xmath298 , causing the two - point correlation functions for @xmath291 to vanish when @xmath299 . this is a result of the it interpretation we chose in writing the state update equation in eq . . this can also be verified with the discretized version of the path integral and is presented in appendix [ app - heaviside ] . to compute quantities such as an expectation value @xmath300 where @xmath156 is an arbitrary function written in the form of eq . , one needs to expand the exponential @xmath301 in a power series of @xmath201 , and looks for terms that contribute to its result . fortunately , from the two - point correlation functions eq . , we know that system variables @xmath291 can only connect with the their conjugate variables at earlier times , e.g. , @xmath302 can only connect with @xmath303 if @xmath304 . knowing this helps predict which terms will contribute to the sum of the expansion . for example , to show that @xmath305 , we look at terms in the interaction action eq . and see that all of them contain exactly one conjugate variable in each . looking at the propagators in eqs . , we know that it is impossible for any higher order terms in the expansion of @xmath301 to have all conjugate variables matched with system variables at their later times . this is because every time we want to find terms with system variables at later times , there will be at least one more unmatched conjugate variable appearing . therefore , there are no other contributions apart from 1 , resulting in @xmath306 . in order to make the calculation of moments @xmath307 more systematic , we develop diagrammatic rules to ease the process of keeping track of the perturbative expansion . each term in the expansion of @xmath301 is a combination of 12 additive terms shown in the interaction action eq . , with repeated appearance also possible . the 12 interaction terms ( _ interaction vertices _ ) are shown in table [ tb - vertices ] with their labeling names and examples of their full integral forms , where we use the latter to characterize the vertices into three types shown as the three rows . at the end , all combinations of the interaction vertices ( all additive terms in the expansion of @xmath301 ) are then multiplied with the function of interest @xmath156 before taking the free expectation @xmath308 . the functional @xmath156 is a function of system variables and noise variables ( i.e. , @xmath309 ) at all times . these variables are called _ ending _ vertices . let us use the word _ combination _ for an individual term combining interaction vertices ( from additive terms in the expansion @xmath301 ) and the ending vertices ( from the function @xmath156 ) . one can determine which combinations are non - vanishing and compute their values by following these diagrammatic rules : * a non - vanishing combination of vertices should contain equal number of each of the system variables ( @xmath291 ) and their conjugates ( @xmath310 ) , and there should be even number of the noise variables ( @xmath71 ) . * the system variable can only be connected to its conjugate at earlier time , while the noise variable @xmath71 connects to another noise variable at the same time . the connections are presented as ` edges ' , or lines joining vertices . the number of edges for each vertex is equal to the number of variables it contains ( graphical representations are shown in table [ tb - vertices ] , in the column of ` diagrams ' ) . a non - vanishing combination can have any number of vertices , but edges have to all be connected . * a non - vanishing combination is presented by connected diagrams with the ending vertices ( variables with their time arguments being in the range of @xmath311 $ ] ) on the most left and their time arguments clearly stated . the time ordering decreases from left to right , with variables at the same time lining up in the same vertical line , and the vertices with initial conditions ( @xmath287 ) on the most right . * a result of the free expectation value ( @xmath312 ) for each combination is computed by taking into account the constants ( i.e. , @xmath313 ) and integrals attached to each of the interaction vertices , the propagators ( the green s function ) , and the numbers of possible ways to connect diagrams . from the point of view of graphical diagrams , edges represent propagators , and points common to more than one edge correspond to integrals over time . moreover , if there are @xmath166 repeated vertices in a combination , it needs to be multiplied by a factor @xmath314 ( resulting from coefficients in the expansion of @xmath301 ) . however , these factors usually cancel with the number of possible ways to connect the repeated vertices . ) ( a ) an ending vertex connects with an initial vertex ( @xmath315 ) . ( b ) two ending vertices connect with two interaction vertices ( @xmath316 and @xmath317 ) . ( c ) two ending vertices connect with six interaction vertices ( from left - right , top - bottom order : @xmath318 , @xmath317 , @xmath317 , @xmath319 , @xmath320 and @xmath320 ) . the unequal length of horizontal edges following the ending vertices @xmath321 and @xmath322 indicate that @xmath323.,width=219 ] in figure [ fig - exdiagram ] , we show examples of non - vanishing diagrams showing connections of one and two ending vertices . for each of the diagrams , we can determine its contribution by explicitly writing out its full form and computing its integrals . in the following subsections , we present some of the calculations using these diagrams and rules to estimate statistical averages of the qubit trajectories . we note that it is also possible to construct different diagrammatic rules based on different formations of the path integrals . for example , one can perform integrals over the readout noise ( e.g. , the noise @xmath71 ) , transforming the action in eq . and to a new action that contains only the system variables @xmath324 . in this case , the diagrams will be completely different from the one we presented above , but still giving exactly the same results for the statistical quantities related to the system variables . here , we choose to present the diagrammatic rules with explicit noise variables because of the ability to compute correlation functions between the noise and the system states , as shown in section [ sec - noisesyscorr ] . the diagrammatic rules help in determining non - vanishing terms , but we need a systematic approach to keep track of the order of expansion , especially when there are infinitely many ways to construct diagrams . we consider the small noise approximation around the saddle point solution mentioned in section [ sec - briefintro ] . to obtain the action of the form in eq . , we introduce a small parameter denoted by @xmath226 to the noise term of eqs . , controlling the noise variance in the qubit dynamics , leading to a modified action , @xmath325 leaving all terms in the interaction action intact . we then rescale the conjugate variables by replacing @xmath326 , @xmath327 and @xmath328 ( as well as changing variables in the source term , e.g. , @xmath329 ) , noting that the rescaling of the auxiliary variables does not affect the qubit dynamics . since all terms in the action eq . , except for the noise term , have exactly one conjugate variable ( see eqs . ) , the resulting action is then in the desired form , @xmath330,\end{aligned}\ ] ] where @xmath331 are the rescaled conjugate variables . following the discussion in section [ sec - briefintro ] and the diagrammatic rules presented in the previous section , the order of expansion can be determined by counting the number of edges and vertices of a diagram . each vertex in a diagram contributes a factor of @xmath228 to its result , while each edge ( which represents a propagator ) contributes a factor of @xmath226 . therefore , each term ( diagram ) in the expansion carries a factor of @xmath332 , where @xmath333 , @xmath334 , and @xmath335 are numbers of its external edges ( edges that connect with ending vertices ) , internal edges , and vertices , respectively . this corresponds to the loop expansion where one instead counts the number of loops from @xmath336 , which leads to another way of writing the order of the expansion @xmath337 , noting that the number of external edges @xmath333 is fixed for each set of ending vertices . we show how to count the power of @xmath226 of the diagrams shown in figure [ fig - exdiagram ] . the numbers of edges , vertices , and loops are as follows : ( a ) @xmath338 , @xmath339 , @xmath340 , @xmath341 , ( b ) @xmath342 , @xmath343 , @xmath344 , @xmath341 , ( c ) @xmath342 , @xmath345 , @xmath346 , @xmath347 , giving the zeroth(zeroth ) , zeroth(first ) , and first(second ) orders of loop(@xmath226 ) , respectively . as an example of how to use the diagrammatic rules to compute statistical expectation values , we show a derivation of @xmath348 , an average of the @xmath90-coordinate of the quantum trajectories . from the variable transformation in eq . , we know that @xmath349 , therefore we can write the expectation value in terms of the free moments and then compute the diagrams from the vertices in table [ tb - vertices ] , @xmath350 \coordinate[label = below:$v_t$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$p_{v0}$ ] ( bp ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw ( bp ) circle ( .06 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \end{tikzpicture}\ , + \,\begin{tikzpicture}[node distance=0.6 cm and 0.8 cm ] \coordinate[label = below:$w_t$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$p_{w0}$ ] ( bp ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw ( bp ) circle ( .06 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \end{tikzpicture},\\ { \nonumber}=&\left\langle { v}(t)\,{v}_i \!\!\!\int \!\!\!{\mathrm{d}}t'\ , { p_v}(t')\delta ( t ' ) \right\rangle_f \\ & { \nonumber}+ \left\langle { w}(t)\,{w}_i \!\!\!\int \!\!\!{\mathrm{d}}t'\ , { p_w}(t')\delta ( t ' ) \right\rangle_f,\\ { \nonumber}= & \,{v}_i \!\!\ ! \int \!\!{\mathrm{d}}t ' g_{{v}}(t , t')\delta(t')+{w}_i \!\!\!\int\!\ ! { \mathrm{d}}t ' g_{{w}}(t , t')\delta(t ' ) , \\ = & \ , { v}_i e^{{\lambda}_2 t } + { w}_i e^{{\lambda}_3 t } , \quad \text{where $ t \ge 0$},\end{aligned}\ ] ] where the green s functions are from eqs . . from the types of vertices in table [ tb - vertices ] , there is only one possibility connecting the ending vertices @xmath351 and @xmath352 with @xmath315 and @xmath320 , respectively . this is because choosing other vertices will lead to at least one unmatched noise variable @xmath71 , which , when trying to pair it with another vertex containing @xmath71 , will result in more unmatched conjugate variables . as a result , the expectation value eq . is explicitly found by transforming back to the system variables @xmath255 , @xmath353 which is exactly the same as the solution we would get from averaging over all possible noise realizations in the it stochastic master equations in eqs . and solving for an average of @xmath354 . the averages of @xmath92 and @xmath93 can be computed in the similar way . we note that the exact solution for the average trajectory involves only the diagrams with zeroth order of the small parameter @xmath226 ( @xmath355 ) . particularly , one can show that the average trajectory coincides with a saddle point solution of the action eq . when @xmath356 . the vanishing functional derivatives of the action over the variables @xmath331 gives the exact same equations eqs . ( though , the noise variable @xmath71 is no longer a stochastic function ) , while the vanishing functional derivatives over @xmath357 leads to the differential equations for conjugate variables , [ eq - saddlepointp ] @xmath358 where the noise variable is now a solution of extremizing the action over @xmath71 giving , @xmath359 as the noise parameter @xmath226 decreases to zero , as does the noise term @xmath71 , the ordinary differential equations of @xmath357 are uncoupled from eqs . and their solution ( a saddle point solution ) is then the same as the average trajectory , the solution of eqs . when @xmath360 . another example is a correlation function between the system variable @xmath90 and the noise variable @xmath71 , given by @xmath361 . since there are now two ending vertices , the situation is more complicated and there are infinitely many ways to connect the diagrams . however , we can compute terms to the lowest order of the loop expansion , which in this case is the zeroth loop ( tree - level diagrams ) . there are in total six contributions to the correlation function , as shown in the following , @xmath362 \coordinate [ ] ( b2 ) ; \coordinate[right = of b2,label = left:$\xi_{t_2}$ ] ( bp ) ; \coordinate[above = of b2,label = above:$v_{t_1}$](a1 ) ; \coordinate[above = of bp , label = above:$p_{v}$](ap ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( bp ) circle ( .05 cm ) ; \end{tikzpicture}\,+ \!\!\!\!\begin{tikzpicture}[node distance=0.6 cm and 0.8 cm ] \coordinate [ ] ( b2 ) ; \coordinate[right = of b2,label = left:$\xi_{t_2}$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$v$,label = above:$v$ ] ( ap ) ; \coordinate[left = of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of ap](cc ) ; \coordinate[above=0.3 cm of cc , label = right:$v_0$](cp ) ; \coordinate[below=0.1 cm of cc , label = right:$v_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( ap ) sin ( dp ) ; \draw[particle ] ( ap ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( bp ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture}+ \!\!\!\!\begin{tikzpicture}[node distance=0.6 cm and 0.8 cm ] \coordinate [ ] ( b2 ) ; \coordinate[right = of b2,label = left:$\xi_{t_2}$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$w$,label = above:$v$ ] ( ap ) ; \coordinate[left = of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of ap](cc ) ; \coordinate[above=0.3 cm of cc , label = right:$v_0$](cp ) ; \coordinate[below=0.1 cm of cc , label = right:$w_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( ap ) sin ( dp ) ; \draw[particle ] ( ap ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( bp ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture}\\ { \nonumber}&+\!\!\!\begin{tikzpicture}[node distance=0.6 cm and 0.8 cm ] \coordinate [ ] ( b2 ) ; \coordinate[right = of b2,label = left:$\xi_{t_2}$ ] ( bp ) ; \coordinate[above = of b2,label = above:$w_{t_1}$](a1 ) ; \coordinate[above = of bp , label = above:$p_{w}$](ap ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( bp ) circle ( .05 cm ) ; \end{tikzpicture}\,+\!\!\!\ ! \begin{tikzpicture}[node distance=0.6 cm and 0.8 cm ] \coordinate [ ] ( b2 ) ; \coordinate[right = of b2,label = left:$\xi_{t_2}$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$w$,label = above:$w$ ] ( ap ) ; \coordinate[left = of ap , label = above:$w_{t_1}$ ] ( a1 ) ; \coordinate[right = of ap](cc ) ; \coordinate[above=0.3 cm of cc , label = right:$w_0$](cp ) ; \coordinate[below=0.1 cm of cc , label = right:$w_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( ap ) sin ( dp ) ; \draw[particle ] ( ap ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( bp ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture}+ \!\!\!\!\begin{tikzpicture}[node distance=0.6 cm and 0.8 cm ] \coordinate [ ] ( b2 ) ; \coordinate[right = of b2,label = left:$\xi_{t_2}$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$v$,label = above:$w$ ] ( ap ) ; \coordinate[left = of ap , label = above:$w_{t_1}$ ] ( a1 ) ; \coordinate[right = of ap](cc ) ; \coordinate[above=0.3 cm of cc , label = right:$w_0$](cp ) ; \coordinate[below=0.1 cm of cc , label = right:$v_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( ap ) sin ( dp ) ; \draw[particle ] ( ap ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( bp ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture},\\ { \nonumber}= & \,\kappa_2 \!\!\int \!\!{\mathrm{d}}t ' g_{{v}}(t_1,t ' ) g_{\xi}(t',t_2)\\ { \nonumber}&+\alpha { v}_i^2\!\ ! \int\!\ ! { \mathrm{d}}t ' g_{{v}}(t_1 , t')g_{\xi}(t',t_2)g_{{v}}(t ' , 0)^2 + \cdots,\\ { \nonumber}=&\ , e^{{\lambda}_2 t_1}\left(\kappa_2 e^{-{\lambda}_2 t_2}+\alpha { v}_i^2 e^{{\lambda}_2 t_2}+\alpha { v}_i { w}_i e^{{\lambda}_3 t_2}\right)\\ & + e^{{\lambda}_3 t_1}\left(\kappa_3 e^{-{\lambda}_3 t_2}+\alpha { w}_i^2 e^{{\lambda}_3 t_2}+\alpha { v}_i { w}_i e^ { { \lambda}_2 t_2}\right),\end{aligned}\ ] ] where the superscript @xmath363 in the first line indicates the number of loops in the expansion , and , in the third line , we show only the integral form of the first two diagrams . this result involves diagrams with first order of the small noise parameter @xmath226 . the definitions of the parameters such as @xmath364 are defined in the discussion of eqs . - . we note that this result eq . is valid for @xmath304 and it is zero otherwise , because of the special properties of the left continuous heaviside step function @xmath296 . this can be interpreted as the noise being correlated with the qubit state at later times but not with the state at earlier times . this property of the system variable and noise the correlation function is also mentioned in ref . @xcite . let us next consider correlation functions between system variables at two points in time , such as @xmath365 and @xmath366 . both of them can be written in terms of correlation functions between the transformed system variables @xmath367 , for example , the @xmath90-@xmath90-correlation is , @xmath368 . as in the previous example , we keep terms up to the lowest order of loops , which is the zeroth order . each term of the correlation functions , for example @xmath369 , involves 10 different diagrams , thus there are in total @xmath370 diagrams for computing the @xmath90-@xmath90 correlation function . we present in the following three samples of diagrams and their integral forms for the correlation function @xmath371 , @xmath372 \coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$p_{v0}$ ] ( bp ) ; \coordinate[above = of b2,label = above:$v_{t_1}$](a1 ) ; \coordinate[above = of bp , label = above:$p_{v0}$](ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw ( bp ) circle ( .06 cm ) ; \draw ( ap ) circle ( .06 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \end{tikzpicture}\,+\begin{tikzpicture}[node distance=0.6 cm and 0.8 cm ] \coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$p_v$ ] ( bp ) ; \coordinate[above = of bp , label = above:$p_v$ ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \end{tikzpicture}\,+ \,\,\begin{tikzpicture}[node distance=0.6 cm and 0.8 cm ] \coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$p_v$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$v$,label = above:$v$ ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of ap](cc ) ; \coordinate[above=0.3 cm of cc , label = right:$v_0$](cp ) ; \coordinate[below=0.1 cm of cc , label = right:$v_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( ap ) sin ( dp ) ; \draw[particle ] ( ap ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture}\\ { \nonumber } & + \text{(7 more similar diagrams)},\\ { \nonumber}= & \,\la { v}(t_1 ) \ra \la { v}(t_2 ) \ra \\ { \nonumber}&+\kappa_2 ^ 2\!\ ! \int\!\ ! { \mathrm{d}}t ' { \mathrm{d}}t '' g_{{v}}(t_1 , t')g_{{v}}(t_2 , t'')g_{\xi}(t ' , t'')\\ { \nonumber}&+\alpha \kappa_2 { v}_i^2 \!\!\int\!\ ! { \mathrm{d}}t ' { \mathrm{d}}t''\big\{g_{{v}}(t_1,t')g_{{v}}(t_2,t'')\\ & \qquad \quad g_{\xi}(t ' , t'')g_{{v}}(t',0)g_{{v}}(t',0)\big\ } + \cdots,\end{aligned}\ ] ] where , apart from the first trivial diagram , each initial vertex ( @xmath373 or @xmath374 ) can connect to three possible interaction vertices ( @xmath245 , @xmath239 , or @xmath240 ) . as a result , there are total of @xmath375 possible ways , and they are presented in appendix [ app - fullform ] ( all diagrams except the first one are of the first order of @xmath226 ) . the similar kind of calculation is applied to the other correlation functions @xmath376 , @xmath377 and @xmath378 . the correlation function between @xmath93 and @xmath90 can be calculated in a similar way , with the same types of diagrams , but with extra prefactors @xmath379 and @xmath380 , @xmath381 where the coefficients @xmath382 and @xmath383 come from the transformation of variables shown in eq . . we compare the theoretical results , specifically the @xmath90-@xmath90 correlation computed in the previous section , with numerically simulated quantum trajectories . although , the solution of @xmath365 is not explicitly presented here in this paper , as it is too lengthy , it can be computed in a similar way as shown in eq . and also in appendix [ app - fullform ] . we show in figure [ fig - corrlimit](a ) and [ fig - corrlimit](d ) sampled individual trajectories generated from the monte carlo method ( appendix [ app - numer ] ) , in the regime where @xmath384 . the theoretical variances are computed from the @xmath90-@xmath90 correlation function , and they are shown in panels ( b ) and ( e ) along with the variances from the numerical trajectories . in the top panel , for efficient detection @xmath385 , the numerical variance increases in time and gets saturated ( not shown ) at around a value of @xmath386 for a long enough time , indicating that the qubit states at the later times are distributed throughout the perimeter of the @xmath93-@xmath90 plane of the bloch sphere . the variance from the tree - level diagram approximation fails to exactly capture the long - time behaviour , as we can see that the discrepancy in the panel ( b ) starts to grow as time increases . however , for the inefficient detection case , @xmath387 in figure [ fig - corrlimit](e ) , we can see that the theoretical approximation can explain the behaviour quite well , predicting the saturated variance at a value of @xmath388 . to compute this quantity theoretically , one takes a limit @xmath389 of the calculated variance @xmath390 . for the correlation function at two different times , we define a covariance as @xmath391 \equiv \langle z(t_1)z(t_2 ) \rangle - \langle z(t_1 ) \rangle \langle z(t_2 ) \rangle$ ] , and show in figure [ fig - corrlimit](c ) and [ fig - corrlimit](f ) its numerical and theoretical comparisons . we plot the covariance for @xmath392 and @xmath393 where @xmath394 has its value ranged from @xmath395 to @xmath396 . for the efficient detection case , shown in the panel ( c ) , the agreement between the theory and the simulation is better for the short time case , @xmath397 . for the inefficient detection case , panel ( f ) , the agreement is excellent for both @xmath398 and @xmath399 . since our theoretical results are derived with the small noise expansion around the saddle point solution ( or the average solution , see section [ sec - averagesol ] ) using @xmath226 as an expansion tracking parameter , the diagrammatic approximation method works well whenever the qubit trajectories are narrowly distributed around its average . this happens in the short - time regime when the diffusion is still small from the initial state , as well as in the regime when the dynamics is strongly suppressed by the dephasing mechanism ( when the detection efficiency is low ) . we note that for the long - time limit , one needs to compute higher order terms which contain more complex diagrams . however , there are other approaches , such as using the qubit master equation to approximate the correlation functions in the stationary limit and calculate spectral densities of the measurement readouts . these can be found in the works on continuous measurement of mesoscopic electronics such as in refs . @xcite . comparing to these approaches , our results give much more accurate correlation functions at short times . we can extend the discussion of the stochastic path integral formalism to its application to the system under continuous measurement with feedback control . the feedback loop consists of getting information about the system state via the measurement , and feeding back a control signal to the system in order to alter the state as desired . one of the most intuitive models of the feedback loop is that the system hamiltonian changes as a function of the system state , which then can be written as a function of the most recent measurement readout . for example , in our qubit measurement case , taking into account additional time delay @xmath400 , the hamiltonian at any time @xmath217 can be written as a function of the measurement readout in the past , @xmath401 , where the parameters @xmath101 and @xmath402 , as defined in section [ sec - qubit ] , are now functions of the measurement readout @xmath403 at time @xmath404 . we consider an ideal case with instantaneous feedback @xmath405 , i.e. , the measurement readout at time @xmath217 immediately changes the system parameters which are used in computing the state update at the time . this way , the formulation of the stochastic path integral as time - local , its action s extremization equations , and the diagrammatic expansion we presented so far are perfectly applicable . the only modification needed is to treat the readout - dependent hamiltonian parameters as functions of @xmath406 such as @xmath407 and @xmath408 ( or as functions of the quantum state at the time ) . in this section , we present a few examples of the continuous measurement of a qubit with feedback , one with a linear feedback in the form @xmath409 , and another with a state - dependent linear feedback to stabilize a qubit s oscillation . let us consider an instantaneous feedback loop in a qubit measurement introduced in section [ sec - qubit ] , with @xmath410 and the qubit rabi frequency being a function of the measurement readout . we assume a linear form of the rabi frequency as @xmath411 where @xmath406 is the readout as a function of time and @xmath412 are constant parameters characterizing the bare rabi oscillation and the linear feedback , respectively . we will see later in this subsection that this type of feedback can stabilize pre - determined ( arbitrary ) quantum states . this feedback loop in eq . has the advantage of fast processing because the qubit hamiltonian depends directly on the value of the measurement readout , requiring no knowledge of the qubit state . for simplicity , we limit our discussion to an ideal case in which the qubit is measured with an efficient detector and no extra environment dephasing . in this regime , we can re - parametrize the qubit bloch vector into a single state parameter @xmath413 where @xmath414 and @xmath415 ( we set @xmath416 in this case ) . we note that , with this linear feedback eq . , a typical step size for the qubit state defined as @xmath417 has average drift @xmath418 and diffusion @xmath419 of the first order of the time step . thus , the step size is still small . the statistics of quantum trajectories with this instantaneous feedback can be investigated following the same procedure presented in previous sections for the no - feedback qubit measurement . we start with writing the joint probability distribution of the trajectories , and then transform it into a path integral form . the action of the stochastic path integral with the feedback rabi frequency @xmath420 is given by , @xmath421 where @xmath422 , @xmath413 , and @xmath94 are functions of time , omitting the time argument . this is a generalization of the action given in ref . @xcite where the quantum jump in qubit measurement is analyzed . we note here that in eq . , we have chosen to use the @xmath70-expansion in both the state update and the readout probability distribution , because we are interested in investigating the action - extremized solutions , the optimal paths of the system . the optimal paths of the action in eq . are obtained by extremizing the action over all variables @xmath422 , @xmath413 , and @xmath94 , leading to two ordinary differential equations and one constraint , [ eq - fbodes ] @xmath423 with arbitrary boundary conditions on the variable @xmath413 and @xmath422 . the solutions of these equations are the most likely paths of the system . by writing the action in the form @xmath424)$ ] , where @xmath425 is a _ stochastic hamiltonian _ , we can examine the most likely paths by using phase space analysis @xcite , motivated by the phase space concept in classical mechanics . substituting the @xmath94 constraint in eqs . into the action ( equivalent to integrating the path integral over the variable @xmath426 ) , we obtain the action and the stochastic hamiltonian in terms of only the system variable @xmath413 and its conjugate @xmath422 , @xmath427 this quantity is explicitly time - independent and so it is a constant of motion for the most likely path , a solution of the ordinary differential equations in eqs . let us define the stochastic energy @xmath428 , and then solve for the conjugate variable @xmath429 as a function of @xmath413 and @xmath333 from eq . . each value of @xmath333 will then correspond to a curve in the phase space portrait , describing dynamics of the most likely paths , the same way the individual classical trajectories are depicted as constant - energy curves on a phase space plot . as an example , we show in figure [ fig - phasespace](a ) the phase space portrait for the pure linear feedback case ( i.e. , @xmath430 ) , where we plot the conjugate variable @xmath422 as a function of @xmath413 , @xmath431 for different values of @xmath333 . this is an interesting case . we can see from the phase space plot in figure [ fig - phasespace](a ) that there exists some attractors to which all states eventually limit to . these attractors coincide with the divergence of the conjugate variable @xmath429 in eq . , and also appear as stationary points where @xmath432 in eqs . . solving these equations , we obtain the attractors are located at @xmath433 and @xmath434 where @xmath435 is an even integer . these attractors may be interpreted as stabilized states achieved by turning on a linear feedback @xmath436 and turning off the bare qubit frequency @xmath437 . they are effectively the new collapse points , rather than at the poles of the bloch sphere , i.e. , @xmath438 ( @xmath439 ) . considering only angles between @xmath395 and @xmath440 , for the positive feedback @xmath441 , as @xmath442 increases from zero , the stabilized states @xmath443 move from @xmath395 and @xmath444 toward each other and coalesce at @xmath445 when @xmath446 ; whereas in the negative feedback @xmath447 , as @xmath442 grows , the stabilized states move toward each other in the region of @xmath444 and @xmath448 and coalesce at @xmath449 when @xmath450 . we simulate numerical qubit trajectories using the monte carlo method ( appendix [ app - numer ] ) , showing that individual trajectories initialized at different states are eventually pinned to the stabilized states as predicted from the most likely path phase space . this is shown in figure [ fig - phasespace](b ) for three different initial conditions , along with their most likely paths . in this subsection , we show an example using the path integral to compute a correlation function for a system with linear feedback loop . the feedback loop of this example resembles the one used in the solid - state qubit measurement in ref . @xcite which has been adapted and realized in the transmon qubit experiment @xcite . the theoretical setup is an oscillating qubit ( a double quantum dot ) continuously monitored by a near quantum - limited detector ( a quantum point contact ) , of which the hamiltonian and the measurement operator are in the same form as presented in section [ sec - qubit ] . we as before consider the ideal symmetric qubit case where @xmath410 , and the qubit is measured with an efficient detector with no extra environment dephasing . the qubit state is represented by a single parameter @xmath413 where @xmath451 and @xmath452 . the stochastic master equation for the qubit state , in the @xmath70-expansion similar to eqs . ( stratonovich form , as discussed in ref . @xcite ) , is given by , @xmath453 where @xmath413 and @xmath71 are functions of time and @xmath454 is the feedback rabi frequency , which is assumed to be a function of the qubit state @xmath413 . we later neglect the second term on the right side of eq . because we consider a diffusive rabi limit @xmath455 . the feedback protocol considered in this subsection ( and also in ref @xcite ) is a linear feedback designed to stabilize the quantum oscillation of the qubit state , against the random phase kicks due to the measurement . the desired qubit evolution is described by @xmath456 and @xmath457 where we define @xmath458 as the target oscillation frequency . the difference between the actual phase @xmath413 and the target phase @xmath459 , denoted as @xmath460 , is used to control the oscillating part of the qubit hamiltonian . therefore , we write the feedback rabi frequency as , @xmath461 , where @xmath462 is the dimensionless feedback factor . this feedback loop continuously corrects the random changes of the state made by the measurement so that the outcome trajectory closely follows the desired qubit oscillation . let us assume that the phase difference @xmath463 is a slowly changing variable as compared to the oscillation with the desired frequency @xmath458 . therefore , we can average the fluctuating process described in eq . over the oscillation period @xmath464 , taking @xmath463 to be constant during this period . the period @xmath465 will eventually be our new time scale . we define a new noise @xmath466 as a time - average of the last term of eq . over the oscillation period , @xmath467 with its zero ensemble average @xmath468 and its variance being @xmath469 . we then can simplify the differential equation eq . to one in terms of the phase difference @xmath463 , @xmath470 with the new time scale @xmath465 . here we will use the above differential equation eq . to compute a correlation function @xmath471 using the path integral approach . following the derivation of the action in eq . and where @xmath472 and @xmath473 are now our new system variable and a noise probability density function , we obtain the action in this form , @xmath474 omitting the boundary terms . note that we have written the pure imaginary conjugate variable ( @xmath475 ) explicitly with @xmath476 . as before , we can compute an average quantity by integrating the paths @xmath477 . so , we first write the correlation function @xmath478 in terms of the phase difference @xmath463 , then average over the oscillation period @xmath465 , getting rid of the fast fluctuating parts , and we are left with @xmath479\ra \cos ( \delta_d \tau)/2 + \sin[\delta \theta(t)-\delta \theta(t+\tau)]\ra \sin ( \delta_d \tau)/2 $ ] . the correlation function apparently can be written in terms of the real part and imaginary part of the following quantity , @xmath480 where we have used the same notations , such as @xmath481 for the integral over all possible paths @xmath482 , and @xmath38 for a normalized factor . the gaussian integral over @xmath466 in eq is quite straightforward resulting in a bilinear term in @xmath483 , which then leads to another gaussian integral of @xmath483 . as one would expect , these integrals generate another prefactor that cancels the normalized factor @xmath38 . consequently , the last integral over @xmath463 is left as @xmath484 where the exponent @xmath485 is given by , @xmath486 this effective action can be transformed further using the integration by parts , for example , @xmath487 , assuming that the boundary conditions for @xmath463 at both end points vanish . we then write the action in terms of the inverse green s function and the source term as discussed in section [ sec - briefintro ] , @xmath488 where the inverse green s function , the green s function , and a particular form of the source term are given by , @xmath489 performing the last gaussian functional integral over @xmath463 gives , @xmath490 for the positive value of the time difference @xmath394 . this quantity is a real number , therefore we obtain the correlation function , @xmath491 assuming that @xmath492 . this result agrees with the solution found in ref . @xcite which is presented in different notation . we have developed and extended the stochastic path integral technique to study statistical behaviour of a quantum system under weak continuous measurement , as well as measurement with feedback , presented with several qubit examples . the path integral approach is constructed based on the joint probability distribution of the measurement records , which is then extended to the distribution of quantum states , describing all possible quantum trajectories . we have shown that with this path integral and its action formalism , the optimal dynamics , such as the most likely paths , can be obtained naturally from the extremization of the action , whereas other statistical quantities can be achieved from direct integration or perturbation theory . in the case of plain measurement of a qubit , we have derived analytic solutions for the average trajectory , the variance , and the correlation functions conditioning on the fixed initial and final states , which show an excellent agreement with the numerically simulated data . we have also presented a diagrammatic perturbation method used in computing expectation values and correlation functions of quantum trajectories , and elaborated it with examples of the qubit with rabi oscillation case . the variances and multi - time correlation functions of qubit trajectories in the short - time regime have been revealed using this method given initial conditions , and the results are in good agreement with the numerical simulation . moreover , we have considered quantum measurement with feedback control , using the action principle to investigate the dynamics of the most likely paths of a qubit with linear feedback on its oscillating frequency . we have discovered that the direct linear feedback , manipulating the qubit hamiltonian instantly using the measurement readout , can stabilize the qubit state to arbitrarily chosen pure states . we have also considered the example of the feedback loop stabilizing the qubit rabi frequency introduced in ref . @xcite , and we have computed the correlation function for the qubit trajectory using the path integration method . so far , the stochastic path integral formalism in the context of continuous quantum measurement has been proven to be useful in studying the statistics of quantum trajectories ; however , there are some unsolved issues that need to be further explored . one is the limitation of the statistical average solutions derived from the perturbation expansion theory . only the first few orders of the expansion have been computed , resulting in the solutions that are valid only in the certain parameter regimes . we hope to find solutions in an arbitrary regime , possibly with some modifications of our approach . another issue is the assumption of the instantaneous feedback , which can be difficult to realize in experiments . feedback loops modelled with time delays will be taken into account in future work . we thank j. dressel for discussions , helpful comments on the manuscript , and for collaborating in our first paper on this subject @xcite . we thank i. siddiqi and s. g. rajeev for discussions . this work was supported by us army research office grants no . w911nf-09 - 0 - 01417 and no . w911nf-15 - 1 - 0496 , by national science foundation grant dmr-1506081 , by john templeton foundation grant i d 58558 , and by development and promotion of science and technology talents project thailand . there are interesting comparisons between the path integral formalism we introduced in the main text and other path integral approaches built upon feynman path integral in quantum mechanics @xcite . as we have shown , our formalism is aimed at describing the probability distribution of quantum trajectories , paths of state of a system under continuous measurement , on its quantum state space ( such as hilbert space for pure states ) . each individual quantum trajectory in the path integral is realizable and tractable , as demonstrated experimentally , such as , in solid - state systems @xcite . however , for other path integral formalisms developed from the feynman path integral to investigate quantum systems under measurement @xcite , the evolution of the quantum state ( or wavefunction ) is based on interference of all possible classical paths in measurable configuration space ( such as the system s positions or spin states ) . thus , in this latter case , the integrations are over configuration coordinates of the measured system . despite the differences in forms , the two approaches mentioned above can be related . the path integral via feynman s concept can be used to compute the probability distribution of the measurement results , which is then , as shown in the main text , one of the most important ingredients in constructing our stochastic paths in quantum state space . to elaborate this connection in more detail , we consider the path integral method presented by caves @xcite , for measurements providing information about the position @xmath493 of a nonrelativistic , one - dimensional quantum system , evolved in time . in that approach , the effect of measurements is to restrict the sum over paths by weighting each path differently depending on the measurement results . these weights appear in the path integral in ref . @xcite as the ` resolution amplitude ' , @xmath494 , which also accounts for the imprecision of the measurements . let us assume instantaneous position measurements that are equally distributed in times , at @xmath495 where @xmath496 , giving measurement readouts denoted by @xmath497 . the joint probability amplitude @xmath498 of the measurement records , and that the system is at @xmath499 at time @xmath500 , given an initial wavefunction @xmath501 , is in this form , @xmath502 where the position coordinates are denoted by @xmath503 at time @xmath504 , and the integral measure is defined as @xmath505 . we note here that we have modified the ordering of the measurements from the original version by caves , in which he assumed that the measurements start only after the initial state has evolved for time @xmath70 . we assume that this change has an infinitesimal effect on the probability amplitude as @xmath60 . the first bracketed term on the right hand side of eq . describes the influence of the measurement . without it , the path integral will be the usual feynman path integral , exactly equal to @xmath506 or the wave function at the final time @xmath150 . from eq . , the joint probability distribution function ( pdf ) of the sequence of measurement readouts can be obtained by integrating the square of the probability amplitude over the final coordinate @xmath499 , @xmath507 given the initial wavefunction @xmath501 . in order to see that this joint pdf in eq . is the same as what we have earlier in the main text ( in this special case ) , we first rewrite this term @xmath508 as a probability amplitude right after one measurement ( @xmath509 ) and one unitary transformation ( @xmath510 ) . then , we obtain a probability density function for the first measurement outcome as @xmath511 . we further introduce an updated wavefunction @xmath512 , a result of a state @xmath513 that has gone through one measurement @xmath514 and one unitary transformation @xmath510 , @xmath515 where the denominator assures the correct norm of the wavefunction . then , the joint distribution of the measurement readouts of the first two times is given by @xmath516 . repeating this procedure further to the next measurement readouts @xmath517 , at the end , we obtain the full joint pdf , @xmath518 as a product of the conditional probability density functions , defined in a general form as @xmath519 . this is the joint pdf of the measurement readouts , the main component of the joint pdf in eq . . to obtain an analogous form of the joint pdf in eq . , we add probability density functions of the quantum state ( or wavefunction ) as delta functions to every single time step . we obtain the full joint pdf , @xmath520 where the update state in this case is written as @xmath521 as in eq . . the integer @xmath6 is the dimension of the vector @xmath522 ( which maybe generalized to infinite dimension via a functional form of the @xmath523-function ) , and @xmath524 is called a joint probability density function of the measurement outcomes @xmath525 and the wavefunctions @xmath526 of the measured system . we note that our path integral presented in the main text deals with mixed states instead of pure states , and the system we consider is the qubit ( or spin ) system with the discrete basis , i.e. , @xmath527 . therefore , the operator @xmath514 in the discussion above is equivalent to what we have as the measurement operator @xmath528 , defined in section [ sec - qubit ] , and the resolution function is a gaussian function , @xmath529 for @xmath530 and @xmath531 . it is also worth mentioning that in our approach , the stochastic trajectories in quantum state spaces can be thought of as classical trajectories in configuration space , such as trajectories on the bloch sphere can be considered as three - dimensional random walks in a unit - radius sphere . then , one could find connections between our path integral and the formalism in classical stochastic processes , such as the pilgram - sukhorukov - jordan - bttiker path integral @xcite , the martin - siggia - rose formalism , the wiener integral , or feynman - kac path integral ( see eqs . , , and ) , however , we do not cover the discussion in this paper . we show the derivation here , how the action in eq . can be written in this form , @xmath532 = { \cal s}[\bar{u } ] - \frac{\tau_m}{2 \delta t } \sum_{k=0}^{n-1}(\eta_{k+1}-\eta_k)^2,\end{aligned}\ ] ] where @xmath533 is the optimal path , extremizing the action @xmath128 $ ] . to see this , one can taylor expand the action eq . in discrete forms , such as @xmath128 = { \cal s}[\bar{u } + \eta ] = { \cal s}[\bar{u } ] + { \cal s}'[{\bar u}]\eta + \frac{1}{2}{\cal s}'[{\bar u}]\eta^2 + o(\eta^3)$ ] , and show that higher order terms vanish ( @xmath534 ) . however , there is a simpler way to see this , using the action in a continuous form , @xmath535 which is the same action as in eq . , with definitions of the time integral , @xmath536 , and the derivative , @xmath537 . in this form , one can see that the last two terms in the action can be integrated easily . the second term can be written in this general form , @xmath538 where @xmath539 . the contribution of this term to the action is only dependent on the boundary terms . therefore , we can write the last two terms in eq . as equivalent to @xmath540 . after this simplification , the first term in eq . can be written as @xmath541 , performing the integration by parts giving @xmath542 . the action is then given by , @xmath532 = { \cal s}[{\bar u } ] - \frac{\tau_m}{2}\!\!\ ! \int \!\!\ ! { \mathrm{d}}t \,{\dot \eta}(t)^2,\end{aligned}\ ] ] where , @xmath543 = - \int_0^t \!\!\!{\mathrm{d}}t \left\ { \frac{\tau_m}{2 } { \dot { \bar u}}(t)^2 - \tanh { \bar u}(t ) { \dot { \bar u}}(t ) + \frac{1}{2 \tau_m } \right\},\end{aligned}\ ] ] is the action in terms of the optimal solution @xmath544 . we show here the inverse of the matrix @xmath144 in eq . , assuming that it is a square matrix with @xmath6 dimensions . the inverse matrix @xmath167 is given by , @xmath545 where the diagonal elements are of the form @xmath546 , where @xmath547 , and the other off - diagonal elements are @xmath176 for @xmath177 as verified by direct calculation . the solutions of the preselected and postselected average and variance in @xmath90 to infinite order are shown here . the average is given in terms of the infinite summation as , @xmath548 where the differentials in the bracket are evaluated at the optimal path @xmath533 . similarly , the variance is given by , @xmath549 the average @xmath550 are given explicitly in eq . in the main text . the numerical data presented in figure [ fig - qnd ] , [ fig - corrlimit ] and [ fig - phasespace ] are from the simulation of quantum trajectories using monte carlo method . the numerical trajectories are generated in @xmath0-discrete steps of @xmath70 . starting from an initial state @xmath16 , each step , we randomly generate a measurement readout from a distribution , for example , @xmath551 , and compute a quantum state from the update equation @xmath552 $ ] ( or @xmath109 $ ] ) , repeating the procedures from @xmath553 to @xmath28 to get a full trajectory @xmath554 . in figure [ fig - qnd ] , we simulate the data using @xmath555 time steps , with the step size @xmath556 and @xmath557 . in figure [ fig - corrlimit ] , we use @xmath558 , @xmath559 , and @xmath560 , where the rabi frequency @xmath402 is set to @xmath448 . we note that in the main text we present these numbers in the unit of @xmath80 . for the data presented in figure [ fig - phasespace ] , where the rabi oscillation is linearly dependent on the highly fluctuating measurement readouts , the state update equation needs to be modified to minimize numerical errors in each time step of the calculation . the update state is then computed from @xmath561 $ ] , where the measurement operator @xmath562 and unitary operator @xmath563 are each divided into 10 pieces and are operated onto the qubit state @xmath564 alternately . we simulate the data using @xmath565 time steps , with the step size @xmath559 and @xmath557 . the feedback rabi oscillation is @xmath566 , where @xmath2 is a generated measurement readout at time @xmath3 . in the main text , the state updating procedure is based on the assumption that a measurement outcome is randomly generated from a readout distribution @xmath19 , which is a function of a state right before the measurement . however , in the limit when the measurement readout is highly fluctuating , for example , the standard deviation of the readout distribution is much larger than the measurement response , @xmath567 , we can approximate the readout as a sum of two parts , one being its average ( which is related to the measured system state ) and another being a zero - mean independent fluctuating noise . following the readout distribution in eq . , the average of the readout at a time @xmath3 is exactly @xmath568 , therefore , we write @xmath569 where @xmath570 is a zero - mean gaussian white noise . the next step is to find a probability distribution for the independent noise @xmath570 , which we then approximate as having the same variance as the original distribution , which in this case the variance is @xmath571 . this approximation is valid as long as the variance is much larger than the separation between two outcomes @xmath572 and @xmath573 . a more rigorous proof can be done by writing @xmath574 where @xmath71 is a gaussian white noise with variance @xmath252 . note that @xmath71 is the time - derivative of the wiener increment , scaling as @xmath575 ( it calculus ) @xcite , therefore , in any expansion , we need to keep terms containing @xmath576 . using these rules , we can write , @xmath577 where these two equations are exactly equal in the expansion up to the first order of @xmath70 . the second line confirms that the distribution of @xmath570 is gaussian with the same variance @xmath571 and the mean @xmath123 . let us assume that @xmath92 is the only system variable we consider , where an update equation is written as @xmath578 and the probability density function for a measurement readout is proportional to @xmath579 . we then write the action for this system as , @xmath580 where @xmath184 is the conjugate variable . in this case , we could write the first term , @xmath581 , where @xmath582 , however , the matrix @xmath583 would not be a square matrix , and its inverse is not simply defined . therefore , we separate out one term , @xmath584 , from the sum , which symmetrizes the double sum , @xmath585 resulting in a square matrix @xmath586 , @xmath587 with the row index , @xmath588 to @xmath28 , and the column index , @xmath589 to @xmath0 . for the measurement readout term , we define its square matrix , @xmath590 with the row and column index being @xmath591 to @xmath28 . as a result , we can rewrite the action @xmath34 into two separated terms , a free action @xmath592 and an interaction action @xmath201 , where @xmath593 the appearance of the first term in the interaction action , @xmath594 , is the reason we have the extra terms ( we represented them as @xmath281 ) in eq . . we now show why the two - point correlation functions derived from our path integrals , such as the one derived from the free action in eq . , that is @xmath595 consist of a left continuous heaviside step functions @xmath296 that behaves differently from the usual heaviside step function . this left continuous heaviside step function has the properties , @xmath297 and @xmath596 , while the usual heaviside step function has properties , @xmath597 , @xmath596 , and @xmath598 . this is true from a point of view of the discrete form of a correlation function , @xmath599 , that the propagator has this property @xmath600 when @xmath601 and it vanishes otherwise . let us start from writing a free generating function in a discrete form , @xmath602 $ ] @xmath603= & \frac{\left(\frac{\delta t}{2 \pi}\right)^{\frac{n}{2}}}{(2 \pi i)^{n}}\int \!\!{\mathrm{d}}[x_k]_1^n \int \!\!{\mathrm{d}}[p_k]_0^{n-1 } \exp\left\{-\sum_{i=0}^{n-1}\sum_{j=1}^{n } p_{i}(g^{-1}_x)_{i , j } x_j + \sum_{k=0}^{n-1 } p_k ( j_p)_k \delta t + \sum_{k=1}^{n } ( j_x)_k x_k \delta t\right\}\\ & \times \int { \mathrm{d}}[\xi_k]_0^{n-1 } \exp\left\{- \frac{1}{2}\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}\xi_i ( g^{-1}_\xi)_{i , j } \xi_j + \sum_{k=0}^{n-1 } \xi_k ( j_\xi)_k \delta t \right\}.\end{aligned}\ ] ] these matrix integrals can be carried out using multi - dimensional gaussian integrals . the first one is , @xmath604},\end{aligned}\ ] ] where @xmath184 is a pure imaginary @xmath6-dimensional vector , @xmath605 , are real vectors , and @xmath606 is a real and invertible matrix . another is , @xmath607)^{1/2}},\end{aligned}\ ] ] where @xmath71 is a real @xmath6-dimensional vector and @xmath281 is a real and symmetric matrix . after integrating over all variables , the free generating function is given by , @xmath608 we note that the integrals generate a prefactor @xmath609 , knowing that @xmath610 , and another prefactor @xmath611 , which both are then canceled with the prefactor in eq . . also , from eqs . - , we can compute the inverses of them as , @xmath612 with the row index , @xmath613 to @xmath0 , and the column index , @xmath614 to @xmath28 , and @xmath615 with the row and column indices , @xmath591 to @xmath28 . now , let us compute the two - point correlation function @xmath616 , @xmath617_1^n \,{\mathrm{d}}[p_k]_0^{n-1 } { \mathrm{d}}[\xi_k]_0^{n-1 } ( x_a p_b ) \exp\left\{-\sum_{i=0}^{n-1}\sum_{j=1}^{n } p_{i}(g^{-1}_x)_{i , j } x_j - \frac{1}{2}\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}\xi_i ( g^{-1}_\xi)_{i , j } \xi_j \right\ } \\ \nonumber = & \frac{1}{\delta t}\frac{\partial}{\partial ( j_x)_a}\frac{1}{\delta t } \frac{\partial}{\partial ( j_p)_b } z_f[j_x , j_p , j_\xi ] \bigg |_{j_x , j_p , j_\xi=0 } \\ \nonumber = & \frac{1}{\delta t}\frac{\partial}{\partial ( j_x)_a } \frac{1}{\delta t } \frac{\partial}{\partial ( j_p)_b}\exp\left[\sum_{i=1}^{n } \sum_{j=0}^{n-1 } ( j_x)_i ( g_x)_{i , j } ( j_p)_j \delta t^2\right ] \bigg |_{j_x , j_p=0}\\ = & ( g_x)_{a , b } = \begin{cases } 1 & \text { , if , } a \ge b+1 \text { , or , } a > b , \\ 0 & \text { , otherwise . } \end{cases}\end{aligned}\ ] ] where @xmath618 . the conclusion in the last line comes from the fact that the square matrix , @xmath619 , has the row and column indices different by one time step ( see eq . ) , which actually originates from the indices of , @xmath620 , and @xmath621 . continued from eq . , we presented all 10 diagrams here , @xmath623\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$v_0 $ ] ( bp ) ; \coordinate[above = of b2,label = above:$v_{t_1}$](a1 ) ; \coordinate[above = of bp , label = above:$v_0$](ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw ( bp ) circle ( .06 cm ) ; \draw ( ap ) circle ( .06 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \end{tikzpicture}\,+ \begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2 ] ( bp ) ; \coordinate[above = of bp ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \end{tikzpicture}\,+ \begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2 ] ( bp ) ; \coordinate[above = of bp , label = below right:$v$,label = above:$v$ ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of ap](cc ) ; \coordinate[above=0.3 cm of cc , label = right:$v_0$](cp ) ; \coordinate[below=0.1 cm of cc , label = right:$v_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( ap ) sin ( dp ) ; \draw[particle ] ( ap ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture } + \begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2 ] ( bp ) ; \coordinate[above = of bp , label = below right:$w$,label = above:$v$ ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of ap](cc ) ; \coordinate[above=0.3 cm of cc , label = right:$v_0$](cp ) ; \coordinate[below=0.1 cm of cc , label = right:$w_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( ap ) sin ( dp ) ; \draw[particle ] ( ap ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture } + \begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$v$,label = above right:$v$ ] ( bp ) ; \coordinate[above = of bp ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of bp](cc ) ; \coordinate[above=0.1 cm of cc , label = right:$v_0$](cp ) ; \coordinate[below=0.3 cm of cc , label = right:$v_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( bp ) sin ( dp ) ; \draw[particle ] ( bp ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture } \,+ \begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$v$,label = above right:$w$ ] ( bp ) ; \coordinate[above = of bp ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of bp](cc ) ; \coordinate[above=0.1 cm of cc , label = right:$w_0$](cp ) ; \coordinate[below=0.3 cm of cc , label = right:$v_0$](dp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle2 ] ( bp ) sin ( dp ) ; \draw[particle ] ( bp ) sin ( cp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \end{tikzpicture}\\ { \nonumber}&+\begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$v$,label = above right:$v$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$v$,label = above:$v$ ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of bp](cc ) ; \coordinate[above=0 cm of cc , label = right:$v_0$](cp ) ; \coordinate[below=0.4 cm of cc , label = right:$v_0$](dp ) ; \coordinate[right = of ap](dd ) ; \coordinate[above=0.4 cm of dd , label = right:$v_0$](ep ) ; \coordinate[below=0 cm of dd , label = right:$v_0$](fp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle ] ( bp ) sin ( dp ) ; \draw[particle ] ( bp ) sin ( cp ) ; \draw[particle ] ( ap ) sin ( ep ) ; \draw[particle ] ( ap ) sin ( fp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \draw ( ep ) circle ( .06 cm ) ; \draw ( fp ) circle ( .06 cm ) ; \end{tikzpicture}\ , + \begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$v$,label = above right:$w$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$v$,label = above:$v$ ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of bp](cc ) ; \coordinate[above=0 cm of cc , label = right:$w_0$](cp ) ; \coordinate[below=0.4 cm of cc , label = right:$v_0$](dp ) ; \coordinate[right = of ap](dd ) ; \coordinate[above=0.4 cm of dd , label = right:$v_0$](ep ) ; \coordinate[below=0 cm of dd , label = right:$v_0$](fp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle ] ( bp ) sin ( dp ) ; \draw[particle ] ( bp ) sin ( cp ) ; \draw[particle ] ( ap ) sin ( ep ) ; \draw[particle ] ( ap ) sin ( fp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \draw ( ep ) circle ( .06 cm ) ; \draw ( fp ) circle ( .06 cm ) ; \end{tikzpicture}\ , + \begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$v$,label = above right:$v$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$v$,label = above:$w$ ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of bp](cc ) ; \coordinate[above=0 cm of cc , label = right:$v_0$](cp ) ; \coordinate[below=0.4 cm of cc , label = right:$v_0$](dp ) ; \coordinate[right = of ap](dd ) ; \coordinate[above=0.4 cm of dd , label = right:$w_0$](ep ) ; \coordinate[below=0 cm of dd , label = right:$v_0$](fp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle ] ( bp ) sin ( dp ) ; \draw[particle ] ( bp ) sin ( cp ) ; \draw[particle ] ( ap ) sin ( ep ) ; \draw[particle ] ( ap ) sin ( fp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \draw ( ep ) circle ( .06 cm ) ; \draw ( fp ) circle ( .06 cm ) ; \end{tikzpicture}\,+ \begin{tikzpicture}[node distance=0.6 cm and 0.8cm]\coordinate[label = below:$v_{t_2}$ ] ( b2 ) ; \coordinate[right = of b2,label = below:$v$,label = above right:$w$ ] ( bp ) ; \coordinate[above = of bp , label = below right:$v$,label = above:$w$ ] ( ap ) ; \coordinate[left=1.1 cm of ap , label = above:$v_{t_1}$ ] ( a1 ) ; \coordinate[right = of bp](cc ) ; \coordinate[above=0 cm of cc , label = right:$w_0$](cp ) ; \coordinate[below=0.4 cm of cc , label = right:$v_0$](dp ) ; \coordinate[right = of ap](dd ) ; \coordinate[above=0.4 cm of dd , label = right:$w_0$](ep ) ; \coordinate[below=0 cm of dd , label = right:$v_0$](fp ) ; \draw[particle ] ( a1 ) -- ( ap ) ; \draw[particle ] ( b2 ) -- ( bp ) ; \draw[gluon ] ( ap ) -- ( bp ) ; \draw[particle ] ( bp ) sin ( dp ) ; \draw[particle ] ( bp ) sin ( cp ) ; \draw[particle ] ( ap ) sin ( ep ) ; \draw[particle ] ( ap ) sin ( fp ) ; \fill[black ] ( a1 ) circle ( .05 cm ) ; \fill[black ] ( b2 ) circle ( .05 cm ) ; \draw ( cp ) circle ( .06 cm ) ; \draw ( dp ) circle ( .06 cm ) ; \draw ( ep ) circle ( .06 cm ) ; \draw ( fp ) circle ( .06 cm ) ; \end{tikzpicture},\\ { \nonumber}= & { v}_i^2 \,g_{{v}}(t_1,0 ) g_{{v}}(t_2,0 ) + \int { \mathrm{d}}t ' \left[g_{{v}}(t_1,t ' ) g_{{v}}(t_2,t')\left\{\kappa_2+\alpha { v}_i^2 g_{{v}}(t',0)^2 + \alpha { v}_i { w}_i g_{{v}}(t',0)g_{{w}}(t',0 ) \right\}^2\right],\\ = & { v}_i^2 e^{{\lambda}_2 t_1}e^ { { \lambda}_2 t_2}+ \int { \mathrm{d}}t ' \left [ e^{{\lambda}_2(t_1-t')}e^{{\lambda}_2(t_2-t')}\left\{\kappa_2 + \alpha { v}_i^2 \left(e^{{\lambda}_2 t'}\right)^2 + \alpha { v}_i { w}_i e^{{\lambda}_2 t ' } e^{{\lambda}_3 t ' } \right\}^2 \right],\end{aligned}\ ] ] 70ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop _ ( , ) @noop _ _ ( , ) @noop _ _ ( , ) @noop _ _ ( , ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physreva.87.032115 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.97.166805 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.200401 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.080501 [ * * , ( ) ] link:\doibase 10.1103/physrevb.67.241305 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.112.170501 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physreva.49.2133 [ * * , ( ) ] link:\doibase 10.1103/physreva.62.012105 [ * * , ( ) ] link:\doibase 10.1103/physrevx.3.021008 [ * * , ( ) ] link:\doibase 10.1103/physrevb.66.041401 [ * * , ( ) ] link:\doibase 10.1103/physrevb.72.245322 [ * * , ( ) ] link:\doibase 10.1103/physrevb.71.201305 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physreva.67.030301 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevlett.100.160503 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevd.33.1643 [ * * , ( ) ] link:\doibase 10.1103/physrevd.35.1815 [ * * , ( ) ] link:\doibase 10.1103/physreva.36.5543 [ * * , ( ) ] link:\doibase 10.1007/bf02894935 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.93.260604 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physreve.75.051919 [ * * , ( ) ] link:\doibase 10.1186/s13408 - 015 - 0018 - 5 [ * * , ( ) , 10.1186/s13408 - 015 - 0018 - 5 ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physreva.8.423 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.107.240501 [ * * , ( ) ] link:\doibase 10.1103/physreva.47.1652 [ * * , ( ) ] @noop _ _ ( , ) link:\doibase 10.1103/physrevb.63.085312 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevb.67.075303 [ * * , ( ) ] @noop _ _ , ed . ( , ) \doibase http://dx.doi.org/10.1006/aphy.1996.0052 [ * * , ( ) ]

massive stars dominate the feedback to the local interstellar medium ( ism ) in star - forming galaxies via their stellar winds and ultimate death as core - collapse supernovae . in particular , wolf - rayet ( wr ) stars typically have wind densities an order of magnitude higher than massive o stars . they contribute to the chemical enrichment of galaxies , they are the prime candidates for the immediate progenitors of long , soft gamma ray bursts ( grbs , woosley & bloom 2006 ) , and they provide a signature of high - mass star formation in galaxies ( schaerer & vacca 1998 ) . spectroscopically , wr stars are spectacular in appearance , with strong , broad emission lines instead of the narrow absorption lines which are typical of normal stellar populations ( e.g. beals 1940 ) . the class are named after wolf & rayet ( 1867 ) who identified three stars in cygnus with such broad emission lines . it was immediately apparent that their spectra came in two flavours , subsequently identified as those with strong lines of helium and nitrogen ( wn subtypes ) and those with strong helium , carbon , and oxygen ( wc and wo subtypes ) . gamov ( 1943 ) first suggested that the anomalous composition of wr stars was the result of nuclear processed material being visible on their surfaces , although this was not universally established until the final decade of the 20th century ( lamers et al . 1991 ) . specifically , wn and wc stars show the products of the cno cycle ( h - burning ) and the triple-@xmath2 ( he - burning ) , respectively . in reality , there is a continuity of physical and chemical properties between o supergiants and wn subtypes . typically , wr stars have masses of 1025 @xmath1 , and are descended from o - type stars . they spend @xmath010% of their @xmath05myr lifetime as wr stars ( meynet & maeder 2005 ) . at solar metallicity the minimum initial mass for a star to become a wr star is @xmath025 @xmath1 . this corresponds closely to the humphreys & davidson ( 1979 ) limit for red supergiants ( rsg ) , according to a comparison between the current temperature calibration of rsg and stellar models that allow for mass - loss and rotation ( e.g. levesque et al . 2005 ) . consequently , some single wr stars are post - red supergiants within a fairly limited mass range of probably 2530@xmath1 . evolution proceeds via an intermediate luminous blue variable ( lbv ) phase above 30@xmath1 . for close binaries , the critical mass for production of a wr star has no such robust lower limit , since roche lobe overflow or common envelope evolution could produce a wr star instead of an extended rsg phase . the strong , broad emission lines seen in spectra of wr stars are due to their powerful stellar winds . the wind is sufficiently dense that an optical depth of unity in the continuum arises in the outflowing material . the spectral features are formed far out in the wind and are seen primarily in emission . the line and continuum formation regions are geometrically extended compared to the stellar radii and their physical depths are highly wavelength dependent . the unique spectroscopic signature of wr stars has permitted their detection individually in local group galaxies ( e.g. massey & johnson 1998 ; massey 2003 ) , collectively within knots of local star forming galaxies ( e.g. hadfield & crowther 2006 ) , and as significant contributors to the average rest - frame uv spectrum of lyman break galaxies ( shapley et al . 2003 ) . the present review focuses on observational properties of classical wolf - rayet stars in the milky way and beyond , plus physical and chemical properties determined from spectroscopic analysis , plus comparisons with interior evolutionary models , and provides revisions to the topic with respect to the excellent abbott & conti ( 1987 ) review . low mass ( @xmath3 ) central stars of planetary nebulae displaying a wolf - rayet spectroscopic appearance ( denoted [ wr ] ) are not considered . nevertheless , analysis tools discussed here are common to both types of star ( e.g. crowther et al . visual spectral classification of wr stars is based on emission line strengths and line ratios following smith ( 1968a ) . wn spectral subtypes follow a scheme involving line ratios of niii - v and hei - ii , ranging from wn2 to wn5 for ` early wn ' ( wne ) stars , and wn7 to wn9 for ` late wn ' ( wnl ) stars , with wn6 stars either early or late - type . a h suffix may be used to indicate the presence of emission lines due to hydrogen ( smith , shara & moffat 1996 ) . complications arise for wn stars with intrinsically weak emission lines . for example , wr24 ( wn6 ha ) has a heii @xmath44686 emission equivalent width that is an order of magnitude smaller than those in some other wn6 stars ; the ` ha ' nomenclature indicates that hydrogen is seen both in absorption and emission . from a standard spectroscopic viewpoint , such stars possess mid to late wn spectral classifications . however , their appearance is rather more reminiscent of of stars than classic wn stars , since there exists a continuity of properties between normal o stars and late - type wn stars . these stars are widely believed to be massive o stars with relatively strong stellar winds at a rather early evolutionary stage . they are believed not to represent the more mature , classic he - burning wn stars . smith , crowther & prinja ( 1994 ) extended the wn sequence to very late wn1011 subtypes in order to include a group of emission line stars originally classified as ofpe / wn9 ( bohannan & walborn 1989 ) . wn11 subtypes closely resemble extreme early - type b supergiants except for the presence of heii @xmath44686 emission . a quantitative comparison of optical line strengths in of and wnl stars is presented in figure 8 of bohannan & crowther ( 1999 ) . r127 ( wn11 ) in the large magellanic cloud ( lmc ) was later identified as a lbv ( stahl et al . 1983 ) , whilst a famous galactic lbv , ag car exhibited a wn11-type spectrum at visual minimum ( walborn 1990 ; smith et al . 1994 ) . various multi - dimensional classification systems have been proposed for wn stars ; they generally involve line strengths or widths , such that strong / broad lined stars have been labelled wn - b ( hiltner & schild 1966 ) , wn - s ( hamann , koesterke & wessolowski 1993 ) or wnb ( smith , shara & moffat 1996 ) . of these , none have generally been adopted . from a physical perspective , strong- and weak - lined wn stars do form useful sub - divisions . therefore we shall define weak ( -w ) and strong ( -s ) wn stars as those with heii @xmath45412 equivalent widths smaller than or larger than 40 . an obvious limitation of such an approach is that intrinsically strong - lined wn stars would be diluted by binary companions or nearby stars in spatially crowded regions and so might not be identified as such . wne - w stars tend to exhibit triangular line profiles rather than the more typical gaussian lines of wne - s stars ( marchenko et al . 2004 ) , since one observes material much closer to the stellar core that is being strongly accelerated . wc spectral subtypes depend on the line ratios of ciii and civ lines along with the appearance of o iii - v , spanning wc4 to wc9 subtypes , for which wc46 stars are ` early ' ( wce ) and wc79 are ` late ' ( wcl ) . rare , oxygen - rich wo stars form an extension of the wce sequence , exhibiting strong ovi @xmath53811 - 34 emission ( kingsburgh , barlow & storey 1995 ) . the most recent scheme involves wo1 to wo4 subtypes depending on the relative strength of ov - vi and c iv emission lines ( crowther , de marco & barlow 1998 ) . finally , civ @xmath45801 - 12 appears unusually strong in an otherwise normal wn star in a few cases , leading to an intermediate wn / c classification ( conti & massey 1989 ) . wn / c stars are indeed considered to be at an intermediate evolutionary phase between the wn and wc stages . representative examples of wn and wc stars are presented in figure [ wnc - montage ] . various x - ray to mid - ir spectroscopic datasets of galactic wolf - rayet stars are presented in table [ atlas ] , including extreme ultraviolet synthetic spectra from model atmospheres ( smith , norris & crowther 2002 ; hamann & grfener 2004 ) . wr stars can not be distinguished from normal hot stars using ubv photometry . broad - band visual measurements overestimate the true continuum level in extreme cases by up to 1 magnitude , or more typically 0.5 mag for single early - type wr stars due to their strong emission - line spectra . consequently , westerlund ( 1966 ) introduced narrow - band @xmath6 filters that were specifically designed to minimize the effect of wr emission lines ( although their effect can not be entirely eliminated ) . these passbands were later refined by smith ( 1968b ) and by massey ( 1984 ) , such that most photometry of wr stars has used the @xmath7 filter system , which is compared to johnson ubv filters in fig . [ wnc - montage ] . as with normal stars , @xmath8 photometry permits a determination of the interstellar extinction , @xmath9 . let us adopt a typical ratio of total , @xmath10 to selective , @xmath11 extinction , @xmath12 . following turner ( 1982 ) , the broad - band and narrow - band optical indices for wr stars are then related by : @xmath13 a direct determination of wr distances via stellar parallax is only possible for @xmath14 vel ( wc8+o ) using _ hipparcos _ , and even that remains controversial ( millour et al . otherwise , cluster or association membership is used to provide an approximate absolute magnitude - spectral type calibration for milky way wr stars . the situation is much better for wr stars in the magellanic clouds , although not all subtypes are represented . typical absolute magnitudes range from @xmath15 = 3 mag at earlier subtypes to 6 mag for late subtypes , or exceptionally 7 mag for hydrogen - rich wn stars . the typical spread is @xmath160.5 mag at individual subtypes . conti ( 1976 ) first proposed that a massive o star may lose a significant amount of mass via stellar winds , revealing first the h - burning products at its surface , and subsequently the he - burning products . these evolutionary stages are spectroscopically identified with the wn and wc types . this general picture has since become known as the ` conti scenario ' . such stars should be over - luminous for their mass , in accord with observations of wr stars in binary systems . massey ( 2003 ) provides a more general overview of massive stars within local group galaxies . wolf - rayet stars are located in or close to massive star forming regions within the galactic disk . a catalogue is provided by van der hucht ( 2001 ) . a quarter of the known wr stars in the milky way reside within massive clusters at the galactic centre or in westerlund 1 ( van der hucht 2006 ) . from membership of wr stars in open clusters , schild & maeder ( 1984 ) and massey , degioia - eastwood & waterhouse ( 2001 ) investigated the initial masses of wr stars empirically . a revised compilation is provided in crowther et al . ( 2006b ) . overall , hydrogen - rich wn stars ( wnha ) are observed in young , massive clusters ; their main - sequence turn - off masses ( based on meynet et al . 1994 isochrones ) suggest initial masses of @xmath17 , and are believed to be core - h burning ( langer et al . 1994 ; crowther et al . lower - mass progenitors of 4050@xmath1 are suggested for classic mid - wn , late wc , and wo stars . progenitors of some early wn stars appear to be less massive still , suggesting an initial - mass cutoff for wr stars at solar metallicity around 25@xmath1 . from an evolutionary perspective , the absence of rsgs at high luminosity and presence of h - rich wn stars in young massive clusters suggests the following variation of the conti scenario in the milky way , i.e. for stars initially more massive than @xmath18 @xmath19 whereas for stars of initial mass from @xmath20 , @xmath21 and for stars of initial mass in the range 2540@xmath1 , @xmath22 indeed , the role of the lbv phase is not yet settled it may be circumvented entirely in some cases ; it may follow the rsg stage , or it may even dominate pre - wr mass - loss for the most massive stars ( langer et al . 1994 ; smith & owocki 2006 ) . conversely , the presence of dense , circumstellar shells around type iin sn indicates that some massive stars may even undergo core - collapse during the lbv phase ( smith et al . remarkably few milky way clusters host both rsg and wr stars , with the notable exception of westerlund 1 ( clark et al . 2005 ) ; this suggests that the mass range common to both populations is fairly narrow . although optical narrow - band surveys ( see below ) have proved very successful for identifying wr stars in the solar neighbourhood , only a few hundred wr stars are known in the milky way , whilst many thousands are expected within the galactic disk ( van der hucht 2001 ) . consequently , near - ir narrow - band imaging surveys together with spectroscopic follow - up may be considered for more extensive surveys to circumvent high interstellar extinction ( homeier et al . . limitations of ir emission - line surveys are that fluxes of near - ir lines are much weaker than those of optical lines , also , no strong wr lines are common to all spectral types in the frequently used k band . an added complication is that some wc stars form dust which heavily dilutes emission line fluxes longward of the visual . nevertheless , infrared surveys are presently underway to get an improved census of wr stars in the milky way . alternatively , wr candidates may be identified from their near- to mid - ir colours , which , as in other early - type supergiants , are unusual due to strong free - free excess emission ( hadfield et al . 2007 ) . wr stars have typically been discovered via techniques sensitive to their unusually broad emission - line spectra , based on objective prism searches or interference filter imaging ( see massey 2003 ) . narrow - band interference filter techniques have been developed ( e.g. moffat , seggewiss & shara 1985 ; massey , armandroff & conti 1986 ) that distinguish strong wr emission lines at heii @xmath44686 ( wn stars ) and ciii @xmath44650 ( wc stars ) from the nearby continuum . such techniques have been applied to regions of the milky way disk , the magellanic clouds and other nearby galaxies . an example of this approach is presented in figure [ ngc300 ] for the spiral galaxy ngc 300 ( @xmath23 2 mpc ) . a wide - field image of ngc 300 is presented , with ob complex iv - v indicated , together with narrow - band images centred at @xmath44684 ( heii 4686 ) and @xmath44781 ( continuum ) . several wr stars are seen in the difference ( heii - continuum ) image , including an apparently single wc4 star ( schild et al . 2003 ) . it is well established that the absolute number of wr stars and their subtype distribution are metallicity dependent . n(wr)/n(o)@xmath00.15 in the relatively metal - rich solar neighbourhood ( conti et al . 1983 ; van der hucht 2001 ) , yet n(wr)/n(o)@xmath00.01 in the metal - deficient smc on the basis of only 12 wr stars ( massey , olsen & parker 2003 ) versus @xmath01000 o stars ( evans et al . it is believed that the majority of galactic wr stars are the result of single - star evolution , yet some stars ( e.g. v444 cyg ) result from close binary evolution ( vanbeveren et al . 1998 ) . similar relative numbers of wn to wc stars are observed in the solar neighbourhood ( hadfield et al . in contrast , wn stars exceed wc stars by a factor of @xmath05 and @xmath010 for the lmc and smc , respectively ( breysacher , azzopardi & testor 1999 ; massey , olsen & parker 2001 ) . at low metallicity the reduced wr population and the relative dominance of wn subtypes most likely result from the metallicity dependence of winds from their evolutionary precursors ( mokiem et al . consequently , only the most massive single stars reach the wr phase in metal - poor environments . single stars reaching the wc phase at high metallicity may end their lives as a rsg or wn stars in a lower metallicity environment . as such , one might suspect that most wr stars at low metallicity are formed via binary evolution . however , foellmi , moffat & guerrero ( 2003a ) suggest a similar wr binary fraction for the smc and milky way . not all wr subtypes are observed in all environments . early wn and wc subtypes are preferred in metal - poor galaxies , such as the smc ( massey et al . 2003 ) , while late wc stars are more common at super - solar metallicities , such as m83 ( hadfield et al . 2005 ) line widths of early wc and wo stars are higher than late wc stars , although width alone is not a defining criterion for each spectral type . the correlation between wc subclass and line width is nevertheless strong ( torres , conti & massey 1986 ) . the subtype distributions of wr stars in the solar neighbourhood , lmc , and smc are presented in figure [ wrpop ] . we shall address this aspect in sect [ metallicity ] . individual wr stars may , in general , be resolved in local group galaxies from ground - based observations , whilst the likelihood of contamination by nearby sources increases at larger distances . for example , a typical slit width of 1@xmath24 at the 2mpc distance of ngc 300 corresponds to a spatial scale of @xmath010 pc . relatively isolated wr stars have been identified , albeit in the minority ( recall figure [ ngc300 ] ) . this is even more problematic for more distant galaxies such as m 83 where the great majority of wr stars are observed in clusters or associations ( hadfield et al . 2005 ) . so - called ` wr galaxies ' are typically starburst regions exhibiting spectral features from tens , hundreds , or even thousands of wr stars ( schaerer , contini & pindao 1999 ) . average milky way / lmc wn or wc line fluxes ( schaerer & vacca 1998 ) are typically used to calculate stellar populations in wr galaxies . these should be valid provided that the line fluxes of wr templates do not vary with environment . however , it is well known that smc wn stars possess weak emission lines ( conti , garmany & massey 1989 ) . in spite of small statistics and a large scatter , the mean heii @xmath44686 line luminosity of wn24 stars in the lmc is 10@xmath25 ergs@xmath26 , a factor of five times higher than the mean of equivalent stars in the smc ( crowther & hadfield 2006 ) . the signature of wn stars is most readily seen in star forming galaxies at heii @xmath41640 , where the dilution from other stellar types is at its weakest ( e.g. hadfield & crowther 2006 ) . the strongest uv , optical , and near - ir lines indicate flux ratios of @xmath27(heii 1640)/@xmath27(heii 4686)@xmath010 and @xmath27(he ii 4686)/@xmath27(heii 1.012@xmath28m)@xmath06 for wn stars spanning smc to milky way metallicities . similar comparisons for wc stars are hindered because the only carbon - sequence wr stars at the low metallicity of the smc and ic 1613 are wo stars . their emission line fluxes are systematically weaker than wc stars in the lmc and milky way ( kingsburgh et al . 1995 ; kingsburgh & barlow 1995 ; schaerer & vacca 1998 ) . the mean civ @xmath55801 - 2 line luminosity of wc4 stars in the lmc is @xmath29 ergs@xmath26 ( crowther & hadfield 2006 ) . again , detection of wc stars is favoured via ultraviolet spectroscopy of civ @xmath41550 . for wc stars , the strongest uv , optical , and near - ir lines possess flux ratios of @xmath27(civ 1548 - 51)/@xmath27(civ 5801 - 12)@xmath06 and @xmath27(c iv 5801 - 12)/@xmath27(civ 2.08@xmath28m)@xmath015 . the observed binary fraction amongst milky way wr stars is 40% ( van der hucht 2001 ) , either from spectroscopic or indirect techniques . within the low metallicity magellanic clouds , close binary evolution would be anticipated to play a greater role because of the diminished role of o star mass - loss in producing single wr stars . however , where detailed studies have been carried out ( bartzakos , moffat & niemela 2001 ; foellmi , moffat & guerrero 2003ab ) , a similar binary fraction to the milky way has been obtained ( recall figure [ wrpop ] ) , so metallicity - independent lbv eruptions may play a dominant role . the most robust method of measuring stellar masses is from kepler s third law of motion , particularly for eclipsing double - lined ( sb2 ) systems , from which the inclination may be derived . orbital inclinations may also be derived from linear polarization studies ( e.g. st - louis et al . 1993 ) or atmospheric eclipses ( lamontagne et al . 1996 ) . masses for galactic wr stars are included in the van der hucht ( 2001 ) compilation , a subset of which are presented in figure [ binary_masses ] together with some more recent results . wc masses span a narrow range of 916@xmath1 , whilst wn stars span a very wide range of @xmath01083@xmath1 , and in some cases exceed their ob companion , i.e. @xmath30 ( e.g. wr22 : schweickhardt et al . 1999 ) . wr20a ( smsp2 ) currently sets the record for the highest orbital - derived mass of any star , with @xmath31 for each wn6 ha component ( rauw et al . as discussed above , such stars are h - rich , extreme o stars with strong winds rather than classical h - poor wn stars . they are a factor of two lower in mass than the apparent @xmath32 stellar mass limit ( figer 2005 ) , such that still more extreme cases may await discovery . spectroscopic measurement of masses via surface gravities using photospheric lines is not possible for wr stars due to their dense stellar winds . rotation is very difficult to measure in wr stars , since photospheric features used to estimate @xmath33 in normal stars are absent . velocities of 200500 kms@xmath26 have been inferred for wr138 ( massey 1980 ) and wr3 ( massey & conti 1981 ) , although these are not believed to represent rotation velocities , since the former has a late - o binary companion , and the absorption lines of the latter are formed within the stellar wind ( marchenko et al . 2004 ) . fortunately , certain wr stars do harbour large scale structures , from which a rotation period may be inferred ( st - louis et al . 2007 ) . alternatively , if wr stars were rapid rotators , one would expect strong deviations from spherical symmetry due to gravity darkening ( von zeipel 1924 ; owocki , cranmer & gayley 1996 ) . harries , hillier & howarth ( 1998 ) studied linear spectropolarimetric datasets for 29 galactic wr stars , from which just four single wn stars plus one wc+o binary revealed a strong line effect , suggesting significant departures from spherical symmetry . they presented radiative transfer calculations which suggest that the observed continuum polarizations for these stars can be matched by models with equator to pole density ratios of 23 . of course , the majority of milky way wr stars do not show a strong linear polarization line effect ( e.g. kurosawa , hillier & schulte - ladbeck ( 1999 ) . ring nebulae are observed for a subset of wr stars . these are believed to represent material ejected during the rsg or lbv phases that is photo - ionized by the wr star . the first known examples , ngc 2359 and ngc 6888 , display a shell morphology , although many subsequently detected in the milky way and magellanic clouds exhibit a variety of spatial morphologies ( chu , treffers & kwitter 1983 ; dopita et al . nebulae are predominantly associated with young wr stars i.e. primarily wn subtypes , with typical electron densities of 10@xmath34 @xmath35 to 10@xmath36 @xmath35 ( esteban et al . 1993 ) . ring nebulae provide information on evolutionary links between wr stars and their precursors ( weaver et al . once a massive star has reached the wr phase , its fast wind will sweep up the material ejected during the immediate precursor ( lbv or rsg ) slow wind . the dynamical evolution of gas around wr stars with such progenitors has been discussed by garca - segura , maclow & langer ( 1996a ) and garca - segura , langer & maclow ( 1996b ) . esteban et al . ( 1993 ) attempted to derive wr properties indirectly from hii regions associated with selected milky way stars ( see also crowther et al . unfortunately , relatively few hii regions are associated with individual wr stars , and for the majority of these , the nebular parameters are insufficiently well constrained to distinguish between different stellar atmosphere models . our interpretation of hot , luminous stars via radiative transfer codes is hindered with respect to normal stars by several effects . first , the routine assumption of lte breaks down for high - temperature stars . in non - lte , the determination of populations uses rates which are functions of the radiation field , itself a function of the populations . consequently , it is necessary to solve for the radiation field and populations iteratively . second , the problem of accounting for the effect of millions of spectral lines upon the emergent atmospheric structure and emergent spectrum known as line blanketing remains challenging for stars in which spherical , rather than plane - parallel , geometry must be assumed due to stellar winds , since the scale height of their atmospheres is not negligible with respect to their stellar radii . the combination of non - lte , line blanketing ( and availability of atomic data thereof ) , and spherical geometry has prevented the routine analysis of such stars until recently . radiative transfer is either solved in the co - moving frame , as applied by cmfgen ( hillier & miller 1998 ) and powr ( grfener , koesterke & hamann 2002 ) or via the sobolev approximation , as used by isa - wind ( de koter , schmutz & lamers 1993 ) . the incorporation of line blanketing necessitates one of several approximations . either a ` super - level ' approach is followed , in which spectral lines of a given ion are grouped together in the solution of the rate equations ( anderson 1989 ) , or alternatively , a monte carlo approach is followed , which uses approximate level populations ( abbott & lucy 1985 ) . stellar temperatures for wr stars are difficult to characterize , because their geometric extension is comparable with their stellar radii . atmospheric models for wr stars are typically parameterized by the radius of the inner boundary @xmath37 at high rosseland optical depth @xmath38 ) . however , only the optically thin part of the atmosphere is seen by the observer . the measurement of @xmath37 depends upon the assumption that the same velocity law holds for the visible ( optically thin ) and the invisible ( optically thick ) part of the atmosphere . the optical continuum radiation originates from a ` photosphere ' where @xmath39 . typical wn and wc winds have reached a significant fraction of their terminal velocity before they become optically thin in the continuum . @xmath40 , the radius at @xmath41 lies at highly supersonic velocities , well beyond the hydrostatic domain . for example , crowther et al . ( 2006a ) obtain @xmath37 = 2.9@xmath42 and @xmath40 = 7.7@xmath42 for hd 50896 ( wn4b ) , corresponding to @xmath43 = 85kk and @xmath44 = 52kk , respectively . in some weak - lined , early - type wn stars , this is not strictly true since their spherical extinction is modest , in which case r@xmath45 ( e.g. hd 9974 , marchenko et al . 2004 ) . stellar temperatures of wr stars are derived from lines from adjacent ionization stages of helium or nitrogen for wn stars ( hillier 1987 , 1988 ) , or lines of carbon for wc stars ( hillier 1989 ) . high stellar wind densities require the simultaneous determination of mass - loss rate and stellar temperature from non - lte model atmospheres , since their atmospheres are so highly stratified . metals such as c , n and o provide efficient coolants , such that the outer wind electron temperature is typically 8kk to 12kk ( hillier 1989 ) . figure [ wr_ross ] compares @xmath37 , @xmath40 , and the principal optical wind line - forming region ( @xmath46 to 10@xmath47 @xmath35 ) for hd 66811 ( @xmath48 pup , o4i(n)f ) , hd 96548 ( wr40 , wn8 ) and hd 164270 ( wr103 , wc9 ) on the same physical scale . some high - ionization spectral lines ( e.g. nv and civ lines in wn8 and wc9 stars , respectively ) are formed at higher densities of @xmath49 @xmath35 in the wr winds . derived stellar temperatures depend sensitively upon the detailed inclusion of line - blanketing by iron peak elements . inferred bolometric corrections and stellar luminosities also depend upon detailed metal line - blanketing ( schmutz 1997 ; hillier & miller 1999 ) . until recently , the number of stars studied with non - lte , clumped , metal line - blanketed models has been embarrassingly small , due to the need for detailed , tailored analysis of individual stars using a large number of free parameters . hamann , grfener & liermann ( 2006 ) have applied their grid of line - blanketed wr models to the analysis of most galactic wn stars , for the most part resolving previous discrepancies between alternate line diagnostics , which were first identified by crowther et al . ( 1995b ) . to date , only a limited number of wc stars in the milky way and magellanic clouds have been studied in detail ( e.g. dessart et al . 2000 ; crowther et al . 2002 ; barniske , hamann & grfener 2007 ) . results for galactic and lmc wr stars are presented in table [ wrtemp ] . these range from 30kk amongst wn10 subtypes to 40kk at wn8 and approach 100kk for early - type wn stars . spectroscopic temperatures are rather higher for wc stars , i.e. 50kk for wc9 stars , increasing to 70kk at wc8 and @xmath50100kk for early wc and wo stars stellar structure models predict radii @xmath51 that are significantly smaller than those derived from atmospheric models . for example , @xmath52 for hd 191765 ( wr134 , wn6b ) in table [ wrtemp ] , versus @xmath53 which follows from hydrostatic evolutionary models , namely @xmath54 for hydrogen - free wr stars ( schaerer & maeder 1992 ) . theoretical corrections to such radii are frequently applied , although these are based upon fairly arbitrary assumptions which relate particularly to the velocity law . consequently , a direct comparison between temperatures of most wr stars from evolutionary calculations and empirical atmospheric models is not straightforward , except that one requires @xmath55 , with the difference attributed to the extension of the supersonic region . petrovic , pols & langer ( 2006 ) established that the hydrostatic cores of metal - rich wr stars above @xmath56 exceed @xmath51 in eqn [ radii ] by significant factors if mass - loss is neglected , due to their proximity to the eddington limit , @xmath57 = 1 . here , the eddington parameter , @xmath57 , is the ratio of radiative acceleration due to thompson ( electron ) scattering to surface gravity and may be written as @xmath58 where the number of free electrons per atomic mass unit is @xmath59 . in reality , high empirical wr mass - loss rates imply that inflated radii are not expected , such that the discrepancy in hydrostatic radii between stellar structure and atmospheric models has not yet been resolved . absolute visual magnitudes of wr stars are obtained primarily from calibrations obtained from cluster or association membership ( van der hucht 2001 ) . inferred bolometric corrections range from @xmath60 = 2.7 mag amongst very late wn stars ( crowther & smith 1997 ) to approximately 6 mag for weak - lined , early - type wn stars and wo stars ( crowther et al . 1995b ; crowther et al . 2000 ) . stellar luminosities of milky way wn stars range from 200,000 @xmath61 in early - type stars to 500,000 @xmath61 in late - type stars . hydrogen - burning o stars with strong stellar winds , spectroscopically identified as wnha stars , have luminosities in excess of @xmath62 . for milky way wc stars , inferred stellar luminosities are @xmath0150,000 @xmath61 , increasing by a factor of two for lmc wc stars ( table [ wrtemp ] ) . systematically higher spectroscopic luminosities have recently been determined by hamann , grfener & liermann ( 2006 ) for galactic wn stars , since they adopt uniformly high @xmath63 mag for all non - cluster member wn69h stars . absolute magnitudes for normal late - type wn stars are subject to large uncertainties since such stars positively shy away from clusters . as a consequence , their results suggest a bi - modal distribution around @xmath64 for early wn stars , and 12@xmath65 for all late wn stars . from stellar structure theory , there is a mass - luminosity relation for h - free wr stars which is described by @xmath66 this expression is effectively independent of the chemical composition since the continuum opacity is purely electron scattering ( schaerer & maeder 1992 ) . spectroscopic luminosities need to be corrected for the luminosity that powers the stellar wind , @xmath67 , in order to determine the underlying nuclear luminosity , @xmath68 . in most cases , the recent reduction in estimates of mass - loss rates due to wind clumping ( see sect . [ clump ] ) , plus the increase in derived luminosities due to metal line - blanketing indicate a fairly modest corrective factor . from table [ wrtemp ] , one expects typical masses of 1015 @xmath1 for hydrogen - free wr stars , which agree fairly well with binary mass estimates ( recall fig . [ binary_masses ] ) . indeed , spectroscopically derived wr masses obtained using this relationship agree well with binary derived masses ( e.g. @xmath14 vel : de marco et al . 2000 ) . lyman continuum ionizing fluxes , n(lyc ) , are typical of mid - o stars in general ( table [ wrtemp ] ) . as such , the low number of wr stars with respect to o stars would suggest that wolf - rayet stars play only a minor role in the lyman continuum ionization budget of young star - forming regions . h - rich late - type wn stars provide a notable exception , since their ionizing output compares closely to o2 stars ( walborn et al . 2004 ) . crowther & dessart ( 1998 ) showed that the wn6 ha stars in ngc 3603 provided @xmath020% of the lyman continuum ionizing photons , based upon calibrations of non - blanketed models for o and wr stars . since wr stars represent an extension of o stars to higher temperatures , significant hei continuum photons are emitted , plus strong heii continua for a few high - temperature , low - density cases . the primary effect of metal - line blanketing is to redistribute extreme uv flux to longer wavelengths , reducing the ionization balance in the wind , such that higher temperatures and luminosities are required to match the observed wr emission line profile diagnostics relative to unblanketed models . recent revisions to estimated temperatures and luminosities of o stars ( as measured from photospheric lines ) have acted in the reverse sense , relative to previous plane - parallel unblanketed model analysis , due to backwarming effects , as shown in a number of recent papers ( e.g. martins , schaerer & hillier 2002 ; repolust , puls & herrero 2004 ) . common techniques are generally now employed for o and wr studies , such that a factor of two increase in n(lyc ) for wr stars plus the reverse for o stars suggests that in such cases wr stars might provide close to half of the total ionizing photons in the youngest starbursts , such as ngc 3603 . the strength of wr winds affects the hardness of their ionizing radiation . atmospheric models for wr stars with dense winds produce relatively soft ionizing flux distributions , in which extreme uv photons are redistributed to longer wavelength by the opaque stellar wind ( schmutz , leitherer & gruenwald 1992 ) . in contrast , for the low wind - density case , a hard ionizing flux distribution is predicted , in which extreme uv photons pass through the relatively transparent wind unimpeded . consequently , the shape of the ionizing flux distribution of wr stars depends on both the wind density and the stellar temperature . we shall show in section [ mdot ] that low - metallicity wr stars possess weaker winds . in figure [ wn_grid ] , we compare the predicted lyman continuum ionizing flux distribution from four 100kk wn models , in which only the low metallicity , low mass - loss rate models predicts a prodigious number of photons below the he@xmath69 edge at 228 . consequently , one expects evidence of hard ionizing radiation from wr stars ( e.g. nebular heii @xmath44686 ) solely at low metallicities . this is generally borne out by observations of hii regions associated with wr stars ( garnett et al . previous studies of metal - rich regions have claimed a low limit to the stellar mass function from indirect hii region studies at high metallicity in which wr stars were spectroscopically detected . gonzalez delgado et al . ( 2002 ) were able to reconcile a high stellar mass limit from uv spectral synthesis techniques with a soft ionizing spectrum for the metal - rich wr galaxy ngc 3049 by applying the smith , norris & crowther ( 2002 ) line blanketed grid of wr models at high metallicity . for wr stars , it has long been suspected that abundances represented the products of core nucleosynthesis , although it has taken the development of non - lte model atmospheres for these to have been empirically supported . balmer - pickering decrement studies by conti , leep & perry ( 1983 ) concluded that hydrogen was severely depleted in wr stars . a clear subtype effect regarding the hydrogen content of galactic wn stars is observed , with late - type wn stars generally showing some hydrogen ( typically @xmath7010% ) , and early - type wn stars being hydrogen - free , although exceptions do exist . this trend breaks down within the lower metallicity environment of the magellanic clouds , notably the smc ( foellmi et al 2003a ) . milky way late - type wn stars with weak emission lines denoted as ` ha ' due to intrinsic absorption lines plus the presence of hydrogen are universally h - rich with @xmath7150% ( crowther et al . 1995a ; crowther & dessart 1998 ) . non - lte analyses confirm that wn abundance patterns are consistent with material processed by the cno cycle in which these elements are used as catalysts , i.e. @xmath72 in which @xmath731% by mass is observed in milky way wn stars . carbon is highly depleted , with typically @xmath74 0.05% . oxygen suffers from fewer readily accessible line diagnostics , but probably exhibits a similarly low mass fraction as carbon ( e.g. crowther , smith & hillier 1995b ; herald , hillier & schulte - ladbeck 2001 ) . non - lte analysis of transition wn / c stars reveals elemental abundances ( e.g. @xmath74 5% , @xmath75 1% by mass ) that are in good agreement with the hypothesis that these stars are in a brief transition stage between wn and wc ( langer 1991 ; crowther , smith & willis 1995c ) . neither hydrogen nor nitrogen are detected in the spectra of wc stars . recombination line studies using theoretical coefficients for different transitions are most readily applicable to wc stars , since they show a large number of lines in their optical spectra . atomic data are most reliable for hydrogenic ions , such as civ and ovi , so early - type wc and wo stars can be studied most readily . smith & hummer ( 1988 ) suggested a trend of increasing c / he from late to early wc stars , with c / he=0.040.7 by number ( 10% @xmath76 60% ) , revealing the products of core he burning @xmath77 although significant uncertainties remain in the rate of the latter nuclear reaction . these reactions compete during helium burning to determine the ratio of carbon to oxygen at the onset of carbon burning . in reality , optical depth effects come into play , so detailed abundance determinations for all subtypes require non - lte model atmosphere analyses . koesterke & hamann ( 1995 ) indicated refined values of c / he=0.10.5 by number ( 20% @xmath76 55% ) , with no wc subtype dependence , such that spectral types are not dictated by carbon abundance , contrary to suggestions by smith & maeder ( 1991 ) . indeed , lmc wc4 stars possess similar surface abundances to milky way wc stars ( crowther et al . 2002 ) , for which the heii @xmath45412 and c iv @xmath45471 optical lines represent the primary diagnostics ( hillier 1989 ) . these recombination lines are formed at high densities of 10@xmath78 to 10@xmath47 @xmath35 at radii of 330 @xmath37 ( recall figure [ wr_ross ] ) . oxygen diagnostics in wc stars lie in the near - uv , such that derived oxygen abundances are rather unreliable unless space - based spectroscopy is available . where they have been derived , one finds @xmath79 510% for wc stars ( e.g. crowther et al . 2002 ) . core he burning in massive stars also has the effect of transforming @xmath80n ( produced in the cno cycle ) to neon and magnesium via @xmath81 and serves as the main neutron source for the s - process in massive stars . neon lines are extremely weak in the uv / optical spectrum of wc stars ( crowther et al . 2002 ) , but ground - state fine - structure lines at [ neii ] 12.8@xmath28 m and [ neiii ] 15.5@xmath28 m may be observed via mid - ir spectroscopy , as illustrated for @xmath14 vel in van der hucht et al . fine - structure wind lines are formed at hundreds of stellar radii since their critical densities are of order 10@xmath82 @xmath35 . barlow , roche & aitken ( 1988 ) came to the conclusion that neon was not greatly enhanced in @xmath14 vel with respect to the solar case ( @xmath00.1% by mass primarily in the form of @xmath83ne ) in @xmath14 vel from their analysis of fine - structure lines . this was a surprising result , since the above reaction is expected to produce @xmath02% by mass of @xmath84ne at solar metallicity . once the clumped nature of wr winds is taken into consideration , neon is found to be enhanced in @xmath14 vel and other wc stars ( e.g. dessart et al . the inferred neon mass fraction is @xmath01% ( see also crowther , morris & smith 2006a ) . meynet & maeder ( 2003 ) note that the @xmath84ne enrichment depends upon nuclear reaction rates rather than stellar models , so the remaining disagreement may suggest a problem with the relevant reaction rates . more likely , a lower metal content is inferred from the neon abundance than solar metallicity evolutionary models ( z=0.020 ) . indeed , if the solar oxygen abundance from apslund et al . ( 2004 ) is taken into account , a revised metal content of z=0.012 for the sun is impled . allowance for depletion of heavy elements due to diffusion in the 4.5 gyr old sun suggests a solar neighbourhood metallicity of z=0.014 ( meynet , private communication ) . it is likely that allowance for a reduced cno content would bring predicted and measured ne@xmath84 abundances into better agreement . wo stars are extremely c- and o - rich , as deduced from recombination analyses ( kingsburgh , barlow & storey 1995 ) , and supported by non - lte models ( crowther et al . further nucleosynthesis reactions produce alpha elements via @xmath85 producing a core which is initially dominated by @xmath86o and @xmath83ne . _ spitzer _ studies are in progress to determine neon abundances in wo stars , in order to assess whether these stars show evidence of @xmath2capture of oxygen ( in which case enhanced @xmath83ne would again dominate over @xmath84ne ) . the existence of winds in early - type stars has been established since the 1960 s , when the first rocket uv observations ( e.g. morton 1967 ) revealed the characteristic p cygni signatures of mass - loss . electron ( thompson ) scattering dominates the continuum opacity in o and wr stars , whilst the basic mechanism by which their winds are driven is the transfer of photon momentum to the stellar atmosphere through the absorption by spectral lines . the combination of a plethora of spectral lines within the same spectral region as the photospheric radiation allows for efficient driving of winds by radiation pressure ( milne 1926 ) . wind velocities can be directly measured , whilst wind density estimates rely on varying complexity of theoretical interpretation . a theoretical framework for mass - loss in normal hot , luminous stars was developed by castor , abbott & klein ( 1975 ) , known as cak theory , via line - driven radiation pressure . the wavelength of the blue edge of saturated p cygni absorption profiles provides a measure of the asymptotic wind velocity . from these wavelengths , accurate wind velocities of wr stars can readily be obtained ( prinja , barlow & howarth 1990 ; willis et al . alternatively , optical and near - ir hei p cygni profiles or mid - ir fine - structure metal lines may be used to derive reliable wind velocities ( howarth & schmutz 1992 ; eenens & williams 1994 ; dessart et al . 2000 ) . in principle , optical recombination lines of heii and c iii - iv may also be used to estimate wind velocities , since these are formed close to the asymptotic flow velocity . however , velocities obtained from spectral line modelling are preferable . for wr stars exhibiting weak winds whose lines are formed interior to the asymptotic flow velocity only lower velocity limits may be obtained . nevertheless , observational evidence suggests lower wind velocities at later subtypes by up to a factor of ten than early subtypes ( table [ wrtemp ] ) . wind velocities of lmc wn stars compare closely with milky way counterparts . unfortunately , uv spectroscopy of smc wn stars is scarce , such that one has to rely on optical emission lines for wind velocity estimates , which provide only lower limits to terminal wind velocities . the current record holder amongst non - degenerate stars for the fastest stellar wind is the galactic wo star wr93b . for it , a wind velocity of 6000 kms@xmath26 has been obtained from optical recombination lines ( drew et al . individual wo stars have now also been identified in a number of external galaxies . one observes a reduction in line width ( and so wind velocity ) for stars of progressively lower metallicity , by a factor of up to two between the milky way and ic 1613 ( crowther & hadfield 2006 ) . although numbers are small , this downward trend in wind velocity with decreasing metallicity is believed to occur for other o and wr spectral types ( e.g. kudritzki & puls 2000 ; crowther 2000 ) . the mass - loss rate relates to the velocity field @xmath87 and density @xmath88 via the equation of continuity @xmath89 for a spherical , stationary wind . wr winds may be observed at ir - mm - radio wavelengths via the free - free ( bremsstrahlung ) continuum excess caused by the stellar wind or via uv , optical or near - ir emission lines . mass - loss rates ( e.g. leitherer , chapman & koribalski 1997 ) follow from radio continuum observations using relatively simple analytical relations , under the assumption of homogeneity and spherical symmetry . the emergent radio flux @xmath90 depends on the distance to the star @xmath91 , mass - loss rate and terminal velocity as follows @xmath92 ( wright & barlow 1975 ) , where @xmath93 in the constant - velocity regime . accurate determinations of wr mass - loss rates depend upon composition , ionization balance and electron temperature at physical radii of @xmath94 . wind collisions in an interacting binary system will cause additional non - thermal ( synchrotron ) radio emission ( sect . [ synch ] ) , so care needs to be taken against overestimating mass - loss rates in this way . optical spectral lines observed in wr stars can be considered as recombination lines , although line formation is rather more complex in reality ( hillier 1988 , 1989 ) . since recombination involves the combination of ion and electron density , the strength of wind lines scales with the square of the density . this explains why only a factor of @xmath010 increase in wind density with respect to the photospheric absorption line spectrum from o supergiants produces an emission line wolf - rayet spectrum . recall the comparison of @xmath48 pup ( o4i(n)f ) to hd 96548 ( wr40 , wn8 ) in figure [ wr_ross ] . there is overwhelming evidence in favour of highly clumped winds for wr and o stars . line profiles show propagating small - scale structures or ` blobs ' , which are turbulent in nature ( e.g. moffat et al . 1988 ; lpine et al . 2000 ) . for optically thin lines , these wind structures have been investigated using radiation hydrodynamical simulations by dessart & owocki ( 2005 ) . alternatively , individual spectral lines , formed at @xmath95 , can be used to estimate volume filling factors @xmath96 in wr winds ( hillier 1991 ) . this technique permits an estimate of wind clumping factors from a comparison between line electron scattering wings ( which scale linearly with density ) and recombination lines ( density - squared ) . this technique suffers from an approximate radial density dependence and is imprecise due to severe line blending , especially in wc stars . nevertheless , fits to uv , optical and ir line profiles suggest @xmath97 . as a consequence , global wr mass - loss rates are reduced by a factor of @xmath98 relative to homogeneous models ( dm / dt @xmath99 ) . representative values are included in table [ wrtemp ] . spectroscopically derived mass - loss rates of milky way wn stars span a wide range of 10@xmath100 to 10@xmath101 yr@xmath26 . in contrast , galactic wc stars cover a much narrower range in mass - loss rate , from 10@xmath102 to 10@xmath101 yr@xmath26 . independent methods support clumping - corrected wr mass - loss rates . binary systems permit use of the variation of linear polarization with orbital phase . the modulation of linear polarization originates from thomson scattering of free electrons due to the relative motion of the companion with respect to the wr star . this technique has been applied to several wr binaries including v444 cyg ( hd 193576=wr139 , wn5+o ) by st - louis et al . ( 1993 ) and has been developed further by kurosawa , hillier & pittard ( 2002 ) using a monte carlo approach . for the case of v444 cyg , polarization results suggest a clumping factor of @xmath103 . the most likely physical explanation for the structure in wr and o star winds arises from theoretical evidence supporting an instability in radiatively - driven winds ( lucy & solomon 1971 ; owocki , castor & rybicki 1988 ) . there is a strong potential in line scattering to drive wind material with accelerations that greatly exceed the mean outward acceleration . simulations demonstrate that this instability may lead naturally to structure . such a flow is dominated by multiple shock compressions , producing relatively soft x - rays . hard x - ray fluxes from early - type stars are believed to be restricted to colliding wind binary systems ( e.g. @xmath14 vel : schild et al . 2004 ) , for which @xmath104 . we shall now consider empirical evidence in favour of metallicity - dependent wr winds . nugis , crowther & willis ( 1998 ) estimated mass - loss rates for galactic wr stars from archival radio observations , allowing for clumped winds . nugis & lamers ( 2000 ) provided empirical mass - loss scaling relations by adopting physical parameters derived from spectroscopic analysis and/or evolutionary predictions . for a combined sample of wn and wc stars , nugis & lamers ( 2000 ) obtained @xmath105 where y and z are the mass fractions of helium and metals , respectively . smith & willis ( 1983 ) compared the properties of wn stars in the lmc and milky way , concluding there was no significant differences between the wind properties of the two samples . these conclusions were supported by hamann & koesterke ( 2000 ) from detailed non - lte modelling , although a large scatter in mass - loss rates within each parent galaxy was revealed . either there is no metallicity dependence , or any differences are too subtle to be identified from the narrow metallicity range spanned by the milky way and lmc . within the milky way , most late - type stars contain hydrogen and most early - type stars do not . in contrast , early - type stars dominate wn populations in the magellanic clouds ( recall figure [ wrpop ] ) , and often contain atmospheric hydrogen ( smith , shara & moffat 1996 ; foellmi , moffat & guerrero 2003ab ) . figure [ mdot_plot ] compares the mass - loss rates of cluster or association member wn stars in the milky way with magellanic cloud counterparts . mass - loss estimates are obtained from their near - ir helium lines ( crowther 2007 , following howarth & schmutz 1992 ) . the substantial scatter in mass - loss rates is in line with the heterogeneity of line strengths within wn subtypes . stronger winds are measured for wn stars without surface hydrogen , in agreement with eqn [ nugis - lamers00 ] ) and recent results of hamann , grfener & liermann ( 2006 ) . from the figure , measured mass - loss rates of hydrogen - rich early - type wn stars in the smc ( 1/5 @xmath106 ) are 0.4 dex weaker than equivalent stars in the milky way and lmc ( 1/2 to 1 @xmath106 ) . this suggests a metallicity dependence of dm / dt @xmath107 for wn stars , with @xmath108 . the exponent is comparable to that measured from h@xmath2 observations of milky way , lmc and smc o - type stars ( mokiem et al . 2007 ) . there are two atmospheric factors which contribute to the observed trend towards earlier wn subtypes at lower metallicities ; 1 . cno compromises @xmath01.1% by mass of the solar photosphere ( asplund et al . 2004 ) versus 0.48% in the lmc and 0.24% in the smc ( russell & dopita 1990 ) . since wn stars typically exhibit cno equilibrium abundances , there is a maximum nitrogen content available in a given environment . for otherwise identical physical parameters , crowther ( 2000 ) demonstrated that a reduced nitrogen content at lower metallicity favours an earlier subtype . this is regardless of metallicity dependent mass - loss rates , and results from the abundance sensitivity of nitrogen classification lines additionally , a metallicity dependence of wn winds would enhance the trend to earlier spectral subtypes . dense wn winds at high metallicities lead to efficient recombination from high ionization stages ( e.g. n@xmath109 ) to lower ions ( e.g. n@xmath110 ) within the optical line formation regions . this would not occur so effectively for low density winds , enhancing the trend towards early - type wn stars in metal - poor environments . consequently , both effects favour predominantly late subtypes at high metallicity , and early subtypes at low metallicity , which is indeed generally observed ( fig . [ wrpop ] ) . it is well established that wc stars in the inner milky way , and indeed all metal - rich environments , possess later spectral types than those in the outer galaxy , lmc and other metal - poor environments ( figure [ wrpop ] ; hadfield et al . this observational trend led smith & maeder ( 1991 ) to suggest that early - type wc stars are richer in carbon than late - type wc stars , on the basis of tentative results from recombination line analyses . in this scenario , typical milky way wc59 stars exhibit reduced carbon abundances than wc4 counterparts in the lmc . however , quantitative analysis of wc subtypes allowing for radiative transfer effects do not support a subtype dependence of elemental abundances in wc stars ( koesterke & hamann 1995 ) , as discussed in sect [ abundances ] . if differences of carbon content are not responsible for the observed wc subtype distribution in galaxies , what is its origin ? let us consider the wc classification lines ciii @xmath45696 and civ @xmath55801 - 12 in greater detail . specifically the upper level of @xmath45696 has an alternative decay via @xmath4574 , with a branching ratio of 147:1 ( hillier 1989 ) . consequently @xmath45696 only becomes strong when @xmath4574 is optically thick , i.e. if the stellar temperature is low _ or _ the wind density is sufficiently high . from non - lte models it has been established that the temperatures of galactic wc57 stars and lmc wc4 stars are similar , such that the observed subtype distribution argues that the wind densities of galactic wc stars must be higher than the lmc stars . figure [ mdot_plot ] also compares clumping corrected mass - loss rates of wc stars in the milky way and lmc , as derived from optical studies . the galactic sample agree well with eqn [ nugis - lamers00 ] from nugis & lamers ( 2000 ) . crowther et al . ( 2002 ) obtain a similar mass - loss dependence for wc4 stars in the lmc , albeit offset by 0.25 dex . the comparison between derived lmc and solar neighbourhood wc wind properties suggests a dependence of dm / dt @xmath107 with @xmath111 . crowther et al . ( 2002 ) argued that the wc subtype distributions in the lmc and milky way resulted from this metallicity dependence . ciii @xmath45696 emission is very sensitive to mass - loss rate , so weak winds for lmc wc stars would produce negligible ciii @xmath45696 emission ( wc4 subtypes ) and strong winds interior to the solar circle would produce strong ciii @xmath45696 emission ( wc8 - 9 subtypes ) , in agreement with the observed subtype distributions . historically , it has not been clear whether radiation pressure alone is sufficient to drive the high mass - loss rates of wr stars . let us briefly review the standard castor , abbott & klein ( 1975 , hereafter cak ) theory behind radiatively driven winds before addressing the question of line driving for wr stars . pulsations have also been proposed for wr stars , as witnessed in intensive monitoring for the most photometrically variable wn8 star with the _ most _ satellite by lefvre et al . interpretation of such observations however remains ambiguous ( townsend & macdonald 2006 ; dorfi , gautschy & saio 2006 ) . the combination of plentiful line opacity in the extreme uv , where the photospheric radiation originates , allows for efficient driving of hot star winds by radiation pressure ( milne 1926 ) . in a static atmosphere , the photospheric radiation will only be efficiently absorbed or scattered in the lower layers of the atmosphere , weakening the radiative acceleration , @xmath112 , in the outer layers . in contrast , atoms within the outer layers of an expanding atmosphere see the photosphere as doppler - shifted radiation , allowing absorption of undiminished continuum photons in their line transitions ( sobolev 1960 ) . the force from optically thick lines , which provide the radiative acceleration by absorbing the photon momentum , scale with the velocity gradient . there can be at most @xmath113 thick lines , implying a so - called single - scattering limit @xmath114 castor , abbott & klein ( 1975 ) and abbott ( 1982 ) developed a self - consistent solution of the wind properties , from which one obtains a velocity ` law ' @xmath115 for which @xmath116=0.8 for o - type stars ( pauldrach , puls & kudritzki 1986 ) . @xmath112 can be written in terms of the thompson ( electron ) scattering acceleration , i.e. @xmath117 . the equation of motion can then be expressed as @xmath118 it is clear that both @xmath57 and @xmath119 need to be large for a hot star to possess a wind . in the cak approach , optically thick lines are assumed not to overlap within the wind . in reality , this is rarely true in the extreme uv where spectral lines are very tightly packed and the bulk of the line driving originates . consequently , another approach is needed for wr stars whose winds exceed the single scattering limit ( lamers & leitherer 1993 ) , namely the consideration of multiple scattering . historically , the strength of wr winds were considered to be mass dependent , but metallicity - independent ( langer 1989 ) . observational evidence now favours metallicity - dependent wr winds , with a dependence of dm / dt @xmath107 , with @xmath120 for wn stars ( sect . [ metallicity ] ) . it is established that o star winds are driven by radiation pressure , with a metallicity dependence that is similar to wn stars ( mokiem et al . consequently , the notion that wr winds are radiatively driven is observationally supported . theoretically , lucy & abbott ( 1993 ) and springmann ( 1994 ) produced monte carlo wind models for wr stars in which multiple - scattering was achieved by the presence of multiple ionization stages in the wind . however , a prescribed velocity and ionization structure was adopted in both case studies , plus the inner wind acceleration was not explained . schmutz ( 1997 ) first tackled the problem of driving wr winds from radiatively - driven winds self - consistently using a combined monte carlo and radiative transfer approach . he also introduced a means of photon - loss from the heii ly@xmath2 303 line via a bowen resonance - fluorescence mechanism . this effect led to a change in the ionization equilibrium of helium , requiring a higher stellar luminosity . the consideration of wind clumping by schmutz ( 1997 ) did succeed in providing a sufficiently strong outflow in the outer wind . photon - loss nevertheless failed to initiate the requisite powerful acceleration in the deep atmospheric layers . subsequent studies have supported the principal behind the photon - loss mechanism for wr stars , although the effect is modest ( e.g. de marco et al . 2000 ) . the next advance for the inner wind driving was by nugis & lamers ( 2002 ) whose analytical study suggested that the ( hot ) iron opacity peak at 10@xmath121 k is responsible for the observed wr mass - loss in an optically thick wind ( a cooler opacity peak exists at 10@xmath122k ) . indeed , grfener & hamann ( 2005 ) established that highly ionized fe ions ( feix - xvii ) provides the necessary opacity for initiating wr winds deep in the atmosphere for wr111 ( hd 165763=wr111 , wc5 ) . the wind acceleration due to radiation and gas pressure self - consistently matches the mechanical and gravitational acceleration in their hydrodynamical model . grfener & hamann ( 2005 ) achieved the observed terminal wind velocity by adopting an extremely low outer wind filling factor of @xmath96=0.02 . this degree of clumping is unrealistic since predicted line electron scattering wings are too weak with respect to observations . a more physical outer wind solution should be permitted by the inclusion of more complete opacities from other elements such as ne and ar . the velocity structure from the grfener & hamann ( 1995 ) hydrodynamical model closely matches a typical @xmath116=1 velocity law of the form in eqn [ velocity_law ] in the inner wind . a slower @xmath116=5 law is more appropriate for the outer wind . indeed , such a hybrid velocity structure was first proposed by hillier & miller ( 1999 ) . theoretically , both the hydrodynamical models of grfener & hamann ( 2005 ) and recent monte carlo wind models for wr stars by vink & de koter ( 2005 ) argue in favour of radiation pressure through metal lines as responsible for the observed multiple - scattering in wr winds . the critical parameter involving the development of strong outflows is the proximity of wr stars to the eddington limit , according to grfener & hamann ( 2007 ) . vink & de koter ( 2005 ) performed a multiple - scattering study of wr stars at fixed stellar temperatures and eddington parameter . a metallicity scaling of dm / dt @xmath107 with @xmath123=0.86 for 10@xmath124 was obtained . this exponent is similar to empirical wn and o star results across a more restricted metallicity range . grfener & hamann ( 2007 ) predict a decrease in the exponent or late - type wn stars at higher @xmath57 , plus reduced wind velocities at lower metallicities . vink & de koter ( 2005 ) predict that the high metal content of wc stars favour a weaker dependence with metallicity than wn stars . a mass - loss scaling with exponent @xmath123=0.66 for @xmath125 is predicted . this is consistent with the observed wc mass - loss dependence between the lmc and milky way . at low metallicity , vink & de koter ( 2005 ) predict a weak dependence of @xmath123=0.35 for @xmath126 providing atmospheric carbon and oxygen abundances are metallicity - independent . the components of a massive binary may evolve independently , as if they were single stars providing their orbital periods are sufficiently large . in contrast , an interacting or close binary system represents the case in which the more massive primary component expands to fill its roche lobe , causing mass exchange to the secondary . for an initial period of several days , the primary will reach its roche lobe whilst still on the h - core burning main sequence ( case a ) and transfer the majority of its hydrogen - rich envelope to the secondary . for initial periods of a few weeks or years , mass transfer will occur during the hydrogen - shell burning phase ( case b ) or he - shell burning ( case c ) , respectively . cases b and c are much more common than case a due to the much larger range of orbital periods sampled . according to vanbeveren , de loore & van rensbergen ( 1998 ) a common envelope will occur instead of roche lobe overflow if there is an lbv phase . close binary evolution will extend the formation of wr stars to lower initial mass , and consequently lower luminosity with respect to single star evolution ( e.g. vanbeveren et al . the observed lower mass limit to wr formation in the milky way is broadly consistent with the single star scenario ( meynet & maeder 2003 ) . therefore , either low - mass he stars are not recognised spectroscopically as wr stars , or they may be too heavily diluted by their brighter o star companions for them to be observed . one natural , albeit rare , consequence of massive close binary evolution involves the evolution of the initial primary to core - collapse , with a neutron star or black hole remnant , in which the system remains bound as a high mass x - ray binary ( hmxb , wellstein & langer 1999 ) . the ob secondary may then evolve through to the wolf - rayet phase , producing a wr plus neutron star or black hole binary . for many years , searches for such systems proved elusive , until it was discovered that cyg x3 , a 4.8 hour period x - ray bright system possessed the near - ir spectrum of a he - star ( van kerkwijk et al . 1992 ) . nevertheless , the nature of cyg x3 remains somewhat controversial . wr plus compact companion candidates have also been identified in external galaxies , ic 10 x1 ( bauer & brandt 2004 ) and ngc 300 x-1 ( carpano et al . 2007 ) . the presence of two early - type stars within a binary system naturally leads to a wind - wind interaction region . in general , details of the interaction process are investigated by complex hydrodynamics . nevertheless , the analytical approach of stevens , blondin & pollock ( 1992 ) provides a useful insight into the physics of the colliding winds . a subset of wr stars display non - thermal ( synchrotron ) radio emission , in addition to the thermal radio emission produced via free - free emission from their stellar wind . consequently , a magnetic field must be present in the winds of such stars , with relativistic electrons in the radio emitting region . shocks associated with a wind collision may act as sites for particle acceleration through the fermi mechanism ( eichler & usov 1993 ) . free electrons would undergo acceleration to relativistic velocities by crossing the shock front between the interacting stellar winds . indeed , the majority of non - thermal wr radio emitters are known binaries . for example , wr140 ( wc7+o ) is a highly eccentric system with a 7.9 year period , in which the radio flux is thermal over the majority of the orbit . between phases 0.55 and 0.95 ( where phase 0 corresponds to periastron passage ) , the radio flux increases dramatically and displays a non - thermal radio index ( williams et al . 1990 ) . one may introduce the wind momentum ratio , @xmath127 @xmath128 where the mass loss rates and wind velocities of components @xmath129 are given by @xmath130 and @xmath131 , respectively . in the simplest case of @xmath127=1 , the intersection between the winds occurs in a plane midway between the two stars . in the more likely situation of @xmath132 1 , the contact discontinuity appears as a cone wrapped around the star with the less momentum in its wind . by way of example , radio emission in the wr147 ( wn8 + b0.5 ) system has been spatially resolved into two components : thermal emission from the wn8 primary wind , plus non - thermal emission located close to the companion ( williams et al . the location of the non - thermal component is consistent with the ram pressure balance of the two stellar winds , from which a momentum ratio of @xmath133 can be obtained . for early - type stars that have reached their terminal velocities of several thousand kms@xmath26 , the post - shock plasma temperature is very high ( @xmath134 k ) . x - rays represent the main observational signature of shock - heated plasma . significantly harder x - ray fluxes are expected from massive binaries with respect to single stars . phase - locked x - ray variability is expected due to the change in opacity along the line - of - sight , or varying separation for eccentric binaries ( stevens , blondin & pollock 1992 ) . a colliding wind system in which substantial phase - locked variability has been observed with _ is @xmath14 vel ( wc8+o ) . the x - ray emission from the shock is absorbed when the opaque wind from the wr star lies in front of the o star . when the cavity around the o star crosses our line - of - sight , x - ray emission is significantly less absorbed ( willis , schild & stevens 1995 ) . the principal sources of interstellar dust are cool , high mass - losing stars , such as red giants , asymptotic giant branch stars , plus novae and supernovae . dust is observed around some massive stars , particularly lbvs with ejecta nebulae , but aside from their giant eruptions , this may be material that has been swept up by the stellar wind . the intense radiation fields of young , massive stars would be expected to prevent dust formation in their local environment . however , allen , swings & harvey ( 1972 ) identified excess ir emission in a subset of wc stars , arising from @xmath01000 k circumstellar dust . williams , van der hucht & th ( 1987 ) investigated the infrared properties of galactic wc stars , revealing persistent dust formation in some systems , or episodic formation in other cases . for a single star whose wind is homogeneous and spherically symmetric , carbon is predicted to remain singly or even doubly ionized due to high electron temperatures of @xmath010@xmath135k in the region where dust formation is observed to occur . however , the formation of graphite or more likely amorphous carbon grains requires a high density of neutral carbon close to the wc star . one clue to the origin of dust is provided by wr140 ( hd 193793 , wc7+o ) which forms dust episodically , near periastron passage . at this phase the power in the colliding winds is at its greatest ( williams et al . usov ( 1991 ) analytically showed that the wind conditions of wr140 at periastron favour a strong gas compression in the vicinity of the shock surface , providing an outflow of cold gas . it is plausible that high density , low temperature , carbon - rich material associated with the bow - shock in a colliding wind wc binary provides the necessary environment for dust formation . in contrast to episodic dust formers , persistent wc systems are rarely spectroscopic wc binaries , for which wr104 ( ve 2 - 45 , wc9 ) is the prototype identified by allen et al . spectroscopic evidence from crowther ( 1997 ) suggested the presence of an ob companion in the wr104 system when the inner wc wind was obscured by a dust cloud , analogous to r coronae borealis stars . conclusive proof of the binary nature of wr104 has been established by tuthill , monnier & danchi ( 1999 ) from high spatial resolution near - ir imaging . dust associated with wr104 forms a spatially confined stream that follows a spiral trajectory ( so - called ` pinwheel ' ) , analogous to a garden rotary sprinker . the cocoon stars after which the quintuplet cluster at the galactic centre was named have also been identified as dusty wc pinwheel stars by tuthill et al . ( 2006 ) . binarity appears to play a key role in the formation of dust in wc stars , providing the necessary high density within the shocked wind interaction region , plus shielding from the hard ionizing photons . the presence of hydrogen from the ob companion may provide the necessary chemical seeding in the otherwise hydrogen - free wc environment . alas , this possibility does have difficulties , since chemical mixing between the wc and ob winds may not occur in the immediate vicinity of the shock region . nevertheless , it is likely that all dust forming wc stars are binaries . the various inputs to stellar interior evolutionary models originate from either laboratory experiments ( e.g. opacities , nuclear reaction rates ) or astronomical observations ( e.g. mass - loss properties , rotation rates ) . indeed , mass - loss ( rather than convection ) has a dominant effect upon stellar models for the most massive stars . here we illustrate one of the potential pitfalls of this approach . according to koesterke et al . ( 1991 ) , stellar luminosities of some weak - lined early - type wn stars were unexpectedly low ( @xmath136 ) . in order to reproduce such results , plus the observed n(wr)/n(o ) ratio , meynet et al . ( 1994 ) adopted higher stellar mass - loss rates , with respect to previous empirical calibrations . improvements in non - lte models have subsequently led to higher derived stellar luminosities ( e.g. hamann , grfener & liermann 2006 ) . revised wr luminosities removed the primary motivation behind elevated mass - loss rates . indeed , allowance for wind clumping has now led to the need to reduce mass - loss rates in evolutionary models . mass - loss and rotation are intimately linked for the evolution of massive stars . stellar winds will lead to spin - down for the case of an efficient internal angular momentum transport mechanism . at solar metallicity , one anticipates rapid spin - down for very massive stars due to their strong stellar winds ( langer 1998 ) . initial rotational velocities are erased within a few million years . in contrast , initial conditions may remain preserved throughout the main sequence lifetime of o - type stars in metal - poor environments due to their weak stellar winds . discrepancies between evolutionary model predictions and a number of observed properties of high mass stars led to the incorporation of rotational mixing into interior models , following the theoretical treatment of zahn ( 1992 ) . rotational mixing reproduces some of the predictions from the high mass - loss evolutionary models of meynet et al . ( 1994 ) , for which two approaches have been developed . meynet & maeder ( 1997 ) describe the transport of angular momentum in the stellar interior through the shear and meridional instabilities . in contrast , momentum is transported radially from the core to the surface in the approach of heger et al . ( 2000 ) . rotation favours the evolution into the wr phase at earlier stages , increasing the wr lifetime . lower initial mass stars also enter the wr phase . for an assumed initial rotational velocity of 300 kms@xmath26 , the minimum initial mass star entering the wr phase is 22 @xmath1 , versus 37@xmath1 for non - rotating models at solar metallicity ( meynet & maeder 2003 ) . evolutionary models allowing for rotational mixing do predict a better agreement with the observed ratio of wr to o stars at low metallicity , the existence of intermediate wn / c stars ( though see langer 1991 ) , and the ratio of blue to red supergiants in galaxies . regarding the initial rotation velocities of massive stars , evolutionary models adopt fairly high values . observationally , @xmath137 kms@xmath26 for young ob stars in the smc cluster ngc 346 ( mokiem et al . 2006 ) , suggesting somewhat lower initial rotation rates , on average . nevertheless , lower mass limits to the formation of wr stars are predicted to lie in the range 42@xmath1 at z=0.004 ( smc ) to 21@xmath1 at z=0.04 ( m 83 ) for models allowing for such high initial rotation rates ( meynet & maeder 2005 ) . hirschi , meynet & maeder ( 2005 ) present chemical yields from rotating stellar models at solar metallicity , revealing increased c and o yields below 30@xmath1 , and higher he yields at higher initial masses . it is possible to predict the number ratio of wr to o stars for regions of constant star formation from rotating evolutionary models , weighted over the initial mass function ( imf ) . for an assumed salpeter imf slope for massive stars , the ratios predicted are indeed in much better agreement with the observed distribution at solar metallicity ( meynet & maeder 2003 ) . since the o star population is relatively imprecise , the predicted wr subtype distributions are often used instead for comparisons with observations . from figure [ wrpop ] , the solar neighbourhood wr subtype distribution contains similar numbers of wc and wn stars , with an equal number of early ( h - free ) and late ( h - rich ) wn stars . from comparison with evolutionary models , the agreement is reasonable , except for the brevity of the h - deficient wn phase in interior models at solar metallicity . this aspect has been quantified by hamann , grfener & liermann ( 2006 ) . synthetic wr populations from the meynet & maeder ( 2003 ) evolutionary tracks predict that only 20% of wn stars should be hydrogen - free , in contrast to over 50% of the observed sample . non - rotating models provide better statistics , although low luminosity early - type wn stars are absent in such synthetic populations . figure [ wcwn ] shows that the ratio of wc to wn stars is observed to increase with metallicity for nearby galaxies whose wr content has been studied in detail . one notably exception is the low - metallicity local group galaxy ic 10 ( massey & holmes 2002 ; crowther et al . the wr population of ic 10 remains controversial , since high galactic foreground extinction favours the detection of wc stars over wn stars . the preferential detection of wc stars arises because the equivalent widths of the strongest optical lines in wc stars are ( up to 100 times ) larger than those of the strongest optical lines in wn stars ( massey & johnson 1998 ) . two evolutionary model predictions are included in fig . [ wcwn ] ; ( a ) allowing for rotational mixing but without a wr metallicity scaling ( meynet & maeder 2005 ) ; ( b ) neglecting rotational mixing , although with a metallicity scaling for wr stars ( eldridge & vink 2006 ) . the latter models , in which convective overshooting is included , agree better with observations at higher metallicities . it should be emphasised that a significant wr population formed via a close binary channel is required to reproduce the observed wr / o ratio across the full metallicity range in the eldridge & vink ( 2006 ) models ( see also van bever & vanbeveren 2003 ) . a significant binary channel is not required for the meynet & maeder ( 2005 ) rotating evolutionary models . these resolve many issues with respect to earlier comparisons to observations , although some problems persist . in very metal - poor environments ( @xmath01/50 @xmath106 ) the wr phase is predicted for only the most massive single stars ( @xmath138 ) according to non - rotating models of de mello et al . nevertheless , wr stars have been observed in uv and optical spectroscopy of such metal - poor regions within izw 18 ( izotov et al . 1997 ; brown et al . 2002 ) and sbs 0335 - 052e ( papaderos et al . 2006 ) . only wc stars have been unambiguously identified spectroscopically , yet wn stars would be expected to dominate the wr population of such metal - poor regions . the strength of wn winds are believed to depend more sensitively upon metallicity than the strength of wc winds ( vink & de koter 2005 ) . therefore , wn stars may be extremely difficult to directly detect . hot weak - lined wn stars are predicted to have hard uv ionizing flux distributions , so they may be indirectly indicated via the presence of strong nebular heii @xmath44686 emission . indeed , strong nebular heii @xmath44686 is observed in izw 18 , sbs 0335 - 052e and other very metal - poor star forming galaxies . potentially large wr populations are inferred in very metal - deficient galaxies , depending upon the exact wr wind dependence upon metallicity ( crowther & hadfield 2006 ) . potentially , single star rotating evolutionary models are unable to reproduce the observed wr distribution in metal - poor galaxies . close binary evolution might represent the primary formation channel for such metal - poor wr stars , unless lbv eruptions provide the dominant method of removing the h - rich envelope at low metallicity . the end states of massive stars have been studied from a theoretical perspective by heger et al . in particular , wn and wc stars are the likely progenitors of ( at least some ) type ib and type ic core - collapse sn , respectively . this arises because , respectively hydrogen and hydrogen / helium are absent in such sne ( woosley & bloom 2006 ) . direct empirical evidence connecting single wr stars to type ib / c sn is lacking , for which lower mass interacting binaries represent alternative progenitors . one would need observations of @xmath139 wr stars in order to firmly establish a connection on a time frame of a few years , since wr lifetimes are a few 10@xmath82 yr ( meynet & maeder 2005 ) . hadfield et al . ( 2005 ) identified @xmath140 wr stars in m83 . narrow - band optical surveys of a dozen other high star - forming spiral galaxies within @xmath010 mpc would likely provide the necessary statistics . however , ground - based surveys would be hindered by the relatively low spatial resolution of 20 pc per arcsec at 5 mpc . nevertheless , the light curves of broad - lined type ic supernovae sn 1998bw , sn 2003dh and sn 2003lw suggest ejected core masses of order 10@xmath1 ( nakamura et al . 2001 ; mazzali et al . 2003 ; malesani et al . these agree rather well with the masses of lmc wc4 stars inferred by crowther et al . ( 2002 ) , if we additionally consider several solar masses which remain as a compact ( black hole ) remnant . tese supernovae were associated with long grbs , namely 980425 ( galama et al . 1998 ) , 030329 ( hjorth et al . 2003 ) , and 031203 ( malesani et al . 2004 ) , in support of the ` collapsar ' model ( macfadyen & woosley 1999 ) . indeed , wr populations have been detected in the host galaxy of grb 980425 , albeit offset from the location of the grb by several hundred pc ( hammer et al . perhaps grbs are produced by runaway wr stars , ejected from high density star clusters ( cantiello et al . such a scenario would appear to contradict fruchter et al . ( 2006 ) , regarding the location of grbs in their host galaxies . the challenge faced by both single and binary evolutionary models is for a rotating core at the point of core - collapse ( woosley & heger 2006 ) . single star models indicate that stars efficiently spin - down during either : ( a ) the slowly rotating rsg stage due to the magnetic dynamo produced by differential rotation between the rotating he - core and non - rotating h - envelope ; ( b ) the mechanical loss of angular momentum from the core during the high mass - loss wr phase . spectropolarimetry does not favour rapid rotation for milky way wc stars , although some wn stars may possess significant rotation rates . typical magnetic fields of neutron stars are of order 10@xmath47 g , so one would expect a field of 10@xmath34 g for their progenitor wolf - rayet stars . the first observational limits are now becoming available , namely @xmath14125 g for hd 50896 ( wr6 , st - louis et al . if wr stars are credible progenitors of ` magnetars ' , a subset of neutron stars that are highly magnetized ( @xmath142 g ) , the required wr magnetic field would be @xmath143 g. initial rapid rotation of a single massive star may be capable of circumventing an extended envelope via chemically homogeneous evolution ( maeder 1987 ) if mixing occurs faster than the chemical gradients from nuclear fusion . at sufficiently low metallicity , mechanical mass - loss during the wr phase would be sufficiently weak to prevent loss of significant angular momentum permitting the necessary conditions for a grb ( yoon & langer 2005 ) . alternatively , close binary evolution could cause the progenitor to spin - up due to tidal interactions or the merger of a black hole and he core within a common envelope evolution ( podsiadlowski et al . 2004 ) . both single and binary scenarios may operate . at present , the single scenario is favoured since long - soft grbs are predominantly observed in host galaxies which are fainter , more irregular and more metal - deficient than hosts of typical core - collapse supernovae ( e.g. fruchter et al . 2006 ) . of course , the ejecta strongly interact with the circumstellar material , probing the immediate vicinity of the grb itself ( van marle , langer & garca - segura 2005 ) . this provides information on the progenitor , for which one expects @xmath144 for wr winds ( eqn [ continuity ] ) . a metallicity - dependence of wr winds suggests that one would potentially expect rather different environments for the afterglows of long - duration grbs , depending upon the metallicity of the host galaxy . indeed , densities of the immediate environment of many grbs suggest values rather lower than typical solar metallicity wr winds ( chevalier , li & frasson 2004 ) . fryer , rockefeller & young ( 2006 ) estimate half of long grbs apparently occur in uniform environments , favouring a post - common envelope binary merger model . the agreement between multi - wavelength spectroscopic observations of wr stars and current non - lte model atmospheres is impressive . h - rich ( core h - burning ) wn stars are readily distinguished from classic h - deficient ( core he - burning ) wn , wc and wo stars . significant progress has been achieved in interior evolutionary models through the incorporation of rotational mixing . contemporary ( i.e. low ) mass - loss rates together with rotational mixing permits many of the observed properties of wr stars to be reproduced , at least those close to solar metallicity . 2 . empirical evidence and theoretical models both favour metallicity - dependent wr wind , providing a natural explanation to the wc ( and wo ) subtype distribution in the milky way and in external galaxies . a metallicity dependence is partially responsible for the wn subtype dependence , although the reduced nitrogen content in metal - poor galaxies also plays a role . the development of consistent radiatively - driven wr winds represents an important milestone . the predicted metallicity dependence of mass - loss rates from radiatively driven wind models agrees with observational estimates . the apparent convergence of spectroscopic and interior models suggests that we can re - assess the contribution of wr stars to the ionizing , mechanical and chemical enrichment of the ism in young star forming regions with respect to the previous annual review article on this subject ( abbott & conti 1987 ) . wr stars emit a high number of lyman continuum photons due to higher inferred stellar luminosities as a result of the inclusion of line blanketing in atmospheric models . inferred wr mass - loss rates have decreased due to overwhelming evidence in favour of clumped winds . the chemical enrichment from wr stars is believed to be significant due to an extended wc phase , as a result of rotational mixing within evolutionary models . current non - lte models rely upon a simplified clumpy wind structure , plus spherical symmetry . theoretical hydrodynamic predictions for the radial dependence of clumping in wr winds is anticipated to represent a major area of development over the next decade together with two dimensional non - lte radiative transfer codes . discrepancies persist between empirical wr populations and predictions from evolutionary models which allow for rotational mixing . metallicity dependent wr winds may improve consistency , together with the incorporation of magnetic fields . spatially resolved clusters rich in wr stars , such as westerlund 1 ( if it is genuinely co - eval ) , provide a direct means of testing evolutionary predictions . observational limits on magnetic fields within wolf - rayet stars are underway . the role of rotation in wr stars at low metallicity is of particular interest . at present , amongst the best means of establishing rapid rotation in wr stars is by measuring departures from spherical symmetry using spectropolarimetry . this technique has been applied to bright wr stars in the milky way , but wr stars within the magellanic clouds are also within the reach of ground - based 8 - 10 m instruments . the evolutionary paths leading to wr stars and core - collapse sn remain uncertain , as is the role played by lbv eruptions for the most massive stars . given the observational connection between type ic supernovae and long - soft grbs , do grbs result from low metallicity massive stars undergoing chemically homogeneous evolution , massive binaries during a common envelope phase , or runaways from dense star clusters ? the presence of wr stars in large numbers within very low metallicity galaxies appears contrary to the expectations of single star evolutionary models neglecting rotational mixing . are such stars exclusively produced by rapid rotation , close binary evolution , or via giant lbv eruptions ? 6 . most late wc stars are known to be dust formers , the universal presence of binary companions is not yet established for all dusty wc stars . in addition , it is not clear how dust grains form within such an extreme environment , the study of which merits further study . thanks to gtz grfener , john hillier , norbert langer , georges meynet , tony moffat , nathan smith , dany vanbeveren and peredur williams for useful comments on an early version of the manuscript . i wish to thank the royal society for providing financial assistance through their wonderful university research fellowship scheme for the past eight years . * narrow - band imaging : * wr candidates may be identified from narrow - band images sensitive to light from strong wr emission lines , after subtraction of images from their adjacent continua . * p cygni profiles * : spectral lines showing blue - shifted absorption plus red - shifted emission . characteristic of stellar outflows , associated with resonance lines of abundant ions ( e.g. civ 1548 - 51 ) . * non - lte : * solution of full rate equations is necessary due to intense radiation field . radiative processes dominate over collisional processes , so local thermodynamic equilibrium ( lte ) is not valid . * radiatively driven winds : * the transfer of photon momentum in the photosphere to the stellar atmosphere through absorption by ( primarily ) metal spectral lines . * monte carlo models : * a statistical approach to the radiative transfer problem , using the concept of photon packets . * clumped winds : * radiatively driven winds are intrinsically unstable , producing compressions and rarefactions in their outflows . * collapsar : * rapidly rotating wr star undergoes core - collapse to form a black hole fed by an accretion disk , whose rotational axis collimates the gamma ray burst jet . * gamma ray burst : * brief flash of gamma rays from cosmological distances . either a merger of two neutron stars ( short burst ) or a collapsar ( long burst ) . * magnetar : * highly magnetized neutron star , observationally connected with soft gamma repeaters and anomalous x - ray pulsars . * grfener & hamann 2005 : * first solution of hydrodynamics within a realistic wolf - rayet model atmosphere . * hillier 1989 * describes the extended atmospheric structure of a wc star . * van der hucht 2001 * catalogue of milky way wr stars , including cluster membership , binarity , masses . * lamers et al . 1991 * summary of key observational evidence in favour of a late stage of evolution for wr stars . * massey 2003 * review article on the broader topic of massive stars within local group galaxies . * meynet & maeder 2005 * comparison between observed wr populations in galaxies with evolutionary model predictions allowing for rotational mixing . * schaerer & vacca 1998 * describes the determination of wr and o star populations within unresolved galaxies . * williams et al . 1987 * describes various aspects of dust formation around wc stars . * grb * gamma ray burst * lbv * luminous blue variable * rsg * red supergiant * wn * nitrogen sequence wolf - rayet * wc * carbon sequence wolf - rayet * wo * oxygen sequence wolf - rayet * lte * local thermodynamic equilibrium * imf * initial mass function luminous blue variables , also known as hubble - sandage or s doradus type variables , share many characteristics of wolf - rayet stars . lbvs are widely believed to be the immediate progenitors of classic wn stars . lbvs possess powerful stellar winds , plus hydrogen depleted atmospheres , permitting similar analysis techniques to be used ( e.g. hillier et al . 2001 ) . lbvs occupy a part of the hertzsprung - russell diagram adjacent to wolf - rayet stars . typical spectral morphologies vary irregularly between a - type ( at visual maximum ) and b - type ( at visual minimum ) supergiants . examples include ag car and p cyg in the milky way , and s dor and r127 in the lmc ( e.g. humphreys & davidson 1994 ) . lbvs undergo occasional giant eruptive events signatures of which are circumstellar nebulae most notably undergone by @xmath145 car during two decades in the 19th century ( @xmath146 ejected ) . giant eruptions are believed to play a major role in the evolution of very massive stars via the removal of their hydrogen - 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c , langer n. 2005 . a&a 443:643 zahn j - p . a&a 265:115 llllll @xmath4 & window & & sp type & reference + & & late & early & + 525 & x - ray & 10@xmath147 & 10@xmath147 & wn & skinner et al . 2002 + & & & & wc & schild et al . 2004 + @xmath148912 & extreme uv & 39% & 69% & wn , wc & smith , crowther & norris 2002 + & & & & wn & hamann & grfener 2004 + 9121200 & far - uv & 21% & 12% & wn , wc & willis et al . 2004 + 12003200 & uv , near - uv & 33% & 16% & wn , wc & willis et al . 1986 + 32007000 & visual & 5% & 2% & wn , wc & conti & massey 1989 + 70001.1@xmath28 m & far - red & 0.9% & 0.3% & wn , wc & conti , massey & vreux 1990 + & & & & wn , wc & howarth & schmutz 1992 + 15@xmath28 m & near - ir & 0.4% & 0.2% & wn , wc & vacca et al . 2007 + 530@xmath28 m & mid - ir & 0.02% & 0.01% & wn & morris et al . 2000 + & & & & wcd & van der hucht et al . 1996 + llcccccl sp & @xmath43 & @xmath149 & @xmath150 & @xmath151 & @xmath152n(lyc ) & @xmath15 & example + type & kk & @xmath153 & @xmath1yr@xmath26 & kms@xmath26 & phs@xmath26 & mag + + 3-w & 85 & 5.34 & 5.3 & 2200 & 49.2 & 3.1 & wr3 + 4-s & 85 & 5.3 & 4.9 & 1800 & 49.2 & 4.0 & wr6 + 5-w & 60 & 5.2 & 5.2 & 1500 & 49.0 & 4.0 & wr61 + 6-s & 70 & 5.2 & 4.8 & 1800 & 49.1 & 4.1 & wr134 + 7 & 50 & 5.54 & 4.8&1300 & 49.4 & 5.4 & wr84 + 8 & 45 & 5.38 & 4.7&1000 & 49.1 & 5.5 & wr40 + 9 & 32 & 5.7 & 4.8 & 700 & 48.9 & 6.7 & wr105 + + 6 ha & 45 & 6.18 & 5.0 & 2500 & 49.9 & 6.8 & wr24 + 9 ha & 35 & 5.86 & 4.8&1300 & 49.4 & 7.1 & wr108 + + ( wo ) & ( 150 ) & ( 5.22 ) & ( 5.0 ) & ( 4100 ) & ( 49.0 ) & ( 2.8 ) & ( bat123 ) + ( 4 ) & ( 90 ) & ( 5.54 ) & ( 4.6 ) & ( 2750 ) & ( 49.4 ) & ( 4.5 ) & ( bat52 ) + 5 & 85 & 5.1 & 4.9 & 2200 & 48.9 & 3.6 & wr111 + 6 & 80 & 5.06 & 4.9 & 2200 & 48.9 & 3.6 & wr154 + 7 & 75 & 5.34 & 4.7 & 2200 & 49.1 & 4.5 & wr90 + 8 & 65 & 5.14 & 5.0 & 1700 & 49.0 & 4.0 & wr135 + 9 & 50 & 4.94 & 5.0 & 1200 & 48.6 & 4.6 & wr103 +

the phases of quantum chromodynamics ( qcd ) at nonzero temperature and density are a subject of continuing interest . while numerical simulations on the lattice can be of use at nonzero temperature when the quark density is small , standard monte carlo techniques are not of use in cold , dense quark matter . one expansion which is of utility is to expand in the limit of a large number of colors @xcite . for cold , dense quark matter quarks in the fundamental representation , coupled to an @xmath12 gauge theory this gives a `` quarkyonic '' phase @xcite . keeping the quark chemical potential , @xmath2 , of order one as the number of color @xmath13 , the free energy for this phase is dominated by that of quarks . nonetheless , excitations near the fermi surface are confined , perhaps baryonic , whence the name . in this paper , we consider chiral symmetry breaking in quarkyonic matter . we consider a phenomenological model for confinement , taking the timelike component of the gluon propagator to be @xmath14 . this is valid in coulomb gauge , for a spatial momentum @xmath15 , and corresponds to a potential which rises linearly in coordinate space . such a propagator was originally suggested by gribov @xcite and zwanziger @xcite . to use such a propagator in cold , dense , quark matter , it is necessary to assume that gluons are insensitive to screening by quarks . for this to be true , the number of flavors , @xmath6 , must be @xmath16 , and the chemical potential must satisfy @xmath17 , where @xmath3 is the renormalization mass scale of qcd @xcite . chiral symmetry breaking in such a model has been studied by glozman and wagenbrunn @xcite and by guo and szczepaniak @xcite , for values of @xmath18 . we work in the extreme quarkyonic limit , @xmath19 , so that the effects of chiral symmetry breaking in vacuum can be ignored . it is possible for chiral symmetry breaking to occur at large @xmath2 , since we are , by assumption , in a confined regime . of course there is no guarantee that our results apply to qcd , where @xmath20 ; nevertheless , there is certainly some range of @xmath1 , @xmath6 , and @xmath2 , where it does . if applicable to qcd , our results are of interest to intermediate densities , where both conventional nuclear physics and perturbative treatments fail . notably , this may include the astrophysics of neutron stars . in vacuum , chiral symmetry breaking occurs through the pairing of a left handed quark with a right handed anti - quark , @xmath21 , and vice versa . this condensate is , of course , spatially uniform , so that the spontaneous breaking of chiral symmetry does not disturb the lorentz invariance of the vacuum . now consider the effects of a fermi sea , where there is a net excess of quarks over anti - quarks . the analogy of the usual condensate is illustrated in fig . [ qqbar ] . energetically , it costs essentially zero energy to excite a quark right at the edge of the fermi sea . on the other hand , it costs at least @xmath22 to pull an anti - quark out from deep in the dirac sea . ( remember that we assume that @xmath2 is very large . ) thus the usual condensate can not be formed spontaneously , and anti - quarks will not enter into our analysis henceforth . there are numerous features which are not captured by the illustration in fig . [ qqbar ] . we really should draw not one , but two fermi seas : one for left handed quarks , and one for right handed quarks . to avoid unnecessary duplication , instead we assume that the quark , denoted by a filled circle , is always left handed , and that the anti - quark , denoted by an open circle , is right handed . the quark and anti - quark are also assumed to have the same color , so that any condensate is @xmath23 , and survives in the limit of large @xmath1 . if the quark has momentum @xmath15 , then the anti - quark , formed by removing a quark with momentum @xmath15 from the dirac sea , has momentum @xmath24 . thus the quark anti - quark pair has no net momentum , and this condensate is spatially uniform , as in vacuum . [ 1.0 ] , while its momentum is @xmath25.,title="fig : " ] in the presence of a fermi sea , though , it is also possible for chiral symmetry to be broken by pairing , say , a ( left - handed ) quark and a ( right - handed ) quark hole . if both the quark and the quark hole are near the edge of the fermi surface , then it costs little energy to excite them , and the energetic penalty paid to excite an antiquark can be avoided . the natural analogy to the condensate in vacuum is illustrated in fig . [ exciton ] , pairing a quark with momentum @xmath15 , and a hole , formed by removing a quark with momentum @xmath15 from the fermi surface . the momentum of the hole is then @xmath24 , so the quark - hole pair has no net momentum , and is spatially constant . in condensed matter physics , an excitation as in fig . [ exciton ] is known as an exciton . naively , we might expect that excitons are suppressed , since the relative momentum between the particle and the hole , @xmath26 , is large . [ 1.0 ] .,title="fig : " ] however , this is not the only way for quarks and their holes to break the chiral symmetry . consider pairing a ( left handed ) quark , with momentum @xmath15 , and the hole formed by removing a ( right handed ) quark with the _ opposite _ momentum , @xmath24 , from the fermi sea . the quark hole then has the same momentum as the quark , @xmath27 , so that the resulting condensate is _ not _ uniform , and has a net momentum @xmath28 ; this is , it varies as @xmath29 , where @xmath30 is the direction along which the pair moves , @xmath31 . such condensates do not occur in vacuum , where they would imply the spontaneous breaking of rotational symmetry . in condensed matter physics , though , such non - uniform condensates are common , and known as density waves @xcite ; this is then a chiral density wave . note that the relative momentum between the quark and its hole is small , so such a condensate may be favored . [ 1.0 ] , @xmath32.,title="fig : " ] in this paper we show that in the gribov - zwanziger model , that the exciton pairing of fig . [ exciton ] is not generated , but that the chiral density wave of fig . [ cdw ] is . again , this is familiar from systems in condensed matter : typically excitons are only created dynamically as resonances , such as by the absorption of light , and usually do not condense . density waves are common , especially for systems in @xmath8 dimensions @xcite . we will investigate all dirac and flavor structures , and show which types are preferred . for completeness , we illustrate the pairing between two quarks which leads to color superconductivity in fig . [ clr_super ] . this is pairing between a quark at one edge of the fermi surface , with momentum @xmath27 , and another quark at the other edge , with momentum @xmath24 . since pairing is between two quarks , the condensate has no net momentum and is spatially uniform . for this reason , pairing can occur over the entire fermi surface , in a spatially symmetric state . so far , we have not emphasized the @xmath1 and @xmath6 dependence of pairing , which is not captured by the illustrations in figs . [ qqbar ] - [ clr_super ] . the pairing in figs . [ exciton ] and [ cdw ] is between a quark and a quark hole of the same color , so the condensate is @xmath23 . further , to the extent that @xmath33 , the condensate is rather insensitive to @xmath6 . in contrast , the diquark pairing of color superconductivity depends upon @xmath1 and @xmath6 in an essential way . fermi statistics greatly constrains the pairing between two quarks ( or two quark holes ) : it is always anti - symmetric in color , so there are strong relations between the spatial wavefunction , flavor , and chirality . for instance , in case of @xmath34 and @xmath35 , spatially symmetric condensates form by anti - symmetrizing in flavor ; this condensate pairs quarks of the same chirality together , and so does not break the chiral symmetry . on the other hand , for @xmath36 and @xmath35 , the preferred condensate does break the chiral symmetry , through color - flavor locking @xcite . for more than three colors , the gaps for color superconductivity depend sensitively upon which representation one assumes the quarks to lie in . if the quarks are in the fundamental representation , then since the pairing for color superconductivity is anti - symmetric in the colors of the two quarks , the gap is not a color singlet , and is suppressed at large @xmath1 . it is also possible , however , to generalize qcd by letting the quarks lie in the two - index , anti - symmetric representation of color @xcite . this limit is rather different from that which we consider in this paper . there are @xmath37 quarks in this limit , so that gluons are affected the quarks , and there is no quarkyonic phase . this is like taking the number of flavors , @xmath6 , to grow with @xmath1 . in such a limit color superconductivity is not suppressed at large @xmath1 . it is not clear which of these two limits is most like qcd , with three colors and three light flavors . we suggest that it is useful to consider all possible limits , and to see what qualitative conclusions might be tested in qcd . [ 1.0 ] .,title="fig : " ] if a channel for color superconductivity exists , then cooper pairs will form for arbitrarily weak coupling . thus color superconductivity is always the dominant pairing mechanism at asymptotically large chemical potential . the essential question is then , how large does the chemical potential have to be for color superconductivity to win out over other pairing mechanisms , such as chiral density waves ? that chiral density waves @xcite dominate at large @xmath1 was first demonstrated by deryagin , grigoriev , and rubakov @xcite . using a perturbative gluon propagator , @xmath38 in momentum space , they find that chiral density waves form , with a condensate @xmath39 in magnitude , where @xmath40 is the qcd fine structure constant , measured at a scale @xmath41 . implicitly , the computation assumes that dense quarks form a fermi liquid , so that pairing is from quarks ( and holes ) within @xmath42 of the edge of the fermi sea . in contrast , quarkyonic matter is not a fermi liquid because of confinement of quarks . low energy excitations , within @xmath43 of the edge of the fermi sea , interact not through the perturbative gluon propagator , but through the gribov - zwanziger form , @xmath0 . a chiral density wave forms , with the same dirac structure as in perturbation theory . because the relevant momenta for pairing is controlled by a confining potential , the quarkyonic condensate for chiral density waves is inevitably @xmath43 in magnitude . our analysis applies in a regime of intermediate @xmath2 , where the quarkyonic gap is greater in magnitude than the perturbative gap . we find that for @xmath44 whenever @xmath45 , using @xmath46 . for the coupling in qcd , this means that @xmath47 gev . at larger values of @xmath41 , @xmath48 . as we discuss at the end of sec . 2.1 , this is a difficult regime to treat , as the effects of both perturbative interactions , and confinement , must be included . it is known that for a fermi liquid of dense quarks , chiral density waves lose out to color superconductivity except for very large values of the color , @xmath49 @xcite . this is because the chiral density waves generated by perturbative interactions are very sensitive to screening by dynamical quarks . in sharp contrast , we expect that quarkyonic chiral density waves are much less sensitive to screening by dynamical quarks . at large @xmath1 , dynamical quarks do not affect a quarkyonic phase until asymptotically large values of the chemical potential , @xmath50 . this power of @xmath51 follows either from considering the debye mass , as in ref . @xcite , or the free energy , as discussed in appendix [ appendixb ] . as long as there is a quarkyonic phase , and @xmath44 , we expect that quarkyonic chiral density waves dominate . the outline of the paper is as follows . for quark and quark hole excitations near the fermi surface , where the magnitude of the transverse momentum @xmath52 , the theory in @xmath7 dimensions reduces to an effective model in @xmath8 dimensions @xcite . in sec . [ dim_red ] we show that in the gribov - zwanziger model , this implies that in the gluon propagator , we can integrate over @xmath53 to obtain @xmath54 this is the gluon propagator in @xmath8 dimensions , so our effective theory is just qcd in @xmath8 dimensions . we discuss how in the gribov - zwanziger model , it is necessary for @xmath55 for this reduction to hold . in sec . [ eff_lag ] we consider how quantum numbers in @xmath7 dimensions map onto those in @xmath8 dimensions . starting with left and right handed massless quarks in @xmath7 dimensions , we find that the reduced model has a doubled flavor symmetry : @xmath6 flavors in @xmath7 dimensions becomes an @xmath9 symmetry in @xmath8 dimensions . this extended symmetry follows immediately from the analysis of shuster and son @xcite , and is very much like the doubling of flavor symmetry which occurs for heavy quarks . in sec . [ 2d_ferm ] we show how in @xmath8 dimensions , through an anomalous redefinition of the quark fields , a theory at nonzero chemical potential can be mapped onto the corresponding theory in vacuum . the net quark number , present in the theory at @xmath56 , is generated by the axial anomaly , as shown in appendix [ appendixa ] . the mapping can then be reversed : knowing results for a gauge theory in @xmath8 dimensions in vacuum , one can read off what happens in a fermi sea , @xmath56 . we show that a constant condensate in vacuum , @xmath57 , produces a condensate @xmath58 , where @xmath10 is the spatial coordinate , and @xmath59 the dirac matrix in @xmath8 dimensions . in the reduced model , the two dimensional @xmath60 , where @xmath61 and @xmath62 are dirac matrices in @xmath7 dimensions . this type of spatially dependent condensate at @xmath56 is familiar from soluble models in @xmath8 dimensions @xcite , where schn and thies termed it a `` chiral spiral '' @xcite . a chiral spiral was also found by bringoltz , in his numerical analysis of heavy quarks at nonzero density in qcd in @xmath8 dimensions @xcite . we thus term our solution a quarkyonic chiral spiral ( qcs ) . while we concentrate on massless quarks , in sec . [ massive_quarks ] we discuss how massive quarks also exhibit chiral spirals . this is because even for massive quarks , excitations about the fermi surface are gapless at tree level . effective theories about a qcs are considered in sec . [ wzw ] . using non - abelian bosonization , the reduced model in @xmath8 dimensions reduces to a wess - zumino - novikov - witten ( wznw ) model @xcite . this model has long range correlations , which should also produce long range correlations in @xmath7 dimensions . we conclude in sec . [ conclusions ] about whether qcs s may be be relevant for cold , dense qcd . we note that qcs s are closely related to pion condensation @xcite . they are not identical , because a pion condensate is a chiral spiral in the four dimensional @xmath63 , while for a qcs , it is in the two dimensional @xmath59 . we also note that if pionic chiral spirals occur , then probably so do kaonic chiral spirals . as a spatially varying condensate , a kaonic chiral spiral differs for the spatially constant condensate of kaon condensation @xcite . chiral density waves also arise in the sakai - sugimoto model @xcite . the physics of qcs s should be especially rich , however , since it includes the spontaneous breaking of translational and rotational symmetries , and a plethora of light modes . such phenomenon should have direct implications for observations of neutron / quarkyonic stars . we start by considering the schwinger - dyson equation for the quark self - energy . in a quarkyonic phase at large @xmath1 , the gluons are unaffected by the quarks , so that corrections to the gluon self - energy , and vertices , can be neglected . for the quark self - energy we take the sum of rainbow diagrams , @xmath64 where @xmath65 is the quark self - energy . at a nonzero chemical potential @xmath41 , the dressed quark propagator , @xmath66 , is @xmath67\gamma_4 + [ k_j + \sigma_j(k)]\gamma_j } \ ; . \label{sd2}\ ] ] where @xmath68 is determined self - consistently through the integral equation , eq . ( [ sd1 ] ) . we work in euclidean spacetime , @xmath69 . in the limit that @xmath55 , we neglect chiral symmetry effects as in vacuum , as illustrated in fig . [ qqbar ] ; such effects have been considered at @xmath56 by refs . consequently , we neglect terms @xmath70 in the quark self - energy and propagator . for the gluon propagator , @xmath71 , we take the gribov - zwanziger form , @xmath72 at the outset , we stress that we are dealing with a _ model _ of confinement . while the gluon propagator , and vertices , are unaffected by quark loops , there is no fundamental justification in taking the vertices to be the same as the bare ones , nor in taking the gluon propagator of eq . ( [ gluon_propagator ] ) . the gluon propagator involves a parameter , @xmath73 , which is the string tension , and has dimensions of mass squared , @xmath74 . numerical factors @xmath75 and @xmath76 are multiplied to reproduce a correct linear potential for the color singlet channel . we stress that the propagator in eq . ( [ gluon_propagator ] ) is valid only for _ small _ momenta , for @xmath77 . for larger momenta , one should use the usual gluon propagator of perturbation theory , @xmath78 . for excitations near the edge of the fermi sea , though , we can neglect the perturbative part of the propagator . this differs , for example , from the computation of the free energy @xcite . that is dominated by momenta transfers within the entire fermi sea , @xmath79 , for which the perturbative gluon ( and quark ) propagators should be used . there are contributions to the free energy from momenta @xmath43 , but this are small , powers of @xmath80 times the perturbative terms @xcite . after summing over the color indices , eq . ( [ sd1 ] ) becomes @xmath81 normalizing the generators as @xmath82 . since the right hand side of this equation is independent of @xmath83 , so is the quark self - energy , @xmath84 . the schwinger - dyson equation simplifies considerably if we consider only excitations near the edge of a fermi sea . for a free massless quark with momentum @xmath85 , in a fermi sea its mass shell is given by @xmath86 assuming that quark is along the @xmath30 direction , so that @xmath87 , @xmath88 thus , as is well known @xcite , in a fermi sea the dispersion relation linearizes in @xmath89 , allowing us to neglect the effects of the transverse momenta for the quarks . since this is the mass shell for a free quark , it neglects the effects of the quark self - energy , @xmath68 . we expect , though , that for quarks and quark holes near the fermi surface , including the quark self - energy does not affect the suppression of fluctuations in @xmath53 . for the gribov - zwanziger potential , the natural scale for the transverse momenta is @xmath90 . thus we can neglect the quark transverse momenta in the extreme quarkyonic limit , where @xmath55 . what happens when @xmath18 is a difficult problem which we do not address here . in the schwinger - dyson equation , the dominant contribution from such an infrared singular gluon propagator is when the gluon momentum is small . this constrains the internal and external momenta of the quarks to be near one another , @xmath91 . neglecting the transverse momenta of the quark , @xmath92 , we need only consider the two components of the momenta along the light cone , @xmath93 and @xmath94 . the schwinger - dyson equation thus reduces to @xmath95 since the transverse momentum @xmath92 enters only through the gluon propagator , we can now integrate it out , @xmath96 where we define a two dimensional gauge coupling constant @xmath97 , and neglect @xmath53 . the reduced schwinger - dyson equation then becomes @xmath98 this integral equation corresponds to qcd in @xmath8 dimensions . in axial gauge , @xmath99 , the two dimensional action for gluons reduces to a free term , @xmath100 in the limit of large @xmath1 , the model in @xmath8 dimensions obviously gives the integral equation of eq . ( [ sd5 ] ) . of course it is necessary to take some care in the reduction of the dirac matrices from @xmath7 to @xmath8 dimensions . we address this in sec . [ eff_lag ] . a more realistic model for the gluon propagator than eq . ( [ gluon_propagator ] ) is to take a sum of the gribov - zwanziger term , @xmath101 , plus a perturbative piece , @xmath78 . the confining term is valid for momenta @xmath102 ; the perturbative term , for momenta @xmath103 . the analysis goes through as above . the dependence of the quark propagators on the transverse momenta can be neglected , so one is left with an integral of the gluon propagator with respect to @xmath92 . integration over the gribov - zwanziger propagator gives @xmath104 , while the integral over the perturbative piece generates a logarithm @xcite , @xmath105 . with only a perturbative propagator , @xmath106 , the analysis is as follows @xcite : assuming that the @xmath107 is a constant gap , @xmath42 , the dimensional reduction can apply to the region of quark momenta , @xmath108 ( @xmath109 is measured from the fermi momentum ) . integrating over @xmath94 of a quark propagator yields a factor of the inverse energy , @xmath110 . the form of self - consistent equation has a similar structure as the bcs gap equation for the constant gap . a dominant contribution to the integral comes from the inverse energy part , which is sensitive to the gap ( an absence of gap yields infrared divergence in the integrand . this is nothing but the instability of the fermi surface . ) integration over @xmath93 at soft momentum region gives a logarithm term , @xmath111 . combining it with the logarithm of the gluon propagator , one find the squared logarithmic form of the gap equation , @xmath112 , with the self - consistent solution of eq . ( [ perturbative_gap ] ) . when we simply add the confining interaction to the self - consistent equation , its nature strongly changes . the self - energy can be no longer identified as the gap from the symmetry breaking , because the single - excitation - energy gap due to the confinement potential is large or divergent . the integral over @xmath93 in the gap equation is @xmath113 , where @xmath114 is a some regular function and @xmath115 is an infrared cutoff . the infrared cutoff @xmath115 for the gap equation for the single particle spectrum should be identified with the debye mass , so that the gap itself is of order @xmath116 . to see this imagine we try to ionize a hadronic state into its constituent single particle quark excitations . this can only happen when the separation between the constituents is of order the debye screening length , and costs an energy @xmath117 . this provides a gauge invariant definition for the single particle excitation gap . it also shows that the gap is large compared to the confinement scale in the quarkyonic phase . there is no weakly coupled solution for the gap equation when in this phase since , if one goes back to the derivation of the perturbative contribution , the derivation of the logarithmic terms is no longer valid . the dominant contribution for the perturbative piece in arises for momenta greater than the gap itself . it is important to understand that the analysis here concerns only the single particle excitation spectrum . the issue of chiral condensation is not directly related . we have provided a self - consistent derivation of the chiral condensate in other parts of this paper , and the condensate arises from non - perturbative effects . in a self - consistent analysis of course one can not rule out the possibility that there is another solution arising from a different kinematic region . as discussed in the introduction , we consider only a region of intermediate @xmath41 , where the @xmath44 . in this region , it is safe to include only the confining propagator , and neglect the perturbative term . when @xmath48 , we enter a more complicated regime , as the large difference in momentum scales required by the perturbative dimensional reduction , between @xmath118 and @xmath42 , is manifestly affected by confinement . confinement can probably be neglected when @xmath119 , but this only occurs when @xmath120 , which is an astronomically large value of @xmath41 , @xmath121gev . in this subsection we show how the reduction of the schwinger - dyson equation for the quark propagator applies as well to the bethe - salpeter equation for mesonic wave functions . henceforth we work in minkowski spacetime , @xmath122 . we consider the homogeneous equation , which in large @xmath1 is of the ladder type , @xmath123_{\alpha \beta}^{ab } d^{ab}_{\mu \nu}(k - q ) ; \ ] ] @xmath124 and @xmath125 denote color indices , @xmath126 and @xmath127 spinor indices . @xmath128 is a bound state wave function , where @xmath129 is the total momentum , @xmath130 the relative momentum . the dressed quark propagator , @xmath131 , is the self - consistent solution to the schwinger - dyson equation , eq . ( [ sd5 ] ) . the bethe - salpeter equation includes both singlet and adjoint channels , @xmath132 ; for the singlet channel , this reduces to ( @xmath133 ) @xmath134_{\alpha \beta } \frac{8\pi \sigma}{((\vec{k}-\vec{q}\,)^2)^2 } \ ; .\ ] ] as in the previous subsection , we concentrate on very soft gluons , @xmath135 . we assume that the both quarks , with momenta @xmath136 and @xmath137 are close to the fermi momentum , @xmath138 . we assume that the quark and quark hole pair have a total momentum along the @xmath10-direction , @xmath139 , and that the transverse momentum of the wavefunction can be neglected . we consider pairing energies that are both of the exciton type , @xmath140 and @xmath141 , and for a chiral density wave , @xmath142 and @xmath143 . neglecting the transverse momenta of the quark and quark hole , as we did for the quark mass shell in eq . ( [ mass_shell2 ] ) , we then introduce a wavefuntion , @xmath144 , which is obtained by averaging @xmath145 over the relative , transverse momenta , @xmath146 : @xmath147 the bethe - salpeter equation satisfied by this averaged wavefunction is @xmath148_{\gamma \delta } \ ; \frac{8\pi \sigma}{((\vec{k}-\vec{q}\,)^2)^2}. \end{split}\ ] ] we take the momenta of the quarks to be near the fermi surface , @xmath149 . thus we introduce two dimensional quark propagators , @xmath150 the bethe - salpeter equation becomes @xmath151_{\gamma \delta } \ ; \int \frac{d^2 \vec{q}_\perp}{(2\pi)^2 } \ ; \frac{8\pi \sigma}{((\vec{k}-\vec{q}\,)^2)^2 } \ ; . \end{split}\ ] ] the wave function on the right hand side depends only upon @xmath152 and @xmath153 , but not upon @xmath130 , and so we can integrate @xmath154 with respect to @xmath92 . that implies that as for the quark self - energy , we can integrate over @xmath146 , leaving @xmath155_{\gamma \delta } \ ; \frac{1}{(k_z - q_z)^2 } \ ; . \end{split}\ ] ] this is the same form as the bethe - salpeter equation for qcd in two dimensions @xcite , in @xmath99 gauge . in the previous section we demonstrated how dimensional reduction , from @xmath7 to @xmath8 dimensions , occurs for both the quark self - energy and the bethe - salpeter wave function . in this section we show how this arises at the level of effective lagrangians . our discussion elaborates upon that by shuster and son @xcite . start with the free quark lagrangian in @xmath7 dimensions . since we concentrate on quarks near the fermi surface , we assume that the spatial momentum is along the @xmath30 direction , @xmath156 for the dirac matrices , we take @xmath157,\ \ \gamma^j = \left [ \begin{array}{ccc } 0 & -\sigma^j \\ \sigma^j & 0 \end{array } \right],\ \ \gamma^5 = \left [ \begin{array}{ccc } \ { \bf 1}\ & \ 0\ \\ 0 & -{\bf 1 } \end{array } \right ] \ ; ; \ ] ] @xmath158 are the pauli matrices , and @xmath159 the unit matrix in two dimensions . we also define spin matrices as @xmath160 = \left [ \begin{array}{ccc } \ \sigma^{i}\ & \ 0\ \\ 0 & \sigma^{i } \end{array } \right ] \ , ; \ ] ] @xmath161 is the totally antisymmetric tensor , @xmath162 . if transverse momenta can be neglected , the reduced lagrangian in @xmath8 dimensions has an extended symmetry , which is related to the spin that quarks carry in @xmath7 dimensions . we introduce projectors for the quark fields , @xmath163 @xmath164 and @xmath165 are right and left handed fields , eigenstates of chirality . we also introduce projectors for spin @xcite along the @xmath30 direction , @xmath166 ; for eigenstates of chirality , this equals @xmath167 . the usual chiral basis is to take @xmath168 \ ; , \ ] ] but for this problem , this is rather inconvenient . to see this , we write the free lagrangian in terms of @xmath169 and @xmath170 , @xmath171 \ , . \end{split}\ ] ] the fields @xmath172 and @xmath173 have spin up along the @xmath174 direction , while @xmath175 and @xmath176 have spin along the @xmath177 direction . in terms of two dimensions , for positive energy @xmath172 and @xmath175 are right moving fields , while @xmath173 and @xmath176 are left moving fields . because there is no spin in two dimensions , we only need two component spinors . thus from one four component spinor in @xmath7 dimensions we obtain two types of two component spinors in @xmath8 dimensions , @xmath178 \;\;\ ; ; \;\;\ ; \varphi_\downarrow = \left [ \begin{array}{cc } \psi_{l+ } \\ \psi_{r- } \end{array } \right ] \ ; .\ ] ] the spinors @xmath179 and @xmath180 are eigenstate of spin along the @xmath30 direction , and act as two `` flavors '' in @xmath8 dimensions . this is valid to the extent that we can neglect the transverse momenta , which couple to the flavor breaking matrix , @xmath181 . we can then combine @xmath179 and @xmath180 into one four component spinor , @xmath182 , that is related to the original quark field , @xmath183 , by a unitary transform , @xmath184 : @xmath185 = u \ ; \psi \;\;\ ; ; \;\;\ ; u = \left [ \begin{array}{cccc } \ 1\ & \ 0\ & \ 0\ & \ 0\ \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{array } \right ] \quad ; \quad u^{\dagger } u = 1 \,.\ ] ] for the dirac matrices in two dimensions , @xmath186 , @xmath187 , we make the obvious choice , @xmath188 in two dimensions , @xmath189 while this identity is trivial mathematically , it plays an important role in the next section . the conjugate is defined as @xmath190 . it is straightforward to rewrite quark bilinears in @xmath7 dimensions in terms of spinors in @xmath8 dimensions . the operators appearing in the action , @xmath191 directly map from four to two dimensions . because of these identities , we see that in the presence of gauge fields , and a nonzero chemical potential , that a gauge theory in @xmath7 dimensions maps into one in @xmath8 dimensions , @xmath192 \phi - \frac{1}{2 } \ ; { { \rm tr}}\ ; g_{\mu\nu}^2 \,.\ ] ] fermions in two dimensions only require two components . thus a single , four component spinor in four dimensions becomes two types of two components spinors in two dimensions : @xmath183 , or equivalently @xmath182 , become @xmath193 , where @xmath194 is the flavor index in two dimensions , generated dynamically by dimensional reduction . we note that the mass term reduces similarly , @xmath195 one can write the complete dictionary to go from quark bilinears in @xmath7 dimensions to those in @xmath8 dimensions . we introduce matrices for the two dimensional flavor , which act in the space of @xmath179 and @xmath180 : @xmath196 , \left [ \begin{array}{ccc } 0 & -i { \bf 1 } \\ i{\bf 1 } & 0 \end{array } \right ] , \left [ \begin{array}{ccc } \ { \bf 1}\ & \ 0\ \\ 0 & -{\bf 1 } \end{array } \right ] \bigg).\ ] ] for operators which are diagonal in flavor , @xmath197 \phi = \bar{\psi } \ ; \gamma^0 \ ; u^\dag \left [ \begin{array}{ccc } \gamma^0 \ , \gamma^a & 0 \\ 0 & \gamma^0 \ , \gamma^a \end{array } \right ] u \ ; \psi \ ; ; \ ] ] where @xmath198 , @xmath59 , @xmath199 , or @xmath200 . for operators which are not diagonal in flavor , we note that the flavors matrices @xmath201 do not mix right and left moving components , so @xmath202 and @xmath201 commute with each other . thus operators which are not flavor singlets transform as @xmath197 \ ; { \bf \tau}_f \ ; \phi = \bar{\psi } \ ; \gamma^0 \ ; u^\dag \left [ \begin{array}{ccc } \gamma^0 \gamma^a & 0 \\ 0 & \gamma^0 \gamma^a \end{array } \right ] { \bf \tau}_f \ ; u \ ; \psi \,.\ ] ] the complete list of mapping for quark bilinears is given in table [ dictionary ] . .transformation between quark bilinears in @xmath8 and @xmath7 dimensions . [ cols="^,^,^,^,^",options="header " , ] we can also use these results to construct the relevant effective lagrangian for a model in which the gluon propagator is a sum of a gribov - zwanziger term , eq . ( [ gluon_propagator ] ) , and a perturbative term . integration over the former gives qcd in two dimensions , while integration over the perturbative gluon propagator give a non - abelian thirring model @xcite . thus the general effective model is a gauged , non - abelian thirring model . as noted in the introduction , we consider only a region of intermediate @xmath41 , where the effects of the thirring model can be neglected . in two spacetime dimensions , for massless quarks one can directly map the theory at @xmath56 onto that in vacuum , @xmath203 . this has been noted before , in various guises , in the literature before @xcite , especially by christiansen and schaposnik @xcite . hopefully our discussion adds clarity . consider the following transformation of the quark fields : @xmath204 this transformation is defined to be the same for each of the two dimensional `` flavors '' , @xmath179 and @xmath180 , and so preserves the flavor symmetry . under this transformation , the lagrangian becomes @xmath205 \phi = \overline{\phi } ' \big [ i \ , \gamma^\mu \ , ( \partial_\mu + i g_{{\rm 2d}}\ , a_\mu ) \big ] \phi ' . \label{trans_massless_qks}\ ] ] that is , by redefining the quark fields , we have _ completely _ eliminated the chemical potential ; one has transformed the theory from one in the presence of a fermi sea to that in vacuum . this happens because when @xmath206 acts upon @xmath207 , it equals @xmath208 , which by eq . ( [ twodim_identity ] ) , equals @xmath209 , and so cancels the term for the quark chemical potential in the original lagrangian . as we discuss in sec . [ massive_quarks ] , this is special to massless quarks , and does not hold for massive quarks . it also does not hold in higher dimensions , where the effects of transverse fluctuations obviate any such correspondence . we then have a quandry : there is a nonzero density of quarks in the original theory , with @xmath56 . the vacuum has no such density , so where did it go ? the answer is that the transformation of eq . ( [ anom_transf ] ) is anomalous , involving the dirac matrix @xmath59 . one can show that the correct quark density is given , precisely , by the anomaly . there are many ways of doing the calculation ; in appendix [ appendixa ] we give the computation based upon operator regularization . the computation also shows that the only quark bilinear to receive an anomalous contribution is that for quark number . for other operators , the transformation from @xmath182 to @xmath210 can be computed naively . the most interesting transformation is for the chiral condensate . using eq . ( [ mass_transf ] ) , we write the a chiral condensate for @xmath182 , in terms of @xmath211 : @xmath212 assume that there is chiral symmetry breaking in vacuum , so that @xmath213 . actually , in @xmath8 dimensions fluctuations disorder the system , and only leave quasi long range order @xcite . since this is due to fluctuations , at large @xmath1 such disorder only occurs over distances exponential in @xmath1 @xcite . neglecting such details , if chiral symmetry breaking occurs in vacuum , then it also occurs in the presence of a fermi sea , in the following manner : @xmath214 this is the strict analogy of migdal s pion condensation @xcite in @xmath8 dimensions . chiral symmetry is broken , but by a spiral in the two possible directions , between @xmath215 and @xmath216 . schn and thies @xcite termed this as a `` chiral spiral '' , and we adopt their evocative name , and so refer to our result as a quarkyonic chiral spiral . we can also understand why exciton pairing is not favored . an exciton condensate corresponds to @xmath217 , which is a spin @xmath218 operator . in the effective theory , the corresponding operators are @xmath219 and @xmath220 . these operators are flavor non - singlet . in two dimensions in vacuum , though , it is expected that the spontaneous breaking of chiral symmetry proceeds through condensates which are flavor singlets , and not through condensates which carry flavor . we conclude this section by noting that the extended flavor symmetry in @xmath8 dimensions is special to the vector - like interactions of qcd , and is not a generic property of theories in @xmath7 dimensions . consider , for example , a general nambu - jona - lasino ( njl ) model , with interactions such as @xmath221 . this is the square of a mass term , and so is not invariant under the anomalous chiral transformation of eq . ( [ anom_transf ] ) . in accord with this , nickel found that in njl models at nonzero density , the thermodynamically favored ground state is a crystal , but not a chiral spiral @xcite . in this section we show how for massive quarks , excitations near the fermi surface can be mapped onto a ( modified ) theory of the vacuum . for massless quarks , this could be done for the entire theory ; here , it is only for fluctuations near the fermi surface . let the quark mass be @xmath222 , so the fermi momentum is related to the chemical potential as @xmath223 . we work in the extreme quarkyonic limit , where @xmath55 . starting with the theory in @xmath7 dimensions , by neglecting the transverse momenta we obtain an effective theory in @xmath8 dimensions , @xmath224 \phi \,.\ ] ] in this case , we could perform the transformation of eq . ( [ anom_transf ] ) , and so eliminate the term @xmath225 from the action , as in eq . ( [ trans_massless_qks ] ) . the chemical potential is still in the action , though , through the transformation of the mass term into a complicated , position dependent `` mass '' , as in eq . ( [ trans_mass ] ) . while one can not make an exact correspondence to the vacuum theory , one can still make an interesting , if more limited , correspondence . consider not all fluctuations in the theory , but just those near the fermi surface . as is well known @xcite , even massive particles have a massless dispersion near the fermi surface , with a modification to the speed of light : @xmath226 as for the massless case , we can neglect anti - quarks , deep in the fermi sea . first , let us decompose the spectrum of quarks . we have two fermi surfaces , @xmath227 , and have particle and hole excitations at each fermi sea , with energy @xmath228 . we identify the excitation around the fermi sea with @xmath229 as a right moving fermion , and with @xmath230 as a left moving fermion . the effective lagrangian becomes @xmath231 \phi \ , , \label{massive_eff_lag}\ ] ] where we have assumed @xmath99 gauge . this is the same theory as for the massless case , eq . ( [ trans_massless_qks ] ) , except the speed of light is not one , but @xmath232 . these elementary manipulations can be used to explain the nature of chiral spirals in exactly soluble models in @xmath8 dimensions @xcite . for massless quarks , one maps the complete theory , in the presence of a fermi sea , onto the vacuum . thus the computation of @xmath233 is complete , and both @xmath234 and @xmath235 oscillate about zero . there are several examples , such as the gross - neveu model for a large number of flavors , in which this can be computed analytically @xcite . bringoltz considered a nonzero density of massive quarks for qcd in @xmath8 dimensions @xcite . by numerical analysis of the theory in the canonical ensemble , he showed that chiral spirals also arise for massive quarks . these are due to quark and quark hole excitations about the edge of the fermi surface , eq . ( [ massive_eff_lag ] ) . as the density increases , @xmath236 , and the massive theory approaches the massless limit . the massless excitations near the fermi surface can be described in terms of a wess - zumino - novikov - witten ( wznw ) theory @xcite . the dictionary between bosonic and fermionic currents for color , flavor , and ( baryon ) charge elements is @xmath237 = { \ : \!\!}\psi_+^\dag t_a \psi_+ { \!\!:\ } , \ j_-^a = \frac{i}{2\pi } { { \rm tr}}[h ( \partial_- h^{-1 } ) t_a ] = { \ : \!\!}\psi_-^\dag t_a \psi_- { \!\!:\ } , \nonumber \\ j_+^f = & \frac{i}{2\pi } { { \rm tr}}[g^{-1 } ( \partial_+ g ) \tau_f ] = { \ : \!\!}\psi_+^\dag \tau_f \psi_+ { \!\!:\ } , \ j_-^f = \frac{i}{2\pi } { { \rm tr}}[g ( \partial_- g^{-1 } ) \tau_f ] = { \ : \!\!}\psi_-^\dag \tau_f \psi_- { \!\!:\ } , \nonumber \\ j_+ = & \sqrt{\frac{{n_{\rm c}}{n_{\rm f}}}{2\pi } } \partial_+ \phi = { \ : \!\!}\psi_+^\dag \psi_+ { \!\!:\ } , \ j_- = \sqrt{\frac{{n_{\rm c}}{n_{\rm f}}}{2\pi } } \partial_- \phi = { \ : \!\!}\psi_-^\dag \psi_- { \!\!:\ } \ ; ; \end{aligned}\ ] ] @xmath238 is the color matrix , and @xmath239 an element of @xmath12 ; @xmath240 is the flavor matrix , and @xmath241 an element of @xmath9 . normal ordering of a composite operator @xmath242 is denoted by @xmath243 . for free dirac fermions with @xmath1 colors and @xmath244 flavors , @xmath245 \,,\ ] ] the bosonized version is @xmath246 + s_{2 { n_{\rm f}}}^\text{color}[h ] + s_{{n_{\rm c}}}^\text{flavor}[g ] \label{wzw_action}\ ] ] with @xmath247 = & { n_{\rm c}}{n_{\rm f}}\int d^2 x \ ; ( \partial_\mu \phi)^2 , \nonumber \\ s_{k}[l ] = & k\ { { \rm tr}}\bigg [ \frac{1}{16\pi } \int d^2x \ ; \partial_\mu l \partial^\mu l^{-1 } + \frac{1}{24\pi } \int d^3x \ ; \epsilon^{\mu \nu \lambda } ( l^{-1 } \partial_\mu l ) ( l^{-1}\partial_\nu l ) ( l^{-1 } \partial_\lambda l ) \bigg ] . \nonumber \\\end{aligned}\ ] ] this is a sum of a free massless scalar , for the @xmath248 of baryon number , a @xmath12 wznw model with level @xmath249 , and a @xmath9 wznw model with level @xmath1 . a similar form can be derived for the theory of dirac fermions coupled to a gauge field @xcite . the flavor part of the action , @xmath250 $ ] , is completely unchanged , because the currents which define the flavor matrix @xmath241 are color singlets . the color dependent part of the action , @xmath251 $ ] , becomes that of a gauged wznw action . the spectrum of a gauged wznw model is involved @xcite . however , what we are most concerned about are excitations near the fermi surface ; namely , are there gapless modes . in this context , what is of greatest concern is that is that the massless correlations of the _ flavor _ wznw action , @xmath250 $ ] , dominate correlations over large distances . note that this is true for _ both _ massless and massive quarks . in either case , there are numerous gapless modes about the fermi surface . in this paper we approximate the confining potential in a quarkyonic phase as in eq . ( [ gluon_propagator ] ) . we then find that quarkyonic chiral spirals ( qcs s ) arise naturally , and can be expressed in terms of an effective model in @xmath8 dimensions . our analysis is valid in the extreme quarkyonic limit , where @xmath55 , within a narrow skin of the surface of the fermi sea , @xmath43 . in this work we only considered the formation of a single chiral density wave , in a fixed direction . it is most likely , though , that the entire fermi surface is covered with patches of chiral density waves , in different directions @xcite . the detailed manner in which the fermi surface is covered with such patches will be discussed separately @xcite . quarkyonic chiral spirals are reminiscent of the pion condensates of migdal @xcite . pion condensates arise in effective models of nucleons interacting with pions : they are chiral spirals which oscillate as @xmath252 , where @xmath253 is a pure number and @xmath254 is the pion decay constant . in contrast , a qcs arises from the interactions of quarks and gluons ; it is a chiral spiral not in chirality , @xmath63 , but in spin , @xmath255 . it is of great interest that we find a qcs for massive quarks . kaplan and nelson suggested that in dense matter , effective models of nucleons and kaons indicate that there is a condensate for the @xmath256 field @xcite . this is constant in space , and so is unlike the chiral spiral which we would expect in strange quarkyonic matter . for distances @xmath257 , qcs s have numerous modes with long ranged correlations , from the correlations of the wznw model , sec . [ wzw ] . it is possible that the long range correlations of the modes of the wznw model acquired finite range over distances @xmath258 . even so , this predicts _ many _ more light modes than expected otherwise . the formation of qcs s has strong implications for the phase diagram of a gauge theory . the usual expectation is that one goes from a phase where chiral symmetry breaking is broken through a constant condensate , @xmath57 , directly to a phase where it vanishes , @xmath259 . if a qcs forms , chiral symmetry is broken , but through an order parameter which differs from that in vacuum . as for a pion condensate , a qcs spontaneously breaks both translational and rotational symmetries . thus there is a strict order parameter which differentiates between ordinary chiral symmetry breaking , with a constant chiral order parameter , and a qcs . it also implies that there are exactly massless goldstone bosons in a qcs , from the spontaneously broken symmetries of translation and rotation . similarly , there is an order parameter which differentiates between a qcs , and a phase which is chirally symmetric ( at least for massless quarks ) . this implies that at zero temperature , as @xmath41 increases there is first a fermi sea . there is then a well defined phase transition from a phase with constant @xmath57 to a qcs . there is then a second phase transition , from the qcs to a chirally symmetry phase , with @xmath259 . this differs from models which exclude spatially dependent condensates , ref . @xcite , but similar to models which allow for then , ref . @xcite . thus at @xmath260 and @xmath56 , we predict the existence of an intermediate phase , with a qcs . in appendix [ appendixb ] we show that quarks affect gluons at asymptotically large @xmath261 . at such large @xmath41 , deconfinement and chiral symmetry breaking surely occur . thus suggests that at least for large @xmath1 , the region with a qcs is also large . further , it is very possible that the phase transition from a qcs , to a chirally symmetry phase , occurs before the theory deconfines . we conclude by suggesting that dynamical quarks do _ not _ easily wash out a quarkyonic phase , nor related effects , such as quarkyonic chiral spirals . in qcd , it is known that the effects of screening , from dynamical quarks , are not strong . notably , the linear term in the quark anti - quark potential persists to rather short distances , @xmath262 fm . this distance is comparable to the short range repulsion experienced by nucleons . short distances then translates into high densities . it would be very interesting to estimate this effect within effective models , such as approximate solutions to the schwinger - dyson equations . our analysis clearly raises more questions than it answers . however , we hope that we have provided a different way for thinking about cold , dense quark matter , and about the surprises which it might provide . in the end , this is of direct relevance to neutron ( quarkyonic ? ) stars , for which a wealth of experimental data will be forthcoming in the next few years . the research of r. d. pisarski and l. mclerran is supported under doe contract no . de - ac02 - 98ch10886 . r. d. pisarski also thanks the alexander von humboldt foundation for their support . this research of y. hidaka is supported by the grant - in - aid for the global coe program `` the next generation of physics , spun from universality and emergence '' from the ministry of education , culture , sports , science and technology ( mext ) of japan . t. kojo is supported by special posdoctoral research program of riken . l. mclerran gratefully acknowledges conversation with thomas schfer , who insisted that chiral symmetry breaking must occur in the form of chiral density waves at large @xmath1 . we also thank gokce basar , barak bringoltz , michael buchoff , aleksey cherman , thomas cohen , gerald dunne , robert konik , alex kovner , dominik nickel , and alexei tsvelik for useful discussions and comments . in this appendix we show that at a nonzero temperature , @xmath263 , screening by dynamical quarks enters when @xmath264 . consider , for example , the square of debye screening mass at one loop order @xcite . gluons contribute @xmath265 , quarks give @xmath266 . balancing the two terms , quarks start affecting gluons when when @xmath267 . next consider the same analysis for the free energy . contributions from gluons are @xmath268 , while those from quarks are @xmath269 times powers of @xmath41 and @xmath263 : @xmath270 , @xmath271 , and @xmath272 , see eq . ( [ semi3 ] ) below . of course at @xmath273 and beyond , the quark and gluon contributions get mixed up together , but this does not affect our power counting in @xmath1 . if we then balance the leading term for gluons , @xmath268 , against that of quarks , @xmath274 , we estimate that quarks dominate when @xmath275 @xcite , and not @xmath276 , as for the debye mass . with the free energy , however , one must take more care . one can not simply equate the magnitude fo the free energies , but remember that deconfinement is only defined by a change in the relevant order parameter , which is the ( renormalized ) polyakov loop @xcite . this is generated by a nontrivial distribution in the eigenvalues of the thermal wilson line . this can be modeled by expanding about a constant expectation for the timelike component of the vector potential , @xmath277 @xmath278 , is a diagonal matrix in color space , and is traceless , as a sum over elements of @xmath12 . @xmath278 has dimensions of mass , where the mass scale is set by the temperature , @xmath263 . we do not need to know the explicit distribution of the @xmath278 s in order to estimate how large @xmath41 must be for quarks to affect the @xmath278-distribution . at very large @xmath41 we can compute the quark determinant , in the presence of this background field , to one loop order . a constant field for @xmath279 acts like an imaginary chemical potential for color , and so it is natural to introduce a color dependent chemical potential @xcite , @xmath280 at one loop order one obtains the usual result for the pressure , with @xmath41 replaced by @xmath281 : @xmath282 for large @xmath41 , the dominant contribution is from the expansion of the first term , @xmath283 , as we estimated above . while this is as large as the gluon contribution when @xmath275 , that does not matter , since this term is _ independent _ of @xmath278 . this holds order by order in perturbation theory , simply because @xmath278 has dimensions of mass . the next term is from the expansion of @xmath284 , equal to @xmath285 ; this vanishes , though , because @xmath278 is a traceless matrix . the leading term which is @xmath278-dependent is the next term in the expansion of @xmath286 , which is @xmath287 . since the trace is @xmath23 , this is as large as the gluon contribution , @xmath37 , when @xmath267 . this agrees with our estimate from the debye mass . at large @xmath1 , since the quarks do not affect gluons until @xmath267 , the boundary from the confining , to the deconfining , phase is a straight line in the plane of @xmath263 and @xmath41 @xcite . the pure glue theory has a global symmetry of @xmath288 , which for large @xmath1 is approximately @xmath248 . at large @xmath41 , the leading term from quarks is @xmath289 , and breaks the @xmath288 symmetry . like other terms from quarks , this favors a real expectation value for the polyakov loop . such a term acts to wash out the line of first order transitions . there are two possibilities . one is that the quarks produce a critical endpoint for deconfinement . the other is that the first line for deconfinement bends and meets the axis for @xmath290 . since all of our analysis depends upon @xmath291 , we favor the former . in this appendix we compute the baryon number generated by the anomalous transformation in eq . ( [ anom_transf ] ) . consider the operator for baryon number , computed with point splitting : @xmath292 } } \over { z[\mu \neq 0 ] } } \\ = & \lim_{\epsilon \to 0 } { { \int { \cal d}\overline { \phi } ' { \cal d } \phi ' \ \overline{\phi}'(x+\epsilon ) e^{i\mu \epsilon_z \gamma^5 } \gamma^0 \phi'(x ) e^{is[\phi';\mu = 0 ] } } \over { z[\mu=0 ] } } \\ = & 2 \ ; \lim_{\epsilon \to 0 } \bigg(\frac{i}{2\pi \epsilon^2 } \bigg ) \bigg [ { { \rm tr}}[\gamma^0 { { \ooalign{\hfil/\hfil\crcr$\epsilon$ } } } ] + i\mu \epsilon_z { { \rm tr}}[\gamma^5\gamma^0 { { \ooalign{\hfil/\hfil\crcr$\epsilon$ } } } ] + o(\epsilon^3 ) \bigg ] \\ = & 2 \ ; \lim_{\epsilon \to 0 } \bigg [ \frac{i \epsilon_0}{\pi \epsilon^2 } - \frac{\mu \epsilon_z^2 } { \pi \epsilon^2 } \bigg ] \ , . \label{nb } \end{split}\ ] ] we take the symmetric limit to preserve lorentz symmetry in the ultraviolet regime , @xmath293 the first term is odd in @xmath294 , and so vanishes after averaging over both directions . hence @xmath295 this equals the baryon number density for a gas of two flavors of free quarks , with chemical potential @xmath41 . the computation can be repeated for other operators , such as @xmath234 , @xmath235 , and @xmath296 . these operators do not have anomalous terms , since @xmath297 = 0 , \ \ { { \rm tr}}[\gamma^5{{\ooalign{\hfil/\hfil\crcr$\epsilon$ } } } ] = 0,\ \ \frac{\epsilon_z}{\epsilon^2 } { { \rm tr}}[\gamma^5\gamma^z { { \ooalign{\hfil/\hfil\crcr$\epsilon$ } } } ] = \frac{\epsilon_z \epsilon_0}{\epsilon^2 } \rightarrow 0 \ ; 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thermochemical models have been used to describe alkali @xcite , titanium and vanadium @xcite , carbon , nitrogen , and oxygen @xcite , sulfur and phosphorus @xcite , condensate ( e.g. , * ? ? ? * ; * ? ? ? * ) chemistry in the atmospheres of gas giant planets , brown dwarfs , and low - mass dwarf stars . here we continue and extend these previous studies by using thermochemical equilibrium calculations to model the chemical behavior of fe , mg , and si in substellar objects . iron , magnesium , and silicon are the most abundant rock - forming elements in a solar composition gas , and condensed as iron metal ( fe ) , and forsterite ( mg@xmath0sio@xmath1 ) and enstatite ( mgsio@xmath2 ) will produce the most massive cloud layers in substellar atmospheres . cloud formation strongly affects the optical and infrared spectra of substellar objects by removing gases from the overlying atmosphere and by introducing solid or liquid cloud particles ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the large and growing number of discovered brown dwarfs ( @xmath3750 ) and extrasolar planets ( @xmath3400 ) makes it impractical to model the thermochemistry of all objects individually @xcite , as was done for jupiter and saturn @xcite and gliese 229b @xcite . instead , we adopt an approach similar to that of our previous papers @xcite and determine the abundance of each chemical species as a function of pressure , temperature , and metallicity , plotted in abundance contour diagrams . our results are thus independent of any particular pressure - temperature profile , and in principle , the atmospheric profile for any object may be superimposed on the abundance diagrams to determine its equilibrium atmospheric chemistry . in some instances , the behavior of key gases may be diagnostic of atmospheric temperature and/or tracers of weather in substellar objects . we begin with a brief description of our computational method ( @xmath4[s computational method ] ) , and then present our results for iron chemistry in substellar atmospheres in @xmath4[s iron chemistry ] . we first give an overview of iron chemistry in a solar composition gas and identify important gases and condensates ( @xmath4[ss overview of iron chemistry ] ) . this is followed by more detailed discussion of the chemical behavior individual fe - bearing gases as a function of temperature , pressure , and metallicity ( @xmath4[ss chemical behavior of iron - bearing gases ] ) . wherever possible , we note relevant spectroscopic observations of fe - bearing gases in substellar objects . our results for iron are then summarized by illustrating fe gas chemistry along the atmospheric profiles of representative substellar objects ( @xmath4[ss iron chemistry in substellar objects ] ) . a similar approach to magnesium and silicon chemistry follows in @xmath4[s magnesium chemistry ] and @xmath4[s silicon chemistry ] , respectively . we conclude with a brief summary in @xmath4[s summary ] . thermochemical equilibrium calculations were performed using a gibbs free energy minimization code , previously used for modeling the atmospheric chemistry of saturn @xcite and sulfur and phosphorus chemistry in substellar objects @xcite . thermodynamic data for the equilibrium calculations were taken from the compilations of @xcite , @xcite , the fourth edition of the janaf tables @xcite , and the thermodynamic database maintained in the planetary chemistry laboratory @xcite . this database includes additional thermodynamic data from the literature for compounds absent from the other compilations . all calculations were conducted using elemental abundances from @xcite for a solar system ( i.e. , protosolar ) composition gas . the effect of metallicity on sulfur and phosphorus chemistry was examined by running computations at [ fe / h ] = -0.5 dex ( subsolar ) , [ fe / h ] = 0 dex ( solar ) , and [ fe / h ] = + 0.5 dex ( enhanced ) metallicities . the metallicity factor , @xmath5 , is defined as @xmath6 $ ] . we assume that the elemental abundance ratios for mg , si , and other elements of interest vary similarly with [ fe / h ] ( e.g. , [ mg / h ] @xmath7 [ si / h ] @xmath7 [ fe / h ] ) over the range of metallicities considered here ( see * ? ? ? * ; * ? ? ? * ; * ? ? ? when considering the chemical behavior of individual gases , we focus on higher temperatures ( 800 k and higher ) , where thermochemical processes are expected to dominate over disequilibrium processes such as photochemistry or atmospheric mixing ( e.g. , see * ? ? ? we assume that condensates settle gravitationally into a cloud layer and are removed from the cooler , overlying atmosphere . this equilibrium cloud condensate scenario for the deep atmospheres of giant planets and brown dwarfs is supported by several lines of evidence ( e.g. , see * ? ? ? * and references therein ) . first , the presence of germane ( geh@xmath1 ) and the absence of silane ( sih@xmath1 ) in the upper atmospheres of jupiter and saturn ( even though si is expected to be much more abundant than ge ) can be explained by the removal of si from the gas into silicate clouds deeper in the atmosphere , whereas ge remains in the gas phase @xcite . secondly , the detection of h@xmath0s in jupiter s troposphere by the _ galileo _ entry probe indicates that fe must be sequestered into a cloud layer at deep atmospheric levels , because the formation of fes would otherwise remove h@xmath0s from the gas above the @xmath8 k level @xcite . third , absorption from monatomic k gas in the spectra of t dwarfs @xcite requires the removal of al and si at deeper atmospheric levels , because k would otherwise be removed from the observable atmosphere by the condensation of orthoclase ( kalsi@xmath2o@xmath9 ) @xcite . the presence of monatomic na gas in brown dwarfs @xcite also suggests al and si removal , because albite ( naalsi@xmath2o@xmath9 ) condensation would otherwise effectively remove na from the observable atmosphere . furthermore , the removal of na by na@xmath0s cloud formation is consistent with the observed weakening of na atomic lines throughout the l dwarf spectral sequence and their disappearance in early t dwarfs ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? finally , as we note below , the disappearance of iron , magnesium , and silicon spectral features in later spectral types is consistent with removal of these elements into cloud layers . in our thermochemical model , the abundances of fe- , mg- , and si - bearing gases above the clouds are computed assuming saturation ( equilibrium ) vapor pressure . if supersaturation occurs , a condensate will form and settle toward the cloud layer to restore equilibrium . figure [ figure iron chemistry overview ] illustrates model atmospheric profiles for an m dwarf ( @xmath10 = 2600 k , @xmath11 ; * ? ? ? * ) , an l dwarf ( @xmath10 = 1800 k , @xmath11 ; * ? ? ? * ) , the hot , close - in ( pegasi ) planet hd209458b ( @xmath10 = 1350 k ; * ? ? ? * ) , the t dwarf gliese 229b ( @xmath10 = 960 k ; * ? ? ? * ) , and jupiter ( @xmath10 = 124 k ) , indicated by dashed lines . we note that jovian atmospheric chemistry differs slightly than that for a solar - metallicity gas because jupiter has a heavy element enrichment comparable to [ fe / h ] @xmath7 + 0.5 dex @xcite . also shown in figure [ figure iron chemistry overview ] are lines indicating where a(ch@xmath1 ) = a(co ) and a(h@xmath0 ) = a(h ) . these boundaries are important because carbon and oxygen affect the chemical behavior of many fe , mg , and si - bearing gases . methane is the dominant carbon - bearing gas in jupiter and t dwarfs ( such as gliese 229b ) whereas co is the dominant carbon - bearing gas in l dwarfs and pegasi planets ( such as hd209458b ) . molecular hydrogen dissociates into monatomic h at high temperatures and low pressures ( lower right corner of figure [ figure iron chemistry overview ] ) . however , h@xmath0 is the dominant form of hydrogen in substellar objects , and we therefore take @xmath12 throughout the following . this approximation holds for metallicities up to [ fe / h ] @xmath7 + 0.5 dex ; at higher metallicities the h@xmath0 mole fraction abundance decreases as the relative abundance of heavy elements increases ( e.g. , @xmath13 at [ fe / h ] @xmath7 + 1.0 dex ) . the dotted line in figure [ figure iron chemistry overview ] shows the condensation curve for fe metal , with an open circle denoting its normal melting point ( 1809 k ) . iron condensation occurs via @xmath14 the equilibrium condensation temperature ( @xmath15 ) of fe is approximated by @xmath16,\ ] ] where higher pressures and/or metallicities lead to higher fe condensation temperatures . iron cloud formation effectively removes nearly all iron from the atmosphere , and the abundances of fe - bearing gases above the clouds rapidly decrease with altitude . monatomic fe gas is the dominant fe - bearing gas in a protosolar composition gas until it is replaced by fe(oh)@xmath0 at low temperatures and high pressures . the conversion between fe and fe(oh)@xmath0 is represented by the net thermochemical reaction @xmath17 and the solid line dividing the fe and fe(oh)@xmath0 fields in figure [ figure iron chemistry overview ] indicates where these gases have equal abundances a(fe ) = a(fe(oh)@xmath0 ) @xmath18fe@xmath19 , where @xmath20fe@xmath19 is the total amount of iron in the gas . the position of this line is given by @xmath21,\ ] ] showing that an increase in metallicity will shift the fe - fe(oh)@xmath0 boundary to higher temperatures and lower pressures . in other words , an increase in metallicity increases the stability field of fe(oh)@xmath0 for otherwise constant conditions . the mole fraction abundance of monatomic fe as a function of pressure and temperature is shown in figure [ figure iron gas 1]a . as mentioned above , monatomic fe is the dominant fe - bearing gas over a wide range of pressures and temperatures in a solar system composition gas . below the fe clouds , the abundance of fe gas is representative of the total iron content of the atmosphere ( @xmath22 ) , and its abundance is given by @xmath23.\ ] ] upon fe metal condensation , the amount of iron in the gas rapidly decreases and the equilibrium fe gas abundance is governed by its vapor pressure over solid or liquid iron , represented in reaction ( [ reaction fe condensation ] ) . the mole fraction abundance of monatomic fe above the clouds is given by @xmath24 inversely proportional to @xmath25 . the fe gas abundance is independent of metallicity in this region because it depends solely on the temperature - dependent vapor pressure of iron . as described below ( see @xmath4[sss iron hydride , feh ] and @xmath4[sss iron monohydroxide , feoh ] ) , this expression can be used with chemical equilibria to determine the abundances of other iron gases in substellar atmospheres . equations giving the abundance of fe ( and other fe - bearing gases ) as a function of temperature , pressure , and metallicity below and above the fe clouds are listed in table [ table iron reactions ] . neutral monatomic fe possesses several hundred spectral lines in the @xmath26 and @xmath27 bands @xcite , and fe i features are observed in the spectra of brown dwarfs and low - mass dwarf stars ( e.g. * ? ? ? the strong fe feature at 1.189 @xmath28 m weakens in mid- to late - type m dwarf spectra and generally disappears in mid - type l dwarfs @xcite . this trend is consistent with the removal of iron from the gas into an fe metal cloud deck located deeper below the photosphere in objects with low effective temperatures ( e.g. * ? ? ? the chemical behavior of feh in a protosolar composition gas is illustrated in figure [ figure iron gas 1]b . the conversion between fe and feh occurs via the net thermochemical reaction @xmath29 here we show how chemical equilibria may used to derive equations giving the abundance of fe - bearing species as a function of temperature , pressure , and metallicity . rearranging the equilibrium constant expression for reaction ( [ reaction fe : feh ] ) yields @xmath30 using the fe abundance from equation ( [ equation fe below clouds ] ) , the temperature dependence of @xmath31 ( @xmath32 from 800 to 2500 k ) , and the hydrogen abundance ( @xmath33 ) , the feh abundance between the h@xmath0-h boundary and the fe cloud deck is given by @xmath34,\ ] ] proportional to @xmath35 and @xmath5 . at high temperatures ( @xmath36 k ) and low pressures ( @xmath37 bar ) as monatomic h becomes increasingly abundant the h@xmath0 abundance begins to decrease near the h@xmath0-h boundary . the reduced h@xmath0 abundance , in turn , reduces feh formation via reaction ( [ reaction fe : feh ] ) and changes the chemical behavior of feh at high temperatures and low pressures ( i.e. , lower right corner of figure [ figure iron gas 1]a ) . above the iron clouds , the fe abundance in equation ( [ equation fe above clouds ] ) is used in equation ( [ equation feh abundance generic ] ) to give @xmath38 where @xmath39 is proportional to @xmath40 and independent of metallicity . the differences in chemical behavior of feh below ( equation [ equation feh abundance below clouds ] ) and above ( equation [ equation feh abundance above clouds ] ) the fe cloud deck are illustrated in the shape of the feh abundance contours in figure [ figure iron gas 1]b . the sharp bends in the contours correspond to the fe condensation curve in figure [ figure iron chemistry overview ] . absorption bands from feh are common in the near - infrared spectra of brown dwarfs ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the most prominent feh feature is the band located at 0.9896 @xmath28 m , which weakens throughout the l dwarf spectral sequence and in early t dwarfs @xcite , consistent with the removal of iron into a cloud located deeper and deeper in the atmosphere . this feature unexpectedly strengthens again in mid - type t dwarf spectra @xcite , prompting different explanations for the shape of the color - magnitude diagram for brown dwarfs near the l - t transition . @xcite suggested that the strengthening feh bands are caused by upward convective mixing of feh gas from deeper levels where it is more abundant . however , as pointed out by @xcite and demonstrated @xcite , the fragile fe@xmath41h bond is unlikely to survive convective upwelling in a t dwarf atmosphere . instead , the observations are plausibly explained by cloud disruption and clearing ( in a @xmath421 @xmath28 m window ) which allows the observation of feh gas at deep atmospheric levels @xcite . mole fraction contours of feoh are illustrated in figure [ figure iron gas 1]c . the equilibrium between fe and feoh is represented by the net thermochemical reaction @xmath43 and expressions giving the feoh abundance as a function of temperature , pressure , and metallicity are listed in table [ table iron reactions ] . above the iron clouds , the curvature in the feoh abundance contours along the ch@xmath1-co boundary results from the effect of carbon chemistry on the h@xmath0o abundance in reaction ( [ reaction fe : feoh ] ) ( e.g. * ? ? ? for example , the atmospheric water abundance may be written as @xmath44,\ ] ] where @xmath45 is the water abundance in a solar - metallicity gas , which is @xmath46 inside the ch@xmath1 field and @xmath47 inside the co field in figure [ figure iron chemistry overview ] . at temperatures and pressures near the ch@xmath1-co boundary , @xmath45 may be derived from ch@xmath1-co equilibria @xcite . rearranging the equilibrium constant expression for reaction ( [ reaction fe : feoh ] ) gives @xmath48 substituting for the fe abundance from equation ( [ equation fe above clouds ] ) , the h@xmath0o abundance from equation ( [ equation h2o abundance generic ] ) , the h@xmath0 abundance ( @xmath49 ) , and the temperature dependence of @xmath50 ( @xmath51 from 800 to 2500 k ) , equation ( [ equation feoh abundance generic ] ) becomes @xmath52 + \log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'.\ ] ] this expression gives the feoh abundance above the iron clouds and includes the effect of metallicity on the atmospheric water abundance . the kinks in the feoh contours denote the position of the fe condensation curve . as demonstrated in equation ( [ equation feoh abundance above clouds ] ) and shown in figure [ figure iron gas 1]c , the mole fraction abundance of feoh is proportional to @xmath40 throughout this region . the chemical behavior of fe(oh)@xmath0 as a function of temperature and pressure is illustrated in figure [ figure iron gas 1]d . the equilibrium conversion between fe and fe(oh)@xmath0 is represented by reaction ( [ reaction fe : fe(oh)2 ] ) . below the fe clouds ( kinks in abundance contours ) , the fe(oh)@xmath0 abundance increases with total pressure , and has a strong @xmath53 dependence on metallicity ( see table [ table iron reactions ] ) . the fe(oh)@xmath0 abundance in reaction ( [ reaction fe : fe(oh)2 ] ) is very sensitive to the h@xmath0o abundance since the formation of one mole of iron hydroxide requires two moles of water . as a result , there is a large shift in the fe(oh)@xmath0 abundance contours above the clouds ( inflections in figure [ figure iron gas 1]d ) when moving between the ch@xmath1 and co fields . within each field , the fe(oh)@xmath0 abundance above the clouds is essentially pressure - independent ( see table[table iron reactions ] ) and is therefore , in principle , diagnostic of atmospheric temperature . as shown in figure [ figure iron chemistry overview ] , fe(oh)@xmath0 becomes the most abundant fe - bearing gas at low temperatures and high pressures in a solar composition gas . iron hydroxide is therefore expected to be the most abundant iron gas at @xmath54 on jupiter ( @xmath55\approx+0.5 $ ] ) and @xmath56 on gliese 229b . however , we emphasize that the amount of iron remaining in the gas is greatly diminished by the condensation of fe metal at deeper atmospheric levels . for example , even where it is the most abundant fe - bearing gas , the predicted fe(oh)@xmath0 mole fraction is @xmath57 at the 1000 k level in jupiter s atmosphere . mole fraction abundance contours for feo are shown in figure [ figure iron gas 2]a for a solar composition gas . monatomic iron reacts with water to form feo via the net reaction @xmath58 between the h@xmath0-h boundary and the fe cloud deck , @xmath59 is proportional to @xmath60 and is independent of pressure ( see table [ table iron reactions ] ) . in principle , the feo abundance in this region is therefore diagnostic of either temperature or metallicity . above the clouds , the feo abundance decreases with pressure and increases with metallicity . as for feoh and fe(oh)@xmath0 , the feo abundance is sensitive to the h@xmath0o abundance and the feo abundance contours display a subtle shift along the ch@xmath1-co equal abundance boundary . the abundance of fes gas as a function of temperature and pressure is illustrated in figure [ figure iron gas 2]b . the chemical behavior of fes is governed by the reaction between fe and h@xmath0s : @xmath61 above the iron clouds , h@xmath0s is the dominant sulfur - bearing gas ( @xmath62 $ ] ) , and the fes abundance is proportional to @xmath63 and @xmath5 ( see table [ table iron reactions ] ) . below the fe cloud deck , the fes abundance is @xmath64 ppb and decreases at higher temperatures and lower pressures as h@xmath0s is replaced by sh and s @xcite . iron sulfide is predicted to be the second or third most abundant iron gas throughout l dwarf atmospheres and in the upper atmospheres ( @xmath65 k ) of pegasi planets . figures [ figure iron chemistry summary]a-[figure iron chemistry summary]d summarize the iron gas chemistry in four representative substellar objects : jupiter , the t dwarf gliese 229b , the pegasi planet hd209458b , and an l dwarf ( @xmath66 k ) . iron chemistry in substellar objects is strongly affected by fe metal condensation at deep atmospheric levels . monatomic fe is the dominant fe - bearing gas in the deep atmospheres of jupiter and gliese 229b , and throughout the atmospheres of hd209458b and the l dwarf . a number of fe - bearing gases become relatively more abundant in objects with lower effective temperatures . on jupiter and gliese 229b , fe(oh)@xmath0 replaces fe at lower temperatures . the fe(oh)@xmath0 and feo abundances are pressure - independent and thus potentially diagnostic of atmospheric temperature , respectively , above and below the iron clouds . for all four objects , iron hydride ( feh ) is the second most abundant fe - bearing gas at deep atmospheric levels , until it is replaced at lower temperatures by fe(oh)@xmath0 , feoh , or fes . because of its strong absorption and relatively high abundance near the iron condensation level , feh is a tracer of weather in brown dwarfs ( e.g. * ? ? ? other fe - bearing gases ( e.g. fe , feoh , fes ) are potential tracers of weather since they typically achieve their maximum abundance near iron cloud base . figure [ figure magnesium chemistry overview ] gives an overview of mg chemistry as a function of pressure and temperature in a protosolar composition gas . the dotted lines labeled mg@xmath0sio@xmath1(s , l ) , mgsio@xmath2(s , l ) , and mgo(s ) show the equilibrium condensation curves for forsterite , enstatite , and periclase , respectively , and the open circles denote the normal melting temperatures for forsterite ( 2163 k ) and enstatite ( 1830 k ) . forsterite ( mg@xmath0sio@xmath1 ) condenses via the net thermochemical reaction @xmath67 and its condensation temperature as a function of pressure and metallicity is approximated by @xmath68,\ ] ] at slightly lower temperatures , enstatite ( mgsio@xmath2 ) condensation occurs via the net reaction @xmath69 the enstatite condensation curve is approximated by @xmath70,\ ] ] where higher pressures and/or metallicities lead to higher condensation temperatures . at very high pressures , forsterite and enstatite condensation temperature are depressed as sio is replaced by sih@xmath1 ( see @xmath4[ss overview of silicon chemistry ] ) . periclase ( mgo ) condenses via the net thermochemical reaction @xmath71 at pressures greater than @xmath72 bar . the condensation curve for akermanite ( ca@xmath0mgsi@xmath0o@xmath73 ) is not shown here because most ca is expected to be removed at deeper atmospheric levels by the condensation of refractory calcium aluminates ( e.g. , * ? ? ? if no calcium is removed , ca@xmath0mgsi@xmath0o@xmath73 condensation would consume @xmath74 of the total atmospheric mg inventory . in the same way , olivine ( ( mg , fe)@xmath0sio@xmath1 ) and fayalite ( fe@xmath0sio@xmath1 ) are not expected in substellar atmospheres because nearly all fe is removed from the gas phase by iron metal condensation at higher temperatures ( e.g. , * ? ? ? * ; * ? ? ? * ) magnesium - silicate cloud formation is very effective at removing nearly all ( @xmath399% ) magnesium from the atmosphere , and the abundances of mg - bearing gases rapidly decrease with altitude above the clouds . this behavior is generally consistent with the disappearance of mg spectral features by early - type l dwarfs ( see [ sss monatomic magnesium , mg ] ) . furthermore , @xcite find a si@xmath41o absorption feature at 10 @xmath28 m in mid - type l dwarfs which is consistent with the presence of silicate grains and a weak 9.17 @xmath28 m feature tentatively attributed to crystalline enstatite . monatomic mg is the dominant mg - bearing gas in substellar atmospheres until it is replaced by mg(oh)@xmath0 at low temperatures and high pressures . the conversion between mg and mg(oh)@xmath0 is represented by the net thermochemical reaction @xmath75 and the solid line in figure [ figure magnesium chemistry overview ] indicates where mg and mg(oh)@xmath0 have equal abundances a(mg ) = a(mg(oh)@xmath0 ) @xmath18mg@xmath19 , where @xmath20mg@xmath19 is the total amount of magnesium in the gas . the position of the equal - abundance line is given by @xmath76,\ ] ] where an increase in metallicity shifts the mg - mg(oh)@xmath0 boundary to higher temperatures and lower pressures . figure [ figure magnesium gas 1]a shows the chemical behavior of monatomic mg as a function of pressure and temperature in a solar - metallicity gas . the chemical behavior of mg is strongly affected by silicate cloud formation . below the forsterite clouds , the abundance of mg gas is given by @xmath77,\ ] ] and comprises nearly 100% of the total elemental mg content in the atmosphere ( @xmath78 ) . upon forsterite and enstatite condensation , the magnesium gas abundance is governed by its vapor pressure over rock , represented by reactions ( [ reaction forsterite condensation ] ) and ( [ reaction enstatite condensation ] ) . curvature in the mg mole fraction contours in figure [ figure magnesium gas 1]a occurs along the ch@xmath1-co boundary , which affects the h@xmath0o abundance in reactions ( [ reaction forsterite condensation ] ) and ( [ reaction enstatite condensation ] ) . the mg abundance is therefore @xmath79.\ ] ] in ch@xmath1-dominated objects , and @xmath80.\ ] ] in co - dominated objects . we can use these expressions along with chemical equilibria to determine the abundances of other mg - bearing gases in substellar atmospheres . equations giving the abundance of mg - bearing gases as a function of pressure , temperature , and metallicity in a protosolar composition gas are listed in table [ table magnesium reactions ] . several mg absorption lines are present in the near infrared spectra of m dwarfs , including the prominent feature at 1.183 @xmath28 m @xcite . these features weaken in mid- to late - type m dwarfs and generally disappear by @xmath42 l1 @xcite , consistent with the removal of magnesium into mg@xmath0sio@xmath1 and mgsio@xmath2 clouds located at increasingly greater depths below the observable atmosphere ( e.g. , see * ? ? ? * ; * ? ? ? mole fraction contours for mgh are illustrated in figure [ figure magnesium gas 1]b . the mgh abundance is governed by equilibrium with monatomic mg , via the net thermochemical reaction @xmath81 between the h@xmath0-h boundary and the mg - silicate cloud base , the mgh abundance is proportional to @xmath35 and @xmath82 ( see table [ table magnesium reactions ] ) . in contrast , the mgh abundance above the magnesium - silicate clouds is proportional to @xmath40 and @xmath83 . the sharp bends in the mgh abundance contours occur at the condensation temperature of forsterite . slight inflections in the mgh abundance occur along the ch@xmath1-co boundary , because the mg abundance in reaction ( [ reaction mg : mgh ] ) is sensitive to the h@xmath0o abundance in reactions ( [ reaction forsterite condensation ] ) and ( [ reaction enstatite condensation ] ) . magnesium hydride is an important opacity source from 0.44 to 0.56 @xmath28 m @xcite , and mgh bands at 0.48 and 0.52 @xmath28 m have been found in the optical spectra of bright l dwarfs and extreme subdwarfs @xcite . the chemical behavior of mgoh as a function of pressure and temperature is shown in figure [ figure magnesium gas 1]c . the equilibrium abundance of mgoh is governed by the reaction @xmath84 decreasing with temperature and proportional to @xmath35 and @xmath60 below the mg - silicate clouds . the kinks in the mgoh contours occur where forsterite condenses . above the clouds , @xmath85 is proportional to @xmath40 ( see table [ table magnesium reactions ] ) . interestingly , the mgoh abundance in this region is independent of metallicity and shows no shift at the ch@xmath1-co boundary because these effects cancel out in reaction ( [ reaction mg : mgoh ] ) . for example , the h@xmath0o abundance slightly decreases when moving from the ch@xmath1 to the co field ( e.g. * ? ? ? however , as can be seen by comparing equations ( [ equation mg above clouds ch4 ] ) and ( [ equation mg above clouds co ] ) , there is a corresponding increase in the mg abundance , so the resulting mgoh abundance in reaction ( [ reaction mg : mgoh ] ) remains unaffected by carbon chemistry . magnesium monohydroxide is typically the third most abundant mg - bearing gas in substellar atmospheres . mole fraction abundance contours for mg(oh)@xmath0 are shown in figure [ figure magnesium gas 1]d for a solar - metallicity gas . the equilibrium conversion between mg and mg(oh)@xmath0 is represented by reaction ( [ reaction mg : mg(oh)2 ] ) . below the mg - silicate clouds , the mg(oh)@xmath0 abundance decreases with temperature and pressure , and has a very strong ( @xmath86 ) dependence on metallicity . above the magnesium - silicate clouds ( the kinks in figure [ figure magnesium gas 1]d ) , the mg(oh)@xmath0 abundance contours show inflections along the ch@xmath1-co boundary , because mg(oh)@xmath0 is sensitive to the water abundance in reaction ( [ reaction mg : mg(oh)2 ] ) . within either the ch@xmath1 or co field , the mg(oh)@xmath0 abundance is pressure - independent . in principle , the abundance of mg(oh)@xmath0 is therefore diagnostic of atmospheric temperature if the object s metallicity is known . at low temperatures and high pressures , mg(oh)@xmath0 becomes the most abundant mg - bearing gas , as illustrated in figure [ figure magnesium chemistry overview ] . magnesium hydroxide is thus expected to be the dominant magnesium gas below @xmath421550 k in the atmosphere of jupiter ( cf . * ) and below @xmath42980 k on gliese 229b . the abundance of mgo as a function of pressure and temperature is illustrated in figure [ figure magnesium gas 2]a . magnesium monoxide forms via the net thermochemical reaction @xmath87 below the mg - silicate cloud deck , at pressures less than @xmath64 bar , the mgo abundance is effectively pressure - independent and is therefore potentially diagnostic of atmospheric temperature if the metallicity is known . for similar reasons as for mgoh ( see [ sss magnesium monohydroxide , mgoh ] ) , the mgo abundance above the magnesium - silicate clouds is independent of metallicity and is unaffected by the ch@xmath1-co boundary . for example , the mg abundance decreases with metallicity , as shown by equations ( [ equation mg above clouds ch4 ] ) and ( [ equation mg above clouds co ] ) , whereas the h@xmath0o abundance increases with metallicity . these effects cancel out in reaction ( [ reaction mg : mgo ] ) , with the result that mgo is unaffected by changes in metallicity or the prevailing carbon chemistry . figure [ figure magnesium gas 2]b displays mole fraction contours for mgs in protosolar composition gas . the equilibrium abundance of mgs is governed by the reaction @xmath88 near the silicate cloud base , the mgs abundance is @xmath64 ppb in a solar - metallicity gas and decreases at higher temperatures and lower pressures as h@xmath0s is replaced by monatomic s. above the mg - silicate clouds , @xmath89 is proportional to @xmath63 and is metallicity - independent ( see table [ table magnesium reactions ] ) . magnesium sulfide is typically among the more abundant mg - bearing gases in brown dwarfs and extrasolar giant planets , and becomes the second most abundant magnesium gas at temperatures below @xmath421600 k in the atmospheres of l dwarfs and pegasi planets . magnesium gas chemistry along the pressure - temperature profiles of jupiter , gliese 229b , hd209458b , and an l dwarf ( @xmath66 k ) are illustrated in figures [ figure magnesium chemistry summary]a-[figure magnesium chemistry summary]d . monatomic mg is the dominant mg - bearing gas throughout the atmospheres of pegasi planets and l dwarfs , and in the deep atmospheres of giant planets and t dwarfs . furthermore , the mg abundance below the mg - silicate cloud deck is essentially constant and representative of the total mg abundance in the atmosphere . upon condensation , the abundances of magnesium - bearing gases rapidly decrease with decreasing temperature above the magnesium - silicate clouds . in objects with lower effective temperatures , a number of other mg - bearing gases become relatively abundant and mg(oh)@xmath0 replaces mg as the most abundant magnesium gas at the @xmath90 k level on jupiter and the @xmath91 k level on gliese 229b . magnesium hydride ( mgh ) is the second most abundant magnesium gas in the deep atmospheres of substellar objects until it is replaced at lower temperatures by mg(oh)@xmath0 and mgoh ( in giant planets and t dwarfs ) or mgs ( in l dwarfs and pegasi planets ) . magnesium hydroxide ( mg(oh)@xmath0 ) and mgo are potential atmospheric temperature probes , respectively , above and below the magnesium - silicate clouds . an overview of silicon chemistry as a function of pressure and temperature is illustrated in figure [ figure silicon chemistry overview ] . the dotted lines labeled mg@xmath0sio@xmath1(s , l ) and mgsio@xmath2(s , l ) show the condensation temperatures of forsterite and enstatite , which together remove nearly all silicon from the overlying atmosphere . @xcite found that silicate absorption features near 10 @xmath28 m in mid - type l dwarf spectra are consistent with the presence of these magnesium silicates , but noted the possibility of additional absorption by quartz ( sio@xmath0 ) grains based upon the predictions of @xcite . in contrast to the models of @xcite , we find that quartz will not condense in the atmospheres of substellar objects unless enstatite condensation is suppressed . this is demonstrated in figure 10 , which shows the elemental distribution of si in condensed phases at 1 bar total pressure in a solar - metallicity gas with ( figure [ figure enstatite]a ) and without ( figure [ figure enstatite]b ) enstatite condensation . as shown in figure [ figure enstatite ] , mg@xmath0sio@xmath1 formation consumes nearly half of the total si abundance because the solar elemental abundances of mg and si are approximately equal . enstatite formation plausibly proceeds via reactions between sio gas and pre - existing forsterite grains and continues until nearly all silicon is consumed . thus , quartz ( @xmath92 k ) can only form in the absence of enstatite ( @xmath93 k ) , because mgsio@xmath2 otherwise efficiently removes silicon from the gas phase . even in the absence of gas - grain reactions between sio and mg@xmath0sio@xmath1 , the vapor pressures of mg and sio above forsterite ( reaction [ reaction forsterite condensation ] ) remain high enough to drive enstatite cloud formation via the net thermochemical reaction ( [ reaction enstatite condensation ] ) , so that mgsio@xmath2 condenses instead of sio@xmath0 . we therefore conclude that sio@xmath0 will not condense within the silicate cloud . the most abundant si - bearing gas over a wide range of pressures and temperatures is sio , until it is replaced at higher pressures by silane , sih@xmath1 ( see figure [ figure silicon chemistry overview ] ) . the equilibrium conversion between sio and sih@xmath1 is @xmath94 the position of the line where sio and sih@xmath1 have equal abundances a(sio ) = a(sih@xmath1 ) @xmath95 ( where @xmath96 is the total amount of silicon in the gas ) is given by @xmath97.\ ] ] as temperatures decrease , sih@xmath1 is replaced by sih@xmath2f via the net thermochemical reaction @xmath98 this reaction is independent of pressure and the position of the sih@xmath1-sih@xmath2f boundary is given by @xmath99),\ ] ] and occurs at @xmath100 960 k in a solar - metallicity gas . meanwhile , the equilibrium between sio and sih@xmath2f is @xmath101 where the position of the sio - sih@xmath2f line is given by @xmath102 independent of the metallicity . the sih@xmath1-sio , sih@xmath1-sih@xmath2f , and sio - sih@xmath2f equal abundance lines intersect to form a `` triple point '' at @xmath100 960 k and @xmath103 bar in solar - metallicity gas , where all three gases have equal abundances [ a(sio ) = a(sih@xmath1 ) = a(sih@xmath2f ) @xmath104 . at lower temperatures , sih@xmath0f@xmath0 replaces sih@xmath2f via the net thermochemical reaction @xmath105 this reaction is also independent of pressure and the sih@xmath2f - sih@xmath0f@xmath0 boundary is given by @xmath106),\ ] ] and is located at @xmath100 917 k in a solar - metallicity gas . the conversion between sio and sih@xmath0f@xmath0 takes place by the reaction @xmath107 where the position of the sio - sih@xmath0f@xmath0 boundary is given by @xmath108.\ ] ] in a solar - metallicity gas , equations ( [ boundary line sio : sih3f ] ) and ( [ boundary line sio : sih2f2 ] ) intersect to form the sio - sih@xmath2f - sih@xmath0f@xmath0 `` triple point '' at @xmath109 k and @xmath110 bar , where all three gases have equal abundances ( @xmath111 for [ fe / h ] = 0 ) . however , the abundances of sih@xmath2f and sih@xmath0f@xmath0 in this region are extremely low because most silicon is removed from the atmosphere by cloud formation at deeper levels . the chemical behavior of sio in a protosolar composition gas is illustrated in figure [ figure sio ] as a function of pressure and temperature . within the sio field , the sio abundance below the magnesium - silicate clouds is given by @xmath112,\ ] ] and sio contains @xmath42100% of the atmospheric silicon inventory ( @xmath113 ) . upon mg - silicate condensation , silicon is efficiently removed from the gas phase and the sio abundance rapidly decreases with decreasing temperature . above the clouds , the sio abundance is governed by its vapor pressure over rock . curvature in the sio contour lines along the ch@xmath1-co equal abundance boundary results from the effect of carbon chemistry on the h@xmath0o abundance in reactions ( [ reaction forsterite condensation ] ) and ( [ reaction enstatite condensation ] ) . the sio abundance is thus @xmath114,\ ] ] in ch@xmath1-dominated objects and @xmath115,\ ] ] in co - dominated objects , inversely proportional to pressure and metallicity . these expressions , along with chemical equilibria , are used to determine the equilibrium abundances of other si - bearing gases . expressions giving the abundances of silicon species as a function of temperature , pressure , and metallicity in a protosolar composition gas are listed in table [ table silicon reactions ] . we note that the types of cloud condensates present will affect the gas chemistry of sio and subsequently all other si - bearing gases . for example , as shown in figure [ figure sio condense ] , sio mole fraction abundances above the clouds are @xmath420.5 dex higher if enstatite formation is suppressed and replaced by sio@xmath0 condensation ( see [ ss overview of silicon chemistry ] above ) . the sio abundance is therefore potentially diagnostic of weather and cloud composition in brown dwarf atmospheres . silicon monoxide has not yet been detected in the atmospheres of brown dwarfs or giant planets , but has been observed in numerous objects including molecular clouds , circumstellar envelopes , the photospheres of late - type stars , and sunspots ( e.g. , * ? ? ? * and references therein ) . abundant sio gas was recently detected in the circumstellar disk of the @xmath116 pic analog hd172555 @xcite . mole fraction contours for sis are shown in figure [ figure silicon gas 1]a for a solar - metallicity gas . silicon monosulfide is formed by the reaction between sio and h@xmath0s : @xmath117 below the mg - silicate cloud deck , @xmath118 - 10 ppm in a solar - metallicity gas . the sis abundance decreases at higher temperatures and lower pressures as h@xmath0s is replaced by sh and monatomic s ( e.g. * ? ? ? * ) . above the clouds , the sis abundance is inversely proportional to metallicity and total pressure ( see table [ table silicon reactions ] ) . the sis abundance contours in figure [ figure silicon gas 1]a also display curvature along the ch@xmath1-co boundary . this shift is more pronounced for sis than for sio because the sis abundance in reaction ( [ reaction sio : sis ] ) depends on the sio and h@xmath0o abundances , both of which are affected by carbon chemistry . for example , when reaction ( [ reaction sio : sis ] ) is at equilibrium , lechtelier s principle shows that more sis is produced either by adding sio ( or h@xmath0s ) or removing h@xmath0o . when moving from the ch@xmath1 field to the co field in a protosolar composition gas , the sio abundance increases and the h@xmath0o abundance decreases , yielding a correspondingly large increase in the sis abundance . silicon sulfide is expected to be the second most abundant silicon - bearing gas ( after sio ) in the atmospheres of brown dwarfs and extrasolar giant planets . furthermore , it is a potential tracer of weather in these objects because its maximum abundance is typically achieved near the magnesium - silicate cloud base . figure [ figure silicon gas 1]b illustrates the chemical behavior of monatomic si gas as a function of pressure and temperature . the abundance of si is governed by the net thermochemical reaction @xmath119 below the mg - silicate cloud deck and at pressures less than @xmath64 bar , the si abundance is effectively independent of pressure and metallicity and thus potentially diagnostic of atmospheric temperature . above the clouds , the si abundance rapidly decreases with decreasing temperature and is proportional to @xmath63 and @xmath120 ( see table [ table silicon reactions ] ) . a number of si absorption bands are observed in the near infrared spectra of low - mass dwarf stars . these features generally weaken and disappear in late - type m dwarfs @xcite , consistent with the removal of silicon into mg@xmath0sio@xmath1 and mgsio@xmath2 clouds deeper in the atmosphere . the abundance of sih as a function of pressure and temperature is illustrated in figure [ figure silicon gas 2]a . the sih abundance is governed by the net thermochemical reaction @xmath121 below the mg - silicate clouds , the sih abundance is proportional to @xmath35 and is independent of metallicity . above the clouds , the sih abundance is proportional to @xmath40 and @xmath120 ( see table [ table silicon reactions ] ) . when moving from the ch@xmath1 to the co field , the sio abundance increases and the h@xmath0o abundance decreases in reaction ( [ reaction sio : sih ] ) . both effects serve to increase the sih abundance , resulting in a shift in the sih contour lines along the ch@xmath1-co boundary . the @xmath120 dependence on metallicity means that for every @xmath55=+1 $ ] dex increase in metallicity , the sih abundance decreases by a factor of 100 at a given pressure and temperature . mole fraction contours for sih@xmath0 are shown in figure [ figure silicon gas 2]b . the equilibrium between sio and sih@xmath0 is given by the reaction @xmath122 beneath the magnesium - silicate cloud deck , the metallicity - independent silylene abundance increases with total pressure . upon rock condensation , the amount of sih@xmath0 remaining in the gas rapidly decreases with decreasing temperature . as for sih , the sih@xmath0 abundance is sensitive to the sio and h@xmath0o abundances in reaction and thus a shift occurs in the sih@xmath0 contour lines when moving between the ch@xmath1 and co fields . within each field , the sih@xmath0 abundance is pressure - independent ( see figure [ figure silicon gas 2 ] and table [ table silicon reactions ] ) , and thus potentially diagnostic of temperature if the metallicity is known and sih@xmath0 is thermochemical in origin . the chemical behavior of the sih@xmath2 radical in a solar - metallicity gas is illustrated in figure [ figure silicon gas 2]c . the abundance of sih@xmath2 is governed by the reaction @xmath123 between the h@xmath0-h boundary and the mg - silicate cloud base , the sih@xmath2 is proportional to @xmath124 . above the clouds , the sih@xmath2 abundance is proportional to @xmath124 and @xmath120 ( see table [ table silicon reactions ] ) . there is curvature in the sih@xmath2 contour lines along the ch@xmath1-co boundary because the prevailing carbon chemistry affects both the sio and h@xmath0o abundances in reaction ( [ reaction sio : sih3 ] ) mole fraction abundance contours for sih@xmath1 are illustrated in figure [ figure silicon gas 2]d . the equilibrium conversion between sio and sih@xmath1 is represented by reaction ( [ reaction sih4:sio ] ) , and expressions giving the sih@xmath1 abundance as a function of pressure , temperature , and metallicity are listed in table [ table silicon reactions ] . below the clouds , the silane abundance has a strong ( @xmath125 ) dependence on total pressure . upon mg - silicate cloud formation , there is an inflection in the sih@xmath1 contour lines , and the sih@xmath1 abundance above the clouds is proportional to @xmath25 and @xmath120 . curvature in the sih@xmath1 contour lines along the ch@xmath1-co boundary results from the effect of carbon chemistry on the sio and h@xmath0o abundances in reaction ( [ reaction sih4:sio ] ) . at high pressures and temperatures ( @xmath126 k for @xmath55=0 $ ] ) , sih@xmath1 becomes the most abundant silicon - bearing gas . we thus expect sih@xmath1 to be the dominant silicon gas below the 1031 k level in the deep atmosphere of jupiter ( @xmath55\approx+0.5 $ ] ) . however , we again emphasize the efficiency with which rock condensation removes silicon from the gas , and we expect an abundance of @xmath127 at the 1031 k level on jupiter . at lower temperatures , sih@xmath1 is surpassed by sih@xmath2f and sih@xmath0f@xmath0 . silane is expected to be the third most abundant si - bearing gas in the deep atmosphere of gliese 229b . we also point out the trend in pressure dependence for the silicon hydrides below ( @xmath128 , @xmath129 , @xmath130 , @xmath131 ) and above the silicate clouds ( @xmath132 , @xmath133 , @xmath134 , @xmath135 ) , which is evident in the shapes of the contour lines in figure [ figure silicon gas 2 ] . the abundance of each of the silicon hydrides is metallicity - independent below the mg - silicate clouds and proportional to @xmath120 above the clouds . figures [ figure silicon chemistry summary]a-[figure silicon chemistry summary]d summarize the chemical behavior of silicon gases along the pressure temperature profiles of jupiter , the t dwarf gliese 229b , the pegasi planet hd209458b , and an l dwarf ( @xmath66 k ) . there is a clear trend in silicon chemistry as a function of effective temperature . on jupiter , sih@xmath1 is the dominant si - bearing gas throughout the deep atmosphere , and is replaced by sih@xmath2f and sih@xmath0f@xmath0 at lower temperatures ( see figure [ figure silicon chemistry overview ] ) . the second most abundant si - bearing gas is sio , followed by sis and a number of other silicon gases . in the warmer atmosphere of gliese 229b , sio is the dominant si - bearing gas , followed by sis and sih@xmath1 . the relative importance of sih@xmath1 decreases with increasing effective temperature , and sio and sis are the most important silicon gases throughout the atmospheres of pegasi planets and l dwarfs . the abundances of all the silicon gases shown in figure [ figure silicon chemistry summary ] rapidly decrease with decreasing temperature above the silicate clouds , which explains the non - detection of sih@xmath1 and other silicon species in the atmospheres of jupiter and saturn . below the cloud base , the important silicon gases sio and sis reach their maximum abundance and are therefore potential tracers of weather in brown dwarfs and pegasi planets . silylene ( sih@xmath0 ) and monatomic silicon ( si ) are potentially diagnostic of atmospheric temperature , respectively , above and below the magnesium silicate clouds . the abundant silicon gases sio and sis typically achieve their maximum abundance near the cloud base and are potential tracers of weather in brown dwarfs and pegasi planets . the chemical behavior of iron species in substellar atmospheres is strongly affected by fe metal condensation , which efficiently removes most iron from the gas phase . similarly , most magnesium and silicon is removed from the gas by forsterite ( mg@xmath0sio@xmath1 ) and enstatite ( mgsio@xmath2 ) cloud formation . the equilibrium abundances of fe- , mg- , and si - bearing gases rapidly decrease with increasing altitude ( and decreasing temperature ) above the clouds . monatomic iron is the dominant fe - bearing gas throughout the atmospheres of l dwarfs and pegasi planets . other less abundant iron gases become increasingly important in objects with lower effective temperatures , and fe(oh)@xmath0 replaces fe at low temperatures in t dwarfs and giant planets . magnesium gas chemistry is similar to that of iron . monatomic mg is the most abundant magnesium gas throughout the atmospheres of l dwarfs and pegasi planets and in the deep atmospheres of giant planets and t dwarfs , where mg is replaced by mg(oh)@xmath0 at lower temperatures . a number of mg - bearing gases become relatively abundant with decreasing effective temperature . silicon monoxide ( sio ) is the most abundant si - bearing gas , followed by sis , throughout the atmospheres of l dwarfs and pegasi planets and in the deep atmospheres of t dwarfs . in objects with lower effective temperatures , a number of other silicon gases become increasingly important and sih@xmath1 is the dominant silicon gas in the deep atmosphere of jupiter . at high pressures and low temperatures sih@xmath1 and sio are replaced by sih@xmath2f and/or sih@xmath0f@xmath0 . the abundances of several fe- , mg- , and si - bearing gases are pressure - independent and thus , in principle , diagnostic of atmospheric temperature . these include fe(oh)@xmath0 , mg(oh)@xmath0 , and si below the clouds and feo , mgo , and sih@xmath0 above the clouds . in addition , a number of gases ( e.g. fe , feh , feoh , fes , mg , mgh , mgoh , mgs , sio , sis ) may serve as indicators of weather since they generally reach their maximum abundance just below the iron metal or magnesium - silicate cloud decks . this may be particularly useful for late m dwarfs and early l dwarfs in which the metal and silicate clouds are located at relatively shallow depths below the photosphere . this research was conducted at washington university in st . louis and was supported by the nasa planetary atmospheres program ( nng06gc26 g ) . support for k. lodders was also provided by nsf grant ast-0707377 . final preparation of the manuscript was supported by the lunar and planetary institute / usra ( nasa cooperative agreement ncc5 - 679 ) . lpi contribution no . xxxx . llc fe & @xmath137 $ ] & + feh & @xmath138 $ ] & [ reaction fe : feh ] + feoh & @xmath139 $ ] & [ reaction fe : feoh ] + fe(oh)@xmath0 & @xmath140 $ ] & [ boundary line fe : fe(oh)2 ] + feo & @xmath141 $ ] & [ reaction fe : feo ] + fes & @xmath142 $ ] & [ reaction fe : fes ] + fe & @xmath143 & [ reaction fe condensation ] + feh & @xmath144 & [ reaction fe : feh ] + feoh & @xmath145 + \log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction fe : feoh ] + fe(oh)@xmath0 & @xmath146 + 2\log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction fe : fe(oh)2 ] + feo & @xmath147 + \log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction fe : feo ] + fes & @xmath148 $ ] & [ reaction fe : fes ] + [ table iron reactions ] llc mg & @xmath149 $ ] & + mgh & @xmath150 $ ] & [ reaction mg : mgh ] + mgoh & @xmath151 $ ] & [ reaction mg : mgoh ] + mg(oh)@xmath0 & @xmath152 $ ] & [ reaction mg : mg(oh)2 ] + mgo & @xmath153 $ ] & [ reaction mg : mgo ] + mgs & @xmath154 $ ] & [ reaction mg : mgs ] + mg & @xmath155- \log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction forsterite condensation ] , [ reaction enstatite condensation ] + mgh & @xmath156- \log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction mg : mgh ] + mgoh & @xmath157 & [ reaction mg : mgoh ] + mg(oh)@xmath0 & @xmath158 + \log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction mg : mg(oh)2 ] + mgo & @xmath159 & [ reaction mg : mgo ] + mgs & @xmath160 & [ reaction mg : mgs ] + [ table magnesium reactions ] llc sio & @xmath161 $ ] & + sis & @xmath162 $ ] & [ reaction sio : sis ] + si & @xmath163 & [ reaction sio : si ] + sih & @xmath164 & [ reaction sio : sih ] + sih@xmath0 & @xmath165 & [ reaction sio : sih2 ] + sih@xmath2 & @xmath166 & [ reaction sio : sih3 ] + sih@xmath1 & @xmath167 & [ reaction sih4:sio ] + sih@xmath2f & @xmath168 $ ] & [ reaction sio : sih3f ] + sih@xmath0f@xmath0 & @xmath169 $ ] & [ reaction sio : sih2f2 ] + sio & @xmath170 - \log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction forsterite condensation ] , [ reaction enstatite condensation ] + sis & @xmath171 - 2\log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction sio : sis ] + si & @xmath172 - 2\log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction sio : si ] + sih & @xmath173 - 2\log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction sio : sih ] + sih@xmath0 & @xmath174 - 2\log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction sio : sih2 ] + sih@xmath2 & @xmath175- 2\log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction sio : sih3 ] + sih@xmath1 & @xmath176- 2\log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction sih4:sio ] + sih@xmath2f & @xmath177 - 2\log x_{\textrm{\scriptsize{h}}_{2}\textrm{\scriptsize{o}}}'$ ] & [ reaction sio : sih3f ] + sih@xmath0f@xmath0 & @xmath178 & [ reaction sio : sih2f2 ] + [ table silicon reactions ]

the antarctic neutrino telescope amanda @xcite and the future km@xmath0 icecube @xcite are designed to observe high - energy neutrinos from astrophysical sources . the ice is instrumented with photomultipliers to pick up the cherenkov light from secondary charged particles . in order to reach the large volume needed to detect the expected small fluxes at high energies , the density of optical modules is far too sparse to measure , for example , solar neutrinos . however , it has been recognized for a long time that these instruments can detect a supernova ( sn ) neutrino burst because the cherenkov glow of the ice can be identified as time - correlated noise among all phototubes @xcite . this approach has been used by amanda to exclude the occurrence of a galactic sn over a recent observation period @xcite . for amanda the physics potential of a possible sn observation is essentially limited to its detection , notably in the context of the supernova early warning system ( snews ) that would alert the astronomical community several hours before the optical explosion @xcite . for the future icecube with 4800 optical modules , however , the number of detected cherenkov photons would be of order @xmath2 and thus so large that several interesting physics questions could be addressed in earnest . the observed quantity is the number of cherenkov photons caused by the sn neutrinos as a function of time , i.e. a measure of the energy deposited by the neutrinos in the ice . therefore , the information about the sn signal is far more limited than what can be extracted from a high - statistics observation in super - kamiokande or other low - energy experiments that detect individual events . however , galactic sne are so rare , perhaps a few per century , that the chances of observing one depend crucially on the long - term stability of the neutrino observatories . once icecube has been built it may well operate for several decades , backing up the low - energy experiments . besides the detection and associated early warning one could measure important details of the neutrino light - curve , for example the existence and duration of the initial sn accretion phase , the overall duration of the cooling phase , and so forth . such an observation would provide a plethora of astrophysically valuable information . however , from the perspective of neutrino physics a _ simultaneous _ observation in both icecube and another large detector such as super - kamiokande or hyper - kamiokande would be especially useful . assuming that the neutrinos have traversed significantly different paths through the earth , the two signals could well show measurable differences caused by neutrino oscillations in matter . as this earth effect shows up only for certain combinations of neutrino mixing parameters , a dual observation may well distinguish , for example , between the normal and inverted neutrino mass hierarchy . it is well known that observing sn neutrinos with two or more detectors with different earth - crossing lengths is extremely useful , but icecube s potential has not been explored in this context . with all the low - energy observatories being in the northern hemisphere , icecube s location in antarctica is uniquely complementary for this purpose . any oscillation signature depends on the small flavor - dependent differences between the fluxes and spectra at the source . if these differences were as large as had been assumed until recently there would be little question about icecube s usefulness for co - detecting the earth effect . however , a more systematic study of the flavor - dependence of the sn neutrino fluxes and spectra reveals that these differences are more subtle , although by no means negligible @xcite . we evaluate icecube s potential as a co - detector from the perspective of these `` pessimistic '' assumptions about the primary fluxes and spectra . this paper is organized as follows . in sec . [ at - icecube ] , we show that the neutrino signal from a galactic supernova can be measured at icecube with a sub - percent statistical precision . in sec . [ earth - effects ] , we calculate the earth matter effects on this signal and illustrate that it is possible to detect them in conjunction with another high statistics experiment . sec . [ concl ] concludes . the sn neutrinos streaming through the antarctic ice interact according to @xmath3 and some other less important reactions . the positrons , in turn , emit cherenkov light that is picked up by the optical modules ( oms ) frozen into the ice . while the expected number of detected photons per om was calculated in refs . @xcite , we revisit their estimate for two reasons . first , the sn signal was directly scaled to the historical sn 1987a observation in kamiokande ii so that the exact assumptions about the neutrino flux are not directly apparent . second , the expected number of cherenkov photons detected by one om was based on estimating an effective ice volume seen by one om . however , it is much simpler to work in the opposite direction and start with the homogeneous and isotropic cherenkov glow of the ice caused by the sn neutrinos . the om is immersed in this diffuse bath of photons and picks up a number corresponding to its angular acceptance and quantum efficiency . as a first simplification we limit ourselves to the signal caused by the inverse @xmath4 reaction @xmath3 . the @xmath5 fluence ( time - integrated flux ) at earth is @xmath6 we define the `` sn fudge factor '' as @xmath7 where @xmath8 is the total energy leaving the sn in the form of @xmath5 after flavor oscillations have been included , @xmath9 is the average @xmath5 energy , and @xmath10 the distance . the energy deposited in the ice per target proton is @xmath11 . for the inverse @xmath4 cross section we ignore weak - magnetism and recoil corrections and also the difference between @xmath5 and positron energy so that @xcite @xmath12 for the neutrino flux of each neutrino and anti - neutrino species we assume a distribution of the form @xcite @xmath13\,,\ ] ] where @xmath14 is the average energy , @xmath15 a parameter that typically takes on values 2.55 depending on the flavor and the phase of neutrino emission , and @xmath16 the overall flux at the detector in units of @xmath17 . this distribution implies @xmath18 altogether we thus find @xmath19 with @xmath20 this fudge factor can also be taken to include deviations from the simplified energy dependence of the cross section and deviations from the assumed spectral shape . the cherenkov angle for photon emission by a charged particle is @xmath21 where @xmath22 is the medium s refractive index and @xmath4 the particle s velocity . with @xmath23 for ice , neglecting the @xmath24-dependence , and @xmath25 we have @xmath26 . a particle with unit charge produces cherenkov photons per unit path length and per unit wavelength band according to @xmath27 where @xmath28 is the fine - structure constant . assuming that @xmath22 and thus @xmath29 are independent of wavelength we integrate over @xmath24 and find @xmath30 taking the useful wavelength range to be 300600 nm this translates into 319 photons per cm pathlength . taking the positron mean free path to be 12 cm for an energy of 20 mev , and taking it to be proportional to its energy , the number of useful cherenkov photons per deposited neutrino energy is @xmath31 with yet another fudge factor @xmath32 . the density of ice is @xmath33 , corresponding to about @xmath34 proton targets . therefore , the sn neutrinos produce @xmath35 useful cherenkov photons per unit volume of ice . multiplying this number with the speed of light and dividing by @xmath36 gives us the resulting diffuse photon flux in units of @xmath37 . however , the average lifetime of these photons is @xmath38 with @xmath39 the absorption length . therefore , the neutrino - induced photon fluence is found by multiplying the flux with @xmath38 , @xmath40 where @xmath41 . the number of events produced by this fluence in a given om depends on the average quantum efficiency taken to be @xmath42 . in addition , it depends on the angular acceptance , i.e. the effective photo cathode detection area @xmath43 times the angular acceptance range @xmath44 . therefore , in one om we expect @xmath45 with @xmath46 this result is independent of the presence of bubbles in the ice that scatter the photons . the cherenkov glow of the ice represents an isotropic and homogeneous distribution that is not changed by elastic scattering . in order to compare our result with the one derived in ref . @xcite we need to translate their assumptions into our fudge factors . the @xmath5 distribution was taken to follow a fermi - dirac spectrum with @xmath47 mev , implying @xmath48 mev . the distance of the sn was taken to be 10 kpc , and the total energy release was scaled to the kamiokande ii signal for sn 1987a . with our choice of the @xmath4 cross section these assumptions correspond to @xmath49 erg , i.e. to @xmath50 . these authors also used a quadratic energy dependence of the cross section . integrating over their fermi - dirac spectrum they effectively used @xmath51 . further , they assumed 3000 useful cherenkov photons for 20 mev deposited energy , i.e. effectively @xmath52 . for the absorption length they used 300 m , i.e. @xmath53 . finally , they assumed a quantum efficiency of 25% , a cathode area of 280 @xmath54 , and an acceptance range of @xmath55 , i.e.@xmath56 . altogether , we find for these assumptions @xmath57 per om . this compares with 273 in ref . @xcite , i.e. our result is larger by a factor 1.5 . the result in ref . @xcite was backed up by a detailed monte carlo treatment of the production and propagation of cherenkov photons in the amanda detector . therefore , the difference may well relate to details of the om acceptance and wavelength - dependent quantum efficiency and photon propagation . many of these details will be different in icecube where 10 inch photomultiplier tubes and different regions of ice will be used . detailed values for the detector - dependent fudge factors must be determined specifically for icecube once it has been built . the main difference between the assumptions in ref . @xcite and our estimate is the absorption length . when using amanda as a sn observatory a realistic value was taken to be around 100 m @xcite . the vast difference between these estimates is that the former was based on the measured absorption length in a dust - free region of the ice . for our further estimates we stick to 100 m as a conservative assumption . in our derivation we have used the time - integrated neutrino flux , amounting to the assumption of a stationary situation . the absorption time for photons is very small , @xmath58 . the sn signal will vary on time scales exceeding 10 ms . therefore , the cherenkov glow of the ice follows the time - variation of the sn signal without discernible inertia . hence one may replace the neutrino fluence with a time - dependent flux and @xmath59 with an event rate @xmath60 . moreover , for our further discussion it will be useful to consolidate our fudge - factors into one describing the detector response , and others characterizing the neutrino flux . therefore , we summarize our prediction for the event rate per om in the form @xmath61 where @xmath62 is the @xmath5 luminosity after flavor oscillations and @xmath63 here , @xmath64 also includes corrections for the energy dependence of the @xmath4 cross section . we stress that our simple estimate of the counting rate primarily serves the purpose of determining its magnitude relative to the background . the important feature is that the signal relative to the background can be determined with a good statistical precision . of course , for an absolute detector calibration a detailed modeling would be necessary . for our present purpose , however , even an uncertainty of several 10% in our estimated counting rate is irrelevant . icecube will have 4800 oms so that one expects a total event number of @xmath65 , taking all fudge factors to be unity . assuming a background counting rate of 300 hz per om over as much as 10 s this compares with a background rate of @xmath66 . assuming poisson fluctuations , the uncertainty of this number is @xmath67 , i.e. 0.25% of the sn signal . therefore , one can determine the sn signal with a statistical sub - percent precision , ignoring for now problems of absolute detector calibration . in order to illustrate the statistical power of icecube to observe a sn signal we use two different numerical sn simulations . the first was performed by the livermore group @xcite that involves traditional input physics for mu- and tau - neutrino interactions and a flux - limited diffusion scheme for treating neutrino transport . the great advantage of this simulation is that it covers the full evolution from infall over the explosion to the kelvin - helmholtz cooling phase of the newly formed neutron star . we show the livermore @xmath5 and @xmath68 lightcurves in fig [ fig : simulations ] ( left panels ) . here and in the following we take @xmath68 to stand for either @xmath69 or @xmath70 . apart from very small differences the sn fluxes and spectra are thought to be equal for @xmath71 , @xmath72 , @xmath69 , and @xmath70 . our second simulation was performed with the garching code @xcite . it includes all relevant neutrino interaction rates , including nucleon bremsstrahlung , neutrino pair processes , weak magnetism , nucleon recoils , and nuclear correlation effects . the neutrino transport part is based on a boltzmann solver . the neutrino - radiation hydrodynamics program allows one to perform spherically symmetric as well as multi - dimensional simulations . the progenitor model is a 15@xmath73 star with a @xmath74 iron core . the period from shock formation to 468 ms after bounce was evolved in two dimensions . the subsequent evolution of the model is simulated in spherical symmetry . at 150 ms the explosion sets in , although a small modification of the boltzmann transport was necessary to allow this to happen @xcite . unmanipulated full - scale models with an accurate treatment of the microphysics currently do not obtain explosions @xcite . this run will be continued beyond the current epoch of 750 ms post bounce ; we here use the preliminary results currently available @xcite . we show the garching @xmath5 and @xmath68 lightcurves in fig [ fig : simulations ] ( right panels ) . we take the livermore simulation to represent traditional predictions for flavor - dependent sn neutrino fluxes and spectra that were used in many previous discussions of sn neutrino oscillations . the garching simulation is taken to represent a situation when the @xmath68 interactions are more systematically included so that the flavor - dependent spectra and fluxes are more similar than had been assumed previously @xcite . we think it is useful to juxtapose the icecube response for both cases . another difference is that in livermore the accretion phase lasts longer . since the explosion mechanism is not finally settled , it is not obvious which case is more realistic . moreover , there could be differences between different sne . the overall features are certainly comparable between the two simulations . in fig . [ fig : simulationsicecube ] we show the expected counting rates in icecube on the basis of eq . ( [ eq : rateprediction ] ) for an assumed distance of 10 kpc and 4800 oms for the livermore ( left ) and garching ( right ) simulations . we also show this signal in 50 ms bins where we have added noise from a background of 300 hz per om . the baseline is at the average background rate so that negative counts correspond to downward background fluctuations . one could easily identify the existence and duration of the accretion phase and thus test the standard delayed - explosion scenario . one could also measure the overall duration of the cooling phase and thus exclude the presence of significant exotic energy losses . therefore , many of the particle - physics limits based on the sn 1987a neutrinos @xcite could be supported with a statistically serious signal . if the sn core were to collapse to a black hole after some time , the sudden turn - off of the neutrino flux could be identified . in short , when a galactic sn occurs , icecube is a powerful stand - alone neutrino detector , providing us with a plethora of information that is of fundamental astrophysical and particle - physics interest . in addition , icecube is extremely useful as a co - detector with another high - statistics observatory to measure neutrino oscillation effects , a topic that we now explore . neutrino oscillations are now firmly established by measurements of solar and atmospheric neutrinos and the kamland and k2k long - baseline experiments . evidently the weak interaction eigenstates @xmath75 , @xmath71 and @xmath72 are non - trivial superpositions of three mass eigenstates @xmath76 , @xmath77 and @xmath78 , @xmath79 where @xmath80 is the leptonic mixing matrix that can be written in the canonical form @xmath81 here @xmath82 and @xmath83 etc . , and @xmath84 is a phase that can lead to cp - violating effects , that are , however , irrelevant for sn neutrinos . the mass squared differences relevant for the atmospheric and solar neutrino oscillations obey a hierarchy @xmath85 . this hierarchy , combined with the observed smallness of the angle @xmath86 at chooz @xcite implies that the atmospheric neutrino oscillations essentially decouple from the solar ones and each of these is dominated by only one of the mixing angles . the atmospheric neutrino oscillations are controlled by @xmath87 that may well be maximal ( 45@xmath88 ) . the solar case is dominated by @xmath89 , that is large but not maximal . from a global 3-flavor analysis of all data one finds the 3@xmath90 ranges for the mass differences @xmath91 and mixing angles summarized in table [ tab : mixingparameters ] . @lll observation&mixing angle&@xmath92 [ mev@xmath93 + sun , kamland & @xmath94 27@xmath8842@xmath88 & @xmath95 55190 + atmosphere , k2k & @xmath96 32@xmath8860@xmath88 & @xmath97 14006000 + chooz & @xmath98 & @xmath99 + a sn core is essentially a neutrino blackbody source , but small flavor - dependent differences of the fluxes and spectra remain . we denote the fluxes of @xmath100 and @xmath101 at earth that would be observable in the absence of oscillations by @xmath102 and @xmath103 , respectively . in the presence of oscillations a @xmath5 detector actually observes @xmath104 f_x^0\ , , \label{fedbar}\ ] ] where @xmath105 is the @xmath5 survival probability after propagation through the sn mantle and perhaps part of the earth before reaching the detector . a significant modification of the survival probability due to the propagation through the earth appears only for those combinations of neutrino mixing parameters shown in table [ tab : earthcases ] . the earth matter effect depends strongly on two parameters , the sign of @xmath106 and the value of @xmath107 @xcite . the `` normal hierarchy '' corresponds to @xmath108 , i.e. @xmath109 , whereas the `` inverted hierarchy '' corresponds to @xmath110 , i.e.@xmath111 . note that the presence or absence of the earth effect discriminates between values of @xmath112 less or greater than @xmath113 , i.e. @xmath86 less or larger than about @xmath114 . thus , the earth effect is sensitive to values of @xmath86 that are much smaller than the current limit . @lll 13-mixing&normal hierarchy&inverted hierarchy + @xmath115&@xmath75 and @xmath5 & @xmath75 and @xmath5 + @xmath116&@xmath5&@xmath75 + let us consider those scenarios where the mass hierarchy and the value of @xmath86 are such that the earth effect appears for @xmath5 . in such cases the @xmath5 survival probability @xmath105 is given by @xmath117 where the energy dependence of all quantities will always be implicit . here @xmath118 is the mixing angle between @xmath100 and @xmath119 in earth matter while @xmath120 is the mass squared difference between the two anti - neutrino mass eigenstates @xmath121 and @xmath119 in units of @xmath122ev@xmath123 , @xmath124 is the distance traveled through the earth in units of 1000 km , and @xmath125 is the neutrino energy in mev . we have assumed a constant matter density inside the earth , which is a good approximation for @xmath126 , i.e. as long as the neutrinos do not pass through the core of the earth . in order to calculate the extent of the earth effect for icecube , we will assume that the relevant mixing parameters are @xmath127 and @xmath128 . we further assume that the source spectra are given by the functional form eq . ( [ eq : spectralform ] ) . the values of the parameters @xmath15 and @xmath129 for both the @xmath100 and @xmath68 spectra are in general time dependent . in fig . [ fig : exampleeartheffect ] we show the variation of the expected icecube signal with earth - crossing length @xmath124 for the two sets of parameters detailed in table [ tab : examplecases ] . the first could be representative of the accretion phase , the second of the cooling signal . we use the two - density approximation for the earth density profile , where the core has a density of @xmath130 and a radius of 3500 km , while the density of the earth mantle was taken to be @xmath131 . we observe that for short distances , corresponding to near - horizontal neutrino trajectories , the signal varies strongly with @xmath124 . between about 3,000 and 10,500 km it reaches an asymptotic value that we call the `` asymptotic mantle value . '' for case ( a ) , this value corresponds to about 1.5% depletion of the signal , whereas for ( b ) it corresponds to about 6.5% depletion . @llllllll example&phase&@xmath9 & @xmath132 & @xmath133&@xmath134 & @xmath135 & asymptotic + & & [ mev]&[mev]&&&&earth effect + ( a)&accretion&15&17&4&3&1.5&@xmath136% + ( b)&cooling & 15&18&3&3&0.8&@xmath137% + beyond an earth - crossing length of @xmath13810,500 km , the neutrinos have to cross the earth core with another large jump in density . the core effects change the asymptotic mantle value by @xmath139% as can be seen in fig . [ fig : exampleeartheffect ] . we neglect the core effects in the following analysis , and the `` asymptotic value '' always refers to the asymptotic mantle value . for the largest part of the sky the earth effect either appears with this asymptotic value ( `` neutrinos coming from below '' ) , or it does not appear at all ( `` neutrinos from above '' ) . therefore , we now focus on the asymptotic value and study how the signal modification depends on the assumed flux parameters . in table [ tab : asymptotic1 ] we show the signal modification for @xmath140 , @xmath141 , and @xmath142 as a function of @xmath132 and the flux ratio @xmath135 . in table [ tab : asymptotic2 ] we show the same with @xmath143 . the results are shown in the form of contour plots in fig . [ fig : cont ] . @lllllll flux ratio & + @xmath135 & 15&16&17&18&19&20 + 2.0 & 1.026 & 1.014 & 1.002 & 0.988 & 0.974 & 0.960 + 1.9 & 1.023 & 1.011 & 0.999 & 0.985 & 0.971 & 0.956 + 1.8 & 1.021 & 1.009 & 0.995 & 0.982 & 0.967 & 0.952 + 1.7 & 1.018 & 1.005 & 0.992 & 0.978 & 0.963 & 0.948 + 1.6 & 1.015 & 1.002 & 0.988 & 0.974 & 0.959 & 0.944 + 1.5 & 1.012 & 0.998 & 0.984 & 0.969 & 0.954 & 0.939 + 1.4 & 1.008 & 0.994 & 0.980 & 0.965 & 0.949 & 0.934 + 1.3 & 1.004 & 0.990 & 0.975 & 0.960 & 0.944 & 0.928 + 1.2 & 1.000 & 0.985 & 0.970 & 0.954 & 0.938 & 0.922 + 1.1 & 0.995 & 0.980 & 0.964 & 0.948 & 0.932 & 0.915 + 1.0 & 0.989 & 0.974 & 0.957 & 0.941 & 0.925 & 0.908 + 0.9 & 0.983 & 0.967 & 0.950 & 0.934 & 0.917 & 0.901 + 0.8 & 0.976 & 0.959 & 0.942 & 0.925 & 0.909 & 0.892 + 0.7 & 0.967 & 0.950 & 0.933 & 0.916 & 0.899 & 0.883 + 0.6 & 0.958 & 0.940 & 0.923 & 0.906 & 0.889 & 0.873 + 0.5 & 0.946 & 0.928 & 0.911 & 0.894 & 0.877 & 0.862 + @lllllll flux ratio & + @xmath135 & 15&16&17&18&19&20 + 2.0 & 1.036 & 1.024 & 1.012 & 1.000 & 0.986 & 0.972 + 1.9 & 1.033 & 1.022 & 1.010 & 0.996 & 0.983 & 0.968 + 1.8 & 1.031 & 1.019 & 1.006 & 0.993 & 0.979 & 0.964 + 1.7 & 1.028 & 1.016 & 1.003 & 0.989 & 0.975 & 0.960 + 1.6 & 1.025 & 1.013 & 0.999 & 0.985 & 0.971 & 0.955 + 1.5 & 1.022 & 1.009 & 0.995 & 0.981 & 0.966 & 0.951 + 1.4 & 1.019 & 1.005 & 0.991 & 0.976 & 0.961 & 0.945 + 1.3 & 1.015 & 1.001 & 0.986 & 0.971 & 0.955 & 0.940 + 1.2 & 1.010 & 0.996 & 0.981 & 0.965 & 0.949 & 0.933 + 1.1 & 1.006 & 0.991 & 0.975 & 0.959 & 0.943 & 0.927 + 1.0 & 1.000 & 0.985 & 0.969 & 0.952 & 0.936 & 0.919 + 0.9 & 0.994 & 0.978 & 0.961 & 0.945 & 0.928 & 0.911 + 0.8 & 0.986 & 0.970 & 0.953 & 0.936 & 0.919 & 0.903 + 0.7 & 0.978 & 0.961 & 0.944 & 0.926 & 0.910 & 0.893 + 0.6 & 0.968 & 0.950 & 0.933 & 0.916 & 0.899 & 0.882 + 0.5 & 0.956 & 0.938 & 0.920 & 0.903 & 0.886 & 0.870 + even for mildly different fluxes or spectra the signal modification is several percent , by far exceeding the statistical uncertainty of the icecube signal , although the _ absolute _ calibration of icecube may remain uncertain to within several percent . however , the signal modification will vary with time during the sn burst . during the early accretion phase that is expected to last for a few 100 ms and corresponds to a significant fraction of the overall signal , the @xmath68 flux may be almost a factor of 2 smaller than the @xmath5 flux , but it will be slightly hotter and less pinched @xcite . this corresponds to case ( a ) above ; it is evident from fig . [ fig : exampleeartheffect ] and table [ tab : asymptotic1 ] that this implies that the earth effect is very small . during the kelvin - helmholtz cooling phase the flux ratio is reversed with more @xmath68 being emitted than @xmath5 , but still with the same hierarchy of energies . this corresponds to case ( b ) ; in this case the earth effect could be about 6% . this time dependence may allow one to detect the earth effect without a precise absolute detector calibration . in order to illustrate the time dependence of the earth effect we show in fig . [ fig : percenteffect ] the expected counting rate in icecube for both the livermore ( left panels ) and garching ( right panels ) simulations . in the upper panels we show the expected counting rate with flavor oscillations in the sn mantle , but no earth effect ( solid lines ) , or with the asymptotic earth effect ( dashed lines ) that obtains for a large earth - crossing path . naturally the differences are very small so that we show in the lower panels the ratio of these curves , i.e. the expected counting rate with / without earth effect as a function of time for both livermore and garching . while for the livermore simulation there is a large earth effect even at early times , the change from early to late times in both cases is around 45% . therefore , the most model - independent signature is a time variation of the earth effect during the sn neutrino signal . in order to demonstrate the statistical significance of these effects we integrate the expected signal for both simulations separately for the accretion phase and the subsequent cooling phase ; the results are shown in table [ tab : integratedrates ] . for both simulations the earth effect itself and its change with time is statistically highly significant . based on the livermore simulation , the earth effect is much more pronounced than in garching , the latter involving more up - to - date input physics for neutrino transport . however , the _ difference _ between the earth effect during accretion and cooling is not vastly different between the two simulations . recalling that the absolute detector calibration may be very uncertain so that one has to rely on the temporal variation of the earth effect , the difference between livermore and garching becomes much smaller . we expect that it is quite generic that the temporal change of the earth effect is a few percent of the overall counting rate . @lrrrr & & + & & & & + integration time [ s]&00.500&0.5003&00.250&0.2500.700 + sn signal [ counts ] + no earth effect & 519,080&818,043&173,085&407,715 + asymptotic earth effect&488,093&751,137&171,310&390,252 + difference & 30,987 & 66,906 & 1,775 & 17,463 + fractional difference&@xmath144%&@xmath145%&@xmath146%&@xmath147% + background [ counts]&720,000&4,320,000&360,000&648,000 + @xmath148/signal&0.16%&0.25%&0.35%&0.20% + one can measure the earth effect in icecube only in conjunction with another high - statistics detector . we do not attempt to simulate in detail the sn signal in this other detector but simply assume that it can be measured with a precision at least as good as in icecube . one candidate is super - kamiokande , a water cherenkov detector that would measure around @xmath149 events from a galactic sn at a distance of 10 kpc . therefore , the statistical precision for the total neutrino energy deposition in the water is around 1% and thus worse than in icecube . even though super - kamiokande will measure a larger number of cherenkov photons than icecube , a single neutrino event will cause an entire cherenkov ring to be measured , i.e. the photons are highly correlated . therefore , in the estimated statistical @xmath150 fluctuation of the signal , the fluctuating number @xmath151 is that of the detected neutrinos . if the future hyper - kamiokande is built , its fiducial volume would be about 30 times that of super - kamiokande . in this case the statistical signal precision exceeds that of icecube for the equivalent observable . we denote the equivalent icecube signal measured by super- or hyper - kamiokande as @xmath152 and the icecube signal as @xmath153 . if the distances traveled by the neutrinos before reaching these two detectors are different , the earth effect on the neutrino spectra may be different , which will reflect in the ratio @xmath154 . of course , in the absence of the earth effect this ratio equals unity by definition . the geographical position of icecube with respect to super- or hyper - kamiokande at a latitude of @xmath155 is well - suited for the detection of the earth effect through a combination of the signals . using fig . [ fig : exampleeartheffect ] we can already draw some qualitative conclusions about the ratio @xmath156 . clearly , @xmath157 if neutrinos do not travel through the earth before reaching either detector . if the distance traveled by neutrinos through the earth is more that 3000 km for both detectors , the earth effects on both @xmath152 and @xmath158 are nearly equal and their ratio stays around unity . if the neutrinos come `` from above '' for sk and `` from below '' for icecube , or vice versa , the earth matter effect will shift this ratio from unity . in fig . [ fig : earthmap ] , we show contours of @xmath159 for the sn position in terms of the location on earth where the sn is at the zenith . the map is an area preserving hammer - aitoff projection so that the sizes of different regions in the figure gives a realistic idea of the `` good '' and `` bad '' regions of the sky . in order to generate the contours we use the parameters of case ( b ) in table [ tab : examplecases ] so that the asymptotic suppression of the signal is about 6.5% . the sky falls into four distinct regions depending on the direction of the neutrinos relative to either detector as described in table [ tab : skymapregions ] . when the neutrinos come from above for both detectors ( region d ) there is no earth effects . if they come from below in both ( region c ) , the earth effect is large in both . depending on the exact distance traveled through the earth , the event ratio can be large , but generally fluctuates around 1 . in the other regions where the neutrinos come from above for one detector and from below for the other ( regions a and b ) the relative effect is large . @lllll region&sky fraction & & @xmath156 + & & icecube&super - k & + a&0.35&below&above&1.070 + b&0.35&above&below&0.935 + c&0.15&below&below&fluctuations around 1 + d&0.15&above&above&1 + for assumptions about the flavor - dependent sn neutrino fluxes and spectra that agree with state - of - the - art studies , the earth matter effect on neutrino oscillations shows up in the icecube signal of a future galactic sn on the level of a few percent . if the icecube signal can be compared with another high - statistics signal , notably in super - kamiokande or hyper - kamiokande , the earth effect becomes clearly visible as a difference between the detectors . as one is looking for a signal modification in the range of a few percent , the absolute detector calibration may not be good enough in one or both of the instruments . however , for typical numerical sn simulations the effect is time dependent and most notably differs between the early accretion phase and the subsequent neutron star cooling phase . therefore , one would have to search for a temporal variation of the relative detector signals of a few percent . the large number of optical modules in icecube renders this task statistically possible . in fact depending on the differences in flavor - dependent fluxes , the statistical accuracy of super - kamiokande may turn out to be the limiting factor . this limitation is not significant for hyper - kamiokande . the unique location of icecube in antarctica implies that for a large portion of the sky this detector sees the sn through the earth when super- and hyper - kamiokande sees it from above , or the other way round , i.e. the chances of a relative signal difference between the detectors are large . if both detectors were to see the sn from above there would be no earth effect to detect . assuming that the magnitude of the mixing angle @xmath86 can be established to be large in the sense of @xmath160 by a long - baseline experiment @xcite , observing the earth effect in sn anti - neutrinos implies the normal mass hierarchy . on the other hand , if @xmath161 has been established , the earth effect is unavoidable . not observing it would imply that the primary sn neutrino fluxes and spectra are more similar than indicated by state - of - the - art numerical simulations . if @xmath160 is known , and we do not observe the earth effect , it still does not prove the inverted mass hierarchy . it could also mean that we do not properly understand the flavor - dependent source fluxes and spectra . therefore , even if @xmath160 is known , our method only allows one to detect the normal mass hierarchy , it does not strictly allow one to exclude it . as far as neutrino parameters are concerned , only a positive detection of the earth effect would count for much . of course , a normal mass hierarchy and @xmath160 is certainly a plausible scenario so that expecting a positive identification of the earth effect is not a far - fetched possibility . in summary , even though galactic sne are rare , the anticipated longevity of icecube and the long - term neutrino program at super- or hyper - kamiokande imply that detecting the earth effect in a sn neutrino burst is certainly a distinct possibility . this could identify the normal neutrino mass hierarchy , a daunting task at long - 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the oort cloud @xcite remains the most mysterious part of our solar system , primarily because it can not be directly observed . our only observational clues to the size , shape , mass and composition of the oort cloud come from observations of long - period comets . the demographics of observed long - period comets have been the starting point of almost all attempts to model the oort cloud ( eg . * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . until the last ten or so years , the vast majority of comets were discovered by systematic eyeball searches , using small telescopes @xcite . these surveys have been highly effective at identifying large samples of comets , and in deriving their orbital parameters . they do , however , have three major drawbacks : * unknown selection function : it is very unclear how often different parts of the sky are surveyed , and to what depth . surveys are clearly more sensitive to comets with bright absolute magnitudes and perihelia close to the earth , but the strength of this effect is very hard to estimate @xcite . * limited range of comets observed : eyeball surveys find few comets fainter than an absolute magnitude of 10 and with perihelia beyond 3au . * poorly defined photometry : these surveys quote the `` total brightness '' of a comet . total cometary magnitudes are notoriously unreliable . they are typically measured by defocussing a standard star to the same apparent size as the comet , but this apparent size is heavily dependent on observing conditions and observational set - up . despite these drawbacks , many attempts have been made to derive the basic parameters of the long period comet population from eyeball - selected historic samples . the most heroic and influential attempt was that of @xcite . everhart carried out an exhaustive analysis of the historical circumstances in which comets were discovered , over a 127 year period . he developed a model for the sensitivity of the human eye , and used it to calculate the period over which a given historical comet could have been seen . this was then used to estimate the completeness of the comet sample : if a given type of comet was typically seen early in its visibility window , surveys should be complete for this type of comet . if the mean time to find a given type of comet is , however , comparable to the length of the estimated visibility window , the completeness is probably low . using this method , everhart estimated that for every comet seen , another 31 were missed . a more modest and recent attempt was that of @xcite . he restricted himself to the brightest and nearest comets , for which he claimed ( on the basis of discovery trends ) historical surveys were highly complete . the statistics of these comets were simply extrapolated to larger perihelia and fainter absolute magnitudes , with no correction for observational incompleteness . as one would expect , the flux of long - period comets through the inner solar system estimated by @xcite is much lower that that estimated by @xcite . despite these attempts , several basic questions about the demographics of long - period comets remain unresolved . one question concerns small comets @xcite : those with nuclear radii less than @xmath0 km ( absolute magnitudes @xmath1 ) . extrapolating the everhart data implies that there should be a large population of such comets . @xcite was unable to tell whether his model predicted a large population of such comets or not . a second question concerns the number of comets per unit perihelion . @xcite found that this number rises from the sun out to 1au , but was unable to determine whether it keeps rising at larger perihelia . @xcite found no significant rise , but had large enough error bars to bracket both of everhart s possibilities . the observational situation has changed radically in the last few years . the advent of large format sensitive ccds has allowed automated surveys to supplant eyeball searches as the main mechanism for finding new long - period comets . most long period comets are now being found as by - products of various automated searches for near - earth objects , such as the lincoln near - earth asteroid research ( linear ) project @xcite , the catalina sky survey , loneos and neat @xcite . many are also found by space - based coronagraphs as they approach very close to the sun @xcite , though these are mostly fragments of recently disintegrated larger comets @xcite . in this paper , i attempt to deduce the statistical properties of the long - period comet population from one of these ccd surveys : the linear survey . this has a far better defined selection criterion than any historical eyeball survey , and extends to much larger perihelia and fainter absolute magnitudes . it thus allows both an independent check and an extension of previous estimates of the long - period comet population . near earth asteroid ( neo ) surveys are not optimized for comet detection . while they find many long - period comets , they do not publish their raw data , nor all the details one would like of their exact detection algorithms and sky coverage . in particular , they do not publish on - going photometry of the comets they discover . nonetheless , enough information is available to make a first pass at estimating the true population of long - period comets from their data . there have been previous attempts to use these surveys to detemine the true populations of neos ( eg . * ? ? ? * ) and dormant comets @xcite , but this paper is the first attempt of which i am aware to do this for active comets . in the next few years , the situation should further improve , with the advent of a new generation of wide - field survey telescopes , such as skymapper , pan - starrs @xcite and gaia @xcite . these surveys will predominantly find comets much fainter and more distant than historical surveys . the analysis in this paper allows a first estimate of just how many long - period comets these surveys can find , and how best to identify them . i start off by defining a sample of comets drawn from the linear sample , and examining its properties , which are very different from those of eyeball samples ( [ sample ] ) . a model of the long - period comet population is then generated ( [ model ] ) based on and extrapolating the historical eyeball - selected surveys . a monte - carlo simulation of this comet population as it would be observed by linear is then developed ( [ montecarlo ] ) . the results are compared to the observed sample in [ compmodels ] : i find that the hughes model is quite a good fit to the data , but that the everhart model is not . i derive my own best - fit model of long - period comet demographics . the consequences of this new model are many : i examine them in [ discussion ] before drawing conclusions in [ conc ] . of the several near - earth asteroid surveys now under - way , the lincoln near - earth asteroid research ( linear ) project @xcite was most suitable for constraining the long - period comet population . this is because : * they discover more comets than any other single survey . * they publish sky charts on their web page showing the area of the sky observed during each lunation , with the point - source magnitude limit reached at each location . * their sky coverage and magnitude limit is relatively simple and uniform across this period . the comet sample was defined as follows : 1 . the comet has an orbital period longer than 200 years . 2 . the comet reached perihelion between 2000 jan 1 and 2002 dec 31 . the comet was either discovered by linear between these dates , or could have been discovered by linear between these dates had it not already been discovered by someone else , or discovered prior to 2000 jan 1 . the 2000 - 2002 date range was chosen because comet details ( from the catalog of cometary orbits , * ? ? ? * ) and sky - maps ( including limiting magnitudes ) are available . @xcite listed 25 comets as having been discovered or co - discovered by linear which met our criteria . i needed , however , to add two additional sub - samples : * comets discovered prior to 2000 , but which reach perihelion in the period 2000 - 2002 , and which could have been first discovered by linear within this period , had they not already been found . * comets discovered in 2000 - 2002 inclusive by other surveys , but which would subsequently have been seen by linear during this period . potential members of the two additional sub - samples were selected from @xcite . each candidate was checked for its detectability by linear , using the ephemerides and predicted magnitudes generated by the minor planet center . the predicted positions and brightnesses were compared to the maps of linear sky coverage . these maps show only the integrated coverage per lunation , not the night - by - night or hour - by - hour coverage , but most of these comets move slowly enough that this should nt much matter . this process added another 27 comets to our sample . 8 had been detected by linear during 1999 , but reached perihelion in 2000 or 2001 . most of the remainder were first identified by other near - earth asteroid surveys , particularly the catalina sky survey , loneos and neat . for every comet in our final sample , the original discovery details ( as distributed by the central bureau of astronomical telegrams ) were checked . from these , the discovery date , discovery magnitude @xmath2 and discovery circumstances were noted . the discovery magnitudes are total magnitudes ( m1 ) . it is not clear how reliable and homogeneous these magnitudes are , but no better source of ccd photometry is available . they are based on ccd observations by professional astronomers of typically barely resolved objects , and so should be good to @xmath3mag . absolute magnitudes @xmath4 were computed from these discovery magnitudes @xmath2 . the standard equation was used : @xmath5 ( eg . * ? ? ? * ) , where @xmath2 is the observed total magnitude at discovery and @xmath6 a power - law parameterization of the dependence on heliocentric distance . as is conventional for solar system work , the absolute magnitude is defined as the observed magnitude if the object were at a distance of 1 au from both the earth and the sun . following @xcite , the dynamically new and old comets were treated differently ( the new ones are much brighter at large heliocentric radii , at least on their way in ) . a comet is classed as dynamically new if its original semi - major axis @xmath7 is @xmath8 10,000 au , old if @xmath9 10,000 au , and undetermined if the orbit class in @xcite is ii or worse . for new comets , @xmath10 was used if they are seen pre - perihelion and @xmath11 if seen afterward . for old comets , the values are 5.0 and 3.5 respectively . the canonical value of @xmath12 is used for comets of undetermined orbit type . this is uncertain both because real comets show a dispersion in @xmath6 , and because the @xmath6 values in @xcite are based on observations at smaller heliocentric distances . it is , however , self - consistent with the analysis used in our monte - carlo simulations . our sample is listed in table [ sampletab ] . lcccc c/1999 f1 & 5.7869 & 7.82 & 0.000038 & 1a + c/1999 j2 & 7.1098 & 6.39 & 0.000019 & 1a + c/1999 k5 & 3.2558 & 9.75 & 0.000024 & 1a + c/1999 k8 & 4.2005 & 6.33 & 0.000681 & 1a + c/1999 l3 & 1.9889 & 10.21 & 0.013741 & 1b + c/1999 n4 & 5.5047 & 9.99 & 0.000068 & 1a + c/1999 s4 & 0.7651 & 7.84 & 0.000720 & ii + c/1999 t1 & 1.1717 & 4.36 & 0.000173 & ii + c/1999 t2 & 3.0374 & 6.05 & 0.000596 & 1a + c/1999 t3 & 5.3657 & 5.12 & 0.000231 & 1b + c/1999 u4 & 4.9153 & 7.60 & 0.000037 & 1a + c/1999 y1 & 3.0912 & 9.80 & 0.000044 & 1a + c/2000 a1 & 9.7431 & 8.13 & 0.000044 & 1a + c/2000 ct54 & 3.1561 & 10.71 & 0.000051 & 1a + c/2000 h1 & 3.6366 & 10.29 & & + c/2000 j1 & 2.4371 & 12.62 & 0.001406 & 1a + c/2000 o1 & 5.9218 & 7.03 & 0.000037 & 1a + c/2000 of8 & 2.1731 & 14.07 & 0.000048 & 1b + c/2000 sv74 & 3.5416 & 9.40 & 0.000090 & 1a + c/2000 u5 & 3.4852 & 9.88 & 0.000358 & 1a + c/2000 w1 & 0.3212 & 10.44 & & + c/2000 wm1 & 0.5553 & 6.81 & -0.000459 & ii + c/2000 y1 & 7.9747 & 9.54 & 0.000063 & 1a + c/2000 y2 & 2.7687 & 9.65 & 0.001934 & 1b + c/2001 a1 & 2.4062 & 10.71 & 0.005738 & 2a + c/2001 a2 & 0.7790 & 14.22 & 0.000447 & ii + c/2001 b1 & 2.9280 & 11.14 & 0.000071 & 1b + c/2001 b2 & 5.3065 & 5.60 & 0.000187 & 1b + c/2001 c1 & 5.1046 & 10.30 & 0.000020 & 1a + c/2001 g1 & 8.2356 & 7.45 & 0.000024 & 1a + c/2001 ht50 & 2.7921 & 3.15 & 0.000878 & 1a + c/2001 k3 & 3.0601 & 9.80 & 0.000072 & 1b + c/2001 k5 & 5.1843 & 8.10 & 0.000029 & 1a + c/2001 n2 & 2.6686 & 5.77 & 0.000455 & 1a + c/2001 rx14 & 2.0576 & 6.06 & 0.000776 & 1a + c/2001 s1 & 3.7500 & 11.36 & 0.018168 & + c/2001 u6 & 4.4064 & 7.42 & 0.000998 & 1a + c/2001 w1 & 2.3995 & 14.04 & & + c/2001 x1 & 1.6976 & 12.54 & 0.002285 & 2a + c/2002 b2 & 3.8430 & 10.12 & & + c/2002 b3 & 6.0525 & 7.92 & & + c/2002 c2 & 3.2538 & 8.88 & 0.000393 & 1b + c/2002 e2 & 1.4664 & 10.34 & 0.000173 & 1b + c/2002 h2 & 1.6348 & 13.29 & 0.004024 & 2a + c/2002 k2 & 5.2378 & 7.62 & & + c/2002 l9 & 7.0316 & 5.60 & 0.000035 & 2a + c/2002 o4 & 0.7762 & 13.59 & -0.000772 & 2a + c/2002 p1 & 6.5307 & 8.55 & 0.002023 & 2a + c/2002 q2 & 1.3062 & 16.09 & & + c/2002 q5 & 1.2430 & 16.59 & 0.000058 & 1b + c/2002 u2 & 1.2086 & 14.63 & 0.001075 & 1b + the linear sample has very different properties from historical samples ( as typified by the everhart sample ) . figure [ everhart_compare ] shows that the linear sample extends around 4 magnitudes deeper , and to much larger perihelia . the overlap is small : only @xmath13 of the linear comets lie within the absolute magnitude and perihelion region sampled by historical samples . analysis of the discovery telegrams indicates that almost all of the comets in the sample were originally identified as moving point sources . they were posted on the near earth object ( neo ) confirmation page at the minor planet center . follow - up observations then determined that the sources were spatially extended and hence comets . 77% were discovered before reaching perihelion ( fig [ discover_hist ] ) , and 73% were were first detected when more than 3au from the sun . the necessity for follow - up potentially introduces two sources of incompleteness into this sample . firstly , some fraction of objects posted on the neo confirmation page are never followed up in enough detail to determine whether they are comets or not . timothy spahr kindly provided records of all objects posted to the neo confirmation page in 2000 - 2002 . only 11% of these were not followed - up well enough to determine an orbit : this places an upper limit on the fractional incompleteness of our sample due to failed follow - up . this is probably a conservative upper limit : most of these lost objects were most likely either not real to begin with or fast - moving objects only visible for a short window of time . secondly , some comets might have been inactive at these large heliocentric distances , and hence classified as minor planets . the minor planet centre database was checked for non - cometary objects on long - period , highly eccentric orbits , but only one was found which reached perihelion within the period 2000 - 2002 : 2002 rn109 . thus this too is not a major source of incompleteness . it also shows that most comets down to the linear magnitude limit are still active out to 10au from the sun . fig [ q_a ] shows an intriguing correlation between perihelion distance and semi - major axis in the linear sample . this correlation was first noted by @xcite at smaller perihelia . they suggested that it was a selection effect . dynamically new comets are brighter at large heliocentric radii ( eg . * ? ? ? * ) , presumably due to extra outgassing at large heliocentric distances from their relatively pristine surfaces , due perhaps to @xmath14 or a water ice phase transition . the whipple data did not extend to distances beyond 4au from the sun . if this trend continues to larger heliocentric distances , however , it would make dynamically new comets far brighter than older comets with the same absolute magnitude . this could thus , in principle , bias the sample heavily towards new comets , and hence larger semi - major axes . the distribution of perihelion positions is shown in fig [ peripos ] . the comets are weakly concentrated at intermediate galactic latitudes ( fig [ bhist ] ) , consistent with the galactic tide playing a major part in making them observable @xcite . the galactic latitude distribution is not , however , significantly different from the predictions of a best - fit model ( [ best ] ) assuming a random distribution , as measured by the kolmogorov - smirnov ( ks ) test or the kuiper statistic . there are also no significant great - circle alignments @xcite . the linear comet sample was compared against a monte - carlo simulation of the long period comet distribution . in this section i discuss the parameters used in this simulation . the @xcite and @xcite studies are based on comets with a limited range of perihelion distances q and hence give only weak constraints on this distribution . everhart found a factor of two rise in the number of comets per unit perihelion between 0 and 1 au , but beyond that the data are consistent either with a continuing rise or a flat distribution . hughes found no significant trend in number of comets against perihelion , but his data are quite consistent with such a trend . on theoretical grounds , however , a gentle rise in the number of comets as a function of perihelion is expected , as comets diffuse into the solar system past the barrier of giant planet perturbations ( eg . * ? ? ? * ; * ? ? ? as a first guess , i chose to model the perihelion distribution as an unbroken straight line : @xmath15 where @xmath16 gives a reasonable fit to the @xcite distribution . the @xcite slope is shallower , but only shown out to 3au . the distribution of semi - major axes @xmath7 makes no significant difference to the conclusions , as the comets are all very close to being on parabolic orbits . i chose to randomly class 40% of comets as dynamically new and given them all @xmath1720,000 , while the remainder were given a value of @xmath7 randomly and uniformly distributed between 1000 and 20,000 au . the time of perihelion passage , orbital inclination and perihelion direction were randomly chosen to give a uniform distribution on the celestial sphere . this ignores possible great - circle alignments @xcite and galactic tidal effects @xcite . @xcite found that the absolute magnitude distribution of comets was best fit by a broken power - law , with the break at @xmath18 . @xcite also found a break at about the same absolute magnitude , but was unable to decide whether it was a real break or simply the effect of increasingly incomplete samples at fainter magnitudes . to bracket the possibilities , i use a broken power - law of the form . @xmath19 where @xmath20 is the break magnitude , @xmath21 is the bright end slope and @xmath22 is the faint end slope . everhart gives @xmath23 , @xmath24 and @xmath25 . hughes gives @xmath26 and @xmath27 . if i assume that his observed break is real and not an artefact of sample incompleteness , his plots imply a faint end slope of @xmath28 , which i adopt to bracket the possibilities . this version of the hughes formulation thus predicts dramatically fewer faint comets , as would be expected as these are the ones for which everhart applies the largest incompleteness correction fraction . @xcite and @xcite give different estimates of the long - period comet flux through the inner solar system . everhart estimates a flux of 8000 comets with @xmath29 and @xmath30 over 127 years . hughes estimates a flux of 0.53 comets per year brighter than @xmath31 per unit perihelion . i ran the simulations using both . for a given model comet population , the aim is to simulate the observable properties of a sample that matches the selection effects of the linear sample . the simulation starts off by generating a set of comets that reach perihelion within a three year period . the comets are randomly generated using the model distributions in the previous section . the model extends down to @xmath32 and out to @xmath33 . i generated two model populations : one using the everhart absolute magnitude distribution and flux , the other using the hughes absolute magnitude distribution and flux . in the everhart model 260,000 comets are generated , while only 4,000 are needed in the hughes model . the position of each comet is then calculated at 24 hour intervals throughout the three year period , and its heliocentric distance @xmath34 , distance from the earth @xmath35 , apparent celestial coordinates and apparent angular velocity written to file . pure elliptical orbits are used : no attempt is made to allow for planetary perturbations . at each location , the apparent total magnitude is then calculated . comets are notoriously variable in how rapidly their apparent magnitude varies as a function of heliocentric distance . i parameterize this , as is conventional , using equation [ eqabsmag ] . two values of @xmath6 are randomly assigned to each comet : one for before perihelion and another for after . for the 40% of our simulated comets that i set as dynamically new , the pre - perihelion value of @xmath6 is chosen from a gaussian distribution of mean 2.44 and standard deviation 0.3 . post - perihelion , the mean is 3.35 with a scatter of 0.27 . for the remaining comets , the pre - perihelion numbers are 5.0 with a scatter of 0.8 , and after perihelion 3.5 with a scatter of 0.5 . all these values are taken from @xcite . at large distances from the sun , cometary activity will presumably stop , and a bare nucleus will have @xmath36 . the near - ubiquitous detection of fuzz around the linear comets implies , however , that this only happens further from the sun than our models reach . this approach can only be a rough approximation to the real radial brightness dependence . the value of @xmath6 for an individual comet is typically time dependent , and all the tabulated values are for comets within @xmath37 au of the sun , whereas our simulation tracks them out beyond 10au . in addition , comets show occasional flares above and beyond this power - law behavior , which i have not attempted to model . such flares might introduce an amplification bias , with comets being pushed over the detection threshold . as we will see , however , the slope of the absolute magnitude distribution is so gentle that this is unlikely to be a major effect . the apparent total magnitude of each simulated comet can now be calculated at any given point in its orbit . unfortunately , in any ccd - based survey , it is the peak surface brightness of the coma that determines whether something has been seen , not the total magnitude . the linear skymaps , furthermore , list only the magnitude limit _ for a point source _ at any given location on the sky ( typically around 19 ) . as discussed in the introduction , total cometary magnitudes are notoriously unreliable . quantitative studies prefer more reproducible and physically meaningful parameters such as @xmath38 ( eg . * ? ? ? unfortunately , not enough long period comets have been studied in this way to derive the @xmath39 distribution . we are therefore forced to attempt some conversion between total magnitudes and point - source equivalent magnitudes . for bright and near - by comets , this correction can be as large as @xmath13 magnitudes ( eg . * ? ? ? the comets in the linear sample were , however , typically first seen when very faint ( fig [ maghist ] ) , and were generally mistaken for point sources in the initial observation . we might therefore expect the correction factor to be much smaller , at least when close to the detection threshold . the histogram of detection magnitudes ( fig [ maghist ] ) climbs steeply down to @xmath40 , and then falls off fast ( the one comet discovered when fainter than 20th mag was found by spacewatch , which has a fainter magnitude limit ) . this fall - off occurs at almost exactly the same magnitudes as linear s point source limit , which ranged from around 18 to 20 . i thus conclude that near the linear detection threshold , total magnitudes and point source equivalent magnitudes are similar . when generating mock samples , it is only the magnitude near the detection threshold that determines whether or not a given model comet is included in the mock catalog . the exact value of this correction value was set iteratively . i initially guess that the point - source equivalent ( pse ) magnitude and total magnitude ( tm ) are the same , and run the simulations of the comet sample . i use the model that best fits the data ( [ best ] ) to calculate the predicted discovery magnitude distribution , and compare this to the observed distribution . i then tweak the pse@xmath41tm correction to bring the histograms into agreement . the best match is obtained when pse@xmath41tm@xmath42 ( fig [ maghist ] ) . i use a value of @xmath43 throughout this paper , except where otherwise noted . the predicted magnitude is corrected for the effects of trailing . linear exposure times vary from 3 to 12 sec : the latter was used in the correction as it minimised the predicted number of very faint comets . 2 seeing ( fwhm ) was assumed . trailing makes very little difference , except for the very faintest comets . linear uses unfiltered ccd magnitudes while the historical surveys use unfiltered visual magnitudes . these will be somewhat different , due to the different wavelength sensitivity of the human eye and of the linear ccds , but the discrepancy should only be a few tenths of a magnitude at most , and hence is not a dominant source of error . anoher possible worry : the absolute magnitudes i quote for the comets in the linear sample ( table [ sampletab ] ) are derived from total magnitudes measured when the comets were barely resolved and far from the sun , using a model for the heliocentric brightness variation . the absolute magnitudes fit by everhart and hughes are based on observations of highly extended comets observed close to the sun and earth . these are thus very different quantities , and might well be systematically different , if there is some error in our heliocentric brightness correction , if total magnitudes for barely resolved comets are systematicaly different from total magnitudes for greatly extended comets , or if there is some systematic bias in the discovery magnitudes reported to the central bureau of astronomical telegrams . to test this , i picked out the five comets in the linear sample which were discovered when far ( more than 3.5 au ) from the sun , but which subsequently passed close enough to the earth and sun for traditional small telescope visual magnitude estimates ( within 2au ) . dan green kindly provided me with compilations of visual magnitude estimates of these comets while they were close to the earth and sun , taken from the archives of the international comet quarterly . these visual / small telescope magnitude estimates should be broadly comparable to the data on which the everhart and hughes papers were based . i then compared the predicted magnitudes when close to the sun ( based on the discovery magnitude and the model in this paper ) with the tabulated observations . there was a considerable scatter in the measured visual magnitudes for each comet : i simply averaged all visual small telescope magnitudes made when the comet was as close as possible to 1au from both sun and earth . my predicted magnitudes were consistent with the observed values , albeit with a large scatter . the mean difference ( predicted magnitude minus observed magnitude ) was @xmath44 , where the error indicates the @xmath45 dispersion of the mean . this is not , alas , a strong constraint , but does indicate that the two magnitude scales are not grossly discrepant . the final step is to determine whether linear imaged a part of the sky in which the comet was detectable and within its magnitude limit . the published linear skymaps show that during each dark period in 2000 - 2002 , they attempted to survey the region whose midnight hour angle is in the range @xmath46 , and in the declination range @xmath47 . in winter months with good weather , they surveyed more than 90% of this whole region down to a point - source magnitude limit of better than 19 . in bad months , this dropped to a magnitude limit of around 18.5 over 60% of this region , and occasionally worse . in the mean month , 72% of this region was surveyed to a visual magnitude limit of 18.5 or better . the exact pattern surveyed was complex and variable : the only constant was that the densest regions of the galactic plane were avoided . each field was imaged five times in succession , with 3 12 sec per exposure , once in every dark period . this sky coverage was approximated as follows . each comet that enters the @xmath46 , @xmath47 region at any point , with a point - source equivalent ( pse ) magnitude brighter than 19 is considered to have been potentially observable , unless it was within ten degrees of the galactic plane . if a comet is predicted to be detectable for an entire lunation , it is given a 80% chance of having been detected during that lunation . if it was predicted to be visible for less than the whole lunation , it is given a probability of having been detected equal to 80% of the fraction of the lunation for which is was potentially observable . does this approximation match the real , more complex selection function ? this was tested by manually checking 100 simulated comet ephemerides , containing monthly positions and magnitudes , against the real linear sky - maps . the approximation was found to give a number and absolute magnitude distribution of detected comets indistinguishable from the manually checked sample . this approach should slightly overestimate the probability of a comet being observed , as comets could be blended with star or galaxy images . experience suggests that at this relatively bright magnitude limit , this is only a few percent effect at worst , at least away from the denser regions of the galactic plane , which the survey did not cover . another possible source of error is sky subtraction . it is unclear exactly how the linear survey do their sky subtraction , but if some of the extended coma emission is included in the sky value , this will artificially suppress the point source equivalent magnitude . the worst case sky subtraction algorithm would be to measure the sky brightness from an annulus close to the comet nucleus . if i assume that the sky is measured only 5 from the nucleus ( unlikely ) , we can used the observed @xmath48 surface brightness profiles of cometary comae @xcite to show that this sky subtraction algorithm would reduce the measured comet brightness by @xmath49 mag . more plausible sky subtraction schemes would reduce it by less , or not at all . thus this too is not a dominant source of error . in this section , i compare the data to the monte - carlo simulations of what linear would have seen over a three year period . in the scatter plots , the data are compared to a single run of the simulation . in all histograms and quoted statistics , however , the data are compared against the average or sum ( as appropriate ) of five monte - carlo runs based on the same comet population model . this summation should suppress the error due to small number statistics in the simulated samples to well below that of the observed sample . the observed sample properties are first compared against the monte - carlo prediction using the @xcite flux normalization and absolute magnitude distribution . figs [ complot_eve ] , [ qhist ] and [ hhist ] compare the distribution of model and observed comets in perihelion @xmath50 and absolute magnitude @xmath4 . the upper boundary to the locus of points is set by the magnitude limit , and seems a reasonable fit to the data . but the model clearly predicts far too many comets : 2228 as compared to the 52 observed . the discrepancy is primarily at fainter absolute magnitudes : brighter than @xmath51 the model and data are consistent . the worst discrepancy is for comets fainter than @xmath52 : ie . fainter than the data on which everhart based his model . it is thus a test of the power - law extrapolation . the model predicts that linear should have seen 1848 comets fainter than @xmath53 , whereas only 12 were seen . irrespective of the flux normalization , the shape of the absolute magnitude distribution ( fig [ hhist ] ) is wrong : a ks - test comparison with the observed distribution shows that they are inconsistent with @xmath54 confidence . is this discrepancy real , or is there some reason why linear would miss faint comets close to the earth ? the model comets with @xmath55 are predicted to be observable for a median period of 52 days , at a median distance from the earth of @xmath56 au . their typical apparent angular velocity is predicted to be @xmath57 degrees per day . their observational properties are thus typical of near earth objects , which linear finds in profusion . it is thus hard to see what selection effect could prevent their detection . grant stokes ( personal communication ) confirms that there is nothing in their data analysis which should preclude the discovery of comets like these . image trailing and short observability windows do reduce the number of these comets seen , but these are already taken into account by the monte - carlo simulation . could the discrepancy be an artifact of the various approximations made in the model ? the discrepancy is insensitive to assumptions about the dependence of brightness or comet number on heliocentric distance , as these sources are observed at close to 1au . one possibility is that i have incorrectly estimated the effective magnitude limit of linear for sources with these apparent total magnitudes . to test this , i repeated the analysis with a magnitude limit set two magnitudes brighter than my best estimate . this reduced the discrepancy but did not remove it : the prediction dropped to 577 observed comets fainter than @xmath58 , still more than two orders of magnitude above the data . none of the plausible incompletenesses in the data , nor other assumptions in the model can come close to removing this discrepancy . i therefore conclude that the everhart model can not be extrapolated to absolute magnitudes fainter than @xmath58 . even at brighter absolute magnitudes , however , there remains a substantial discrepancy . the model predicts that 188 comets with @xmath59 and @xmath60 should have entered the solar system within the three year period , and that 86% of them ( 162 ) would have been detected by linear . only 21 such comets were observed . it is hard to see that linear could have missed many comets this bright passing this close to the sun . the model predicts that these comets should remain visible for a median 208 days ( 7 lunations ) , so almost regardless of position on the sky , they should have had several opportunities to be observed . they spend much of this time many magnitudes above the survey detection limit . indeed , the 21 comets observed with these properties were discovered a median 11 months before perihelion , at a median heliocentric distance of 4.3au , confirming that these are easy targets . the discrepancy occurs mostly at the fainter magnitudes within this range : brighter than @xmath51 there is no significant difference between the everhart predictions and the linear observations . i conclude , therefore , that the everhart model fails in two ways . firstly , the quoted normalization of 8,000 comets per 127 years with @xmath29 and @xmath60 is too high by a factor of @xmath61 . secondly , the faint end slope of the everhart absolute magnitude relation is much too steep , and immensely over - predicts the number of faint comets . this second conclusion was first reached by @xcite : the current paper independently confirms their result . both discrepancies suggest that everhart overestimated the incompleteness of his sample of long - period comets . the discrepancy goes away where the incompleteness correction is small , but is largest at the faint magnitudes where the correction is large . a discrepancy here is perhaps not surprising , as the correction factors calculated by everhart were so large : he corrected the 256 observed comets to a flux of 8000 : a factor of 31 . my analysis reduces this correction factor to only @xmath62 . one possible reason for the difference : everhart calculated the detection threshold for typical historical comet searchers , and assumed that the same threshold applied when searching for comets initially , and when making follow - up observations of known comets . his model was validated by noting that the last observations of comets occurred close to the time when his model suggested that they dropped below detectability . but let us hypothesize that comets just above the detection threshold might be missed as the telescope speeds past during a scan for new comets , even though they could be detected when looking hard for an already known comet at a known position . this would reduce the length of time over which a given comet could have been detected . detections would thus have occurred earlier in the detectability window , and the correction factor for incompleteness would thus decrease . i now compare the data against a monte - carlo simulation using my version of the @xcite absolute magnitude distribution and normalization , combined with the linearly rising perihelion distribution ( figs [ qhist ] , [ hhist ] , [ complot_hug ] ) . once again , the overall envelope of points agrees well with the model , giving some confidence that the selection effects have been modeled correctly . the predictions from the hughes model are in much better agreement with the data . there is no vast excess of faint predicted comets , implying that the break seen in hughes data was real , and hence that the flatter faint - end slope of the absolute magnitude distribution is more accurate . neither the perihelion nor absolute magnitude distributions , however , are formally consistent with the observations at the 99% confidence level , as measured by the ks - test . the overall flux normalization is also too high : the model predicts that linear should have seen 171 comets , rather than the 52 observed . there is no significant discrepancy within the region in which hughes quoted his flux normalization ( 0.54 comets per year with @xmath63 per unit perihelion ) : the discrepancy is at fainter absolute magnitudes and larger perihelia . once again the discrepancy is fairly robust against the exact detection limit : dropping the detection threshold by a magnitude reduces the predicted comet numbers to 148 - still too high . how can this discrepancy be addressed ? possible incompletenesses in the linear sample were discussed in [ prop ] and they can at best increase the observed numbers by @xmath64% . the hughes flux normalization is unlikely to be too low , as it was based on observed counts of very bright comets and made no correction for incompleteness . i therefore tried to improve the match by tinkering with the absolute magnitude and perihelion distributions . i first tried reducing the faint end slope of the absolute magnitude relation . if we assume that the linear sample is 20% incomplete , we need to reduce the faint end slope from 1.07 down to 0.8 to bring the number of predicted comets down to the observed number . unfortunately , this changes the observed absolute magnitude distribution too much : a ks test shows that a model with this slope predicts an observed @xmath4 distribution inconsistent with the data with greater than 99.99% confidence . i then tried changing the perihelion distribution . decreasing the slope @xmath65 in eqn [ perieq ] to zero brought the predicted number of comets down to 95 , but the perihelion distribution is now too skewed towards small values of @xmath50 ( 99.96% confidence ) . i next tried combining both approaches . decreasing the faint end slope to 1.0 , combined with a flat perihelion distribution , brought the predicted numbers into line with the observed numbers . both the perihelion and absolute magnitude distributions were individually marginally acceptable ( ks - test gave 8 and 5% probabilities of them coming from the same population as the data ) but the joint probability was still uncomfortably low ( though the magnitude limit cut means that the two distributions are not independent , so this should not be taken too seriously ) . the model predicted too many comets with @xmath66 and @xmath67 , and too few in the middle . moving the location of the break in the absolute magnitude relation to fainter magnitudes was also a failure : given that the comet flux normalisation is at brighter magnitudes , this simply increased the number of fainter comets still further above the observations . i therefore adopted a different perihelion distribution : one that rises from @xmath68 out to @xmath69 , and is a power - law beyond that . this preserves the everhart observation of a drop in comet numbers below @xmath70 , while allowing us to tinker with the distribution further out . the parameterization used was : @xmath71 where @xmath72 controls the behavior at large perihelia . i then ran a grid of models , varying @xmath72 and the faint end slope of the absolute magnitude distribution ( @xmath22 in equation [ heq ] ) . each simulated population was tested against the data in three ways : a ks - test on the perihelion distribution , a ks - test on the absolute magnitude distribution , and a chi squared test on the overall predicted number of comets . the latter was done for both the observed number of comets and a number 10% higher , to allow for possible incompletenesses . the lowest of these three significance values was used to compute the goodness - of - fit conutours in fig [ contour ] . note that these contours include only random errors : the systematic errors are almost certainly larger , especially on @xmath72 . the lowest probability rather than the joint probability was used becuase the three tests are not strictly independent . quite a tight constraint could be placed on @xmath22 : @xmath73 ( 95% confidence , not including systematic errors ) . the constraint on @xmath72 was weaker : @xmath74 . no useful constraint could be placed on the bright - end slope @xmath21 : in the modeling i use the @xcite value of 2.2 , but it makes little difference . note that these slopes are often described in the literature using the @xmath75 parameter : @xmath76 , so our best - fit faint end slope has @xmath77 . the predicted distribution of comets for the best fit model is shown in figs [ qhist ] , [ hhist ] and [ complot_f ] . models generated using this model predict @xmath78 ( @xmath45 ) observed comets , in excellent agreement with the data . this model was used to predict the discovery magnitude distribution ( fig [ maghist ] ) . this comparison was used to set the equivalent point source vs. total magnitude offset , as described in [ mags ] . my preferred model thus approximately preserves the faint - end slope and normalization derived by hughes . a flat perihelion distribution is marginally ruled out , and best fits are obtained for one that rises from the sun out to @xmath79 and is either flat or gently falling beyond that . the best - fit model can be used to estimate the flux of long - period comets through the inner solar system . by definition , this model uses the @xcite flux of 0.53 comets per year per unit perihelion , brighter than @xmath80 . @xcite estimated a flux of 8000 long - period comets per 127 years with @xmath29 and @xmath60 . my model suggests that the flux of comets with these parameters is much lower : 11 long - period comets per year ( 1600 per 127 years ) . the model suggests that linear is picking up over 60% of comets with these parameters . i estimate a true flux of 37 comets / year with @xmath81 and @xmath82 , of which linear is detecting @xmath83% . many published estimates of the number of comets in the oort cloud use the @xcite comet flux as their starting point , and hence should be revised down . estimates include @xcite , @xcite , @xcite and @xcite . the linear sample includes 22 comets with @xmath29 and @xmath60 over a three year period . 5 of these were dynamically new , 11 dynamically old , and 6 had orbit determinations too poor to tell . if we assume that the comets without good orbit determinations break up between new and old in the same ratio as the other comets , we find a flux of 7 dynamically new comets over the three years . my model suggests that the linear sample is @xmath84% complete for finding comets in this range , implying a total flux of dynamically new comets of @xmath37 per year . this corresponds to @xmath85 per unit perihelion per year if a uniform perihelion distribution is assumed . this is a factor of @xmath86 lower than was assumed by @xcite and @xcite . @xcite and @xcite assume long - period comet fluxes ( not just the dynamically new ones ) of @xmath87 per year per au down to @xmath29 , compared to my value of @xmath88 . after correction , all these estimates come out roughly the same : 1 3 @xmath89 in the outer oort cloud , down to @xmath29 . if this is extended to fainter absolute magnitudes , i estimate an outer oort cloud population of @xmath90 comets down to @xmath91 . my model has a much shallower slope of the absolute magnitude distribution than that of @xcite , so the average mass of a comet _ increases_. this cancels out the decreased number of oort cloud comets i predict to give a similar total oort cloud mass to previous estimates ( eg . * ? ? ? * ; * ? ? ? * ) , both of which used a mean mass computed by integrating the everhart curve . i used two suggested mass - brightness relations to estimate masses from the observed absolute magnitudes : one from @xcite and one from @xcite . note that this relation is extremely uncertain - very few long - period comets have had even their nuclear magnitudes measured . the average mass is crucially dependent on the slope of the bright end of the absolute magnitude distribution , which the data in this paper do not constrain . i bracket the possibilities by using both the everhart and hughes values ( 3.54 and 2.2 respectively ) . i use my own estimate of the faint - end slope . for the everhart bright - end slope , the total mass converges as you go to brighter magnitudes : the bulk of the mass resides in comets with @xmath92 . for the hughes bright - end slope , however , the total mass diverges as you count brighter comets . the bright comets are rare , but their mass goes up faster than their number goes down at bright magnitudes . thus for the weissman mass relation , comets with @xmath93 are 10,000 times more massive than those with @xmath94 . the hughes bright - end slope , however , suggests that they are only 2,600 times less common , so the total mass in the brighter comets is actually four times greater . for the everhart bright - end slope , however , the brighter comets would be 300,000 times less common . the everhart slope is thus physically more appealing , as it avoids the need for a bright cut - off . given the success of both the hughes normalization and faint - end slope in fitting our sample , however , his bright - end slope should perhaps be taken seriously , leading to the prospect of an oort cloud dominated ( in mass terms ) by very large comets . in this section , i will cut off the magnitude range of comets at @xmath95 , but this is arbitrary and will have a large effect on the total oort cloud mass if the hughes distribution is assumed . @xcite showed that kuiper belt objects have a break in their mass distribution at sizes of @xmath96 km , which corresponds to @xmath97 ( [ small ] ) . this may or may not apply to oort cloud comets . if i take the everhart slope , the mean mass of comets down to @xmath58 is @xmath98 g for the @xcite mass relation , and @xmath99 g for the @xcite relation . if i take the hughes slope , however , the average masses rise to @xmath100 g and @xmath101 g respectively . these values translate into total outer oort cloud masses of 2 40 earth masses . my model can be used to calculate the probability of a long - period comet impacting the earth . @xcite calculated that the probability of a given long - period comet with @xmath102au impacting the earth is @xmath103 per perihelion passage . my model suggests that the flux of comets brighter than @xmath32 with @xmath66au is 8 per year . the mean time between comet collisions with the earth is thus @xmath83 million years : very comparable to the figure calculated by @xcite , and to the mean time between global extinction events . most of these comets will , however , be quite small . using the conversion between absolute magnitude and radius described in [ small ] , a comet with @xmath104 has a radius of only @xmath64 m : too small to cause a global extinction event . comets with radii of 1 km or greater ( @xmath105 ) are rarer - the mean time between collisions with comets this large is @xmath106 million years . the shallow faint end slope means that collisions with even small long - period comets are rare . if , for example , the tunguska impact was caused by a comet ( eg . * ) , it would have a mass of @xmath107 g @xcite and hence an absolute magnitude of @xmath108 . the probability of such a long - period comet impacting the earth in the last 100 years is thus @xmath109 . the tunguska impactor must therefore be either asteroidal ( eg . * ? ? ? * ; * ? ? ? * ) or associated with a _ short - period _ comet ( eg . * ? ? ? no comet has ever been detected with a strongly hyperbolic original orbit @xcite : ie . a comet that was not gravitationally bound to the solar system . several authors have discussed this ( eg . * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , with many claiming that this is surprising . oort cloud formation models predict that for every comet that reaches the classical outer oort cloud , a factor @xmath110100 more are expelled into interstellar space ( eg . if most stars have planets , and if planetary formation is usually accompanied by comet ejection , then there should be a substantial population of free - floating interstellar comets . by some estimates , we should have expected to have seen one or more such comets by now , passing through the inner solar system . the results in this paper impact upon this question in two ways . firstly , the non - detection of interstellar comets in the linear sample places an upper limit on their space density . secondly , most previous estimates of the expected number of interstellar comets relied upon the @xcite comet flux : our lower comet flux thus leads to smaller predictions of the interstellar comet density . what limit can we place upon the space density of interstellar comets ( those with strongly hyperbolic orbits ) from the non - detection of any by linear ? i will assume that interstellar comets have the same absolute magnitude distribution that i derive for long period comets , and that their apparent brightness varies with heliocentric distance in the same way as dynamically new inbound comets in our model . given these assumptions , and an assumed magnitude limit of 19 ( point source equivalent ) , one can derive the distance @xmath34 out to which a comet with any given absolute magnitude could have been detected . to convert this into the volume surveyed during the three years of the survey , one must allow for the motion of the comets with respect to the solar system , which can carry new comets into range . typical motions of nearby stars with respect to the sun are @xmath111 @xcite . the volume surveyed in a survey of duration @xmath112 is thus @xmath113 the product of this equation and the absolute magnitude distribution ( equation [ heq ] ) was integrated to calculate the number of comets potentially within linear s magnitude limit , for a given assumed space density of interstellar comets ( defined as the number of interstellar comets per cubic astronomical unit brighter than @xmath58 ) . the integral suggests that the bulk of interstellar comets detected will be those with absolute magnitudes near the break at @xmath31 . the results are quite sensitive to the adopted bright - end slope of the absolute magnitude distribution , being 40% lower for the everhart slope as compared to the hughes slope . not all of these comets will be seen : my model suggests that linear finds @xmath114% of the oort cloud comets within its magnitude limit . the fraction may be lower for interstellar comets because they move faster and hence are not observable for long , but the velocity difference is only @xmath115% . furthermore , a larger fraction of interstellar comets will be brght ones seen at large heliocentric distances , where the visibility period is larger . i adopt a conservative 50% detection probability , which should be ample to include comets not being followed up or not having good orbit determinations . the mean number of comets seen over the three years is then evaluated as a function of the assumed average density . if more than 5 comets are predicted to have been observed , the poisson probability of us having not seen any interstellar comets is less than 5% : this is our adopted limit . i thus derive an upper limit on the local space density of interstellar comets of @xmath116 per cubic au ( 95% confidence ) if the hughes bright - end slope is assumed . for the everhart bright - end slope , this limit increases to @xmath117 per cubic au . these limits are very comparable to the best existing limit : that of @xcite . sekanina s limit is , however , based upon everhart s papers and should thus be regarded with some suspicion . i can extend this calculation by noting that linear has not discovered any interstellar comets in other years . from 1999 through to the end of 2004 their monthly sky coverage ( though not available in detail ) is at least comparable to that during my sample period . these extra three years of data reduce our upper limits to 3 4.5 @xmath118 per cubit au . i now evaluate the expected space density of interstellar comets , given our reduced oort cloud population estimate . following @xcite , the number density of interstellar comets @xmath119 is given by : @xmath120 where @xmath121 is the local number density of stars , @xmath122 is the mean number of outer oort cloud comets per star , and @xmath123 is the ratio of comets expelled from a solar system to the number ending up in the outer oort cloud . i adopt @xmath124 per cubic parsec , from the 8pc sample of @xcite : this is consistent with the value used by @xcite but considerably lower than the value used by @xcite . for @xmath122 , i assume that the number of comets generated per star is proportional to its mass and metallicity . this may or may not be true , but is consistent with the observed tendency for low mass stars to lack hot jupiter planets ( eg . * ? ? ? * and refs therein ) . the local stellar population is dominated by low mass dwarfs : the average stellar mass of the nearby stars in the @xcite catalogue is only @xmath125 . the average metallicity of near - by f and g stars is @xmath126 \sim -0.14 $ ] @xcite : no information is available for local dwarf stars , so i assume the same value . assuming that the outer oort cloud population derived in [ number ] is typical of local stars with solar mass and metallicity , i therefore derive @xmath127 . the most uncertain parameter is @xmath123 : literature values range from 3 100 ( eg . * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? . given these parameters , @xmath128 . thus even in the most optimistic case , the predicted flux is an order of magnitude below current limits . could future surveys reach these predicted densities ? i ran the simulation for five year surveys reaching to deeper magnitude limits . a survey reaching 24th magnitude ( perhaps panstarrs ) would place 95% upper limits of @xmath129 , and might hence detect one or two interstellar comets if @xmath123 is very large . @xcite reached similar conclusions . a survey reaching 26th magnitude ( lsst ? ) would push the limit down to @xmath130 , which would be enough to detect or rule out a large value of @xmath123 . this prediction does , however , depend crucially on the assumed brightness behavior of comets a long way from the sun . simple models which assume that oort cloud comets have an isotropic velocity dispersion imply a constant number of comets per unit perihelion . @xmath6-body integrations ( eg . * ? ? ? * ; * ? ? ? * ) , in contrast , imply a rising number of comets at larger perihelia , due to the diffusion process of comets past the perturbations of the giant planets . both @xcite and @xcite , for example , predict an increase in the number of comets per unit perihelion of @xmath131 between @xmath69 and @xmath132 ( though this is an extrapolation of the wiegert & tremaine model which is only shown out to @xmath133 ) . is this consistent with the linear sample ? the best - fit model has the number of comets per unit perihelion ( beyond @xmath134 ) going as a power law of index @xmath74 ( 95% confidence ) , ie . a gentle _ fall_. to get a rise in numbers consistent with the @xcite and @xcite predictions , we require @xmath135 , which is inconsistent with our model with 99% confidence . the measured perihelion distribution is , however , somewhat degenerate with the assumed dependence of comet brightness on heliocentric distance : @xmath6 in equation [ eqabsmag ] . as i have repeatedly noted , the choice of @xmath6 is an approximation based on extrapolations of observations obtained at much smaller heliocentric distances . we can estimate the change in @xmath6 that would be needed to bring our observations into line with the @xcite predictions . we need to drop the predicted brightnesses of comets with @xmath136 by enough to reduce the observed numbers by 50% , to meet our 95% upper limit . given our best - fit absolute magnitude relation , this requires that @xmath6 be increased by @xmath137 . this is quite a small rise - well within the observed scatter of @xmath6 values seen at lower perihelia . if , for example , i had used the canonical value of @xmath12 for all comets , rather than our more complex scheme , this would make dynamically new comets much fainter when far from the sun , as required ( at the expense of the correlation seen in fig [ q_a ] ) . the data in this paper , while suggestive , are not therefore significantly at odds with the theoretical predictions . a better understanding of heliocentric brightness variations when distant from the sun will be needed to see if this anomaly is real . the faint - end slope of the absolute magnitude relation derived in this paper ( @xmath138 ) is very flat : the number of comets per unit magnitude barely increases as you go fainter . this presumably indicates that small comets are not that much more abundant than large ones . if the absolute magnitudes are converted to nuclear masses , using either the @xcite or @xcite relations , i find that the differential number of comets per unit mass @xmath139 goes as : @xmath140 @xcite made the case for the existence of small comets : those with nuclei only meters to tens of meters in radius . if we assume the @xcite relationship between mass and absolute magnitude , and a density of @xmath141 , 100 m radius corresponds to @xmath142 and 10 m to @xmath143 . linear is therefore detecting at least a few comets with nuclei smaller than 100 m . similar small comets are also detected by the lasco instrument on the soho spacecraft @xcite , though these are mostly fragments of recently disintegrated larger comets @xcite . @xcite was unclear on whether the shallow faint - end slope was real or a selection effect : i confirm that it is real . @xcite pointed out that small long - period comets must be rare from statistics of comets passing close to the earth . @xcite found a similar paucity of small jupiter - family comets . in this section , i point out that the shallowness of the faint - end slope is actually quite interesting theoretically . collisions are rare in the oort cloud @xcite , so the nuclear size distribution should remain largely unchanged from when the proto - comets were planetesimals expelled from the protoplanetary disk @xcite . @xcite modeled the size distribution of planetesimals in the protoplanetary disk . these models imply that 100 m diameter objects should outnumber 2 km sized objects by a factor of @xmath144 ( per unit log radius ) in the oldest disks modeled . our faint - end slope , however , if combined with the @xcite mass / absolute magnitude relation , implies a ratio of only @xmath145 . the size distribution of planetesimals may be greatly modified by collisions while they lie within the dense environment of the proto - planetary disk . most models of this collisional evolution also , however , predict much flatter size distributions than we see : they typically predict an increase in comet numbers , even at faint magnitudes , of a factor of @xmath146 per magnitude @xcite . this is much greater than my measurement ( @xmath147 ) . as noted in [ eve ] , our shallow faint - end slope is quite robust to sample incompletenesses and model assumptions . the conversion of absolute magnitudes to radii is , however , highly uncertain even at bright magnitudes , let alone the faint absolute magnitudes of relevance here . theoretical predictions are also quite uncertain @xcite , and may be consistent with this slope . another possibility , however , is that the probability of a planetesimal escaping the protoplanetary disk and reaching the oort cloud is size - dependent . @xcite , for example , suggested that collisions between planetesimals would act to circularize their orbits , and would prevent them from escaping into the oort cloud until the density of planetesimals was greatly depleted ( see also * ? ? ? the square - cube law would suggest that this effect is most serious for smaller planetesimals , which may be ground down to dust before escaping . this could thus explain the deficit of small comets . alternatively , gas drag could play the same role : the largely gaseous nature of the giant planets indicates that the protoplanetary disk was still full of gas when the giant planets formed . no published modelling currently includes this effect . another possibility : the square - cube law implies that small comets may loose their volatiles faster than large ones . fading may thus be more severe for these comets , making them harder to detect . a final possibility is that comets have been exposed to high temperatures at some point in their history . the resultant loss of volatiles could thus destroy small comets without much affecting the numbers of large ones . @xcite tentatively suggest this as a reason for the lack of jupiter - family comets with small nuclei . long period comets might have been exposed to high temperatures while still in the proto - planetary disk . once in the oort cloud , temperatures are much lower . nearby supernovae and o - stars may have temporarily heated them enough to remove some volatiles @xcite , but this should not appreciably affect the numbers of small comets . i note in passing that my models are strongly inconsistent with the claims by @xcite for an immense population of small comets bombarding the earth . these claims have , however , already been ruled out in many other ways ( eg . * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . the model in this paper can be used to guide future automated comet - searches . the most basic conclusion follows from the shallow faint - end slope of the number / absolute magnitude relation . this means that to find more comets , a survey should always try to maximize the area covered rather than going deep in a small area . if linear , for example , exposed for six times as long per field , it would be sensitive to fainter comets , and would hence see @xmath148 times more comets per unit area . but it would cover a six times smaller area . the more frequently a survey covers a given area of sky , the higher the probability of a given comet being detected . in fig [ freq ] , however , i show that this is not an enormous effect : decreasing the survey frequency for a hypothetical survey from weekly to biannual only drops the number of detected comets by @xmath64% . the comets lost are primarily those with fainter absolute magnitudes , because their visibility period is small . the small number lost is a direct consequence of the shallow faint - end slope : there are few small comets to lose . in fig [ faint ] , i show the predicted samples that would be found by telescopes using a similar survey technique to linear , but going deeper . the deeper limit essentially increases the absolute magnitude limit reached at all perihelia , and by increasing the time over which comets are visible , also improves the completeness for brighter comets . over 3 years , a survey to 20th mag would detect 103 comets , to 22nd mag would detect 150 and to 24th mag , 186 comets . this is much fewer than @xcite estimated . the discrepancy is probably due to my shallow inferred absolute magnitude and perihelion distributions , which mean that the dramatic increase in sensitivity of these surveys as compared to linear only yields a relatively small increase in sample size . i conclude that telescopes such as skymapper , panstarrs and lsst , each capable of surveying large areas to deeper than 22nd magnitude , should be capable of detecting more than 50 long - period comets per year . a substantial fraction of these are forecast to have perihelia beyond 10au , though this conclusion relies upon the rather shaky assumptions of how brightness varies with heliocentric distance this far out . thus five years should suffice to build up a quantitatively selected sample of long - period comets equal in size to any historical sample . six main conclusions can be drawn from this analysis : 1 . the outer oort cloud contains @xmath90 comets down to @xmath91 ( @xmath149 comets down to @xmath58 ) . this is 2 10 times fewer comets than previous published estimates . 2 . the average mass of these comets is , however , higher than previous estimates . down to @xmath58 , the average mass is between @xmath99 g and @xmath101 g , leading to a total mass in the outer oort cloud of 2 40 earth masses , comparable to or larger than previous estimates . the mass of the oort cloud may be dominated by a few large comets . small comets do exist , but are rarer than predicted by many models . this may be because they have difficulty escaping from the protoplanetary disk . the probability of the earth being hit by a long - period comet similar in energy to the tunguska impactor is only one in forty million per year . i place an upper limit on the space density of interstellar comets of @xmath150 per cubic au ( 95% confidence ) . this is still an order of magnitude above our revised prediction for the space density of interstellar comets . the number of long - period comets per unit perihelion seems to decline , or at best rise slowly beyond 2au . this does not agree with theoretical predictions . the discrepancy may be resolved if comets are fainter at large heliocentric distances than i assume . future survey telescopes should be able to assemble samples of several hundred long - period comets in a few years of operation . the major weakness in this analysis is in the photometry : in particular in our limited understanding of how the brightness of comets varies with heliocentric distance when far from the sun , and in how to convert ill - defined total magnitudes into more reproducible and physically meaningful parameters such as @xmath39 . the data exist to address these problems , but are not publicly available . i would like to thank grant kennedy , eriita jones and chris weekes for their work on aspects of this paper , stephen pravdo and grant stokes for responding to e - mail questions about their neo surveys , timothy spahr for providing details of objects posted to the neo confirmation page , and dan green for provide magnitude measurements from the archives of the international comet quarterly . paul weissmann was referee on an earlier version of this paper : his detailed comments were invaluable in educating the author ( an extragalactic astronomer by training ) in comet lore .

turbulent aerosols are of interest in a variety of natural and technological systems . two very important examples are water droplets in turbulent rain clouds @xcite and dust grains in turbulent accretion disks around growing stars @xcite . in both of these systems the aerosol is unstable because the suspended particles collide ( leading to aggregation or possibly fragmentation of the aerosol particles ) . the collision processes therefore have significant consequences : the formation of rain in one case , and the widely hypothesised mechanism for the formation of planets in the other . collisions always occur due to molecular diffusion , but ( as pointed out by smoluchowski @xcite ) , macroscopic fluid motion can considerably increase the collision rate . if the suspended particles are sufficiently heavy ( so that their inertia becomes relevant ) , they can move relative to the fluid . in this case the occurrence of caustics will typically increase the collision rate by several orders of magnitude @xcite . if the aerosol particles are sufficiently light , their molecular diffusion can make a significant contribution to the collision rate , which can be estimated by standard kinetic theory . in this paper we are concerned with the effect of macroscopic motion of a fluid on small particles which have insignificant inertia , so that they follow the flow ( advective motion ) . the seminal papers in this area were due to smoluchowski @xcite , who first considered the effect of shear of the fluid flow on collisions , and saffman & turner @xcite , who gave a formula for the collision rate which has formed the basis for most subsequent work on this problem . their paper was motivated by a problem in meteorology argued that small - scale turbulence in convecting clouds can accelerate collisions between microscopic water droplets , thus initiating rain formation : in this case the particles are brought into contact by hyperbolic or shearing motions of the turbulent flow . the formula for the collision rate by saffman & turner @xcite has been used frequently in the past five decades in cloud physics and in chemical engineering problems . it appears to be widely accepted that their expression is an exact relation for the collision rate in a dilute suspension . in the following we show that the saffman - turner estimate describes an initial transient of the problem only . for particles suspended in incompressible flows , the collision rate falls below the initial transient ( which thus constitutes an upper bound ) . for particles advected in a compressible flows , however , homogeneously distributed particles will cluster in a compressible fluid ( see for example @xcite ) . the clustering may increase the collision rate beyond its initial transient . the saffman - turner approximation treats the flow surrounding a test particle as if it were a steady hyperbolic flow , while in reality the flow fluctuates as a function of time . in section [ sec : 3.2 ] below we give an extension of the saffman - turner formula which does give the collision rate exactly . unfortunately the formula contains information about the time - dependence of the flow , and it is impossible to evaluate it in the general case . because of the importance of understanding collision rates for aerosol particles , it is desirable to find exactly solvable cases which can be used as a benchmark for numerical studies . the collision rate must depend on a dimensionless parameter describing how quickly the fluid velocity fluctuates , the kubo number @xcite . we are able to obtain precise asymptotic results on the collision rate in the limit where the kubo number approaches zero . in this case , particle separations undergo a diffusion process . by solving the corresponding fokker - planck equation we can determine the collision rate exactly . the remainder of this paper is organised as follows . in section [ sec : 2 ] we introduce the equations of motion and the dimensionless parameters of the problem . section [ sec : 3 ] discusses the saffman - turner theory and our extension of it . section [ sec : 3.1 ] describes the expression for the collision rate given in @xcite which is the starting point of our discussions . ( some new results on the evaluation of the saffman - turner expression are described in the appendix ) . in section [ sec : 3.2 ] we discuss our exact formula for the collision rate , and explain why the saffman - turner approximation describes an initial transient only . in sections [ sec : 4]-[sec : 6 ] we discuss how exact asymptotic results may be found for the limit of small kubo number . a fokker - planck equation for the probability density of the separation of particles is described in section [ sec : 4 ] . in section [ sec : 5 ] this is used to obtain the steady - state collision rate and section [ sec : 6 ] gives the full time dependence of the collision rate . these results are compared to numerical simulations in section [ sec : 7 ] , which also contains some concluding remarks , discussing scope for further work in this area . finally , we remark that a brief summary of some of the results of this paper has already been published @xcite . here we discuss the problem in greater depth and generality , and derive expressions for the time - dependent collision rate which were not discussed in @xcite . we consider spherical particles of radius @xmath0 in a fluid with velocity field @xmath1 which has an apparently random motion , usually as a result of turbulence . we assume that the suspended particles do not modify the surrounding flow . when the inertia of the particles is negligible , they are advected by the flow : @xmath2 it is assumed that direct interactions between the particles can be neglected until they collide . in other words , the particles follow equation ( [ eq : 2.1 ] ) until their separation falls below @xmath3 . we model the complex flow of a turbulent fluid by a random velocity field @xmath4 . we consider flows in both two and three spatial dimensions and for convenience we use a gaussian distributed field when we carry through concrete computations . in most cases we are concerned with incompressible flow , satisfying @xmath5 . particles floating on the surface of a fluid may experience a partly compressible flow @xcite , as may particles in gases moving with speeds comparable to the speed of sound . for these reasons we also consider partially compressible flows . it is convenient to construct the random velocity field @xmath1 from scalar stream functions or potentials @xcite . in two spatial dimensions we write @xmath6 where @xmath7 is a normalisation factor and @xmath8 and @xmath9 are independent gaussian random functions . we shall use angle brackets to denote averaging throughout . the fields @xmath10 and @xmath11 have zero averages , @xmath12 , @xmath13 and they both have same correlation function , @xmath14 : @xmath15 where @xmath16 and @xmath17 . this two - point correlation function @xmath14 is a smooth function decaying ( sufficiently rapidly ) to zero for large values of @xmath18 and @xmath19 . for @xmath20 , the flow ( [ eq : 2.2 ] ) is incompressible . for finite values of @xmath21 it acquires a compressible component . in the limit of @xmath22 the flow is purely potential . in some cases the physics of a problem dictates that @xmath8 and @xmath9 should have different correlation functions ; many of our results can be generalised in this way . in three spatial dimensions we write @xmath23 where @xmath24 and @xmath9 are four independent scalar fields with zero mean and with the same correlation function @xmath14 ; and @xmath25 is a normalisation factor . in the remainder of this paper we choose the normalisation factors to be of the form @xmath26 where @xmath27 is the standard deviation of the magnitude of the velocity and @xmath28 denotes the second derivative of the correlation function ( [ eq : 2.3 ] ) with respect to its first argument . fully - developed turbulent flows have a power - law spectrum in the inertial range , covering a wide band of wavenumbers @xcite . this feature can be incorporated by giving @xmath14 a suitable algebraic behaviour over a range of values of @xmath18 , as explained in @xcite . the long - ranged behaviour of the velocity field is not , however , relevant to the advective collision mechanism . it therefore suffices to consider a model with a short - ranged velocity correlation : for the numerical work reported in this paper we used following form of the correlation function @xmath29 where @xmath30 is a constant . in our numerical simulations we represent the flow field by its fourier components which are subject to an ornstein - uhlenbeck process as suggested by sigurgeirson & stuart @xcite . lc + & + particle size & @xmath0 + typical velocity fluctuation & @xmath27 + correlation length of the flow & @xmath31 + correlation time of the flow & @xmath32 + number density of particles & @xmath33 + compressibility & @xmath21 + spatial dimension & @xmath34 + + our problem is characterised by the six parameters listed in table [ tab:1 ] . from the parameters in table [ tab:1 ] , three independent dimensionless combinations can be formed : @xmath35 the first parameter characterises the dimensionless speed of the flow and is called kubo number , discussed in @xcite . note that @xmath36 is not possible , because the motion of the fluid places an upper limit on the correlation time . steady , fully developed turbulence corresponds to @xmath37 . the kubo number can be small for randomly stirred fluids . it is only in the limit @xmath38 that we are able to obtain precise and explicit estimates for the collision rate : this case is considered in sections [ sec : 4]-[sec : 6 ] . the second parameter in ( [ eq : 2.7 ] ) is the packing fraction of particles . throughout we assume that this parameter is small . similarly , the third parameter in ( [ eq : 2.7 ] ) is usually taken to be small . we note that kalda @xcite has considered the collision rate for particles in a non - smooth velocity field : this could be relevant to the case where @xmath39 . throughout this paper we consider the rate of collision of a given particle with any other particle , denoting this by @xmath40 . some papers consider the total rate of collision per unit volume . if the spatial density of particles is @xmath33 , the total rate of collision per unit volume is @xmath41 ( the factor of @xmath42 avoids double - counting ) . we will not be concerned with what happens after particles undergo their first collision : in different physical circumstances they may coalesce , scatter , or fragment , but in this paper we are concerned only with their first contact . we assume that the particles are spherical ( or circular , in two - dimensional calculations ) and that they all have the same radius , @xmath0 . we regard the particles as having collided when their separation reaches @xmath3 , and we neglect effects due to the interaction of the particles and the fluid . in practice the fluid trapped between approaching particles may slightly reduce the collision rate @xcite , but this effect can be accounted for by replacing the radius by an effective radius . the problem of calculating the rate of collision therefore reduces to the following problem . we consider a given particle , and transform to a frame where the centre of this particle is at the origin , and the separation of the centre of another particle is denoted by @xmath43 . initially , the reference particle is surrounded by a gas of particles with spatial density @xmath33 . we assume that these are initially randomly distributed , apart from the constraint that none of the particles is in contact with the reference particle . collisions with the test particle occur when other particles come within a radius @xmath3 of the reference particle . the rate of collisions is therefore the rate at which particles cross a sphere of radius @xmath3 centred at the origin in the relative coordinate system . this is obtained by integrating the inward radial velocity over the sphere , and multiplying by the density @xmath33 . this approach gives an expression for the collision rate which we term @xmath44 : @xmath45 where @xmath46 is the radial velocity at spherical coordinate @xmath47 , radius @xmath18 and time @xmath48 . the function @xmath49 is a heaviside step function , which is used to select regions of the surface where the flow is into the sphere of radius @xmath3 . this is the fundamental expression for the collision rate given by saffman & turner @xcite . in section [ sec : 3.2 ] we discuss why this expression is not exact , and give the precise formula . the remainder of this section considers how this expression is evaluated ; some new results are presented in the appendix . the evaluation of ( [ eq : 3.1 ] ) is greatly simplified in the case where the particles are small , in the sense that @xmath50 . in this case the relative velocity is accurately approximated using the velocity gradient of the random velocity field @xmath51 . this approximation was also considered by saffman & turner @xcite . to lowest order in @xmath52 , the relative velocity @xmath53 is approximated as @xmath54 where @xmath55 is the rate of strain matrix of the flow @xmath56 , with elements @xmath57 ( and @xmath58 ) . two particles with radii @xmath59 moving close to each other will thus experience a relative velocity @xmath60 . this is illustrated in figure [ fig : 1 ] for a flow which is hyperbolic in the vicinity of the reference particle . the corresponding relative radial speed @xmath61 is @xmath62 where @xmath63 is the radial unit vector . in particular , at the distance @xmath64 where particles collide , the radial speed is @xmath65 . thus when @xmath50 , equation ( [ eq : 3.1 ] ) reduces to @xmath66 this expression was also obtained in @xcite . the remainder of this subsection is concerned with the evaluation of ( [ eq : 3.2 ] ) . schematic picture of two particles of radius @xmath0 passing each other in a hyperbolic flow . the particle at the origin will see particles move past on hyperbolic trajectories . collisions occur whenever particles approach closer than @xmath3 . the collision rate is thus determined by the influx of particles into a disc of radius @xmath3 ( dashed line ) around the origin.,width=226 ] earlier , smoluchowski @xcite had considered the special case of a fluid flowing with a uniform shear in @xmath67 dimensions : @xmath68 he obtained the collision rate : @xmath69 which is in agreement with the result of evaluating ( [ eq : 3.2 ] ) for this case . for a general strain - rate matrix @xmath70 the evaluation of ( [ eq : 3.2 ] ) is , however , very difficult . in [ sec : appa ] we discuss how this expression is evaluated for a general matrix @xmath71 in two dimensions , and for a general traceless matrix ( representing an incompressible flow ) in three dimensions . in a turbulent flow , saffman & turner @xcite argued that the elements of @xmath71 change as a function of position and one needs to average over the ensemble of strain matrices @xmath70 at different positions in order to estimate the collision rate , so that ( [ eq : 3.2 ] ) is replaced by : @xmath72 at first sight the requirement to average over @xmath71 appears to complicate the problem . however , for a rotationally invariant ensemble of random flows , the problem is considerably simplified by taking the average . in an incompressible flow , for each realisation of @xmath1 , the currents into and out of the collision region ( the disk or sphere of radius @xmath3 ) cancel precisely . therefore ( [ eq : 3.5 ] ) can be written as @xmath73 ) or ( [ eq : 2.4 ] ) , with @xmath74 . this is demonstrated by the following argument ( here we discuss the two - dimensional case ) . if we did not include the factor @xmath75 in ( [ eq : 3.6 ] ) , we would calculate the sum of the collision rate for the flow @xmath51 and for the time - reversed flow @xmath76 . the time reversed flow is generated by reversing the signs of the potentials @xmath9 and @xmath8 in ( [ eq : 2.2 ] ) . the probability density for the gaussian field @xmath77 is the same as for @xmath9 ( and similarly for @xmath8 ) . it follows that the collision rate for the time - reversed flow @xmath76 is the same as for @xmath51 . the expression ( [ eq : 3.6 ] ) is therefore also valid for the cases of compressible flow which we consider in this paper . now using rotational symmetry , one finds the very simple expression @xmath78 where @xmath79 is the the area of the sphere of radius @xmath3 in @xmath34 dimensions ( explicitly @xmath80 and @xmath81 ) . for the gaussian model flow which we consider , equation ( [ eq : 3.7 ] ) gives @xmath82 with @xmath83 the saffman - turner expression for the collision rate , equation ( [ eq : 3.1 ] ) or ( [ eq : 3.7 ] ) , correctly describes the initial collision rate . there are , however , two reasons why the collision rate may approach a significantly different value after an initial transient . the first reason why ( [ eq : 3.1 ] ) may fail arises from the fact that the flow field fluctuates in time . recall that we are concerned with the rate at which pairs of particles collide for the first time if the flow is time - dependent , the relative position coordinate @xmath84 may pass through the sphere of radius @xmath3 more than once . this effect may be accounted for by writing the collision rate in the form @xmath85 as before @xmath86 is the @xmath34-dimensional surface element at @xmath64 and @xmath61 is the radial velocity component ( the relative speed ) and the heaviside step function ensures that only particles entering the sphere contribute to the collision rate . the factor @xmath87 is the density of particles in the neighbourhood of the test particle at time @xmath48 . the function @xmath88 is an indicator function : it is equal to unity if the point reaching the surface element @xmath47 at radius @xmath3 and at time @xmath48 has not previously passed through the sphere of radius @xmath3 , otherwise it is zero . the effect of including the function @xmath88 is illustrated in figure [ fig : 2 ] . the collision rate will reduce below the saffman - turner estimate for times where @xmath88 is no longer unity .. initial positions of particles which have collided with a reference particle at the origin after a certain time ( shaded regions ) . the collision region ( disk of radius @xmath3 ) is bounded by a dashed circle . at short times this region extends along the direction of the stable eigenvector of the initial flow . at intermediate times , we see the effect of this eigenvector having rotated . at long times , the set of colliding initial conditions is stretched and folded . the shaded region then covers most of the surface of the circle , including the region where the radial velocity is inward . the figure shows results of a simulated flow with small kubo number , of the type discussed in section [ sec : 4 ] , at times @xmath89 , @xmath90 and @xmath91 . , width=566 ] in the general case ( [ eq : 3.13 ] ) is difficult to evaluate since the indicator function @xmath88 depends on the history of the flow . if the flow is rapidly fluctuating ( that is , if the kubo number of the flow is small ) , the relative separation of two particles undergoes a diffusion process which makes it possible to exactly evaluate the collision rate for a given ensemble of random flows . a second effect may modify the collision rate from the saffman - turner estimate is that if the flow field is compressible , particles may cluster together , and the particle density in the vicinity of a test particle may be higher than the average density . this effect is expected to cause the collision rate to increase , after an initial transient during which the clustering becomes established . again , this effect is very hard to quantify in the general case , but we can obtain precise asymptotic results in the limit of small kubo number , where a diffusion approximation becomes applicable . because we consider particles which are advected by the flow , there is no clustering effect when the flow is incompressible . we note that for incompressible flow the only correction to the saffman - turner estimate is the occurrence of the factor @xmath88 in ( [ eq : 3.13 ] ) . this implies that for advected particles in an incompressible flow , the saffman - turner estimate of the collision rate is an upper bound . in the limit of a rapidly changing flow , that is when @xmath92 , particle separations @xmath93 undergo a diffusion process ( see @xcite for a review ) . by solving the corresponding fokker - planck equation with the appropriate boundary conditions , we can determine the collision rate exactly in this limit . consider two particles , one at @xmath94 and the other in its vicinity , at @xmath95 . the equation of motion ( [ eq : 2.1 ] ) implies that their separation @xmath96 obeys @xmath97 integrating ( [ eq : 4.1 ] ) over a short time interval @xmath98 we obtain @xmath99 the moments of the components @xmath100 of @xmath101 are : @xmath102 assuming that @xmath103 . we thus obtain the following fokker - planck equation for the density @xmath104 of particle separations @xcite : @xmath105 here @xmath106 is a diffusion matrix with diffusion coefficients @xmath107\,.\end{aligned}\ ] ] in terms of the correlation function @xmath14 introduced in section [ sec : 2 ] we have @xmath108\ , . \end{array}\ ] ] in order to determine the collision rate it is convenient to write the fokker - planck equation in the form of a continuity equation @xmath109 where the components of the probability current can be identified as @xmath110 . the collision rate @xmath111 between particles of radius @xmath0 is given by the rate at which the particle separation @xmath18 decreases below @xmath3 @xcite . this rate equals the radial probability current of particle separations evaluated at @xmath64 , i.e. @xmath112 where @xmath113 is the radial unit vector . we evaluate ( [ eq : 4.8 ] ) in two steps . first ( in section [ sec : 5 ] ) we determine the steady - state collision rate as @xmath114 . second , we determine the full time - dependence of the collision rate ( section [ sec : 6 ] ) , describing how the initial transient discussed in section [ sec : 3 ] approaches the steady state . because of the angular symmetry of the statistics of the fluid velocity , it is convenient to transform ( [ eq : 4.4 ] ) to spherical coordinates . the probability density @xmath115 depends on the separation @xmath18 only and obeys @xmath116 here the radial probability current @xmath117 is given by @xmath118 with @xmath119\,,{\nonumber}\\ g(r)&={\int_{-\infty}^{\infty}{\rm d}t}\big[(d-1+\beta^2)c''(0,t ) -\frac{d-1}{r}c'(r , t)-\beta^2c''(r , t)\big]\,.\end{aligned}\ ] ] consider now the steady - state solution of ( [ eq : 4.4 ] ) , obeying @xmath120 . the steady - state collision rate , denoted by @xmath121 , is determined by ( [ eq : 4.8 ] ) in terms of the steady - state current . the total current @xmath122 entering a sphere of radius @xmath18 and area @xmath123 must be a constant . this gives @xmath124 where @xmath121 is a constant , equal to the collision rate which we wish to determine . the equation of motion ( [ eq : 2.1 ] ) is only physically meaningful when @xmath59 , so that the fluid velocity is approximately the same throughout all of the region occupied by the particle . we will , however , solve the diffusion equation for general @xmath0 . one reason for treating the general case is that it is hard to verify the analytical expressions for the limiting case @xmath125 in numerical investigations . to determine this current , it is necessary to consider the appropriate boundary conditions first , if the particle distribution initially is uniform with density @xmath33 , we expect it to remain so at large separations . thus we use the boundary condition @xmath126 the boundary condition ( [ eq : 5.1 ] ) is implemented as follows . for a specified current density @xmath117 we can solve ( [ eq : 4.10 ] ) by finding an integrating factor @xmath127 : @xmath128 with integration constant @xmath129 and @xmath130 the boundary condition ( [ eq : 5.1 ] ) determines the integration constant in ( [ eq : 5.2 ] ) to be @xmath131 . second , when particles comes closer than the distance of @xmath3 they collide and must be removed . this is taken care of by the following boundary condition at @xmath3 : @xmath132 inserting this boundary condition into ( [ eq : 5.2 ] ) yields @xmath133 using @xmath134 and solving for the constant @xmath121 we find the following expression for the collision rate : @xmath135^{-1}\ ] ] where @xmath136 is the gamma function . equation ( [ eq : 5.6 ] ) is the main result of this section . we continue by discussing a number of limiting cases . for an incompressible flow ( @xmath137 ) , the integrating factor is just unity ( incompressibility @xmath138 implies @xmath139 which in turn implies that the term proportional to @xmath115 in ( [ eq : 4.10 ] ) vanishes ) . in this case , equation ( [ eq : 5.6 ] ) simplifies to @xmath140^{-1}\hspace*{-3mm}\ , . \label{eq : 5.7}\end{aligned}\ ] ] when @xmath142 , the strength of the solenoidal and potential parts of the field are equal . this case may be relevant to the dynamics of particles floating on the surface of a turbulent fluid @xcite . the collision rate can be evaluated exactly in this case , by rewriting ( [ eq : 5.6 ] ) as @xmath143^{-1}\ , . \label{eq : 5.8}\end{aligned}\ ] ] setting @xmath144 we find @xmath145 for @xmath146 . when @xmath147 we obtain @xmath148 what makes it possible to find exact solutions in this case is the fact that when @xmath142 , the functions @xmath149 and @xmath150 in ( [ eq : 4.11 ] ) are related as @xmath151 . note that this relation is always true in one spatial dimension , independent of the value of @xmath21 . note also that when @xmath151 , equation ( [ eq : 4.10 ] ) can be solved for @xmath115 by integration ( starting at e.g. @xmath3 , with @xmath152 ) @xmath153 thus , a non - vanishing steady - state collision rate is obtained only if @xmath154 or when @xmath147 . the collision rate must vanish as the particle size approaches zero . this means that the integral in the denominator in ( [ eq : 5.6 ] ) must diverge for small @xmath0 . thus if @xmath59 is small enough , the major contribution in the integral in ( [ eq : 5.6 ] ) comes from small values of @xmath18 and the relative error will be small if we replace the integrand by its small @xmath18 expansion . it is convenient to change variables according to @xmath155 . we expand @xmath156 and obtain approximately @xmath157 here the parameter @xmath158 is defined as @xmath159 it corresponds to the diffusion constant @xmath160 in equation ( 39 ) of @xcite and in the incompressible case it corresponds to the diffusion constant @xmath158 defined in equation ( 17 ) of @xcite : here @xmath161 is an alternative parametrisation of the compressibility , defined as follows @xmath162 the parameter @xmath161 was also used in @xcite ( and earlier work cited therein ) . it can take values between @xmath163 and @xmath164 . here @xmath165 corresponds to a completely incompressible flow with @xmath137 and @xmath166 corresponds to a purely potential ( @xmath167 ) compressible flow . in particular , when @xmath168 the field strengths of the compressible and incompressible components of the flow are equal , @xmath142 . using equations ( [ eq : 5.13 ] ) , we get the collision rate for small particles @xmath169 this expression was previously derived in @xcite . for an incompressible flow , with @xmath170 , the particle density is uniform and the collision rate is proportional to the packing fraction @xmath171 as expected . by contrast , if the flow is compressible , the particles will cluster on a fractal set @xcite . this is expected to enhance the collision rate because the particle density is large within the clusters . the clustering effect can be characterised by the correlation dimension of the particles , @xmath172 . it is defined by the scaling law @xmath173 , where @xmath174 is the probability that the particle separation is smaller than @xmath175 @xcite . we have @xmath176 and thus @xmath177 . this expression was derived in @xcite . equation ( [ eq : 5.16 ] ) shows that in compressible flows , the collision rate depends upon the correlation dimension @xmath178 . when @xmath179 the collision rate in a compressible flow is therefore much larger than the corresponding rate in an incompressible flow . note finally that the steady - state collision rate ( [ eq : 5.16 ] ) tends to zero when @xmath180 , i.e. when @xmath181 . for values of @xmath21 larger than @xmath182 , no steady - state current satisfying our boundary conditions exists . in one spatial dimension the parameter @xmath182 vanishes : no non - trivial steady state exists because all particle trajectories eventually coalesce ( this effect was termed path - coalescence wilkinson & mehlig @xcite ) . when the correlation function @xmath14 is of the gaussian form ( [ eq : 2.6 ] ) , we obtain from ( [ eq : 5.6 ] ) @xmath183{\nonumber}\\ & \times\exp{\left(}\beta^2{\int_{\bar r}^{\infty}{\rm d}r ' } r ' \frac{2+d - r'^2}{(d-1+\beta^2){\left(}e^{r'^2/2}-1{\right)}+\beta^2r'^2}{\right)}\bigg\}^{-1 } \bigg]^{-1 } \label{eq : 5.18}\end{aligned}\ ] ] where as before @xmath184 . the major contribution to the integral comes from small values of @xmath185 ( except for when @xmath186 and @xmath142 or @xmath187 , when the integral diverges ) and for small values of @xmath188 , the integrand can be expanded in powers of @xmath185 . expanding the exponential function in the integrand yields @xmath189 where the function @xmath127 was defined in ( [ eq : 5.3 ] ) . the diffusion constant @xmath158 is given by ( [ eq : 5.14 ] ) . with ( [ eq : 2.6 ] ) we find @xmath190 for low dimensions , @xmath191 is a fairly good approximation ( a maximum of 3 percent error when @xmath186 ) . substituting ( [ eq : 5.21 ] ) into ( [ eq : 5.19 ] ) gives once more ( [ eq : 5.16 ] ) . so far we have derived the probability densities and collision rates for the steady state . we now wish to derive expressions of these quantities as functions of time . to this end we need to solve the time dependent fokker - planck equation ( [ eq : 4.9 ] ) with the boundary conditions @xmath192 as before we assume a uniform initial scatter of particles @xmath193 the collision rate is given by equation ( [ eq : 4.8 ] ) @xmath194 where we have used that the area of a @xmath34-dimensional sphere of radius @xmath195 is @xmath196 , and the gamma function is denoted by @xmath136 . in order to render the boundary conditions homogeneous , we split @xmath197 and impose the conditions @xmath198 , and that the differential equation homogeneous in @xmath115 is also homogeneous in @xmath199 . thus @xmath200 is uniquely given by the steady - state solutions found in the previous section . now consider the remaining equation for @xmath201 which is identical to ( [ eq : 4.9 ] ) with homogeneous boundary conditions and initial condition @xmath202 separation of variables @xmath203 in ( [ eq : 4.9 ] ) gives ( using the radial current ( [ eq : 4.10 ] ) ) @xmath204{\right)}\ , . \label{eq : 6.5}\ ] ] since the left - hand side depends on @xmath48 only , and the right - hand side depends on @xmath18 only , both sides must be equal to a constant , @xmath205 say . choosing @xmath206 , where @xmath207 is a positive dimensionless constant , we obtain @xmath208 note that the solution corresponding to @xmath209 has already been taken care of in @xmath210 . therefore we can restrict ourselves to considering @xmath211 in the following . to solve the radial part of ( [ eq : 6.5 ] ) , we consider the limit of @xmath212 , where @xmath149 and @xmath150 are simple power laws ( this follows from ( [ eq : 5.13 ] ) ) . using the dimensionless variable @xmath155 gives the following approximate equation for small values of @xmath185 @xmath213 this is an euler equation . its solution is obtained by the variable substitution @xmath214 , where the factor @xmath215 is included in for later convenience and @xmath216 . we find @xmath217 where @xmath218 and @xmath219 and @xmath220 are integration constants . for all values of @xmath221 in the allowed range @xmath222 , the boundary condition @xmath223 is automatically fulfilled . the boundary condition @xmath224 gives @xmath225 , and thus @xmath226 the boundary conditions do not constrain the eigenvalue @xmath207 and we need to consider a continuous superposition of eigenfunctions @xmath227 here @xmath228 and @xmath229 are functions to be determined by the initial conditions . it turns out that it is sufficient to consider @xmath228 for our initial condition and take @xmath230 from now on . at time @xmath231 we make the change of variables @xmath232 as before . we set @xmath233 and find @xmath234 comparing this to the required initial distribution @xmath235 gives @xmath236 the left hand side is of the form of a fourier sine transform with inverse @xmath237 we insert this expression into the ansatz for @xmath199 . upon changing order of the integrations ( that is we perform the @xmath238-integral first ) , we find @xmath239 to calculate the collision rate we need to know the current @xmath240 at radius @xmath3 . to this end we just require the derivative of @xmath115 evaluated at @xmath64 ( since @xmath115 itself is constrained to vanish there ) . we find @xmath241 equation ( [ eq : 6.3 ] ) now allows us to calculate the time - dependent collision rate . we split @xmath240 [ see equation ( [ eq : 4.10 ] ) ] into two parts depending on @xmath199 and @xmath210 respectively . the second part gives the steady - state collision rate @xmath121 , see equation ( [ eq : 5.6 ] ) . we find ( expanding in powers of @xmath185 ) @xmath242 this is our main result for the time - dependent collision rate , valid for correlations with non - vanishing fourth order derivative at @xmath243 and for small particles . an approximation to this result can be obtained by approximating the steady - state probability density @xmath244 by expanding in powers of @xmath185 . since the @xmath245-integral extends to infinity and since the gaussian contribution in the integrand becomes small for large values of @xmath246 , we must have that @xmath247 goes to @xmath33 as @xmath245 goes to @xmath248 , which is the case for the original @xmath247 . to accomplish this , we match the small-@xmath185 expression of @xmath247 at @xmath249 to the large-@xmath185 expression . by using ( [ eq : 5.13 ] ) and ( [ eq : 5.16 ] ) and matching at @xmath249 we find @xmath250 where the last term is not cut off at @xmath249 as it vanishes sufficiently quickly . the time - dependent collision rate can now be evaluated from the sum of three integrals of the type @xmath251\end{aligned}\ ] ] where @xmath252 , @xmath253 and @xmath254 . putting everything together we find @xmath255{\right)}{\nonumber}\\ & + \bar a^{d}\bigg ( \frac{1}{\sqrt{\pi{{\cal d}}t}}e^{-\mu_0 ^ 2{{\cal d}}t}+(d+\mu_0)e^{d(d+2\mu_0){{\cal d}}t}{\nonumber}\\ & \hspace*{1mm}\times{\left\ { } \newcommand{\rbp}{\right\}}\mathrm{erf}\left[(d+\mu_0)\sqrt{{{\cal d}}t}\right]-\mathrm{erf}\left[(d+\mu_0)\sqrt{{{\cal d}}t}+\frac{\ln\bar a}{2\sqrt{{{\cal d}}t}}\right]\rbp\bigg ) \bigg\}\end{aligned}\ ] ] valid for @xmath256 . for times less than @xmath257^{-1}$ ] the above expression simplifies to @xmath258 for intermediate times , @xmath257^{-1 } \ll t \ll \ln(\bar a)/(2\mu_0{{\cal d}})$ ] , we find approximately @xmath259-\mathrm{erf}\left[(d+\mu_0)\sqrt{{{\cal d}}t}+\frac{\ln\bar a}{2\sqrt{{{\cal d}}t}}\right]\rbp\bigg)\,,\end{aligned}\ ] ] while for large times , @xmath260 , where @xmath261{\right)}\,.\end{aligned}\ ] ] consider finally evaluating ( [ eq : 6.22 ] ) in the incompressible limit , we have @xmath262 and the terms involving the integral cutoffs @xmath263 cancel ( @xmath264 in ( [ eq : 6.21 ] ) ) . thus in the case @xmath137 the collision rate simplifies to @xmath265\bigg\}.\ ] ] equations ( [ eq : 6.22 ] ) and ( [ eq : 6.26 ] ) are compared to results of numerical simulations in the following section . we performed simulations of the collision rate as a function of time at small kubo numbers , for both incompressible and compressible flows . these illustrate ( figure [ fig : 3 ] ) the very complex behaviour of the model . for example , in the case of compressible flows the collision rate at first decreases below the value determined by the saffman - turner approximation due to the effect illustrated in figure [ fig : 2 ] , before rising again as particle clustering becomes apparent . collision rate for particles advected in a two - dimensional flow of the form ( [ eq : 2.2 ] ) with ( [ eq : 2.6 ] ) and @xmath266 . particles are initially randomly distributed , initially overlapping particle pairs are removed , as are particles which have collided . parameters : @xmath267 , @xmath268 , @xmath269 and @xmath270 for all graphs and for * a * @xmath137 and @xmath271 , * b * @xmath137 and @xmath272 , * c * @xmath273 and @xmath271 , * d * @xmath273 and @xmath272 , * e * @xmath274 and @xmath272 , * f * @xmath275 and @xmath272 . the collision rate is approximated by the cumulative sum of all collisions up to @xmath48 , divided by @xmath48 . the light blue areas are the intervals @xmath276 , where @xmath277 is the standard deviation of the rate , and @xmath278 is the number of realisations of the flow . the latter decreases over time ( it is typically @xmath279 for the first two decades in @xmath48 , and @xmath280 for the last decade ) . the saffman - turner estimates ( [ eq : 3.8],[eq : 3.9 ] ) are shown as red dashed lines . our own result ( [ eq : 6.22 ] ) is shown as red solid lines . to correspond to the simulated collision rate , the plotted theoretical collision rates has been integrated to and then divided by @xmath48 . due to the assumptions in the fokker - planck theory , the long time rate ( [ eq : 6.22 ] ) is not valid for @xmath281 . the plotted long time theory is a combination of the short time rate ( [ eq : 7.1 ] ) and the long time rate ( [ eq : 6.22 ] ) , matched at the time at which the long time rate drops below the short time rate ( which is an upper bound ) . , width=453 ] figure [ fig : 3 ] summarises our results for the collision rate of particles advected in flows with small kubo numbers . shown are numerical simulations for a two - dimensional random flow of the form ( [ eq : 2.2 ] ) with correlation function ( [ eq : 2.6 ] ) . panels * a * and * b * show the collision rate in an incompressible flow ( @xmath137 ) . as expected ( see section [ sec : 3.2 ] ) , the collision rates drops below the initial transient given by the saffman - turner approximation , ( [ eq : 3.8],[eq : 3.9 ] ) with @xmath20 . for the particular choice ( [ eq : 2.6 ] ) with @xmath266 , equations ( [ eq : 3.8],[eq : 3.9 ] ) become in two spatial dimensions @xmath282 also shown is our own theory valid for small kubo numbers ( equation ( [ eq : 6.22 ] ) in section [ sec : 6 ] ) . the agreement between the theory and the simulations is good in all cases , but slightly better for the smaller kubo number ( panel * b * ) . as discussed in section [ sec : 3.2 ] , the initial transient constitutes an upper bound to the collision rate . panels * c * to * f * in figure [ fig : 3 ] show the collision rate in compressible flow . now , the saffman & turner theory is no longer an upper bound ( see for example figure [ fig : 3]*f * ) , because initially homogeneously distributed particles in an incompressible flow cluster together . the corresponding density fluctuations increase the collision rate , as our exact result shows ( equation ( [ eq : 6.22 ] ) in section [ sec : 6 ] ) . again we observe good agreement between the simulations and our analytical result . in this paper we have concentrated upon the solvable case of advective collisions in flows with small kubo number , which provides considerable physical insight . we conclude by commenting on the relation between these results and collisions in a turbulent flow field which satisfies the navier - stokes equation . the standard approach , based upon the saffman - turner formula , predicts a collision rate @xmath283 , where @xmath32 is the kolmogorov timescale @xcite of the turbulent flow . the calculation based upon the diffusion equation gives a collision rate @xmath284 . given that @xmath285 for a turbulent flow , we see that the saffman - turner and diffusive expressions are of the same order . these observations are consistent with the hypthesis that the collision rate for small particles in a turbulent flow is @xmath286 where @xmath287 is the rate of dissipation per unit mass , @xmath288 is the kinematic viscosity , and @xmath289 is a universal constant ( depending only on the dimension ) . it would be a valuable addition to the literature on aerosols and suspended particles to determine the value of @xmath290 from numerical simulations using a navier - stokes flow . _ acknowledgments . _ we acknowledge support from vetenskapsrdet and from the research initiative nanoparticles in an interactive environment at gteborg university . + r. a. shaw , _ annu . fluid mech . _ , * 35 * , 183 , ( 2003 ) . m. wilkinson , b. mehlig , and v. uski , _ astrophys . j. suppl . _ , in press , ( 2008 ) . m. v. smoluchowski , _ zeitschrift f. physik . * xcii * , 129 - 168 , ( 1917 ) ; see eq . ( 28 ) on p. 156 . p. g. saffman and j. s. turner , _ j. fluid mech . _ , * 1 * , 16 - 30 , ( 1956 ) . g. falkovich , a. fouxon and g. stepanov , _ nature _ , * 419 * , 151 - 154 , ( 2002 ) . m. wilkinson , b. mehlig and v. bezuglyy , _ phys . _ , * 97 * , 048501 , ( 2006 ) . e. balkovsky , g. falkovich , and a. fouxon , _ phys . _ , * 86 * , 2790 , ( 2001 ) ; cond - mat/9912027 . g. falkovich , k. gawedzki , and m. vergassola , _ rev . * 73 * , 913 , ( 2001 ) . m. wilkinson , b. mehlig , s. stlund and k. p. duncan , _ phys . fluids _ , * 19 * , 113303 , ( 2007 ) . b. andersson , k. gustavsson , b. mehlig , and m. wilkinson , _ europhys _ , * 80 * , 69001 , ( 2007 ) . j. r. cressman , j. davoudi , w. i. goldberg and j. schumacher , _ new j. phys . _ , * 6 * , 53 , ( 2004 ) . falkovich _ et al . _ , _ nature _ , * 435 * , 1045 , ( 2005 ) . u. frisch , _ turbulence _ , cambridge university press , ( 1997 ) . h. sigurgeirson and a. m. stuart , _ phys . fluids _ , * 14 * , 4352 , ( 2002 ) . j. kalda , _ phys . _ , * 98 * , 064501 , ( 2007 ) . j. sommerer and e. ott , _ science _ , * 359 * , 334 , ( 1993 ) . p. grassberger and i. procaccia , _ physica d _ , * 9 * , 189 , ( 1983 ) . m. wilkinson and b. mehlig , _ phys . e _ , * 68 * , 040101(r ) , ( 2003 ) . in this appendix we show how to evaluate the expression ( [ eq : 3.2 ] ) for a general matrix @xmath55 with elements @xmath291 ( we drop the subscript @xmath292 ) . it is convenient to decompose @xmath55 into a symmetric part @xmath293 and an antisymmetric part @xmath294}$ ] @xmath295 the collision rate is independent of the antisymmetric part @xmath294}$ ] ( because rotations do not contribute to the collision rate ) . the symmetric part @xmath296 can be diagonalised by an orthogonal transformation , @xmath297 , where @xmath298 is diagonal with the eigenvalues @xmath299 of @xmath296 . in evaluating ( [ eq : 3.2 ] ) we may write @xmath300 , where @xmath301 . since @xmath302 is just a rotation of @xmath63 which is integrated over all directions , we can replace ( [ eq : 3.2 ] ) by @xmath303 we now show how to perform the integral in ( [ eq : nsn ] ) in two spatial dimensions . note that if the flow determined by @xmath55 is not area preserving , the particle density will change as a function of time . in this case @xmath33 in ( [ eq : 3.2 ] ) must be replaced by @xmath304\,.\ ] ] at short times we approximate @xmath305 and find @xmath306 where @xmath307 and @xmath308 . this is the final result in two spatial dimensions , expressed in terms of the eigenvalues @xmath309 of the symmetric part @xmath296 of the strain matrix . in three spatial dimensions we only consider incompressible shear flows , that is a general , three - dimensional traceless matrix @xmath296 . it has eigenvalues @xmath310 , obeying the relations @xmath311 , @xmath312 and @xmath313 . we use a spherical coordinate system , @xmath314 and find @xmath315 substitute @xmath316 and @xmath317 we obtain @xmath318 where the transformed integrand @xmath319 is smaller than @xmath292 for @xmath320 , where @xmath321 is given by @xmath322 and @xmath323 . because the values @xmath324 can take are limited by @xmath325 , @xmath326 must be in the range @xmath327 and @xmath321 in the range @xmath328 . performing the @xmath48-integral from @xmath292 to @xmath321 gives @xmath329 where @xmath330 , if @xmath331 and @xmath332 , if @xmath333 . expanding the integrand around in @xmath326 around the point @xmath334 ( to be determined below ) we find @xmath335 if we choose @xmath336 , which corresponds to @xmath337 , we can perform the integration after changing the order of summation and integration . the collision rate becomes @xmath338 only even powers contribute to this sum : we have not used that @xmath339 and could thus as well have expanded the starting equations using @xmath340 and @xmath325 , with the only difference that @xmath324 would be replaced by @xmath341 in ( [ eq : a.11 ] ) . we thus find that @xmath342 is symmetric around @xmath343 , i.e. @xmath344 . to obtain an expression valid for @xmath331 , we reorder the eigenvalues of the matrix @xmath296 as @xmath345 and @xmath346 , giving the eigenvalue ranges @xmath347 and @xmath348 . the integrand analogous to ( [ eq : a.5 ] ) becomes @xmath349 where @xmath340 has opposite sign as @xmath325 before . this integrand is smaller than @xmath292 for @xmath350 , where @xmath321 is given by @xmath351 where @xmath352 lies in the interval @xmath353 and @xmath321 lies in @xmath354 . performing the @xmath48 integral from @xmath321 to @xmath355 gives @xmath356 where we have proceeded as in deriving ( [ eq : a.12 ] ) with @xmath357 , if @xmath358 and @xmath332 , if @xmath359 . we obtain the final result @xmath360 where @xmath361 this is the final result for incompressible flows in three spatial dimensions . for the particular case ( [ eq : 3.3 ] ) considered by smoluchowski @xcite we have @xmath362 with eigenvalues @xmath363 . substituting @xmath364 and @xmath365 into ( [ eq : a.15 ] ) we obtain ( [ eq : 3.4 ] ) . when @xmath366 this expression agrees with the classical result ( [ eq : 3.4 ] ) due to smoluchowski @xcite .

a central issue in the field of spintronics is the design of spin - based electronic devices.@xcite they may involve ferromagnets or external magnetic fields to control the spin degree of freedom.@xcite but recently , all - electric spintronic devices also have gained interest.@xcite they rely on spin - orbit ( so ) interaction , the strength of which is tunable via external gates in semiconductor heterostructures,@xcite a basic requirement for the realization of a spin field - effect transistor . a spin - polarized current in a semiconductor can be generated by spin injection . here we focus on an alternative route that relies on pumping . by varying the parameters of a mesoscopic system periodically in time , a finite charge or spin current can be sustained . experimental studies have investigated charge pumping in several mesoscopic devices.@xcite spin pumping has been experimentally realized in the presence of an external magnetic field.@xcite theoretical studies of spin pumping involve external magnetic fields,@xcite ferromagnetic leads,@xcite and also so coupling.@xcite in the present paper , we consider the minimal model that contains so interaction : a quantum dot with two spin - degenerate orbital levels . such a two - level quantum dot with more than two leads has been suggested as a spin filter.@xcite we focus on the _ adiabatic _ limit of pumping , i.e. , the parameters are varied slowly in time compared to the dwell time of the mesoscopic system.@xcite adiabatic pumping of charge and spin through such a two - level dot has been considered in the limit of vanishing charging energy.@xcite it was found that this system can act as an _ all - electric spin battery _ , i.e. , a finite spin current can be achieved without ferromagnets by electrically controlling the dot parameters . for specific symmetries in the tunnel coupling of the dot to the leads even pure spin currents have been suggested . from the analysis of ref . , which was based on a scattering - matrix approach,@xcite it is not clear whether and how the conclusions can be transferred to quantum dots with non vanishing coulomb interaction . to answer this question is the main goal of the present paper . in order to take the coulomb interaction into account , we use a diagrammatic real - time approach@xcite that allows for arbitrary strengths of the coulomb interaction . we focus on the limit of weak tunnel coupling , for which we perform a systematic perturbation expansion to lowest order . to emphasize the role of coulomb interaction , we compare the limit of vanishing coulomb interaction with the limit of an infinitely large charging energy . the paper is organized as follows . in sec . [ model ] we introduce the model that describes the so interaction in a two - level quantum dot with coulomb interaction . section [ method ] deals with the technique to calculate the pumped charge and pumped spin during one pumping cycle . to study the dependence of the pumped charge ( spin ) on the four tunnel - matrix elements in a transparent way , we introduce in sec . [ isospin ] an isospin representation of the orbital degree of freedom . finally , in sec . [ res ] we present the results for the pumped charge and pumped spin . ( color online ) energy scheme of the two - level quantum dot . the two orbital , spin - degenerate levels can be varied in time . they are tunnel coupled to the left ( l ) and the right ( r ) lead , with tunnel - matrix elements @xmath0 . the leads have the same chemical potential @xmath1 . ] we consider a quantum dot with two spin - degenerate orbital levels @xmath2 ( with labels @xmath3 for the orbital and @xmath4 for the spin ) , tunnel coupled to the left ( l ) and the right ( r ) lead ( see fig . [ fig : model ] ) . the system is described by the hamiltonian @xmath5 here , @xmath6 is the hamiltonian of the isolated dot , @xmath7 of the leads , and @xmath8 of the tunneling between dot and leads . the hamiltonian for the isolated quantum dot contains two parts . the single - particle contribution for the two orbitals @xmath9 with energy @xmath10 , which are coupled by so interaction , can be cast in the @xmath11 matrix @xmath12 for the basis @xmath13 , where the spin quantization axis is chosen arbitrarily . here , @xmath14 denotes the vector of pauli matrices , @xmath15 is the identity matrix , and @xmath16 is a real vector describing the so coupling . the matrix in eq . has the most general form that allows time - reversal symmetry . it has been used in the context of pumping@xcite and was also recently applied to electron - transport in the presence of a magnetic field@xcite and to study the josephson current through a double - dot structure.@xcite in the following , we choose the spin quantization axis parallel to @xmath16 so the matrix becomes diagonal in spin space . the second part of the dot hamiltonian accounts for the charging energy @xmath17 , where @xmath18 is the total number of dot electrons and @xmath19 an external gate charge . without loss of generality , we can choose @xmath20 ( any other value can be achieved by a constant shift of the energies @xmath10 ) . this leads ( up to an additive constant ) to the dot hamiltonian @xmath21 where the operator @xmath22 creates an electron in state @xmath2 and the corresponding number operator is @xmath23 . we used the notation @xmath24 for spin parallel ( antiparallel ) to @xmath16 , @xmath25 , and @xmath26 . the leads are modeled as reservoirs of noninteracting electrons , @xmath27 where @xmath28 is the creation operator for an electron with spin @xmath29 and momentum @xmath30 in lead @xmath31 . tunneling between dot and leads is described by the hamiltonian @xmath32 with ( spin - independent ) tunnel - matrix elements @xmath0 for tunneling between lead @xmath31 and orbital @xmath9 . pumping is achieved by varying system parameters periodically in time . in this paper , we assume that the energy levels @xmath33 can be changed in time via external gates capacitively coupled to the system . in principle , the external gates also may affect the so coupling , the tunnel couplings , and the electro chemical potential of the leads ( via parasitic capacitances ) . to simplify the discussion , however , we assume for the following these parameters to be constant in time . we focus on the regime of adiabatic pumping , which is achieved for pumping frequencies @xmath34 smaller than the inverse of the dwell time . this is valid for @xmath35 , where @xmath36 is the tunnel - coupling strength , @xmath37 , with @xmath38 . the density of states @xmath39 is assumed to be flat and equal for the left and right leads . we choose a gauge where all four tunnel - matrix elements are real . to study the effect of coulomb interaction , we compare results for the limit of noninteracting ( @xmath40 ) and infinitely strong interacting ( @xmath41 ) electrons on the dot . in the latter case , the total number of electrons in the quantum dot can only be zero or 1 . to calculate the pumped charge and pumped spin , we use a diagrammatic real - time approach to adiabatic pumping through quantum - dot systems . @xcite for the present context , we extend the analysis of ref . to allow for a time - dependent transformation of the basis states . this is necessary since the so coupling couples time - dependent orbital levels , which , in turn , makes the dot eigenstates time dependent . we start in sec . [ density ] with the kinetic equation for the reduced density matrix in its general form , which describes the time evolution of the dot s degrees of freedom . subsequently , we perform both an adiabatic expansion , i.e. , a perturbation expansion in the pumping frequency ( sec . [ adiabatic ] ) and a perturbation expansion in the tunnel - coupling strength ( sec . [ tunnel ] ) to describe the limit of weak tunnel coupling . the pumped charge and pumped spin currents to lowest order in @xmath36 and @xmath34 are derived in sec . [ pumped ] . finally , in sec . [ weak ] , we perform the limit of weak pumping which assumes small amplitudes of the pumping parameters . the main idea of the diagrammatic real - time technique is based on the fact that the leads are described as large reservoirs of noninteracting electrons which can be integrated out in order to arrive at a reduced density matrix @xmath42 for the dot degrees of freedom only . for a matrix representation with matrix elements @xmath43 ( for the diagonal elements we introduce the notation @xmath44 ) , it is convenient to use the eigenstates @xmath45 with corresponding eigenenergies @xmath46 as a basis . for this , we employ a time - dependent unitary transformation @xmath47 acting on the dot hamiltonian @xmath6 , such that @xmath48 is diagonal . the time evolution of the reduced density matrix is given by the kinetic equation @xmath49\nonumber\\&+ \int^{t}_{-\infty}{\ : \mathrm{d}t^\prime \ : } { \bm{\mathcal{w}}}(t , t^\prime)\ \bm{p}(t^\prime)\ , . \label{master}\end{aligned}\ ] ] the bold face indicates tensor notation . the reduced density matrix @xmath42 and @xmath50 are tensors of rank 2 , while @xmath51 and @xmath52 are tensors of rank 4 , i.e. , @xmath53 the kernel element @xmath54 describes the transition from @xmath55 at time @xmath56 to @xmath57 at time @xmath58 . it is given by the sum over all irreducible blocks on the keldysh contour which correspond to the described transition . the elements of @xmath51 are differences of the eigenenergies defined as @xmath59 . the second term @xmath60 $ ] originates from the time dependence of the transformation @xmath61 , and @xmath62 denotes the time - derivative of @xmath61 . in the following adiabatic expansion and the expansion in the tunnel - coupling strength , we follow the lines of ref . . in the limit of slow variation of the system parameters , such that the duration of one pumping cycle , @xmath63 , is much larger than the dwell time of an electron in the quantum dot , we can perform an adiabatic expansion of eq . , which is equivalent to an expansion of all time dependencies around the final time @xmath58 and to systematically keep all contributions that contain one time derivative . for this , we first do a taylor expansion of the reduced density matrix around the finite time @xmath58 , i.e. , @xmath64 . we then expand the kernel and the density matrix in the pumping frequency , i.e. , @xmath65 and @xmath66 . the instantaneous order , indicated by the index @xmath67 , describes the limit where all system parameters are frozen at time @xmath58 . the adiabatic correction , labeled by @xmath68 , contains one time derivative , i.e. , it collects all contributions to first order in the pumping frequency @xmath34 . the difference in the eigenenergies of the isolated dot , @xmath69 , is of instantaneous order , while @xmath70 belongs to the adiabatic correction . since both @xmath71 and @xmath72 depend only on the difference @xmath73 , it is convenient to perform the laplace transform @xmath74 . using the short notations @xmath75 and @xmath76 , the kinetic equation reads @xmath77 in instantaneous order , and @xmath78\nonumber\\&+{\bm{\mathcal{w}}}^{(a)}_t \bm{p}^{(i)}_t+\partial{\bm{\mathcal{w}}}^{(i)}_t{\ : \frac{\mathrm{d}}{\mathrm{d}t}}\bm{p}^{(i)}_t \label{meqa}\end{aligned}\ ] ] for the adiabatic correction . the normalization condition for the density matrix is expressed as @xmath79 and @xmath80 . in addition to the adiabatic expansion , we perform a perturbation expansion in the tunnel - coupling strength @xmath36 . for a systematic expansion of the kinetic equations , we need to analyze the term @xmath81 . it vanishes for all diagonal matrix elements of @xmath82 . the off - diagonal matrix elements , associated with coherent superpositions , are only nonzero when the superposition is not forbidden by conserved quantum numbers and when the energy difference of the corresponding states is smaller or of the order of @xmath36 . therefore , we count all contributing matrix elements of @xmath51 to be of the order of @xmath36 . the expansion of the kernels , @xmath83 , starts to first order in @xmath36 and the instantaneous order of the reduced density matrix to zeroth order , @xmath84 . to properly match the powers of @xmath36 in eq . , the adiabatic correction of the reduced density matrix , @xmath85 , has to start to minus first order.@xcite in the following , we consider the limit of weak tunnel coupling , @xmath86 , for which we restrict ourselves to the lowest - order contributions in @xmath36 . the instantaneous part of the kinetic equation starts to first order in @xmath36 , @xmath87 with normalization @xmath88 . for the adiabatic correction , the expansion of eq . ( [ meqa ] ) to lowest ( zeroth ) order in @xmath36 yields @xmath89 with @xmath90 . all other terms appearing on the right - hand side of eq . ( [ meqa ] ) are of higher order in @xmath36 and drop out . this is immediately obvious for the last two terms in eq . ( [ meqa ] ) . but also @xmath91 $ ] drops out in the absence of any bias voltage . in this case , @xmath92 is given by the equilibrium distribution , which is diagonal with matrix elements being determined by boltzmann factors . since energy differences , @xmath51 , of states for which coherent superpositions are allowed are of the order of @xmath36 , the difference of the corresponding occupation probabilities for these states is also of the order of @xmath36 and , therefore , vanishes in the perturbation expansion . this means that the matrix elements @xmath93\right)^{\chi_1}_{\chi_2}=\left(p^{(i,0)}_{t\ \chi_2}-p^{(i,0)}_{t\ \chi_1}\right)\left(\bm{t}^\dagger\dot{\bm{t}}\right)^{\chi_1}_{\chi_2}\end{aligned}\ ] ] vanish for all combinations of @xmath94 and @xmath95 which are needed in the kinetic equation . the pumped current and the pumped spin current from the dot into the left lead are given by @xmath96},\\ s_\text{l}(t)&= \frac{\hbar}{2 } \int_{-\infty}^t{\ : \mathrm{d}t^\prime \ : } { \ : \mathrm{tr}\ ! \left [ { \bm{\mathcal{w}}}^{\text{l}}_s(t , t{^\prime})\:\bm{p}(t{^\prime } ) \right]},\end{aligned}\ ] ] respectively . here , we have introduced @xmath97 where @xmath98 only contains those diagrams of @xmath99 in which @xmath100 electrons with spin @xmath29 enter the left lead , i.e. , in which the number of lines for lead @xmath101 and spin @xmath29 going from the upper to the lower contour minus the number of lines from the lower to the upper contour is @xmath100 . analogously to the expansion of the kinetic equation , we perform the adiabatic expansion and the perturbation expansion in the tunnel - coupling strength for the pumped charge and spin current . to lowest order we get @xmath102 } , \label{eq : current1}\\ s^{(a,0)}_\text{l}(t)&= \frac{\hbar}{2 } { \ : \mathrm{tr}\ ! \left [ \left({\bm{\mathcal{w}}}^{\text{l}}_{s , t}\right)^{(i,1)}\:\bm{p}_t^{(a,-1 ) } \right]}.\label{eq : current2}\end{aligned}\ ] ] the pumped charge and the pumped spin per pumping cycle is obtained by integration , @xmath103 and @xmath104 . the diagrammatic rules to calculate analytically @xmath105 can be found in appendix [ rules ] . @xcite after having determined @xmath105 , we obtain the adiabatic correction to the reduced density matrix , @xmath106 , by solving the kinetic equations and . those , then , enter eqs . and for the pumped charge and spin currents . we split the energy of the orbital levels into the time - averaged part , @xmath107 , and the deviation @xmath108 : @xmath109 in the limit of weak pumping , the time - dependent part of the pumping parameters is small compared to other energy scales of the system such as tunnel - coupling strength and temperature , @xmath110 . hence , we can expand the pumped charge , @xmath111 , and pumped spin , @xmath112 , to lowest ( bilinear ) order in @xmath113 and @xmath114 . for adiabatic pumping , the pumped charge ( spin ) is proportional to the area enclosed by the path of @xmath115 in the parameter space during one pumping cycle . therefore , a phase difference is necessary to gain finite pumped charge ( spin ) . the enclosed area is given by @xmath116 . all results in sec . [ res ] are calculated in the weak - pumping limit . for each matrix element @xmath117 that needs to be considered ( all diagonal ones and those off - diagonal ones that describe possible coherent superpositions ) , there is one kinetic equation . it is often convenient to transform the reduced density matrix such that only linear combinations of the @xmath117 s appear , which allows for a straightforward physical interpretation . in the context of spin transport through a single - level quantum dot with ferromagnetic leads , it is advantageous to formulate the kinetic equations separately for the occupation probability of zero , one or two electrons on the quantum dot , and the three components of the spin on the dot . @xcite the vector character of the spin accounts for both a spin imbalance along a given axis and the coherent dynamics of the accumulated spin . one virtue of such a transformation lies in the fact that it is possible to write the kinetic equation in a ( spin-)coordinate - free form , which does not depend on the choice of the spin quantization axis . a similar transformation can also be used for the orbital degree of freedom in systems in which coherent superpositions of the occupation of different orbitals appear . these superpositions are conveniently described by defining an isospin . this has been done before for several double - dot systems . @xcite we will introduce such an isospin description now for the system under consideration . in this paper , we focus on the limits of @xmath40 and @xmath41 . in the first case , @xmath40 , the hilbert space is @xmath118-dimensional , i.e. , the reduced density matrix is a @xmath119 matrix . however , since we choose the spin - quantization axes along the direction of the so field , the hamiltonian divides into two independent spin channels . as a result , the reduced density matrix can be written as a direct product of the @xmath120 density matrices for spin up and spin down , @xmath121 . in the basis @xmath122 that corresponds , for each spin @xmath29 , to the occupation of none of the orbitals , of orbital 1 , of orbital 2 , and of both orbitals , respectively , the reduced density matrix reads @xmath123 note that , in order to keep the notation simple , we put the index @xmath29 only once at the matrix indicating the @xmath29-dependence of each of the matrix elements . for @xmath41 the dot is either singly occupied or empty , i.e. , the hilbert space is five - dimensional . the reduced density matrix takes the form @xmath124 note that , here , @xmath125 is the probability that the dot is not occupied with either spin , while for @xmath40 we used @xmath126 for the probability that the dot is not occupied with spin @xmath29 , irrespective of the occupation of spin @xmath127 . to describe the coherent superposition associated with the off - diagonal matrix elements , it is convenient to introduce , for each physical spin , an isospin operator @xmath128 with quantum - statistical expectation value @xmath129 . choosing the coordinate system for the isospin such that @xmath130 and @xmath131 are the eigenstates of @xmath132 , we get @xmath133 , @xmath134 , and @xmath135 . since ultimately we aim at a coordinate - free form of the kinetic equations , we abbreviate the @xmath136-axis chosen above by the normalized vector @xmath137 , i.e. , @xmath130 and @xmath131 are the eigenstates of @xmath138 . the isospin direction @xmath137 characterizes the eigenstates of the isolated quantum dot in the absence of so coupling . the so coupling , however , couples the two orbitals . as a consequence , the dot eigenstates @xmath139 for single occupation with spin @xmath29 are linear combinations of the two orbitals @xmath130 and @xmath131 , given by the transformation @xmath140 the corresponding eigenenergies are @xmath141 , with the mean dot level @xmath142 and @xmath143 . the transformation depends on the spin @xmath29 , the level spacing of the both orbitals , @xmath144 , and the strength of the so coupling , @xmath145 . we consider the regime where @xmath145 and @xmath146 are of order @xmath36 . therefore , the level spacing @xmath147 of the eigenenergies @xmath148 is also of order @xmath36 . as we pump on both energies , @xmath113 and @xmath114 , the transformation @xmath149 and the eigenenergies @xmath150 are time dependent . the unitary transformation @xmath151 corresponds to a rotation about the @xmath152-axis with the spin - dependent angle @xmath153 in isospin space . this means that the dot eigenstates @xmath154 are eigenstates to the isospin projection @xmath155 along the direction @xmath156 that is obtained from @xmath137 by the above mentioned rotation ( see fig . [ fig : mn ] ) . the tunneling hamiltonian couples the lead - electron states to both orbitals , i.e. , to a linear combination of @xmath130 and @xmath131 . to diagonalize the tunneling from and to lead @xmath31 , we employ the unitary transformation @xmath157 in isospin space , this transformation corresponds to a rotation about the @xmath158-axis with angle @xmath159 applying this rotation on @xmath137 generates the direction @xmath160 ( see fig . [ fig : mn ] ) which has the following physical interpretation : only dot electrons with @xmath161 isospin projection along @xmath162 couple to reservoir @xmath31 , while the @xmath163 isospin projection is decoupled from the lead.@xcite therefore , in a ferromagnetic analogy , the leads are full isospin polarized with polarization along @xmath160 . ( color online ) scheme of different relevant isospin quantization axes . the vector @xmath137 represents the quantization where the orbital levels @xmath130 and @xmath131 are the eigenstates of the @xmath164 operator of the isospin . the two axes @xmath165 are the quantization axes where the eigenstates of @xmath164 are the eigenstates of @xmath166 for single occupation . in a ferromagnetic analogy , the leads are fully isospin polarized along the axes @xmath160 . ] first , we write the kinetic equations and in the basis @xmath167 for the @xmath40 limit and @xmath168 for @xmath41 . those kinetic equations are treated perturbatively to first order in the tunnel - coupling strength @xmath36 . as described above , we count both @xmath145 and @xmath146 as one order in @xmath36 . the elements of the kernel @xmath105 are calculated by the rules in appendix [ rules ] . including the isospin in the formulation of the kinetic equation , the system is fully described by the occupation probabilities of the dot and the expectation values of the isospins . in particular , we perform the transformation from @xmath169 to @xmath170 in the limit of vanishing coulomb interaction . the probabilities describing the occupation of the dot with spin @xmath29 are @xmath171 for empty , @xmath125 , single , @xmath172 , and double occupation , @xmath173 . in the limit of strong coulomb interaction , @xmath41 , the transformation reads @xmath174 to @xmath175 . the relevant occupation probabilities are @xmath176 with @xmath177 being the possibility that the dot is occupied by a single electron with spin @xmath29 . we identify in the resulting kinetic equations the vectors @xmath160 and @xmath178 and get , thus , a representation that is independent of the choice of basis . in the limit of @xmath40 , we get [ eq : dia : mequ0 ] @xmath179 where @xmath180 is the fermi function at energy @xmath181 . as the difference of the eigenenergies , @xmath147 , is of order @xmath36 , we have to drop @xmath147 in terms which are already linear in @xmath36 . therefore , the fermi function , @xmath182 , depends here only on the mean level position , @xmath181 , since every term which includes the fermi function is linear in @xmath36 . in the equations for the probabilities , the isospin projections along the directions defined by the leads enter in the weighted average @xmath183 with @xmath184 . the isospin projection direction given by the so coupling , on the other hand , gives rise to a precession term about the effective field @xmath185 in the equation for the isospin . this effective field is the only place where the so coupling enters the kinetic equations . equations and represent both the instantaneous order and the adiabatic correction of the kinetic equation . for the first case , one needs to set the left - hand side to zero and add the index @xmath186 to the isospin and the occupation probabilities on the right - hand side . ( note that the instantaneous part of the isospin vanishes in lowest order in @xmath36 , @xmath187 . ) for the the second case , we need to add the index @xmath186 on the left - hand side and @xmath188 on the right - hand side , respectively . in the limit of strong coulomb interaction , @xmath41 , the kinetic equations read [ eq : dia : mequi ] @xmath189 in addition to the effective field @xmath190 generated by the so coupling , we identified here another effective field @xmath191 acting on the isospin . the latter appears as a consequence of the interplay between tunneling and coulomb interaction . it is formally identical to the exchange field acting on the physical spin in quantum dots attached to ferromagnetic leads.@xcite in the limit of @xmath192 , it is given by @xmath193 \ , , \label{eq : dia : bufeld}\end{aligned}\ ] ] where @xmath194 is the digamma function and we used @xmath195 . the high - energy cutoff @xmath196 appearing in the second line guarantees convergence of the energy integral.@xcite physically it is provided by the smaller of the band width of the leads and the charging energy . for practical calculations it is not necessary to use the basis - independent form of this isospin representation . it allows for a better physical understanding of the systems dynamics but for evaluating the pumped charge and spin as described in sec . [ method ] , it is convenient to use the basis @xmath122 in the @xmath40 limit and @xmath168 for @xmath41 . in this section , we present the results for the adiabatically pumped charge ( spin ) in the weak - pumping regime . to calculate those , we use the formalism that has been introduced in sec . [ method ] . we integrate the pumped charge and spin currents , eqs . and , over one pumping cycle and obtain the pumped charge and pumped spin per pumping cycle . in order to simplify the time dependence of the pumped currents , we make use of the weak pumping limit ( see sec . [ weak ] ) and expand the integrand up to bilinear order in the pumping parameters , @xmath113 and @xmath114 . in this case , all results are proportional to the area , @xmath197 , enclosed in the pumping - parameter space . we normalize our results by @xmath197 and , thus , they are independent of the exact path in parameter space . to analyze the effect of coulomb interaction , we compare results for noninteracting electrons , @xmath40 , with the limit of strong coulomb interaction . the latter is realized by setting @xmath41 in the hamiltonian and , thereby , suppressing occupation of the quantum dot with more than one electron . furthermore , finite coulomb interaction influences the amplitude of the exchange field , @xmath191 , via the high - energy cutoff , @xmath196 . in all calculations , we set @xmath198 . we assume weak tunnel coupling between quantum dot and leads , @xmath86 , i.e. , we restrict the calculation to lowest order in the tunnel - coupling strength @xmath36 . if not stated otherwise , the so - coupling strength is @xmath199 . for @xmath40 , the results of this paper can be compared with calculations that include higher orders in @xmath36 . for example , brosco _ et al . _ have studied the two - level quantum dot with so coupling and vanishing coulomb interaction in the limit of zero temperature.@xcite calculations to all orders in @xmath36 can be done , e.g. , with a scattering matrix approach,@xcite which is equivalent to an approach that is based on a formula relating the pumped current to the instantaneous dot greens functions.@xcite the latter is , in general , extendable to finite interaction . in this section , we study the dependence of the pumped charge ( spin ) on various parameters : the strength of the so coupling , @xmath145 , the tunnel coupling to the leads , @xmath0 , and the time - averaged dot levels , @xmath200 . it is convenient to parametrize the latter by the time - averaged mean dot level , @xmath201 , and the averaged spacing of both orbital levels , @xmath202 . in the regime under consideration , the temperature appears only in two ways . first , since the mean energy level only appears in the combination @xmath203 , the temperature provides the energy scale on which variation of the mean level energy changes the pumped charge ( spin ) . second , the absolute value of the pumped charge and pumped spin are proportional to @xmath204 . therefore , all plots are normalized accordingly . the dependences of the pumped charge ( spin ) on the other parameters are not affected by temperature . the tunnel coupling of the two dot orbitals to the left and the right lead is defined by four real tunnel - matrix elements , @xmath0 . if the tunnel - matrix elements are equal for the coupling to the left and right lead , @xmath205 , then , for symmetry reasons , there will be no pumping transport via variation of the quantum dot s levels . to achieve pumping , the left - right symmetry needs to be broken by changing either the magnitude or the sign of one the tunnel couplings . we find it convenient to parametrize the tunnel - matrix elements by angles @xmath206 , which have been introduced in the previous section ( see eq . ) . the tunnel - matrix elements then are given by the relations @xmath207 and @xmath208 . for @xmath209 both orbital levels are coupled symmetrically to lead @xmath31 , i.e. , @xmath210 , and for @xmath211 the orbitals are coupled antisymmetrically , @xmath212 . the necessary condition to get a finite pumped charge ( spin ) is @xmath213 , since @xmath214 ( even for @xmath215 ) leads automatically to an effective one parameter pumping without any finite pumped charge ( spin ) in the adiabatic limit . motivated by the previous discussion , we first focus on a tunnel - coupling configuration with @xmath216 but @xmath213 , where the pumped charge and pumped spin are , in general , finite . both depend on the mean dot - level positions , @xmath217 and @xmath218 , which is shown in fig . [ fig : c ] . those orbital energies of the quantum dot can , in principal , be adjusted by capacitively coupled gate votages . figures [ fig : qu0]-[fig : sui ] illustrate the pumped charge ( spin ) for @xmath40 and @xmath41 and for a tunnel - coupling configuration where the coupling to the left lead is symmetric regarding the orbitals , @xmath219 , while the coupling to the right lead is given by @xmath220 , i.e. , @xmath221 . in the case where orbital @xmath222 is symmetrically @xmath223 and orbital @xmath224 is antisymmetrically @xmath225 coupled to the left and right leads , the pumped spin is in general finite while the pumped charge vanishes for this configuration . in figs . [ fig : psu0 ] and [ fig : psui ] the pumped spin is exemplarily calculated for @xmath226 , which is equivalent to @xmath227 . each plot in fig . [ fig : c ] shows a maximum and a minimum value . for no coulomb interaction , the maximum value is located at @xmath228 ( relative to the chemical potential @xmath1 of the leads ) . in the limit of strong coulomb interaction , the extrema positions are shifted to values of @xmath217 , whose order of magnitude is given by the temperature . the @xmath218-position of the maximum pumped charge ( spin ) depends on the tunnel coupling to the leads and the so - coupling strength . increasing @xmath145 also increases the maximum s position with respect to @xmath218 . furthermore , the pumped charge is in general larger for no coulomb interaction apart from special tunnel - coupling configurations discussed in detail in the next section . that is not surprising since the coulomb interaction reduces the possible transport channels through the dot by suppressing occupations of the dot with more than one electron . both limits @xmath40 and @xmath41 show different symmetries with respect to @xmath229 . in the limit @xmath40 , the pumped charge ( spin ) is exactly antisymmetric in @xmath218 . the antisymmetry with respect to @xmath230 originates from the particle - hole symmetry . the antisymmetry in @xmath218 alone , on the other hand , is a non - trivial result and only valid for the lowest order contribution in @xmath36 . in the limit of strong coulomb interaction , the symmetry in @xmath218 differs from the @xmath40 limit . the exchange field @xmath231 , which interacts with the isospin , leads to a contribution of the pumped charge ( spin ) that is not antisymmetric in @xmath218 . therefore , the antisymmetry is , in general , broken . to point out the symmetry characteristics , we study the pumped charge ( spin ) in two different tunnel - coupling configurations @xmath232 and @xmath233 , for both @xmath234 , @xmath235 which is equivalent to ( 1 ) : @xmath236 , @xmath237 in the first case , and ( 2 ) : @xmath238 , @xmath221 in the second one . tunnel couplings @xmath232 and @xmath233 show that the exchange field can affect the pumped charge and the pumped spin differently , and the effect , thus , depends on the tunnel - coupling parameters . that is accounted for by fig . [ fig : de ] , where the cut through the contour plot ( of fig . [ fig : c ] but with coupling configurations @xmath232 and @xmath233 ) for fixed @xmath217 is shown . ( color online ) pumped charge ( spin ) with finite so coupling , @xmath199 , in the @xmath40 and the @xmath41 limit depending on the averaged level - spacing of both orbital levels . in the @xmath40 limit we choose @xmath228 , which is the position of the maximum value . in the limit of strong coulomb interaction , we use @xmath239 as an approximation to the position of the maximum value . the two sets of coupling parameters ( 1 ) and ( 2 ) are the ones given in the text ( see eq . ) . the vertical dotted lines indicate _ pure spin pumping_. in the weak - coupling limit , this is only possible for pumping with coulomb interaction . ] the fixed value of @xmath217 is @xmath228 in the limit of vanishing coulomb interaction and , for comparison , @xmath239 in the limit of strong coulomb interaction . for configuration @xmath232 , the exchange field leads to a peak located at @xmath240 which has a nearly symmetric behavior in @xmath218 . the pumped spin , on the other hand , is still approximately antisymmetric in @xmath218 . furthermore , without @xmath231 , the pumped charge ( spin ) is usually smaller for @xmath41 , compared to @xmath40 , because of the reduced number of transport channels through the dot , but the exchange field can enhance the pumped charge . there are sets of parameters where the charge transport is even larger for finite coulomb interaction than for @xmath40 . for tunnel coupling @xmath233 , the symmetric part of the exchange - field contribution is less important . the pumped charge , in this case , is not dominated by a symmetric behavior as we observed for coupling @xmath232 . it is , rather , a shift of the point of zero pumped charge to a finite value of @xmath218 similar to the pumped spin . comparing the exchange - field contribution for configurations @xmath232 and @xmath233 , the contribution to the pumped spin reaches its maximum where the contribution to the pumped charge vanishes , and it is approximately half of its absolute maximum value where the contribution to the pumped charge has its maximum . for large values of exchange - field contribution , near its maximum , the pumped charge has a dominant symmetric contribution while the exchange - field contribution to the pumped spin is , in general , too small to generate a peak at @xmath240 . pure spin pumping is achieved whenever the pumped charge vanishes but the pumped spin remains finite . to find such points it is helpful that the pumped charge and pumped spin behave differently in the presence of coulomb interaction , as discussed in the previous section , and that the pumped charge is more sensitive to symmetry in the tunnel - matrix elements than the pumped spin . this defines the two strategies to obtain pure spin pumping : to tune either the orbital energy levels of the dot or the tunnel - matrix elements . for fixed tunnel couplings , we try to tune the orbital energies such that the pumped charge vanishes but the pumped spin remains finite . as discussed above , this is easily possible for strong coulomb interaction , because in this case , the value of @xmath218 at which the pumped charge changes its sign is shifted away from @xmath240 due to the exchange field @xmath191 . in absence of coulomb interaction ( and to lowest order in the tunnel coupling strength ) , this does , in general , not work apart from special coupling configurations , where the pumped charge vanishes independently of the orbital energies , as discussed in the next section . the reason is that both the pumped charge and the pumped spin are , to lowest order in @xmath36 , exactly antisymmetric in @xmath218 , i.e. , the pumped charge and spin vanish simultaneously . the comparison between the two limits is shown in fig . [ fig : de ] . the points of pure spin pumping are indicated by the vertical dotted lines . another interesting feature of the finite difference between the zero - points for the pumped charge and the pumped spin is the possibility to change the sign of the pumped spin , while charge is pumped in the same direction . there are cases in which pure spin current is not only possible for special , fine - tuned orbital energies but for _ all _ values of @xmath217 and @xmath218 . this is illustrated in fig . [ fig : cw ] , which shows the maximum absolute value of the pumped charge ( spin ) in the @xmath241 parameter space , as a function of the coupling parameters @xmath206 for @xmath234 and for both limits @xmath40 and @xmath41 . the plots can be periodically continued . the dotted lines represent coupling configurations where the pumped charge and the pumped spin are zero . along the middle dotted line , @xmath242 , pumped charge and pumped spin vanish due to left - right symmetry as mentioned previously . here , the tunnel - matrix elements are equal for the coupling to the left and the right lead , @xmath205 . the dotted zero - lines @xmath243 for zero pumped charge ( spin ) only exist for lowest order in @xmath36 ; higher - order corrections would lead , in general , to a finite pumped charge ( spin ) . the latter conclusion can be drawn by comparing with calculations for @xmath40 which are exact in @xmath36 , e.g. , by means of a scattering matrix approach,@xcite and it is self - evident that even finite coulomb interaction does not change that significantly . along these lines the tunnel - matrix elements are given by @xmath244 . in any case , these dotted lines do no mark good candidates for pure spin pumping since , there , charge and spin behave similar . the situation differs along the dashed lines . the middle dashed line , @xmath245 , represents a configuration where for each orbital the absolute value of the tunnel - matrix elements is the same , but one element of all four has an opposite sign , i.e. @xmath246 and @xmath247 ( or equivalently @xmath248 ) . here , we find ( to lowest order in @xmath36 ) pure spin pumping for both vanishing and strong coulomb interaction . this generalizes the result found in ref . for the @xmath40 limit to the limit of strong coulomb interaction . the dependence of the pure pumped spin for @xmath249 on @xmath217 and @xmath218 in both coulomb regimes is shown in figs . [ fig : psu0 ] and [ fig : psui ] . the dashed lines @xmath250 ( equivalent to @xmath251 ) indicate a further scenario for pure spin pumping to lowest order in @xmath36 in the @xmath40 limit . for higher orders in @xmath36 , however , the pumped charge becomes finite . it also becomes finite for @xmath41 ( and lowest order in @xmath36 ) as a consequence of the exchange field acting on the isospin . how important is the symmetry @xmath234 ? to answer this question , we calculate the pumped charge and spin for @xmath252 ; see fig . [ fig : cwa ] . as we see , the dependence of the pumped charge and spin on @xmath206 changes substantially for the pumped charge but not so much for the pumped spin . in particular , there are no straight lines with pure spin pumping anymore . for @xmath40 ( and to lowest order in @xmath36 ) , pure spin pumping is still possible on curved lines in the @xmath206 parameter space but not for @xmath41 . therefore , @xmath253 is a necessary requirement for pure spin pumping . the dependence of the pumped charge and pumped spin on the so - coupling strength is visualized in fig . [ fig : aso ] . ( color online ) pumped charge and pumped spin in the @xmath40 and @xmath41 limits depending on the strength of the so coupling . additionally , the green dotted line shows the pumped charge ( spin ) for @xmath41 in the case that the exchange field is turned off by hand . the illustrated functions are the maximum value of the pumped charge ( spin ) for given so strength , @xmath145 , in the @xmath241 parameter space . the coupling parameters are : @xmath234 , @xmath219 , and @xmath254 . ] here , the different functions again show the maximum value of the absolute pumped charge ( spin ) in the @xmath241 parameter space . as can be seen from the upper plot , the pumped charge decreases with increasing so coupling . it also decreases with increasing @xmath218 . in both cases , the pumping is suppressed since the difference of the eigenenergies of the dot hamiltonian becomes large . in general , the coulomb interaction reduces the amount of pumped charge and pumped spin . for small values of @xmath145 compared to @xmath36 , however , the coulomb interaction has the opposite effect on the pumped charge . in this regime , the coulomb interaction increases the pumped charge compared to the limit of @xmath40 . the latter is an effect of the exchange field : without the exchange field , the pumped charge would be reduced due to the coulomb interaction . increasing @xmath145 decreases the influence of the exchange field , i.e. , for large @xmath145 the coulomb interaction again reduces the pumped charge . for the pumped spin , the situation differs : the exchange field reduces the pumped spin even further . the pumped spin , in contrast to the pumped charge , vanishes for @xmath255 . therefore , there is an optimal value of @xmath145 that maximizes the pumped spin ( see fig . [ fig : aso ] ) . this value is smaller than @xmath36 and it depends on the tunnel coupling . we analyze the possibility to build an all - electric spin battery and to generate a pure spin current with a two - level quantum dot in the presence of coulomb interaction . in the limit of vanishing coulomb interaction , both are possible , as has been demonstrated in ref . . here , we show that this is also possible for the experimentally relevant case of a quantum dot with large coulomb interaction . the coulomb interaction changes the pumping characteristics substantially . in particular , symmetries with respect to the orbital energies change as a consequence of an effective exchange field acting on an isospin defined by the orbital level index . the nonvanishing coulomb interaction opens the possibility to achieve a pure spin current by tuning the orbital levels in the weak tunnel - coupling limit . furthermore , we find that a pure spin current is obtained independently of the orbital level energies for a certain configuration of tunnel couplings , where one level is symmetrically and the other one antisymmetrically coupled to the left and right lead , @xmath246 and @xmath247 in terms of tunnel - matrix elements . we acknowledge financial support from the dfg via spp 1285 and the eu under grant no . 238345 ( geomdiss ) . we now specify the diagrammatic rules to calculate the diagrams of the kernels @xmath256 with @xmath257 tunneling lines based on refs . . throughout the presented calculations , only the diagrams with one tunneling line @xmath258 are necessary . 1 . draw all topologically different irreducible diagrams with @xmath257 tunneling lines and the dot eigenstates @xmath259 , for @xmath40 , and @xmath260 , for @xmath41 , contributing to @xmath256 . each segment of the upper and lower contour separated by vertices is assigned with the corresponding eigenenergy @xmath261 . each tunneling line is labeled with the lead @xmath31 , spin @xmath29 and energy @xmath262 . each time segment of the diagram between two vertices at the times @xmath263 and @xmath264 leads to a contribution @xmath265 , where @xmath266 is the difference of left going energies minus right going energies . each tunneling line that goes forward or backward with respect to the keldysh contour contributes with a factor @xmath267 or @xmath268 , respectively , where @xmath268 is the fermi function . furthermore , a tunneling line that begins at a vertex containing a dot operator @xmath269 , with @xmath270 , and ends at a vertex containing @xmath271 introduces a factor @xmath272 . the matrix elements @xmath273 are obtained from the transformation @xmath274 , with @xmath38 , where @xmath275 . each vertex in the @xmath40 limit that connects state @xmath276 with state @xmath277 gives rise to a minus sign . the overall prefactor is @xmath278 , where @xmath279 is the number of vertices on the lower contour line and @xmath280 the number of crossings in the tunneling lines . 6 . integrate over all energies of the tunneling lines and sum over @xmath31 and @xmath29 . sum up all contributing diagrams . 46ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1126/science.1065389 [ * * , ( ) ] link:\doibase 10.1038/nphys551 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.76.323 [ * * , ( ) ] link:\doibase 10.1103/physics.2.50 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.78.1335 [ * * , ( ) ] link:\doibase 10.1126/science.283.5410.2056 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1088/0022 - 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during cell life , membranes are submitted to an inhomogeneous and variable environment . local inhomogeneities can be strongly related to biological processes , which has led to experiments investigating the effect of local modifications on biomimetic membranes @xcite . for instance , in the inner membrane of mitochondria , the enzymes that use the local ph difference across the membrane to synthesize adenosine triphosphate , the cell s fuel , are located in membrane invaginations called cristae @xcite . experiments on model lipid membranes have shown that a local ph change can induce a local dynamical membrane deformation @xcite , and in particular the formation of cristae - like invaginations @xcite : membrane shape is tightly coupled to local ph inhomogeneities . other concentration inhomogeneities in the environment of a cell have a crucial biological role , for instance in chemotaxis or in paracrine signaling . it is therefore of great interest to study the response of a biological membrane to a local modification of its environment . motivated by experiments conducted on biomimetic membranes , we have developed a theoretical description of the dynamics of a lipid bilayer membrane submitted to a local concentration increase of a substance that reacts reversibly and instantaneously with the membrane lipid headgroups . we focus on the regime of small deformations , and we treat linear membrane dynamics in the spirit of ref . @xcite . while our first works focused on the simple case of a constant modification of the membrane involving only one wavelength @xcite , we have recently extended our theoretical description in order to take into account the spatiotemporal profile of the fraction of chemically modified lipids resulting from the local reagent concentration increase @xcite . this profile is determined by the diffusion of the reagent in the solution that surrounds the membrane . in ref . @xcite , we compared the predictions of this theoretical description to experimental measurements of the deformation of the membrane of giant unilamellar vesicles caused by local microinjection of a basic solution , and we obtained good agreement between theory and experiment . in the present article , we pursue the theoretical investigation of the effect of a local chemical modification on a lipid membrane . in general , the dynamics that results from a local reagent concentration increase is quite complex , as it involves the evolution of the reagent concentration profile simultaneously as the response of the membrane . this is the case in the experimental data analyzed in ref . @xcite , which corresponds to microinjection steps lasting a few seconds . here , we show that the effect of the evolution of the reagent concentration profile on the membrane dynamics becomes negligible some time after the beginning of the reagent concentration increase , after what the dynamics corresponds to the response of the membrane to a chemical modification imposed instantaneously . we find that studying this regime enables to extract interesting properties of the membrane response . the article is organized as follows . first , in sec . [ mbr_dyn ] , we review the linear dynamics of a membrane submitted to a local chemical modification . then , in sec . [ res ] , we study separately the dynamics associated with each of the two effects that can arise from a chemical modification , namely a spontaneous curvature change and an equilibrium density change of the external monolayer . we find that a local asymmetric density perturbation between the two monolayers of the membrane relaxes by spreading diffusively in the whole membrane . intermonolayer friction plays a crucial part in this behavior . subsequently , in sec . [ res_b ] , we treat the general case where both effects are present , and we show how the ratio of the spontaneous curvature change to the equilibrium density change induced by the local chemical modification can be extracted from the dynamics . this ratio can not be deduced from the study of global modifications of vesicle equilibrium shapes in light of the area - difference elasticity model @xcite . finally , sec . [ ccl ] is a conclusion . for the article to be self - contained , the present section reviews the linear dynamics of a membrane submitted to a local chemical modification , starting from first principles . the main points of this description were presented in ref . @xcite . in that article , we compared theoretical predictions to experimental measurements of the deformation of a membrane submitted to a local and brief ph increase . here , our aim will be to go further in the analysis of our theoretical description in order to understand the fundamental properties of the response of a membrane to a local chemical modification . our description of the bilayer membrane is based on a local version of the area - difference elasticity membrane model @xcite . we focus on small deformations of an infinite flat membrane , and we denote the upper monolayer by @xmath0 and the lower one by @xmath1 . in the absence of a chemical modification , the local state of each monolayer is described by two variables : the local total curvature @xmath2 defined on the membrane midlayer , which is common to both monolayers , and the local scaled density @xmath3 , defined on the midlayer of the membrane , @xmath4 being a reference density . the sign convention for the curvature is chosen in such a way that a spherical vesicle has @xmath5 . the free energy @xmath6 per unit area in monolayer @xmath7 reads @xcite : @xmath8 where @xmath9 represents the tension of the bilayer and @xmath10 its bending modulus , while @xmath11 is the stretching modulus of a monolayer , and @xmath12 denotes the distance between the neutral surface @xcite of a monolayer and the midsurface of the bilayer . as we assume that the two monolayers of the membrane are identical before the chemical modification , these constants are the same for both monolayers . the spontaneous curvatures of the two monolayers have the same absolute value @xmath13 and opposite signs , since their lipids are oriented in opposite directions . the expression for @xmath6 in eq . ( [ fpm0 ] ) corresponds to a general second - order expansion in the small dimensionless local variables @xmath14 and @xmath15 , around the reference state which corresponds to a flat membrane with uniform density @xmath16 . it is valid for small deformations around this reference state : @xmath17 and @xmath18 , where @xmath19 is a small dimensionless parameter used for bookkeeping purposes , which characterizes the amplitude of the small deformations of the membrane around the reference state . mathematically , @xmath19 is considered infinitesimal . note that in general , both @xmath2 and @xmath14 , which describe local small deformations around the reference state , are functions of time and of position on the membrane . let us now focus on the way the membrane free energy is affected by the local chemical modification . we consider that the reagent source , which corresponds to the micropipette tip in an experiment , is localized in the water above the membrane . besides , membrane permeation and flip - flop are neglected given their long timescales . hence , the chemical modification only affects the upper monolayer , i.e. , monolayer @xmath0 , and not the lower one . let us denote by @xmath20 the local mass fraction of the lipids of the upper monolayer that are chemically modified : @xmath20 depends on time and position since it arises from the local chemical modification . we assume that the reagent concentration is small enough for @xmath20 to remain small at every time and position on the membrane , and we characterize this smallness through @xmath21 . in order to describe the chemically modified membrane , we have to include the third small variable @xmath20 in our second - order expansion of @xmath22 . we obtain @xcite : @xmath23 where the constants @xmath24 , @xmath25 , and @xmath26 describe the response of the membrane to the chemical modification . these constants depend on the reagent that is injected . their physical meaning will be explained in the next paragraph . besides , the non - analytical mixing entropy term @xmath27 has been added to our second - order expansion @xcite . note that we assume that the three small dimensionless local variables @xmath20 , @xmath14 and @xmath28 are of the same order . in fact , in the present work , the deformation of the membrane and the density variation are caused by the local chemical modification , i.e. , they are a response to @xmath20 , which justifies that @xmath28 and @xmath14 are of the same order as @xmath20 . we refer the reader to ref . @xcite for more details on the derivation of eqs . ( [ fpm0 ] ) and ( [ fmod ] ) . the effect of the chemical modification ( i.e. , of @xmath20 ) on the upper monolayer is twofold . first , the scaled equilibrium density on the neutral surface of the upper monolayer is changed by the amount @xmath29 to first order . second , the spontaneous curvature of the upper monolayer is changed by the amount @xmath30 to first order , with @xmath31 . these results are obtained by minimization of the free energy of a homogeneous monolayer with constant mass ( see [ ap_sc_dens ] ) . hence , the constants @xmath24 and @xmath32 describe the linear response of the monolayer equilibrium density and of its spontaneous curvature , respectively , to the chemical modification . this explains the physical meaning of the constants @xmath24 and @xmath26 in eq . ( [ fmod ] ) . note that @xmath25 corresponds to the quadratic response of the membrane to the chemical modification , but it will not have any relevant effect in the following . the elastic force densities in a monolayer described by the free - energy densities in eqs . ( [ fpm0][fmod ] ) have been derived in ref . @xcite to first order in @xmath19 , using the principle of virtual work . as we focus on small deformations of an infinite flat membrane , it is convenient to describe it in the monge gauge by the height @xmath33 , @xmath34 , of its midlayer with respect to the reference plane @xmath35 . then , @xmath36 to second order . such a description is adapted to practical cases where the distance between the reagent source and the membrane is much smaller than the vesicle radius . the force per unit area of the reference plane , which we call `` force density '' , then reads to first order in @xmath19 @xmath37 where @xmath38 is the tangential component of the force density in monolayer @xmath7 " , while @xmath39 is the total normal force density in the membrane . in these formulas , we have introduced the antisymmetric scaled density @xmath40 , and the constant @xmath41 . these expressions show that both the equilibrium density change and the spontaneous curvature change ( i.e. , both @xmath24 and @xmath32 ) can yield a normal force density , and thus a deformation of the membrane , while only the equilibrium density change can yield a tangential force density and induce tangential lipid flow . an illustration of the role of these force densities in the membrane response to a local chemical modification is provided in fig . [ mem_fig ] . ) . ( a ) : membrane at equilibrium in the reference state ( flat shape , uniform lipid densities in both monolayers ) . ( b ) : membrane just after the instantaneous local chemical modification : the membrane has not deformed yet at this stage , and the densities are still uniform . some lipids of monolayer @xmath0 are modified : they are represented in dark blue ( @xmath20 represents their local mass fraction ; for clarity @xmath20 is locally large in the figure , but in our work , @xmath21 ) . here we assume that the result of the chemical modification is to increase the effective size of their headgroups . hence , the equilibrium density in monolayer @xmath0 is locally decreased ( @xmath42 ) , while the spontaneous curvature is increased in absolute value ( @xmath43 ) . this results into a normal force density @xmath44 at the center of the modified zone of the membrane ( see eq . ( [ pn_b ] ) ) , and into a tangential force density in monolayer @xmath0 , oriented toward the exterior of the modified zone ( see eq . ( [ pip_b ] ) ) . both are indicated on the figure by arrows . ( c ) due to the normal force density , the membrane starts deforming upwards . the maximum height @xmath45 and the full width at half maximum @xmath46 of the deformation are indicated . note that tangential flows also appear due to tangential force densities.[mem_fig ] ] using the elastic force densities in eqs . ( [ pip_b][pn_b ] ) , it is possible to describe the dynamics of the membrane to first order in the spirit of ref . let us outline the derivation of these equations @xcite , which is described in more detail in [ apeq ] . we use a normal force balance for the bilayer ( see [ ap_mbdyn ] ) , which involves @xmath47 and the normal viscous stress exerted by the surrounding fluid , which is derived in [ ap_hyd ] . besides , as each monolayer is a two - dimensional fluid , we write down generalized stokes equations ( see [ ap_mbdyn ] ) , which involve @xmath38 , the two - dimensional viscous stress associated with the lipid flow , the tangential viscous stress exerted by the surrounding fluid ( see [ ap_hyd ] ) , and the intermonolayer friction @xmath48 , where @xmath49 is the velocity in monolayer @xmath7 and @xmath50 is the intermonolayer friction coefficient @xcite . this intermonolayer friction is a tangential force that comes into play when one monolayer slides relatively to the other one . finally , we use mass conservation in each monolayer . these dynamical equations are best expressed using two - dimensional fourier transforms of the various fields involved , denoted with hats : for any field @xmath51 which depends on @xmath52 and on time @xmath53 , @xmath54 is such that @xmath55 combining all the above - mentioned equations yields a system of first - order linear differential equations on the two - dimensional variable @xmath56 : @xmath57 where we have introduced the matrix which describes the dynamical response of the membrane @xcite : @xmath58 here , we have assumed that @xmath59 and @xmath60 . this is true for all the wavevectors with significant weight in @xmath61 , if the modified lipid mass fraction @xmath20 has a smooth profile with a characteristic width larger than 1 @xmath62 m . indeed , @xmath63 for water , and typically @xmath64 and @xmath65 @xcite . besides , the forcing term in eq . ( [ ed ] ) reads @xcite : @xmath66 eqs . ( [ ed]-[ed_defs ] ) show that the membrane deformation is coupled to the antisymmetric density : the symmetry breaking between the monolayers causes the deformation of the membrane . here , the symmetry breaking is caused by the chemical modification of certain membrane lipids in the external monolayer , i.e. , to the presence of @xmath20 . and indeed , eq . ( [ ed_defs_2 ] ) shows that the forcing term in eq . ( [ ed ] ) is proportional to @xmath67 . before solving eq . ( [ ed ] ) , we need to determine @xmath68 , which is involved in the forcing term @xmath69 . the profile @xmath70 of the mass fraction of chemically modified lipids in the external monolayer arises from the local reagent concentration increase . we focus on reagents that react reversibly with the membrane lipid headgroups . besides , we assume that the reaction between the lipids and the reagent is diffusion - controlled ( see , e.g. , ref . in other words , the molecular reaction timescales are very small compared to the diffusion timescales . for such a reversible diffusion - controlled chemical reaction , @xmath70 is instantaneously determined by the local reagent concentration on the membrane , which results from the diffusion of the reagent in the fluid above the membrane . note that these hypotheses are verified in the experiments analyzed in ref . @xcite , where the reagent is sodium hydroxide . we consider that the reagent source is localized in @xmath71 , which would represent the position of the micropipette tip in an experiment ( see ref . the cylindrical symmetry of the problem then implies that the fields involved in our description only depend on @xmath72 . we focus on the regime of small deformations @xmath73 , and we work at first order in @xmath74 . besides , we study the linear regime where @xmath75 is proportional to the reagent concentration on the membrane : denoting by @xmath76 the reagent concentration field , we have @xmath77 . to first order in the membrane deformation , this can be simplified into @xmath78 . the field @xmath79 is determined by the diffusion of the reagent from the local source in the fluid above the membrane . in microinjection experiments , there is also a convective transport of the reagent due to the injection , but the pclet number remains so small that diffusion dominates . besides , since the membrane is a surface and @xmath80 , we can neglect the number of reagent molecules that react with the membrane when calculating @xmath79 . hence , @xmath79 can be obtained by solving the diffusion equation @xmath81 where @xmath82 is the diffusion coefficient of the reagent in the fluid above the membrane , and the source term reads @xmath83 where @xmath84 is heaviside s function . this corresponds to a constant injection flow from the source starting at @xmath85 . in addition , the membrane imposes a neumann boundary condition @xmath86 . this relation , which corresponds to a vanishing flux across the membrane , can be simplified to first order into @xmath87 the solution to this diffusion problem reads @xmath88 where the causal green function @xmath89 of our diffusion problem can be expressed using the method of images @xcite : @xmath90 where we have introduced the infinite - volume causal green function of the diffusion equation @xmath91 combining eqs . ( [ cint ] , [ g1 ] , [ g2 ] ) provides an analytical expression for @xmath76 , and for @xmath92 . since @xmath93 , we thus obtain an analytical expression for @xmath61 , which reads @xmath94 where erf denotes the error function . as @xmath95 , @xmath96 converges towards @xmath97 the membrane deformation @xmath98 is given by the solution to eq . ( [ ed ] ) with @xmath61 expressed in eq . ( [ phiq2 ] ) and with the initial condition @xmath99 , corresponding to a non - perturbed membrane . we have @xmath100 . \label{solution}\ ] ] + here , we have introduced the eigenvalues @xmath101 and @xmath102 of the matrix @xmath103 ( see eq . ( [ ed_defs ] ) ) , the associated eigenvectors @xmath104 and @xmath105 , and the solutions @xmath106 and @xmath107 of the linear system @xmath108 the integral in eq . ( [ solution ] ) and the inverse fourier transform of @xmath98 can be calculated numerically . this gives the spatiotemporal evolution of the membrane deformation @xmath109 . both for physical understanding and for computational efficiency , it is interesting to put our description in dimensionless form . using the buckingham pi theorem and choosing @xmath110 as the distance unit , @xmath111 as the time unit and @xmath112 as the mass unit , the ten parameters in our equations yield seven dimensionless numbers : @xmath113 the parameters @xmath114 and @xmath115 compare the characteristic lengthscales @xmath116 and @xmath12 of the membrane to @xmath110 , while @xmath117 is a dimensionless version of the membrane tension @xmath9 . besides , the parameter @xmath118 quantifies the importance of the reagent diffusion on the membrane dynamics . we will discuss briefly the effect of varying @xmath118 in the following . in the case of a lipid membrane in water , the only dimensional parameters that can span various orders of magnitudes are @xmath9 and @xmath110 . for @xmath119 , i.e. , for realistic membrane tensions @xcite , and for @xmath120 m , we will see in the following that the only relevant dimensionless parameters in the dynamics of the membrane are @xmath62 and @xmath121 . the first one , @xmath62 , then corresponds to the ratio of the two eigenvalues of @xmath103 for @xmath122 ( see sec . [ anin ] ) : hence , it is a crucial element of the membrane dynamical response . the second one , @xmath123 , quantifies the relative weight of the spontaneous curvature change and of the equilibrium density change of the external monolayer due to the chemical modification ( see sec . [ res_b ] ) . note indeed that @xmath124 , and that @xmath89 and @xmath125 describe the two effects of the reagent on the membrane , through @xmath32 and @xmath24 . the effect of varying @xmath62 and @xmath123 will thus be discussed in the following . for our numerical calculations , we initially take @xmath126 these values correspond to the injection of naoh in water ( @xmath127 @xcite ) from a source at @xmath128 m above a floppy membrane with typical constants @xmath129 ( see ref . @xcite and references therein ) , @xmath130 , @xmath131 and @xmath132 nm @xcite . the values of the intermonolayer friction coefficient @xmath50 and of the water viscosity @xmath133 are those given above . note that the time unit @xmath111 is then equal to one second . in this section , we will study the two extreme cases @xmath134 and @xmath135 . in other words , we will study separately the dynamics associated with an equilibrium density change and with a spontaneous curvature change of the external monolayer . we will see that these two manifestations of the chemical modification of the membrane due to the reagent concentration increase yield different spatiotemporal evolutions of the membrane deformation . the general case where both effects are present will then be discussed in sec . [ res_b ] . _ a priori _ , the dynamics of the membrane deformation is quite complex , as it involves the evolution of the reagent concentration profile , due to diffusion , simultaneously as the response of the membrane . in order to investigate the effect of the reagent diffusion on the dynamics of the deformation , and to see when the effect of the evolution of the reagent concentration profile becomes negligible , we will compare the two following cases : + ( i ) the realistic case where @xmath96 is given by eq . ( [ phiq2 ] ) , + ( ii ) the theoretical case where the stationary modification @xmath136 in eq . ( [ phis ] ) is imposed instantaneously : @xmath137 . note that this case corresponds to the limit of very large @xmath118 . let us first focus on the case where @xmath134 , in which only the equilibrium density of the upper monolayer is affected by the chemical modification . [ hw_density ] shows the evolution with dimensionless time @xmath138 of the membrane deformation height @xmath139 in front of the source , and of the full width at half - maximum @xmath140 of the deformation ( see fig . [ mem_fig](c ) for an illustration of the definitions of @xmath45 and @xmath46 ) . the cases ( i ) and ( ii ) introduced above are presented . on these graphs , we also show @xmath141 and the full width at half - maximum @xmath142 of @xmath20 in case ( i ) . [ hw_density ] shows that the membrane undergoes a transient deformation that relaxes to zero while getting broader and broader . the local asymmetric density perturbation relaxes by spreading in the whole membrane . however , this process is slowed down by intermonolayer friction , so the density asymmetry is transiently solved by a deformation of the membrane @xcite . this deformation is downwards if @xmath143 , and upwards if @xmath144 . in the realistic case ( i ) , at short times , the membrane dynamics is governed by the evolution of @xmath20 due to the reagent diffusion . conversely , at long times , once @xmath20 is close enough to its steady - state profile @xmath136 , the dynamics of the membrane deformation is similar in case ( i ) and in case ( ii ) . this can be seen in fig . [ hw_density ] : first , @xmath45(i ) and @xmath46(i ) follow @xmath145 and @xmath142 , and then , they have an evolution very similar to those of @xmath45(ii ) and @xmath46(ii ) . thus , after some time , the effect of reagent diffusion on the dynamics of the membrane deformation becomes negligible , and the dynamics of the realistic case ( i ) can be well approximated by that of the simpler case ( ii ) , which corresponds to the response of the membrane to an instantaneously imposed modification . , where only the equilibrium density is changed . the values taken for the other dimensionless numbers are those in eq . ( [ values ] ) . both the realistic case ( i ) where reagent diffusion is accounted for , and the simpler case ( ii ) where the chemical modification is imposed instantaneously , are considered . ( a ) logarithmic plot of the height of the membrane deformation @xmath45 and of the fraction @xmath145 of modified lipids in front of the reagent source versus dimensionless time @xmath146 . both in case ( i ) and in case ( ii ) , @xmath45 is plotted in units of the extremal value it attains in case ( ii ) . ( b ) logarithmic plot of the width @xmath46 of the membrane deformation and of the width @xmath142 of the fraction of modified lipids versus @xmath146 . both @xmath46 and @xmath142 are plotted in units of @xmath110 . it can be seen on graphs ( a ) and ( b ) that cases ( i ) and ( ii ) yield similar dynamics for @xmath147 . the thin red ( gray ) lines correspond to the analytical laws mentioned in the text . [ hw_density ] ] the transition time @xmath148 between the diffusion - dominated regime and the membrane - response dominated regime is determined by the convergence of @xmath20 to @xmath136 . as the reagent takes a dimensionless time @xmath149 to diffuse from the source to the membrane , we expect @xmath150 . we studied the dynamics of the deformation height for @xmath151 $ ] by integrating eq . ( [ ed ] ) numerically , and we found that this law is very well verified . for our standard value of @xmath118 ( see eq . [ values ] ) , used in fig . [ hw_density ] , we have @xmath152 . let us discuss analytically the simple case ( ii ) . the long - time behaviors obtained in this case are especially interesting , since they also apply to the realistic case ( i ) . let us focus on @xmath119 , i.e. , on realistic membrane tensions , and let us keep the standard values of the other parameters involved in @xmath103 ( see below eq . ( [ values ] ) ) , as these parameters can not vary significantly for a membrane in water . the eigenvalues of @xmath103 , which correspond to the two independent relaxation rates of a perturbation of the membrane with wavelength @xmath153 , can then be approximated by @xmath154 for all wavevectors with significant weight in @xmath155 if @xmath120 m . within this approximation , for @xmath122 , we have @xmath156 . this leads to a simple expression of the solution of the dynamical equation eq . ( [ ed ] ) : @xmath157 with @xmath158 . this expression shows that the only parameter that is relevant in the dynamics is @xmath62 . note that , in the realistic case ( i ) , the value of @xmath118 is relevant too . in the long - time limit , as the deformation spreads , we have @xmath159 for all the wavevectors with significant weight in @xmath160 . in this case , calculating the inverse fourier transform of eq . ( [ lim1 ] ) for @xmath161 yields @xmath162 and the numerical results for @xmath46 are in excellent agreement with @xmath163 as can be seen in fig . [ hw_density](b ) . this law can also be obtained analytically if @xmath136 is replaced by a gaussian . in the short - time limit , ( [ lim1 ] ) yields @xmath164 and @xmath165 . these asymptotic behaviors , both for the short - time limit and for the much more interesting long - time limit , are plotted in red ( gray ) lines on fig . [ hw_density ] . the transition between the two asymptotic regimes is determined by the value of @xmath62 , which is the only parameter that controls the dynamics in case ( ii ) . in particular , it is possible to show that the long - term scaling laws are valid for @xmath166 . we studied the dynamics of the deformation height for @xmath167 $ ] by integrating eq . ( [ ed ] ) numerically , and we found that this law is very well verified . the long - term scaling @xmath168 shows that the local antisymmetric density perturbation spreads diffusively . let us emphasize that this diffusive behavior is not related to the diffusion of the reagent in the solution above the membrane , as in the long - term limit , @xmath20 has reached its steady - state profile @xmath136 . the fact that the long - term scaling @xmath168 holds in case ( ii ) , where the profile @xmath136 is established instantaneously , as well as in case ( i ) ( see fig . [ hw_density](b ) ) , illustrates that this scaling law is not related to the reagent diffusion . the long - term diffusive spreading of the antisymmetric density perturbation can be understood as follows . at long times , the difference between the stokes equations for each monolayer eq . ( [ baltg+ ] ) and eq . ( [ baltg- ] ) can be approximated by @xmath169 in real space . to obtain this equation from the difference between eq . ( [ baltg+ ] ) and eq . ( [ baltg- ] ) , we have used @xmath59 and @xmath60 ( see below eq . ( [ ed_defs ] ) ) , and also @xmath170 , which holds for @xmath171 . the last relation can be shown using eq . ( [ lim1 ] ) in the long - time limit . using eq . ( [ balapp ] ) , the antisymmetric convective mass current @xmath172 can be expressed to first order as @xmath173 this current has a diffusive form . combining this with the mass conservation relations in eq . ( [ massc ] ) finally yields the diffusion equation @xmath174 hence , the local asymmetry in density between the two monolayers finally relaxes by spreading diffusively , with an effective diffusion coefficient @xmath175 . this slow relaxation is due to intermonolayer friction . note that we can now interpret the dimensionless number @xmath118 defined in eq . ( [ nbadim ] ) as ( half ) the ratio of the reagent diffusion coefficient @xmath82 to this effective diffusion coefficient . the fact that a local density asymmetry in the two monolayers of a membrane relaxes by spreading diffusively is generic . here , the asymmetry is due to a local chemical modification of one monolayer , but it can also be caused , e.g. , by a sudden flip of some lipids from one monolayer to the other @xcite . our description of this diffusive behavior generalizes that of ref . @xcite , which focused on a perturbation with a spherical cap shape and a uniform density asymmetry . note that this diffusive behavior was first mentioned in ref . @xcite , within a different theoretical framework , and restricting to axially - symmetric deformations . let us now focus on the second extreme case , where @xmath135 , i.e. , where only the spontaneous curvature of the upper monolayer is affected by the chemical modification . [ hw_curv ] shows that the deformation converges to a deformed profile , in contrast with the previous case : while a local density asymmetry spreads on the whole membrane , a local spontaneous curvature modification leads to a locally curved equilibrium shape . the deformation is upwards if @xmath176 , and downwards if @xmath177 . , where only the spontaneous curvature is changed . the values taken for the other dimensionless numbers are those in eq . ( [ values ] ) . both the realistic case ( i ) where reagent diffusion is accounted for , and the simpler case ( ii ) where the chemical modification is imposed instantaneously , are considered . ( a ) logarithmic plot of the height of the membrane deformation @xmath45 and of the fraction @xmath145 of modified lipids in front of the reagent source versus dimensionless time @xmath146 . both in case ( i ) and in case ( ii ) , @xmath45 is plotted in units of the extremal value it attains in case ( ii ) . ( b ) logarithmic plot of the width @xmath46 of the membrane deformation and of the width @xmath142 of the fraction of modified lipids versus @xmath146 . both @xmath46 and @xmath142 are plotted in units of @xmath110 . it can be seen on graphs ( a ) and ( b ) that cases ( i ) and ( ii ) yield similar dynamics for @xmath147 . the thin red ( gray ) lines correspond to the analytical laws mentioned in the text . [ hw_curv ] ] as in sec . [ eqdensc ] , in the realistic case ( i ) , the membrane dynamics is governed by the reagent diffusion at short times @xmath178 , while at long times @xmath179 , it is similar in case ( i ) and in case ( ii ) . the long - time dynamics thus corresponds to the response of the membrane to an instantaneously imposed chemical modification . in the simple case ( ii ) , the approximations on @xmath101 and @xmath102 introduced in sec . [ anin ] yield @xmath180 where we have used the notation @xmath158 as above . calculating the inverse fourier transform of this function for @xmath161 yields @xmath181 for all @xmath182 . this analytical law for the deformation height @xmath45 is plotted in red ( gray ) lines in fig . [ hw_curv](a ) . let us focus on the long - time limit , which is especially interesting , since the results found in case ( ii ) also apply to the realistic case ( i ) . ( [ ancourb ] ) shows that in this limit , @xmath183 , so that the long - time profile of the membrane deformation is fully determined by that of @xmath20 . in particular , for @xmath184 , @xmath185 and @xmath186 . for @xmath187 , we find @xmath165 again . these asymptotic behaviors regarding the deformation width @xmath46 are plotted in red ( gray ) lines in fig . [ hw_curv](b ) , both for the short - time limit and for the much more interesting long - time limit . as in sec . [ eqdensc ] , in case ( ii ) , @xmath62 is the only relevant parameter in the dynamics of the membrane deformation ( see eq . ( [ ancourb ] ) ) , and it determines the transition time between the two asymptotic regimes . in the realistic case ( i ) , the value of @xmath118 is relevant too . generically , a chemical modification will affect both the equilibrium density and the spontaneous curvature of the upper monolayer . the general solution of eq . ( [ ed ] ) is a linear combination of the two solutions obtained for the two extreme cases : @xmath134 ( see sec . [ eqdensc ] ) and @xmath135 ( see sec . [ scc ] ) . in this section , we will only discuss the realistic case ( i ) where the reagent diffusion is taken into account . as in the two extreme cases , the effect on the membrane dynamics of the evolution of @xmath20 due to reagent diffusion is crucial for @xmath178 , while it becomes negligible for @xmath179 . for a lipid membrane in water such that @xmath119 and for @xmath188 m , we have seen that in both extreme cases , the dynamics is influenced only by @xmath62 and @xmath118 ( see secs . [ eqdensc][scc ] ) . hence , in the general case , the dynamics is influenced by @xmath62 , @xmath118 , and by the parameter @xmath189 which quantifies the relative importance of the spontaneous curvature change and of the equilibrium density change of the external monolayer due to the chemical modification . indeed , eq . ( [ pn_b ] ) shows that the ratio of the normal force density arising from the spontaneous curvature change to the normal force density arising from the equilibrium density change is equal to @xmath190 . the effect of varying @xmath118 was discussed in secs . [ eqdensc][scc ] , and in addition , @xmath118 can not vary much for a reagent injected in water above a lipid membrane . hence , we will focus on the influence of @xmath62 and @xmath123 on the membrane dynamics . the value of the ratio @xmath123 of the spontaneous curvature change to the equilibrium density change is _ a priori _ unknown , as @xmath24 and @xmath32 , which are involved in @xmath125 and @xmath89 , respectively ( see eq . [ nbadim ] ) , are unknown . the actual values of @xmath24 and @xmath32 depend on the reagent as well as on the membrane itself , as these two parameters describe the linear response of the membrane to a reagent ( see eq . [ fmod ] ) . let us assume that @xmath191 , i.e. , that the equilibrium density change and the spontaneous curvature change induce deformations in the same direction . this is true , e.g. , for a chemical modification that affects the lipid headgroups in such a way that it yields an effective change of the preferred area per headgroup @xcite . a rough microscopic model , where lipids are modeled as cones that favor close - packing , then yields @xmath192 , which corresponds to the case where the two effects yield destabilizing normal force densities of similar magnitudes @xcite . thus , we expect @xmath193 . [ hw_mixte](a ) shows the evolution of @xmath45 in case ( i ) for three different values of @xmath123 . the deformation height features an extremum @xmath194 , and then a relaxation . this behavior is due to the change of the equilibrium density ( see sec . [ eqdensc ] ) . conversely , the nonzero asymptotic deformation @xmath195 arises from the change of the spontaneous curvature ( see sec . [ scc ] ) . the relative importance of @xmath194 to @xmath195 thus depends on @xmath123 . this means that studying the dynamics of the membrane deformation in response to a local chemical modification provides information on the ratio of the spontaneous curvature change to the equilibrium density change induced by this chemical modification . in fig . [ hw_mixte](b)(c ) , the ratio @xmath196 is plotted versus @xmath123 for different values of @xmath62 . indeed , as mentioned above , the membrane dynamics is determined both by @xmath123 and by @xmath62 ( at constant @xmath118 ) . the order of magnitude of @xmath194 can be estimated assuming an equilibrium density change and/or a spontaneous curvature change of a few percent : @xmath194 is of order @xmath197 m for flaccid membranes , and smaller than @xmath198 m if @xmath199 . hence , we choose @xmath200\,\mathrm{n / m}$ ] , which yields @xmath201 $ ] for @xmath128 m : such values are taken in fig . [ hw_mixte](b)(c ) . in front of the reagent source versus dimensionless time @xmath146 in the realistic case ( i ) , for different values of the parameter @xmath123 , which quantifies the relative importance of the spontaneous curvature change to the equilibrium density change . the values taken for the other dimensionless numbers are those in eq . ( [ values ] ) . in each case , @xmath45 is plotted in units of its asymptotic value in the case @xmath202 . the lines show the asymptotic values @xmath195 of @xmath45 for @xmath184 in each case , and @xmath194 denotes the extremal value of @xmath45 . ( b ) and ( c ) : logarithmic plot of @xmath196 versus @xmath123 in case ( i ) , for different values of the dimensionless parameter @xmath62 , which affects the membrane dynamics as well as @xmath123 . from down to up : @xmath203 . [ hw_mixte ] ] our study shows that one can deduce @xmath123 , and thus the relative importance of the spontaneous curvature change to the equilibrium density change due to a chemical modification , from the measurement of @xmath196 . this is very interesting , given that such information can not be deduced from the study of static and global membrane modifications @xcite . indeed , the equilibrium vesicle shapes in the area - difference elasticity model are determined by the combined quantity @xmath204 which involves both the equilibrium density and the spontaneous curvature . in this expression , @xmath205 is the dimensionless preferred area difference between the two monolayers , which is related to the asymmetric equilibrium density change , while @xmath206 denotes the dimensionless spontaneous curvature of the bilayer , which is related to the asymmetric spontaneous curvature change @xcite . the parameter @xmath207 in eq . ( [ combqty ] ) is a dimensionless number involving the elastic constants of the membrane . note that the vesicle shape variations due to global modifications of the vesicle environment are usually interpreted as coming only from a change of the spontaneous curvature , under the assumption that the preferred area per lipid is not modified @xcite . in order to determine the ratio @xmath123 of the spontaneous curvature change to the equilibrium density change in a practical case , it is necessary to know the value of @xmath62 ( see fig . [ hw_mixte](b)(c ) ) . however , as @xmath62 does not involve any parameter that depends on the reagent ( see eq . ( [ nbadim ] ) ) , it is possible to compare the effects of different reagents on the same membrane , i.e. , their values of @xmath123 , even without knowing the precise value of @xmath62 . we have analyzed theoretically the spatiotemporal response of a membrane submitted to a local heterogeneity , in the realistic case of a local concentration increase of a substance that reacts reversibly and instantaneously with the membrane lipid headgroups . in general , the dynamics of the membrane deformation is quite complex , as it involves the evolution of the reagent concentration profile due to diffusion in the solution above the membrane , simultaneously as the response of the membrane . we have shown that , some time after the beginning of the reagent concentration increase , the effect of the evolution of the reagent concentration becomes negligible . studying this regime enables to extract interesting properties of the membrane response . we have shown that a local density asymmetry between the two monolayers relaxes by spreading diffusively in the whole membrane . intermonolayer friction plays a crucial part in this behavior . in addition , we have shown how the relative importance of the spontaneous curvature change to the equilibrium density change can be extracted from the dynamics of the membrane response to the local chemical modification . this is a significant result since such information can not be deduced from the study of a static and global modification using the area - difference elasticity model . our description provides a theoretical framework for experiments involving the microinjection of a reagent close to biomimetic membranes . in ref . @xcite , we used the theoretical model presented here to analyze experimental results corresponding to brief microinjections of a basic solution in the regime of small deformations , and we obtained good agreement between theory and experiment . the results of the present theoretical work show that it would be interesting to conduct experiments with a continuous injection phase , since the ratio @xmath123 of the spontaneous curvature change to the equilibrium density change could then be determined for various membrane compositions and reagents . in biomimetic membranes as well as in cells , remarkable phenomena occur in the regime of larger deformations : cristae - like invaginations @xcite , tubulation @xcite , pearling @xcite , budding , exo- or endocytosis @xcite , etc . to study such phenomena , it would be useful to pursue our study in the nonlinear regime . we thank miglena i. angelova for motivating us to study this subject . we thank miglena i. angelova and nicolas puff for interesting discussions . the spontaneous curvature and the equilibrium density in monolayer @xmath0 can be obtained by minimizing @xmath208 with respect to @xmath209 and @xmath2 for a homogeneous monolayer with constant mass . first , using the expression of @xmath22 in eq . ( [ fmod ] ) , the minimization with respect to @xmath209 gives , to first order in @xmath19 : @xmath210 then , the minimization with respect to @xmath2 yields to first order , using eq . ( [ min1 ] ) : @xmath211 with @xmath31 ( as defined in the main text ) . note that , since we assume that @xmath212 and @xmath18 , we must have @xmath213 and @xmath214 for our description to be valid for the values of @xmath209 and @xmath2 that minimize @xmath208 . this property has been used to simplify the results of the minimization . the scaled lipid density @xmath215 on the neutral surface of the monolayer is related to @xmath209 through @xmath216 to first order . this relation arises from the geometry of parallel surfaces @xcite , given that the membrane midlayer and the monolayer neutral surface are parallel surfaces separated by a distance @xmath12 . hence , eq . ( [ min1 ] ) can be rewritten as @xmath217 this result is independent of the curvature @xmath2 , contrary to that in eq . ( [ min1 ] ) . indeed , by definition , on the neutral surface , curvature and density are decoupled @xcite , while these two variables are coupled on other surfaces . ( [ min1b ] ) shows that , due to the chemical modification , the scaled equilibrium density on the neutral surface of monolayer @xmath0 is changed by the amount @xmath218 to first order . besides , eq . ( [ min2 ] ) indicates that the spontaneous curvature of monolayer @xmath0 is changed by the amount @xmath219 to first order . one might wonder why the spontaneous curvature @xmath220 found by minimization is not simply @xmath221 for @xmath222 ( see eq . ( [ min2 ] ) ) . this is due to the fact that we work on the membrane midsurface , which is more convenient to study the membrane dynamics . if the monolayer free energy had been originally written using variables and constants defined on the neutral surface , then the spontaneous curvature found by minimization would correspond exactly to the constant @xmath223 which plays the part of @xmath13 when everything is defined on the neutral surface . we wish to determine the velocity field @xmath224 in the fluid above ( @xmath0 ) and below ( @xmath1 ) the membrane . this flow is caused by the deformation of the membrane and by the lateral flow in the membrane : mathematically , it is determined by the boundary conditions corresponding to the continuity of velocity at the interface between the fluid and the membrane . given the short lengthscales considered , the dynamics of @xmath225 can be described using stokes equation . adding the incompressibility condition , we have : @xmath226 where @xmath227 is the excess pressure field in the fluid : the total pressure is @xmath228 , where @xmath229 is a constant and @xmath227 goes to zero far from the membrane . as in the main text , hats indicate two - dimensional fourier transforms . taking the scalar product of @xmath230 and eq . ( [ stokes ] ) , and using eq . ( [ inc ] ) yields @xmath231 taking the scalar product of @xmath232 and eq . ( [ stokes ] ) , and using eq . ( [ interm ] ) yields @xmath233 solving these equations , with the boundary conditions at infinity : @xmath234 for @xmath235 , and at the water membrane interface : @xmath236 and @xmath237 , finally yields @xmath238\,e^{\mp q z}\,,\label{3d_1}\\ \bm{\hat w}^\pm&=\left\{\bm{\hat v}^\pm-\left[i\bm{q } \,(\partial_t \hat h ) \pm q\ , \bm{\hat v}^\pm\right ] z\,\right\}e^{\mp q z}\,,\label{3d_2}\\ \hat w_z^\pm&=\left\{\partial_t \hat h\pm\left[q \,(\partial_t \hat h ) \mp i \bm{q}\cdot\bm{\hat v}^\pm\right ] z\,\right\}e^{\mp q z}\,.\label{3d_3}\end{aligned}\ ] ] for our study of the dynamics of the membrane , what is needed is the stress exerted by the fluid on the membrane , i.e. , since we are working at first order in the membrane deformation , the stress in @xmath35 . the viscous stress tensor of the fluid is defined by @xmath239 where @xmath240 and @xmath241 . hence , using eqs . ( [ 3d_1 ] , [ 3d_2 ] , [ 3d_3 ] ) , we obtain @xmath242 where we have introduced the tangential part @xmath243 of the stress tensor of the fluid above and below the membrane . the first dynamical equation we use is a balance of forces per unit area acting normally to the membrane . it involves the normal elastic force density in the membrane given by eq . ( [ pn_b ] ) and the normal viscous stresses exerted by the fluid above and below the membrane , which are derived in [ ap_hyd ] ( see eq . ( [ st1 ] ) ) . it reads : @xmath244 where @xmath133 denotes the viscosity of the fluid above and below the membrane . besides , as each monolayer is a two - dimensional fluid , we write down generalized stokes equations describing the tangential force balance in each monolayer . the first force involved is the density of elastic forces given by eqs . ( [ pip_b ] ) and ( [ pim_b ] ) . the second one arises from the viscous stress in the two - dimensional flow of lipids . the third one comes from the viscous stress exerted by the water , which is derived in [ ap_hyd ] ( see eq . ( [ st2 ] ) ) . the last force that has to be included is the intermonolayer friction @xcite . we thus obtain : @xmath245 where @xmath49 denotes the velocity in monolayer @xmath7 , while @xmath246 is the two - dimensional viscosity of the lipids and @xmath50 is the intermonolayer friction coefficient @xcite . 25 natexlab#1#1[2]#2 , , , , ( ) . , , , , , , , , , ( ) . , , , , , , ( ) . , , , , , ( ) . , , , , , ( ) . , , , , ( ) . , , , , , , , ( ) . , , , ( ) . , , , , ( ) . , , , , ( ) . , , , , ( ) . , , , ( ) . , , , ( ) . , , , , ( ) . , , ( ) . , , , , , . , , , ( ) . , , , , , , ( ) . , , , . , , ( ) . , , , , , ( ) . , , , , ( ) . , , , , ( ) . , , , , , , ( ) .